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LIBRARY OF THE : LELAND STANFORD JR. UNIVERSITY. Ob. t2436 JUL 20 1900
(The rights of translation and of reproduction are reserved)
INTRODUCTORY
DESCRIPTION OF
CONTENTS.
=o
MATHEMATICAL Psycuics may be divided into two parts— Theoretical and Applied.
In the First Part (1) it is attempted to illustrate the possibility of Mathematical reasoning without numerical data (pp. 1-7); without more precise data than are afforded by estimates of quantity of pleasure (pp. 7-9). (2) An analogy is suggested between the Principles of Greatest Happiness, Utilitarian or Egoistic, which con- stitute the first principles of Ethics and Economics, and those Principles of Maximum Energy which are among the highest generalisations of Physics, and in virtue of | which mathematical reasoning is applicable to physical phenomena quite as complex as human life (pp. 9-15).
The Calculus of Pleasure (Part II.) may be divided into two species—the Economical and the Utilitarian ; the principle of division suggesting an addition to Mr. Sidgwick’s ‘ ethical methods’ (p. 16).
The first species of Calculus (if so ambitious a title may for brevity be applied to short studies in Mathe- matical Economics) is developed from certain Definitions
v1 INTRODUCTORY DESCRIPTION OF CONTENTS.
of leading conceptions, in particular of those connected with Competition (pp. 17-19). Then (a) a mathematical theory of Contract unqualified by Competition is given (pp. 20-30). (8) A mathematical theory of Contract de- termined by Competition in a perfect Market is given, or at least promised (pp. 30-33, and pp. 38-42). Reference is made to other mathematical theories of Market, and to Mr. Sidgwick’s recent article on the ‘ Wages-Fund’ (pp. 32, 33, and Appendix V.) (y) attention is concen- trated on the question— What 1s a perfect Market? It is argued that Market is imperfect, Contract 1s indeter- minate in the following cases :—
(1.) When the number of competitors is limited (pp. 37, 39).
(11.) In a certain similar case likely to occur in con- tracts for personal service (pp. 42, 46).
(1, and 11.) When the articles of contract are not perfectly divisible (p. 42, 46).
(111.) In case of Combination, Unionism ; in which case it is submitted that (in general and abstractly speaking) unionists stand to gain in senses contradicted or ignored by distinguished economists (pp. 44, 47, 48).
(Iv.) In a certain case similar to the last, and likely to occur in Co-operative Association (pp. 45, 49).
The indeterminateness likely from these causes to affect Commercial Contracts, and certainly affecting all sorts of Political Contracts, appears to postulate a prin- ciple of arbitration (pp. 50-52).
It is argued from mathematical considerations that ‘the basis of arbitration between contractors 1s the greatest possible utility of all concerned; the Utilitarian first ‘principle, which can of course afford only a general
INTRODUCTORY DESCRIPTION OF CONTENTS. Vil
direction—yet, as employed by Bentham’s school, has afforded some direction in practical affairs (pp. 53-56).
The Economical thus leads up to the Utiltarian species of Hedonics; some studies in which already published * (under the title of ‘ Hedonical Calculus ’— the species being designated by the generic title) are reprinted here by the kind permission of the Editor of * Mind.’ :
Of the Utilitarian Calculus (pp. 56-82) the central conception is Greatest Happiness, the greatest possible sum-total of pleasure summed through all time and over all sentience. Mathematical reasonings are em- ployed partly to confirm Mr. Sidgwick’s proof that Greatest Happiness is the end of right action; partly to deduce middle axioms, means conducive to that end. This deduction is of a very abstract, perhaps only nega- tive, character ; negativing the assumption that Equality is necessarily implied in Utilitarianism. For, if sentients differ in Capacity for happiness—under similar circum- stances some classes of sentients experiencing on an average more pleasure (¢.g. of imagination and sym- pathy) and less pain (e.g. of fatigue) than others—there is no presumption that equality of circumstances is the most felicific arrangement; especially when account is taken of the interests of posterity.
Such are the principal topics handled in this essay
‘or tentative study. Many of the topics, tersely treated in the main body of the work, are more fully illustrated in the course of seven supplementary chapters, or APPENDICES, entitled :
1 Mind, July 1879.
Vil INTRODUCTORY DESCRIPTION OF CONTENTS.
PAGES
I. On UnnumericaL MaTHEmaTiIcs., . . . 83-93 II. Own THe Importance or HeEponicaL CaLcuus . 93-98 III. On HeEDONIMETRY . . . . . 98-102 IV. On Mixep Mopsgs or Uriirraprantant . . - 102-104 V. Ow Proresson JEvVoNs’s ForMULZ OF ExcHancE . 104-116 VI. On THE ERRors OF THE Gyewperpnrol . ° . 116-125
VII. On toe Present Crisis IN IRELAND . . . 126-148
Discussions too much broken up by this arrangement are re-united by references to the principal headings, in the InDEx; which also refers to the definitions of terms used in a technical sense. The Index also contains the names of many eminent men whose theories, bearing upon the subject, have been noticed in the course of these pages. Dissent has often been expressed. In so terse a composition it has not been possible always to express, what has always been felt, the deference due to the men and the diffidence proper to the subject.
MATHEMATICAL PSYCHICS.
oun QO=——
ON THE APPLICATION OF MATHEMATICS TO THE MORAL SCIENCES.
THE application of mathematics to Belief, the calculus of Probabilities, has been treated by many distinguished writers; the calculus of Feeling, of Pleasure and Pain, is the less familiar, but not in reality’ more paradoxical subject of this essay. |
The subject divides itself into two parts ; concerned respectively with principle and practice, root and fruit, the applicability and the application of Mathematics to Sociology.
PART I.
In the first part it is attempted to prove an affinity between the moral and the admittedly mathematical sciences from their resemblance as to (1) a certain general complexion, (2) a particular salient feature.
(1) The science of quantity is not alien to the study of man, it will be generally admitted, in so far as actions and effective desires can be numerically measured by way of statistics—that 1 is, very far, as Professor Jevons? anticipates. But in so far as our data may consist of ?
1 Cf. Jevons, Theory, p 3 Introduction to Mery of Political Economy.
\, B
2, MATHEMATICAL PSYCHICS.
estimates other than numerical, observations that some conditions are accompanied with greater or less pleasure than others, it is necessary to realise that mathematical reasoning is not, as commonly’ supposed, limited to subjects where numerical data are attainable. Where there are data which, though not numerical are quan- titative—for example, that a quantity is greater or less than another, increases or decreases, is positive or nega- hive, a macimum or minimum,=there mathematical reasoning is possible and may be indispensable. To take a trivial instance: a is greater than 8, and | is greater than c, therefore a is greater than c. Here is mathematical reasoning applicable to quantities which may not be susceptible of numerical evaluation. The following instance is less trivial, analogous indeed to an important social problem. It is required to distribute a given quantity” of fuel, so as to obtain the greatest possible quantity of available energy, among a given set of engines, which differ in efficiency—efficrency being thus defined : one engine is more efficient than another if, whenever the total quantity of fuel consumed by the former is equal to that consumed by the latter, the total quantity of energy yielded by the former is greater than that yielded by the latter.
In the distribution, shall a larger portion of fuel be given to the more efficient engines? always, or only in some cases? and, if so, in what sort of cases? Here is a very simple problem involving no numerical data, yet
1 The popular view pervades much of what Mill (in his Logic), after Comte, says about Mathematics applied to Sociology. There is a good expression of this view in the Saturday Review (on Professor Jevons’s Theory, November 11, 1871.) The view adopted in these pages is expressed by Cournot, Recherches.)
2 Or, & given quantity per unit of time, with corresponding modification of definition and problem.
\
UNNUMERICAL MATHEMATICS. 3
requiring, it may be safely said, mathematics for its | complete investigation.
The latter statement may be disputed in so far as such questions may be solved by reasoning, which, though not symbolical, is strictly mathematical ; answered more informally, yet correctly, by undis- ciplined common sense. But, firstly, the advocate of mathematical reasoning in social science is not con- cerned to deny that mathematical reasoning in social, as well as in physical, science may be divested of symbol. Only it must be remembered that the question how far mathematics can with safety or propriety be divested of her peculiar costume is a very delicate question, only to be decided by the authority and in the presence of Mathematics herself. And, secondly, as ‘to the suf ficiency of common sense, the worst of such unsymbolie, at least unmethodic, calculations as we meet in popular economics is that they are apt to miss the character~ istic advantages of deductive reasoning. He that wi not verify his conclusions as far_as possible by mathe: matics)’ as it were bringing the ingots) of common sens' to, be assayed and coined at the mint of the sovereign sche, at” hardly realize the full value of what he holds will want a measure ‘of what it will be worth in however slightly altered circumstances,*a means of conveying and making it current. When the given conditions are not sufficient to determinate the proble’ —a case of great importance in Political Economy:
he dyewperpyris is less likely to suspect this deficiency & less competent to correct it by indicating what con-' ditions are necessary and sufficient. All this is evident at a glance through the instrument of mathematics; but to the naked eye of common sense partially and ob-' B2
a,
\
Se
4 MATHEMATICAL PSYCHICS.
scurely;-and, as Plato says of unscientific knowledge, in a state between genuine Being and Not-Being.
The preceding problem, to distribute a given quan- tity of material in order 40 a maximum of energy, with its starting point loose quantitative relations rather than numerical data—its slippery though short path almost necessitating the support of mathematics—illustrates fairly well the problem of utilitarian distribution.! To illustrate the economical problem of exchange, the maze of many dealers contracting and competing with each other, it is possible to imagine? a mechanism of many parts where the law of motion?which particular part moves
off with which, is not precisely given—with symbols,aé
arbitrary functions, representing not merely not nu- merical knowledge but * ignorance-where, though the mode of motion towards equilibrium is indeterminate, the position of equilibrium is mathematically deter-
‘mined.
_ Examples not made to order, taken from the common stock of mathematical physics, will of course not fit so exactly. But they may be found in abundance, it is subfaitted, illustrating the property under consideration —mathematical reasoning without numerical data. In Hydrodynamics, for instance, we have a Thomson or Tait * reasoning * principles’ for “determining P and Q will be given later. In the meantime it is obvious that each decreases as X increases. Hence the equations of motion show *—and he goes on to draw a conclusion of
1 See p. 64. * See p. 34. 8 Ignoration of Co-ordinates (Thomson and Tait, Natural Philosophy, 2nd edition), is appropriate in many social problems where we only know in art. P * Thomson and Tait, Treatise on Natural Philosophy, p. 820, 2nd edition. The italics, which are ours, call attention to the unnumerical, loose quanttta- tive, relation which constitutes the datum of the mathematical reasoning.
UNNUMERICAL MATHEMATICS. 5
momentous interest that balls (properly) projected in an infinite incompressible fluid will move as if they, were attracted to each other. And generally in the higher Hydrodynamics, in that boundless ocean o! perfect fluid, swum through by vortices, where the deep first principles of Physics are to be sought, isnot a similar wunnumerical, or hyperarithmetical method there pursued? Ifa portion of perfect fluid so moves at any time that each particle has no motion of rotation, then that portion of the fluid will retain that property for all time*; here is no application of the numerical measuring-rod.
No doubt it may be objected that these hydro- dynamical problems employ some precise data; the very| definition of Force, the conditions of fluidity and con- tinuity. But so also have our sccial problems set precise data: for example, the property of uniformity of price in a market; or rather the (approximately realised) conditions of which that property is the de- ducible effect, and which bears a striking resemblance to the data of hydrodynamics :? (1) the fulness of the market: that there continues to be up to the conclusion of the deal- ing an indefinite number of dealers; (2) the fluidity of the market, or infinite dividedness of the dealers’ interests. Given this property of uniform price, Mr. Marshall and M. Walras deduce mathematically, though not arith- metically, an interesting theorem, which Mill and Thorn- ton failed with unaided reason to discern, though they |
were quite close to it—the theorem that the equation of supply to demand, though a necessary, is not a suffi- cient condition of market price.
To attempt to selectrepresentative instances from each
¥ Stokes, Mathematical Papers, p. 112, 2 See p. 18,
: |
6 MATHEMATICAL PSYCHICS.
recognised branch of mathematical inquiry would exceed the limits of this paper and the requirements of the argu- ment. It must suffice, in conclusion, to direct atten- tion to one species of Mathematics which seems largely affected with the property under consideration, the alculus of Maxima and Minima, or (in a wide sense) of Variations. The criterion of a maximum! turns, not upon the amount, but upon the sign of a certain quan- tity.2 We are continually concerned ® with the ascer- tainment of a certain loose quanttative relation, the decrease-of-rate-of-increase of a quantity. Now, this is the very quantitative relation which it is proposed to employ in mathematical sociology; given in such data as the law of diminishing returns to capital and labour, the law of diminishing utility, the law of increasing fatigue; the very same irregular, unsquared material ‘which constitutes the basis of the Economical and the Utilitarian Calculus.
Now, it is remarkable that the principal inquiries in Social Science may be viewed as mazximum-problems. For Economics investigates the arrangements between agents each tending to his own maximum utility; and
[Politics and (Utilitarian) Ethics investigate the arrange-
ments which conduce to the maximum sum total of
1 Maximum in this paper is employed according to the context for (1) Maximum in the proper mathematical sense ; (2) Greatest possible; (8) sta- tionary; (4) where minimum (or least possible) might have been expected ; upon the principle that every minimum is the correlative of a maximum. Thus Thomson’s Minimum theorem is correlated with Bertrand’s Maximum theorem. (Watson and Burbury.) This liberty is taken, not only for brevity, but also for the sake of a certain suggestiveness. ‘ Stationary,’ for instance, fails to suggest the superlativeness which it connotes.
2 The second term of Variation. It may be objected that the other con- dition of a maximum equation of the first term to zero is of a more precise character, See, however, Appendix I., p. $2.
5 E.g., Todhunter’s Researches on Calculus of Varwations, pp. 21-80, 80, 117, 286, &c.
HEDONIMETRY. 7
utility. Since, then, Social Science, as compared with the Calculus of Variations, starts from similar data— loose quantitative relations—and travels to a similar con- clusion—determination of mazimum—why should it not pursue the same method, Mathematics?
There remains the objection that in Physical Calculus there is always (as in the example quoted above from Thomson and Tait) a potentiality, an expectation, of measurement ; while Psychics want the first condition of calculation, a unit. The following! brief answer is diffidently offered.
Utility, as Professor Jevons’ says, has two dimen-, sions, 2ntensity and time. The unit in each dimension | the just perceivable® increment. The implied equation to each other of each minimum sensibile is a first orinciplel x incapable of proof. It resembles the equation to each other of undistinguishable events or cases,* which con- stitutes the first principle of the mathematical calculus of belief. It is doubtless a principl€ acquired in the course of evolution. The implied equatability of time- intensity units, irrespective of distance in time and + ‘kind of pleasure, is still imperfectly evolved. Such is the unit of economical calculus.
For moral calculus a further dimension is required ; to compare the happiness of one person with the happi- ness of another, and generally the happiness of groups of different members and different average happiness.
Such comparison can no longer be shirked, if there
}
1 For a fuller discussion, see Appendix ITI.
2 In reference to Economics, Theory, p. 51.
8 Cf. Wundt, Physiological Psychology; below, p. 60. Our ‘ ebenmerk- lich’ minim is to be regarded not as an infinitesimal differential, but as a finite small difference ; a conception which is consistent with a (duly cau- tious) employment of infinitesimal notation.
4 Laplace, Essati— Probabilities, p. 7.
8 MATHEMATICAL PSYCHICS.
is to be any systematic morality at all. It is postulated by distributive justice. It is postulated by the population question s that horizon in which every moral prospect terminates {which is presented to the far-seeing at every turn, on the most sacred and the most trivial occasions. ‘)\You cannot spend sixpence utilitarianly, without having considered whether your action tends to increase the ‘\Neomfort of a limited number, or numbers with limited comfort ; without having compared such alternative utilities. —
In virtue of what unit is such comparison possible ? It is here submitted: Any individual experiencing a _ unit of pleasure-intensity during a unit of time is to if ae ‘count for one.’! Utility, then, has three dimensions ;
a mass of utility, ‘lot of pleasure, is greater than another when it has more intensity-time-number units. The third dimension is doubtless an evolutional acqui- sition ; and is still far from perfectly evolved.
Looking back at our triple scale, we find no peculiar difficulty about the third dimension. It is an affair of census. The second dimension is an affair of clock- work ; assuming that the distinction here touched, be- tween subjective and objective measure of time, is of minor importance. But the first dimension, where we leave the safe ground of the objective, equating to unity each minimum sensibile, presents indeed peculiar difficul- ties. Atoms of pleasure are not easy to distinguish and discern ; more continuous than sand, more discrete than liquid ;. as it were nuclei of the just-perceivable, em- bedded in circumambient semi-consciousness.
We cannot count the golden sands of life ; we cannot number the ‘innumerable smile$,of seas of love ; but we
ey
1 In the Pure, for a fraction, in the Impure, imperfectly evolved, Utilita- rianism. See p. 16.
MAXIMUM ENERGY. 9
seem to be capable of observing that there is here a greater, there a less, multitude of pleasure-units, mass of happiness ; and that is enough.
(2) The application of mathematics to the world of:
soul is countenanced by the hypothesis (agreeable to the general hypothesis that every psychical phenomenon is the concomitant, and in some sense the other side of a physical phenomenon), the particular hypothesis adopted in these pages, that Pleasure is the concomitant of Energy. nergy may be regarded as the central idea of Mathematical Physics ; maximum energy the object of the principal investigations in that science. By aid of this conception we reduce into scientific order physical phenomena, the complexity of which may be compared with the. complexity which appears so formidable in Social Science.
Imagine a material Cosmos, a mechanism as com-
posite as possible, and perplexed with all manner of wheels, pistons, parts, connections, and whose mazy complexity might far transcend in its entanglement the webs of thought and wiles of passion; nevertheless, if any given impulses be imparted to any definite points in the mechanism at rest, it is mathematically deducible that each part of the great whole will move off with a velocity such that the energy of the whole may be the greatest possible '—the greatest possible consistent with the given impulses and existing construction. If we know something about the construction of the mechan- ism, if it is ‘a mighty maze, but not without a plan ;’ if we have some quantitative though not numerical datum about the construction, we may be able to deduce a similarly indefinite conclusion about the motion. For instance, any number of cases may be imagined in
i Bertrand’s Theorem.
10 MATHEMATICAL PSYCHICS.
which, if a datum about the construction is that certain parts are less stiff than others, a conclusion about the motion would be that those parts! take on more energy than their stiffer fellows. This rough, indefinite, yet mathematical reasoning is analogous to the reasoning on a subsequent page,” that in order to the greatest possible sum total of happiness, the more capable of pleasure shall take more means, more happiness.
In the preceding illustration the motion of a mechanism was supposed instantaneously generated by the application of given impulses at definite poimts (or over definite surfaces); but similar general views are attainable in the not so dissimilar case in which we suppose motion generated in time by finite forces acting upon, and interacting between, the particles of which the mechanism is composed. This supposition includes the celebrated problem of Many Bodies (attracting each other according to any function of the distance) ; in reference to which one often hears it asked what can be expected from Mathematics in social science, when she is unable to solve the problem of Three Bodies in her own department. But Mathematics can solve the problem of many bodies—not indeed numerically and explicitly, but practically and philosophically, affording approximate measurements, and satisfying the soul of the philosopher with the grandest of generalisaticns. By a principle discovered or improved by Lagrange, each particle of the however complex whole is continually so moving that the accumulation of energy, which 1s consti- tuted by adding to each other the energies of the mechan- ism existing at each instant of time (technically termed Action—the time-integral of Energy) should be a® maxi-
1 Of. Watson and Burbury, Generalised Co-ordinates, Art. 30, and pre- ceding. 2 Pp. 64. 3 See note, p. 6.
MAXIMUM PLEASURE. Il
mum. By the discovery of Sir William Rowan Hamilton’ the subordination of the parts to the whole is more usefully expressed, the velocity of each part is regarded as derivable from the action of the whele; the action is connected by a single, although not an explicit or in gene- ral easily interpretable, relation with the given law of force. The many unknown are reduced to one un- known, the one unknown is connected with the known.
Now this accumulation (or time-integral) of energy which thus becomes the principal object of the physical investigation is analogous to that accumulation of pleasure which is constituted by bringing together in prospect the pleasure existing at each instant of time, the end of rational action, whether self-interested or benevolent. The central conception of Dynamics and (in virtue of pervading analogies it may be said) in general of Mathematical Physics is other-sidedly rdentical with the central conception of Ethics; and a solution practical and philosophical, although not numerical and precise, as it exists for the problem of the inter- action of bodies, so is possible for the problem of the interaction of souls.
This general solution, it may be thought, at most is applicable to the utilitarian problem of which the object is the greatest possible sum total of universal happiness. But it deserves consideration that an object of Eco- nomics also, the arrangement to which contracting agents actuated only hy self-interest tend is capable of being regarded upon the psychophysical hypothesis here entertained as the realisation of the maximum sum- total of happiness, the relative maximum,? or that which is consistent with certain conditions. There is dimly discerned the Divine idea of a power tending to
1 Philosophical Transactions, 1884-5. 3 See pp. 24, 142.
12 MATHEMATICAL PSYCHICS.
the greatest possible quantity of happiness ' under con- ditions; whether the condition of that perfect disinte- gration and unsympathetic isolation abstractedly as- sumed in Economics, or those intermediate ? conditions of what Herbert Spencer might term integration on to that perfected utilitarian sympathy in which the pleasures of another are accounted equal with one’sown. There are diversities of conditions, but one maximum-principle ; many stages of evolution, but ‘one increasing purpose.’
‘Mécanique Sociale’ may one day take her place along with ‘ Mécanique Celeste,’ throned each upon the double-sided height of one maximum principle,’ the supreme pinnacle of moral as of physical science. As the movements of each particle, constrained or loose, in a material cosmos are continually subordinated to one maximum sum-total of accumulated energy, so the movements of each soul, whether selfishly isolated or linked sympathetically, may continually be realising the maximum energy of pleasure, the Divine love of the universe.
‘Mécanique Sociale,’ in comparison with her elder sister, 1s less attractive to the vulgar worshipper in that she is discernible by the eye of faith alone. The statuesque beauty of the one is manifest ; but the fairy- like features of the other and her fluent form are
1 Of. Mill, Essays on Nature and Religion. 3 See p. 16.
5 The mathematical reader does not require to be reminded that upon the principles of Lagrange the whole of (conservative) Dynamics may be pre- sented as a Maximum-Problem; if without gain, at any rate without loss. And the great principle of Thomson (Thomson & Tait, arts. Cf. Theory of Vortices, by Thomson, Royal Society, Edinburgh, 1865), with allied maz:- mum-principles, dominating the theory of fluid motion, dominates Mathema- tical Physics with a.more than nominal supremacy, and most indispensably efficacious power. Similarly, it may be conjectured, the ordinary moral rules are equivalently expressed by the Intuitivist in the (grammatically-speaking),
positive degree, by the Utilitarian in the superlative. But for the higher moral problems the conception of maximum is indispensable.
MBCANIQUE SOCIALE. 13
veiled. But Mathematics has long walked by the evidence of things not seen in the world of atoms (the methods whereof, it may incidentally be remarked, statistical and rough, may illustrate the possibility of social mathematics). The invisible energy of electricity is grasped by the marvellous methods of Lagrange ;* the invisible energy of pleasure may admit of a similar handling.
As in a system of conductors carrying electrical currents the energy due to electro-magnetic force is to be distinguished from the energy due to ordinary dyna- mical forces, ¢.g., gravitation acting upon the conductors, so the energy of pleasure is to be distinguished not only from the gross energy of the limbs, but also from such nervous energy as either is not all represented in con- sciousness (pace G. H. Lewes), or is represented by intensity of consciousness not intensity of pleasure. As electro-magnetic force tends to a maximum energy, so also pleasure force tends to a maximum energy. The energy generated by pleasure force is the physical con- comitant and measure of the conscious feeling of delight.
Imagine an electrical circuit consisting of two rails isolated from the earth connected at one extremity by a galvanic battery and bridged over at the other extremity by a steam-locomotive.? When a current of electricity is sent through the circuit, there is an electro-magnetic force tending to move the circuit or any moveable part of it in sucha direction that the number of lines of force (due to the magnetism of the earth) passing through the circuit in a positive direction may be @ maximum. The electro-magnetic force therefore tends to move the
¥ See Clerk Maxwell, Electricity and Magnetism,on the use of Lagrange's
Generalised Co-ordinates, Part iv., chaps. 5 and 6, 2 Clerk Maxwell has a similar construction.
14 MATHEMATICAL PSYCHICS.
locomotive along the rails in that direction. Now this delicate force may well be unable to move the ponderous locomotive, but it may be adequate to press a spring _and turn a handle and let on steam and cause the loco- motive to be moved by the steam-engine in the direction of the electro-magnetic force, either backwards or forwards according to the direction in which the electrical cur- rent flows. The delicate electro-magnetic force is placed in such a commanding position that she sways the movements of the steam-engine so as to satisfy her own yearning towards maximum.
Add now another degree of freedom ; and let the steam-car governed move upon a plane! in a direction tending towards the position of Minimum Potential Electro-Magnetic Energy. Complicate this conception ; modify it by substituting for the principle of Minimum Force-Potential the principle of Minimum? Momentum- Potential ; imagine a comparatively gross mechanism of i arable degrees of freedom governed, in the sense
y amore delicate system—itself, however inconceivably diversified its degrees of freedom, obedient still to the great Mazimum Principles of Physics, and amenable to mathematical demonstration, though at first sight as hopelessly incalculable as whatever is in life capricious and irregular—as thesmiles of beauty and the waves of passion.
Similarly pleasure in the course of evolution has become throned among grosser subject energies—as it were explosive engines, ready ® to go off at the pressure
1 See p. 24,
2 Momanturn-Potentiat upon the analogy of Velocity-Potential (Thomson on Vortex Motion, § 31); and Mintmum, as I venture to think, in virtue of certain analogies between theories about Energy and about Action. |
5 See the account of the Mechanism of Lsfe, in Balfour Stewart’s Con- aernation of Enerau.
MAN A PLEASURE-MACHINE. 15
of a hair-spring. Swayed by the first principle, she sways the subject energies so as to satisfy her own yearn- ing towards maximum ; ‘her every air Of gesture and least motion’ a law of Force to governed systems—a fluent form, a Fairy Queen guiding a most complicated chariot, wheel within wheel, the ‘ speculative and active instruments,’ the motor nerves, the limbs and the envi- ronment on which they act.
A system of such charioteers and chariots is what constitutes the object of Social Science. The attractions between the charioteer forces, the collisions and com- pacts between the chariots, present an appearance of quantitative regularity in the midst of bewildering com- plexity resembling in its general characters the field of electricity and magnetism. To construct a scientific hypothesis seems rather to surpass the powers of the writer than of Mathematics. ‘Sin has ne possim natures accedere partes Frigidus obstiterit circum pre- cordia sanguis ;’ pleasure machine may justify and facilitate the employ- ment of mechanical terms and Mathematical reasoning
in social science.
PART I.
SucH are some of the preliminary considerations by which emboldened we approach the two fields into which the Calculus of Pleasure may be subdivided,
namely Economics and Utilitarian Ethics. The Econo- |
- mical Calculus investigates the equilibrium of a system of hedonic forces each tending to maximum individual utility ; the Utilitarian Calculus, the equilibrium of a system in which each and all tend to maximum uni-
at least the conception of Man as a.
\
- —_ . ‘.
16 MATHEMATICAL PSYCHICS.
versal utility. The motives of the two species of agents correspond with Mr. Sidgwick’s Egoistic and Universal- istic Hedonism. But the correspondence is not perfect. For, firstly, upon the principle of ‘self limitation’ of a method, so clearly stated by Mr. Sidewick, so persistently misunderstood by critics, the Pure Utilitarian might think it most beneficent to sink his benevolence towards competitors ; and the Deductive Egoist might have need of a Utilitarian Calculus. But further, it is possible that the moral constitution of the concrete agent would be neither Pure Utilitarian nor Pure Egoistic, but poxry mis. For it is submitted that Mr. Sidgwick’s division of Hedonism—the class of ‘Method’ whose principle of action may be generically defined maximising happiness —is not exhaustive. For between the two extremes Pure Egoistic and Pure Universalistic there may be an indefinite number of impure methods; wherein the happiness of others as compared by the agent (in a calm moment) with his own, neither counts for nothing, not yet ‘counts for one,’ but counts for a fraction.
Deferring controversy,} let us glance at the elements of the Economic Calculus; observing that the connota- tion (and some of the reasoning) extends beyond the usual denotation ; to the political struggle for power, as well as to the commercial struggle for wealth.
ECONOMICAL CALCULUS.
Derinitions.—The first principle of Economics? is that every agent is actuated only by self-interest. The
‘workings of this principle may be viewed under two
aspects, according as the agent acts without, or with, the
1 See Appendix IV. 2 Descriptions rather, but sufficient for the purpose of these tentative studies.
ECONOMICAL DEFINITIONS. 17
consent of others affected by his actions. In wide senses, the first species of action may be called war; the second, contract. Examples: (1) A general, or fencer, making moves, a dealer lowering price, without consent of rial. (2) A set of co-operatives (labourers, capital- ists, manager) agreed nem. con. to distribute the joint- produce by assigning to each a certain function of his sacrifice. The articles of contract are in this case the amount of sacrifice to be made by each, and the principle of distribution.
‘Is it peace or war?’ asks the lover of ‘ Maud,’ of economic competition, and answers hastily : It is both, paz or pact between contractors during contract, war, when some of the contractors without the consent of others recontract. ‘Thus an auctioneer having been in contact ‘with the last bidder (to sell at such a price 7f no higher bid) recontracts with a higher bidder. So a landlord on expiry of lease recontracts, it may be, with a new tenant.
The field of competition with reference to a contract, or contracts, under consideration consists of all the individuals who are willing and able to recontract about the articles under consideration. Thus in an auction the field consists of the auctioneer and all who are effectively willing to give a higher price than the last bid. In this case, as the transaction reaches determi- nation, the field continually diminishes and ultimately vanishes. But this is not the case in general. Suppose a great number of auctions going on at the same point ; or, what comes to the same thing, a market consisting of an indefinite number of dealers, say Xs, in commodity x, and an indefinite number of dealers, say Ys, in com- _modity y. In this case, up to the determination of equilibrium, the field continues indefinitely large. To
C
18 MATHEMATICAL PSYCHICS.
be sure it may be said to vanish at the position of equilibrium. But that circumstance does not stultify the definition. Thus, if one chose to define the field of force as the centres of force sensibly acting on a certain system of bodies, then in a continuous medium of attracting matter, the field might be continually of indefinite extent, might change as the system moved, might be said to vanish when the system reached equilibrium.
There is free communication throughout a normal competitive field. You might suppose the constituent individuals collected at a point, or connected by tele-
hones—an ideal supposition, but sufficiently approxi- mate to existence or tendency for the purposes of abstract science.
A perfect field of competition professes in addition certain properties peculiarly favourable to mathematical calculation ; namely, a certain indefinite muliplicity and dividedness, analogous to that infinity and infinitesimality which facilitate so large a portion of Mathematical Physics (consider the theory of Atoms, and all applica- tions of the Differential Calculus). The conditions of a perfect field are four; the first pair referrible’ to the heading multiplicity or continuity, the second to divided- ness or fluidity.
I. Any individual is free to recontract with any out of an indefinite number, ¢.g.,in the last example there are an indefinite number of Xs and similarly of Ys.
IJ. Any individual is free to contract (at the same time) with an indefinite number ; ¢.g., any X (and simi- larly Y) may deal with any number of Ys. This con- dition combined with the first appears to involve
1 See p. 5.
ECONOMICAL DEFINITIONS. 19
the indefinite divisibility of ! each article of contract (if any X deal with an indefinite number of Ys he must give each an indefinitely small portion of x); which might be erected into a separate condition.
If. Any individual is free to recontract with another independently of, without the consent being required of, any third party, ¢.g., there is among the Ys (and simi- larly among the Xs) no combinahon or precontract be- tween two or more contractors that none of them will recontract without the consent of all. Any Y then may accept the offer of any X irrespectively of other Ys.
IV. Any individual is free to contract with another independently of a third party ; ¢.g., in simple exchange each contract is between two only, but secus in the entangled contract described in the example (p. 17), where it may be a condition of production that there should be three at least to each bargain.
There will be observed a certain similarity between the relation of the first to the second condition, and that of the third to the fourth. The failure of the first involves the failure of the second, but not vice versd; - and the third and fourth are similarly related.
A settlement is a contract which cannot be varied with the consent of all the parties to it.
A final settlement is a settlement which cannot be varied by recontract within the field of competition.
Contract is zndeterminate when there are an indefinite number of final settlements.
1 This species of imperfection will not be explicitly treated here; partly because it is perhaps of secondary practical importance; and partly because it has been sufficiently treated by Prof. Jevons (Theory, pp. 185-137). It is important, as suggested in Appendix V., to distinguish the effects of this imperfection according as the competition is, or is not, supposed perfect in other respects.
c 2
20) MATHEMATICAL PSYCHICS.
The PROBLEM to which attention is specially directed in this introductory summary is: How far contract is wndeterminate—an inquiry of more than theoretical im- portance, if it show not only that mdeterminateness tends to prevent widely, but also in what direction an escape from its evils is to be sought.
DemonstrRaTions.'—The generat answer is—(a) Con- tract without competition is indeterminate, (8) Contract with perfect competition is perfectly determinate, (y) Contract with more or less perfect competition is less or more indeterminate.
(a) Let us commence with almost the simplest case of contract,—two individuals, X and Y, whose interest depends on two variable quantities, which they are agreed not to vary without mutual consent. Exchange of two commodities is a particular case of this kind of contract. Let 2and y be the portions interchartged, as in Professor Jevons’s example.? Then the utility of one party, say X, may be written $,(a — 2) + ¥, (y); and the utility of the other party, say Y, $, (x) + ¥, (b—y); where @ and ¥ are the integrals of Professor Jevons’s symbols @ and wp. It is agreed that z and y shall be varied only by consent (not e.g. by violence).
More generally. Let P, the utility of X, one party, = F (zy), and I, the utility of Y, the other party, = @(xy). If now it is inquired at what point they will reach equilibrium, one or both refusing to move further, to what settlement they will consent; the answer is in general that contract by itself does not supply sufficient conditions to determinate the solution; sup- plementary conditions as will appear being supplied by
1 Conclusions rather, the mathematical demonstration of which is not fully exhibited. 2 Theory of Political Economy, 2nd ed., p. 107.
. PURE CONTRACT. 21
competition or ethical motives, Contract will supply only one condition (for the two variables), namely
4P dll @Pdll
dzady adyaz
(corresponding to Professor Jevons’s equation
gp: (a — 2) _ 2 (2) wi(y) (6 —y) Theory p. 108), which it is proposed here to inves- tigate.
Consider P — F (xy) = 0 as a surface, P denoting the length of the ordinate drawn from any point on the plane of z y (say the plane of the paper) to the surface. Consider HW — $(zy)similarly. It is required to find a point (#y) such that, in whatever direction we take an infinitely small step, P and I do not increase together, but that, while one increases, the other decreases. It may be shown from a variety of points of view that the locus of the required point is
dP di dP dil_,.
da zdy dy Ly de ° which locus it is here proposéll to call the contract- curve.
(1) Consider first in what directions X can take an
indefinitely small step, say of length p, from any point (x y). Since the addition to P is
aP dPy . p[ (=) cos 8 + (7, ) sin 6|, p cos @ being = dz, and psin 0 = dy, it is evident that
X will step only on one side of a certain line, the line of indifference, as it might be called; its equation being
(¢-a) (F)+ a-y (F = 0.
22 MATHEMATICAL PSYCHICS.
And it is to be observed, in passing, that the direction in which X will prefer to move, the line of force or line of preference, as it may be termed, is perpendicular to the line of indifference. Similar remarks apply to H. If then we enquire in what directions X and Y will con- sent to move together, the answer is, in any direction between their respective lines of indifference, in a direc- tion positive as it may be called for both. At what point then will they refuse to move at all? When their lines of indifference are coincident (and lines of preference not only coincident, but in opposite directions) ; whereof the necessary " not sufficient) condition is
Ta) (Zy)~ (Gy) (ae) = 9
(2) The same consideration might be thus put. Let the complete variation of P be DP=p (Ss) cos # +
~~) sin 6] and similarly for WH. Thenin general @ can
dy DP *,° _ 9 be taken, so that DH should be positive, say = g’, and so P and II both increase together. aqdP_ .,dail tan @ = de? dz "ap, di dy 7 dy
But this solution fails when (=) (an (dP\ (dit (Gy) (G
' In fact, in this case aes ws the same for all directions.
PURE CONTRACT. 7 23
If, then, that common value of a ‘is negative, motion
is impossible in any direction.
(3) Or, again, we may consider that motion is pos- sible so long as, one party not losing, the other gains. The point of equilibrium, therefore, may be described as a relative maximum, the point at which e.g. IT being constant, P is a maximum. Put P= P—c(1-T7’), where c is a constant and IT’ is the supposed given value of 1. Then P is a maximum ny when —
dP _ ail dx (7 —e7) + ay (F - °F) = 0; whence we have as before the contract-curve.
The same result would follow if we supposed Y in- duced to consent to the variation, not merely by the guarantee that he should not lose, or gain infinitesimally, but by the understanding that he should gain sensibly with the gains of P. For instance, let 1 = k?P where k is a constant, certainly not a very practicable condition. Or, more generally, let P move subject to the condition that DP = 6? x DH, where @ is a function of the co- ordinates. Then DP, subject to this condition, vanishes only when
0= (5) de v+(F ay tet (G,) 42+ (F) ay— ie
(Ge) 42 + (F) ay] J
where c is a constant ;
whence (=) (1 + ¢) — cF (3”) = 0 wa (12) (140) 0 (A)
24 . MATHEMATICAL PSYCHICS.
whence as before (OF 7) _ (7) )(F a7) = 0.
No doubt the one theory which has been thus dif- ferently expressed could be presented by a professed mathematician more elegantly and scientifically. What appears to the writer the most philosophical presenta- tion may be thus indicated.
(4) Upon the hypothesis above shadowed forth,! human action generally, and in particular the step taken by a contractor modifying articles of contract, may be regarded as the working of a gross force governed, let on, and directed by a more delicate pleasure-force. From which it seems to follow upon general dynamical principles applied to this special case that equilibrium is attained when the total pleasure-energy of the contractors is a maximum relative,” or subject, to conditions; the conditions being here (i) that the pleasure-energy of X and Y considered each as a function of (certain values of) the variables 2 and y should be functions of the same values: in the metaphorical language above em- ployed that the charioteer-pleasures should drive their teams together over the plane of zy; (11) that the joint- team should never be urged in a direction con- trary to the preference® of either individual; that the resultant line of force (and the momentum) of the gross, the chariot, system should be continually in- termediate between the (positive directions of the) lines of the respective pleasure-forces. [We may without disadvantage make abstraction of sensible mo- mentum, and suppose the by the condition joint- system to move towards equilibrium along a line of resultant gross force. Let it start from the origin. And
1 See pp. 13-15. 2 See note, p. 11. 5 See p. 22.
PURE CONTRACT. 25.
let us employ an arbitrary function to denote the un- known principle of compromise between the parties ; sup- pose the ratio of the sines of angles made by the resultant line with the respective lines of pleasure- force.]| Then, by reasoning different from the pre-
ceding only in the point of view, it appears that the ' ,
imam at ony | a
total_ utility of _the system 1s a relative maximum at any
point on the pure contract-curve. It “pears from (1) and (2) there is a portion of the
locus =) (Se ae (= EN (& 7) =o where >= is
+, not hevefore indicating immobility, au contraire, the impure (part of the) contract-curve, as it might be called. This might be illustrated by two spheres, each having the plane of the paper as a diametral plane. The contract curve is easily seen to be the line joining the centres. Supposing that the distance between the
contract-curve is impure. If the index, as Mr. Marshall might call it, be placed anywhere in this portion it will run up to a centre. But‘ between the centres the con- tract-curve is pure; the index placed anywhere in this portion is immovable; and if account be taken of the portions of the spheres underneath the plane of the paper, the downward ordinates representing negative pleasures, similar statements hold, mutatis mutandis.
It appears that the pure and impure parts of the contract-curve are demarcated by the points where
DP DP DH changes sign, that is (in general) where either To
or on (do being an increment of the length of the o contract-curve) either vanishes or becomes infinite.
Accordingly the maxima and minima of P and IJ present
I)
centres is less than the less of the radu, part of the |
26 MATHEMATICAL PSYCHICS.
demarcating points; for example, the centre of each sphere, which corresponds to a maximum in reference to the upper hemisphere, a minimum in reference to the lower hemisphere. The impure contract curve is relevant to cases where the commodity of one party is a discommodity to the other.
But even in the pure contract-curve all points do not in the same sense indicate immobility. For, accord- ‘ing to the consideration (3) [above, p. 23], the contract- curve may be treated as the locus where, H being constant, P is stationary, either a mazimum or minimum. Thus any point in our case of two intersecting spheres affords a maximum in relation to the upper hemisphere ; but the same point (it is only an accident that it should be the same point—it would not be the same point if you suppose slightly distorted spheres) affords a mini- mum in relation to the lower hemisphere. This pure, but unstable (part of the) contract-curve is exemplified in certain cases of that’ unstable equilibrium of trade, which has been pointed out by Frincipal Marshall and. Professor Walras.
The preceding theory may easily be extended to several persons and several variables. Let P, =F, (xy z) denote the utility of one of three parties, utility depending on three variables, zyz; and similarly P,=F,, P;=F,;. Then the contract-settlement, the arrangement for the alteration of which the consent of all three parties cannot be obtained, will be (subject to reservations analogous to those analysed in the pre- ceding paragraphs) the Eliminant.
adP, dP, dP, dz dy dz 1 Mr. Marshall's figure 9 but not his figure 8; for the delicate relation
between the conceptions—instability of Zrade (where perfect competition is presupposed) and instability of contact in general—is not one of identity.
PURE CONTRACT. . 27
dP, dP, dP, dz dy az dP, dP, dP; de dy az
In general let there be m contractors and 7 subjects of contract, m variables. Then by the principle (3) [above, p. 23] the state of equilibrium may be con- sidered as such that the utility of any one contractor must be a maximum relative to the utilities of the other contractors being constant, or not decreasing ; which may be thus mathematically expressed :
D(LP,+1,P, +&c.+/,, P,.)=0, where D repre- sents complete increment and /, /, &c., are indeterminate multipliers ; whence, if there be ” variables x, x... Xp; we have nv equations of the form
from which, if » be not less than m, we can eliminate the (m—1 independent) constants / and obtain the con- tract-system consisting of n—(m—1) equations.
The case of n being less than m may be sufficiently illustrated by a particular example. Let the abscissa z represent the single variable on which the utilities P | and II of two persons contracting depend. Then if p and w are the maximum points for the respective plea- sure-curves (compare the reasoning, p. 22) it 1s evident that the tract of abscissa between w and p is of the nature of pure contract-curve; that the index being placed anywhere in that tract will be immovable; secus on either side beyond w and p. Similarly it may be shown that, if three individuals are in contract about two variables z y, the contract locus or region is (the space within) a curvilinear triangle in the plane xy
28 MATHEMATICAL PSYCHICS.
bounded by the three contract-curves presented by suc- cessively supposing each pair of individuals to be in contract with respect to # and y. And similarly for larger numbers in hyperspace.
It is not necessary for the purpose of the present study to carry the analysis further. To gather up and fix our thoughts, let us imagine a simple case—Robinson Crusoe contracting with Friday. The articles of contract : wages to be given by the white, labour to be given by the black. Let Robinson Crusoe =X. Represent y, the labour given by Friday, by a horizontai line measured north- ward from an assumed point, and measure 2, the remu- neration given by Crusoe, from the same point along an eastward line (See accompanying figure 1.). Then
\ Fre. 1.
\
any point between these lines represents a contract. It will very generally be the interest of both parties to vary the articles of any contract taken at random. But there is a class of contracts to the variation of which the consent of both parties cannot be obtained, of settle-
PURE CONTRACT. 29
ments. These settlements are represented by an inde- finite number of points, a locus, the contract-curve CC’, or rather, a certain portion of it which may be supposed to be wholly in the space between our perpendicular lines in a direction trending from south-east to north- west. This available portion of the contract-curve lies between two points, say 7 2 north-west, and y, & south- east ; which are respectively the intersections with the contract-curve of the curves of indifference’ for each party drawn through the origin. Thus the utility of the contract represented by 721s for Friday zero, or rather, the same as if there was no contract. At that point he would as soon be off with the bargain— work by himself perhaps.
This simple case brings clearly into view the charac- teristic evil of indeterminate contract, deadlock, un- decidable opposition of interests, &xpurés? Epis Kal tapaxy. It is the interest of both parties that there should be some settlement, one of the contracts repre- sented by the contract-curve between the limits. But which of these contracts is arbitrary in the absence " arbitration, the interests of the two adversd pugnantia fronte all along the contract-curve, Y desiring to get as far as possible south-east towards ¥,§, X north-west toward yz. And it further appears from the preceding analysis that in the case of any number of articles (for instance, Robinson Crusoe to give Friday in the way of Industrial Partnership a fraction of the produce as well as wages, or again, arrangements about the mode of work), the contract-locus may still be. represented as a sort of line, along which the pleasure-forces of the con- tractors are mutually antagonistic.
An accessory evil of indeterminate contract is the
2 See p. 22. 2 Demosthenes, De Corond.
30 MATHEMATICAL PSYCHICS.
tendency, greater than in a full market, towards dissimu- lation and objectionable arts of higgling. As Professor Jevons! says with reference to a similar case, ‘Such a transaction must be settled upon other than strictly economical grounds. . . . The art of bargaining consists in the buyer ascertaining the lowest price at which the seller is willing to part with his object, without dis- closing, if possible, the highest price which he, the buyer, is willing to give.’ Compare Courcelle-Seneuil’s? account of the contract between a hunter and a wood- man in an isolated region.
With this clogged and underground procedure is contrasted (8) the smooth machinery of the open market. As Courcelle-Seneuil says, ‘& mesure que le nombre des concurrents augmente, les conditions d’échange de- viennent plus nécessaires, plus impersonelles en quelque sorte.’ You might suppose each dealer to write down? his demand, how much of an article he would take at each price, without attempting to conceal his require- ments ; and these data having been furnished to a sort of market-machine, the price to be passionlessly evaluated.
That contract in a state of perfect competition is determined by demand and supply is generally accepted, but is hardly to be fully understood without mathe- matics. The mathematics of a perfect market have been worked out by several eminent writers, in particular Messrs. Jevons, Marshall, Walras ; to whose varied cul-. tivation of the mathematical science, Catallactics, the reader is referred who wishes to digdown to the root of first principles, to trace out all the branches of a com- plete system, to gather fruits rare and only to be reached by a mathematical substructure.
1 Theory, p. 134. 2 Traité, book ii. 8 Cf. Walras, Elements, Art. 50.
PERFECT COMPETITION. 31
There emerges amidst the variety of construction and terminology wo\A@v dvopatwv popdy pia, an essentially identical graphical form or analytical formula express- ing the equation of supply to demand; whereof the simplest type, the catallactic molecule, as it might be called, is presented in the case above described in the definition of perfect competition.’ The familiar pair of equations is deduced* by the present writer from the first principle: Equilibrium is attained when the ex- isting contracts can neither be varied without recontract
with the consent of the existing parties, nor by recon- tract within the field of competition. The advantage of this general method is that it is applicable to the par- ticular cases of imperfect competition; where the con- ceptions of demand and supply at a price are no longer appropriate.
The catallactic molecule is compounded, when we suppose the Xs and Ys dealing in respect each of several articles with several sets of Zs, As, Bs, &c.; a case re- solved by M. Walras.
Thus the actual commercial field might be represented by sets of entrepreneurs Xs, Ys, Zs, each X buying labour from among sets of labourers, As, Bs, Cs, use of capital from among sets of capitalists, Js, Ks, Ls, use of land from among sets of landowners, Ps, Qs, Rs, and selling pro- ducts among a set of consumers consisting of the sum of the three aforesaid classes and the entrepreneurs of a species different from X, the Ys and Zs. As the demand of the labourer is deducible from considering his utility
1 See p. 17. It must be carefully remembered that Prof. Jevons’s Formule of Exchange apply not to bare individuals, an isolated couple, but (as he himself sufficiently indicates, p. 98), to individuals clothed with the properties of a market, a typical couple (see Appendix V.). The isolated couple, the catallactic atom, would obey our (a) law.
2 See p. 38.
32 MATHEMATICAL PSYCHICS.
as a function of wages received and work done, so the demand of the entrepreneur is deducible from consider- ing his utility as a function of (1) his expenditures on the agents of production; (2) his expenditures in the way of consumption ; (3) his receipts from sale of produce ; (4) his labour of superintendence. The last-named variable is not an article of contract ; but there being supposed a definite relation connecting the produce with agents of production and entrepreneur’s labour, the catallactic formulze become applicable. This is a very abstract representation (abstracting e.g. risk, foreign trade, the migration from one employment to another, e.g. Xs becoming Ys,’ &c.), yet more concrete than that of M. Walras, who apparently makes the more abstract supposition of a sort of frictionless entrepreneur, ‘ faisant? ni perte ni bénéfice.’
From the point of view just reached may with ad- vantage be contemplated one of the domains most recently added to Economic Science—Mr. Sidgwick’s contribution to the ‘Fortnightly Review,’ September, 1879. The wndirectness of the relation between wages and interest which Mr. Sidgwick has so clearly demonstrated in words is self-evident in symbols. The predete:minate- ness of the wage-fund, which has received its coup de grace from Mr. Sidgwick, must always, one would think, dave appeared untenable from the humblest mathemati- cal point of view, the consideration of the simplest type * of perfect competition ; from which also it must be added that Mr. Sidgwick’s — perhaps inadvertent, perhaps here misinterpreted—statement, * that contract
1 This permeability between employments (such as explained in Economics of Industry with reference to the supply of unskilled and skilled labour and of business power) tends to a level of utility.
2 Elements, Arts. 231, 242, &c. 3 See pp. 17, 31. 4 Fortnightly Review, 1879, pp. 410 (end) 411 (beginning),
PERFECT COMPETITION. 33
between employer and operative even in the case of _what is here called perfect competition, is indeterminate, does not, it is submitted, appear tenable. It is further submitted that Mr. Sidgwick’s strictures? on Prof. Jevons are hasty; for that by a (compound) employ- ment of the Jevonian (or an equivalent catallactic) for- mula, the complex relations between entrepreneur, capi- talist, and labourer are best made clear. And so ‘ there 1s &@ priort ground for supposing that industrial compe- tition tends to equalize the rate of profit (as well as interest) on capitals of different amount.’® That ‘ the labour of managing capital does not increase in propor- tion to the amount managed’ is so far from creating any peculiar difficulty, that it is rather of the essence of the theory of exchange; quite congruent with the familiar circumstance that the disutility of (common) labour (labour subjectively estimated) does not increase in pro- portion to work done (labour objectively estimated). That the labour of managing capital increases not only not at the same but at a less rate-of-increase than the amount managed, as Mr. Sidgwick seems to imply, is in- deed a peculiar circumstance; but it is of a sort with which the Jevonian formula, the mathematical theory of catallactics, is quite competent to deal, with which in fact Mr. Marshall has dealt in his second class of Demand-Curves.
? See Defin, p. 18
* Fortnightly Review, pp. 411, 412.
$ As the gain per unit of produce is the same for one X as for another X, and the gain per unit of capital lent is the same for one J as for another J; so, if there is in the field in addition to the classes prescinded, a class of capitalist-entrepreneurs, e.g. (J K)s, the gain per unit of produce is the same for one (J K) as for another (JK). But no equation is made between the gain of a (J K) and the sum of the gains of a J and a K; even if to simplify the comparison we abstract rent. (Gain of course in this statement meu- sured objectively, say in money, not subjectively in utility).
D
34 MATHEMATICAL PSYCHICS.
But it is not the purport of the present study to attempt a detailed, much less a polemical, discussion of pure Catallactics, but rather (y) to inquire how far con- tract is determinate in cases of imperfect competition. It is not necessary for this purpose to attack the general problem of Contract qualified by Competition, which is much more difficult than the general problem of un- qualified contract already treated. It is not necessary to resolve analytically the composite mechanism of a competitive field. It will suffice to proceed synthetically, observing in a simple typical case the effect of con- tinually introducing into the field additional com- petitors.
1. Let us start, then, from the abstract typical case above put (p. 28), an X and Y dealing respectively in z and y. Here x represents the sacrifice objectively measured of X; it may be manual work done, or com- modity manufactured, or capital abstained from during a certain time. And y is the objectively measured remuneration of X. Hence it may be assumed, accord- ing to the two first axioms! of the Utilitarian Calculus, the law of increasing labour, and the law of decreas-
ing utility, that P being the utility of X, (1' S* is | _ dP. dP GP dP continually negative, dy positive ; (2) da” dy dady’
continually negative. (Attention is solicited to the inter- pretation of the third condition.) No doubt these latter conditions are subject to many exceptions, especially in regard to abstinence from capital, and in case of pur-
1 See these laws stated in the companion calculus. The proofs were .
offered in Ménd, without acknowledgment, because without knowledge, of the cumulative proofs already adduced by Prof. Jevons. 2 Of. Appendix V.
RS ee
, aie
IMPERFECT COMPETITION. 35
chase not for consumption, but with a view to re-sale ; and in the sort of cases comprised in Mr. Marshall’s Class II. curves. Still, these exceptions, though they destroy the watertightness of many of the reasonings in this and the companion calculus, are yet perhaps of secondary importance to one taking a general abstract view.
This being premised, let us now introduce a second X and asecond Y ; so that the field of competition con- sists of two Xs and two Ys. And for the sake of illus- tration (not of the argument) let us suppose that the new X has the same requirements, the same nature as the old X; and similarly that the new Y is equal- natured with the old.
Then it is evident that there cannot be equilibrium unless (1) all the field is collected at one point ; (2) that point is on the contract-curve. For (1) if possible let one couple be at one point, and another couple at another point. It will generally be the interest of the X of one couple and the Y of the other to rush together, leaving their partners in the lurch. And (2) if the com- mon point is not on the contract-curve, it will be the interest of all parties to descend to the contract-curve.
The points of the contract-curve in the immediate neighbourhood of the limits yo and 42) cannot be final settlements. For if the system be placed at such a point, say slightly north-west of y&,, it will in general be pos- sible for one of the Ys (without the consent of the other) to recontract with the two Xs, so that for all those three parties the recontract is more advantageous than the previously existing contract. For the right line joining the origin to (the neighbourhood, of) yf will in general lie altogether within the indifference- curve drawn from the origin to y&. For the indif-
v2
36 MATHEMATICAL PSYCHICS.
ference-curve is in general convex to the abscissa. For its differential ee 1s
= (Sew) F(z =¥)) Oe) whence =
a -L GE) + G54) (GF r)* (az) [ Ga) + 39 zt (Fy)
which is perfectly positive. Therefore the indifference- curve (so far as we are concerned with it) is convex to the abscissa.
Now, at the contract-curve the two indifference-curves for X and Y touch. Thus the figure 1, page 28, is proved to be a correct representation, indicating that a point 2’7/ can be found both more advantageous for Y than the point on the contract-curve y,&, (on an interior indif- ference-curve, as it may be said), and also such that its co-ordinates are the sums (respectively) of the co-ordi- nates of two other points, both more advantageous for an X. These latter points to be occupied by X, and X, may be properly regarded (owing to the symmetry and competition) as coincident; with co-ordinates - a. Further, it appears from previous reasonings that there
will be a contract-relation between (z’y’) and (F f .
vey) FG §) namely ze . 7 ) = = n(eF 7 9,
; where EF’, is put for the
IMPERFECT COMPETITION. 37
first partially derived function (AEM)
When this relation is satisfied the system of three might remain in the position reached ; but for Y, who has been left out in the cold. He will now strike in, with the result that the system will be worked down to the contract-curve again; to a point at least as favour-
able for the Xs as = v Thus the Ys will have lost some
of their original advantage by competition: And a cer- tain process of which this 1s an abstract typical repre- sentation will go on as long as it is possible to find a point 2 y’ with the requisite properties. Attention to the problem will show that the process will come to a stop at a point on the contract-curve y, &, such that if a line joining it to the origin intersect the curve, the supple- mentary contract-curve as it me be called,
O(c y) *(B5)
5 (zy) y) F, (zy) ($4)
in the point 2’ y’ then ®(& y,) = P (a y’), provided that oY) falls within the indifference-curve for Y drawn
through (&, y,). If otherwise, a slightly different system of equations must be employed.
If now a third X and third Y (still equal-natured) be introduced into the field, the system can be worked down to a point €; 331 whose conditions are obtained
2 x: from those just written by substituting for S ¥ 2° a ay. For this represents the last point at which 2 Ys can re- contract with 3 Xs with advantage to all five. Analyti-
1 Compare the analysis in Appendix VIL
38 ; MATHEMATICAL PSYCHICS.
cal geometry will show that this point is lower down (in respect of the advantage of Y)than €, y,. In the limit, when the Xs and Ys are indefinitely (equally) multiplied, we shall have (2 y’) coincident with (€, y,,), or as we may say for convenience (£ 7), satisfying one or other of the alternatives corresponding to those just mentioned.
In case of the first alternative we have
ED, (Ey) + 7 #, (Ey) = 0
For ® (£ ) = @ (2 7) = ® ((1 + A) E(1 + &) 9). In the limiting case / is infinitesimal. Whence by dif- ferentiating the above equation is obtained. And the eecond alternative (5 ¥ not falling within the indif- ference-curve of Y)1is not to be distinguished from the first in the limiting case.
If this reasoning does not seem satisfactory, it would be possible to give a more formal proof; bringing out the important result that the common tangent to both indifference-curves at the point € 7 is the vector from the origin. |
By a parity of reasoning it may be shown that, if the system had been started at the north-west extremity of (the available portion of) the contract-curve, it would have been worked down by competition between the Xs © to the same point ; determined by the intersection with the contract-curve of € FY « + »F’, = 0; for the same point is determined by the intersection of either curve with the contract-curve. For the three curves evidently intersect in the same point.
Taking account of the two processes which have been described, the competing Ys being worked down for a certain distance towards the north-west, and similarly the competing Xs towards the south-east : we see that
PERFECT COMPETITION. 39
in general for any number short of the practically enjinite (if such a term be allowed) there is a finite length of contract-curve, from & Ym tO 2 Nm, at any point of which if the system is placed, it cannot by contract or recontract be displaced; that there are an indefinite number of final settlements, a quantity continually dimi- nishing as we approach a perfect market. We are brought back. again to case (8), on which some further remarks have been conveniently postponed to this place. (For additional illustrations see Appendix V.)
The two conditions, $7 + »%’, = 0 and €F’, + nF, = 0, just obtained correspond to Professor Jevons’s two equations of exchange. His formule are to be regarded as representing the transactions of two individuals in, or subject to, the law of, a market. Our assumed unity of nature in the midst of plurality of persons naturally brings out the same result. The represented two curves may be called demand-curves, as each expresses the amount of dealing which will afford to one of the dealers the maximum of advantage at a certain rate of
exchange a value of t . This might be elegantly ex-
pressed in polar co-ordinates, tan @ will then be the rate of exchange, and, if P be the utility of X, dP
Zp) == 0 is the demand-curve. By a well known
property of analysis (7) =(Q represents not only
maximum points, but minimum points; the lowest depths of valley, as well as the highest elevations, which one moving continually in a fixed right line from the origin over the utelty-surface would reach. This mini- mum portion of the demand-curve corresponds to Mr. Marshall’s Class 1. We see that the dealer at any given
40 MATHEMATICAL PSYCHICS.
rate of exchange, far from resting and having his end at a pomt on this part of the curve, will tend to move away from it. It has not the properties of a genuine demand- curve.
The dealing of an individual in an open market, in which there prevails what may be called the law of price, the relation between the individual’s require- ments and that quantity collectively-demanded-at-a-price, usually designated by the term Demand, between little d and big D in M. Walras’s terminology, is elegantly exhibited by that author. Compare also Cournot on ‘Concurrence.’
Here it is attempted to proceed without postulating the phenomenon of uniformity of price! by the longer route of contract-curve. When we suppose plurality of natures as well as persons, we have to suppose a plurality of contract-curves (which may be appropriately con- ceived as grouped, according to the well-known loga- rithmic law, about an average). Then, by considerations analogous to those already employed, it may appear that the quantity of final settlements is diminished as the number of competitors is increased. To facilitate conception, let us suppose that the field consists of two Xs, not equally, but nearly equally, natured ; and of two Ys similarly related. And (as in the fifth Appendix) let the indifference curves consist of families of concen-
tric circles. Then, instead of a single contract-curve, we have acontract-region, or bundle of contract-curves ; namely the four lines joming the centres of the circle- systems, the lines C,C’,, C,C’,, C,C’;, C,C’,; wherein (,, C, are the centres of X, and X,, supposed close together ; and similarly C’, and C’, for the Ys.
1 The term will sometimes Le used here for rute of exchange in general, as by M. Walras. —
PERFECT COMPETITION. - 4]
What corresponds here to that settlement of the whole field at a single point in the contract-curve, which we had under consideration in reasoning about equal- natured Xs, may thus be indicated. Take a point £4 on one of the contract-lines, say C,C’,; and let X, and Y, be placed there. Let X,Y, be placed at a neighbouring point, €”\7”,, on the line C,C’,; such that (1) &’1y”; is outside the two indifference curves drawn for X, and Y, respectively through &,7',; (2) &17'/, 1s outside the two indifference-curves drawn for X, and Y, respectively through &’yy”;.
Fi. 2,
, ‘ rf C,
hk | 7 coo
Then the settlement cannot be disturbed by an X and a Y simply changing partners, rushing into each other’s arms, and leaving their deserted consorts to look out for new alliances. Re-contract can now proceed only by one Y moving off with the two Xs, as in the previous case; by which process the system may be worked down to a neighbourhood describable as €,y,. In the limit, when the number of Xs and Ys are increased indefinitely, but not necessarily equally (suppose mX, and nY, where m and » are indefinitely large); if 2,y, represent the dealings of any X, viz. X,, and simi- larly € and » be employed for the dealings of the Ys, we should find for the 2m-+2n variables the following 2m + 2n equations:
(1) m+n equations indicating that each X and each
42 MATHEMATICAL PSYCHICS.
Y is on his individual demand-curve (compare the con- dition stated below, p. 48), e.g.
dF, (x, Y,) dF, (2, ¥,) — * Ga, 1 dy,
(the differentiation being of course partial). (2) m+n—1 equations indicating uniformity of
a= — — 1 — 12 — rice-t= 2 = &. = &e. ° P é”
3) ‘Ale last condition, which might perhaps be called par excellence the equation of Demand to Supply, namely, either Sa2= 3 €,or Sy=Xn. Thus the deal- ings of each and all are completely determinate and de- termined. 7
If we transform to polar co-ordinates, we might write any individual demand-curve, as p = /, (8); and thence obtain two collective demand-curves p = 8 f'(@) and p = 23 ¢(8); substantially identical with those collec- tive demand curves so scientifically developed by M. Walras, and so fruitfully applied by Mr. Marshall.
Thus, proceeding by degrees from the case of two isolated bargainers to the limiting case of a perfect market, we see how contract is more or less indeterminate according as the field is less or more affected with the first wmperfection, limitation of numbers.
i. Let there be equal numbers of equal-natured Xs and equal-natured Ys, subject to the condition that each Y can deal at the same time with only nXs, and similarly’ each X with only n'Ys. First let n=~7’. Then, in the light of the conceptions lately won, it ap- pears that contract is as indeterminate as if the field consisted of only nXs and nYs; that is to say, there are as many and the same final settlements as in that case, represented by the same portion of the contract-curve
COMBINATIONS. : 43 .
between (say) €y and «ry. Let n’ increase. Contract becomes less indeterminate: € moving north-west, and the quantity of final settlements being thereby diminished. The subtracted final settlements are most favourable to the Ys. Let ’ diminish. Contract becomes more inde- terminate; € moving south-east, and the quantity of final settlements being thereby increased. The added final settlements are more favourable to the Ys than those previously existing.
The theorem admits of being extended to the general case of unequal numbers and natures.
m1. Let there be an equal number N of equal- natured Xs and equal-natured Ys, and let each set be formed into equal combinations, there being n Xs in each X combination, and n’ Ysineach Y combination. First, let n =n’. Then contract is as indeterminate as if the
field consisted of * Xs and s Ys; in thesame sense as
that explained in the last paragraph. Let 7’ diminish. Contract becomes less indeterminate, in the same sense as in the last paragraph. Let 7’ increase. Contract be- comes more indeterminate ; the added final settlements being more favourable to the Ys than those previously existing. |
The theorem is typical of the general case in which numbers, natures, and combinations are unequal. Combination tends to introduce or increase indeter- minateness ; and the final settlements thereby added are more favourable to the combiners than the (determinate or indeterminate) final settlements previously existing. Combiners stand to gain in this sense.
The worth of this abstract reasoning ought to be tested by comparison with the unmathematical treat- ment of the same subject. As far as the writer is aware,
44 MATHEMATICAL PSYCHICS.
a straightforward answer has never been offered to the abstract question, What is the effect of combinations on contract in an otherwise perfect state of competition, as here supposed? Writers either! ignore the abstract question altogether, confining themselves to other as- pects of Trade Unionism; its tendency to promote communication, mobility, &c.; in our terms, to render the competition more normal, and more perfect in respect of extent (diminishing our first imperfection, for such is the effect of increased mobility, alike of goods and men). Or, while they seem to admit that unionism would have the effect of raising the rate of wages, they yet deny that the total remuneration of the operatives, the wage-fund (in the intelligible sense of that term), can be increased. But if our reasonings be correct, the one thing from an abstract point of view visible amidst the jumble of catallactic * molecules, the jostle of competitive crowds, is that those who form themselves into compact bodies by combination do not tend to lose, but stand to gain in the sense described, to gain in point of utility, which is a function not only of the (objective) remuneration, but also of the labour, and which, therefore, may increase, although the remuneration decrease; as Mr. Fawcett well sees (in respect to the question of unproductive
1 Mr. Sidgwick indeed (if the passage already referred to, Fortnightly Renew, p. 411, ante, p. 83, might be thus construed ?P)—at any rate some others have observed the momentous dead-lock resulting from the complete solidification of the whole operative-interest and the whole employer-interest ; our (a) case, contract unqualified by competition. But this hardly affords any indication of what would happen, or what the writers suppose would happen, when contract is qualified, however slightly, by competition ; as if, . for instance, there were two or three combinations on one side and two or three on the other; which in view of foreign competition is likely, one might think, to be long the concrete case.
2 Of. Cairnes on Trades Unions (first sections) ; Courcelle-Seneuil on Coalttions. :
COMBINATIONS. 7 45
consumption.—‘ Manual,’ ch. iv.), though he gives so uncertain a sound about Trades Unionism. And if, as seems to be implied in much that has been written on this subject, it is attempted to enforce the argument against Trades Unionism by the consideration that it tends to diminish the total national produce, the obvious reply is that unionists, as ‘economic men,’ are not concerned with the total produce. Because the total produce is diminished, it does not! follow that the labourer’s share is diminished (the loss may fall on the capitalist and the entrepreneur, whose compressibility has been well shown by Mr. Sidgwick in the article already referred to); much less does it follow (as aforesaid) that there should be diminished that quantity which alone the rational unionist is concerned to increase — the labourer’s utility. If this view be correct, it would seem as if, in the matter of wnzonism, as well as in that of the predeterminate wage-fund, the ‘ untutored mind’ of the workman had gone more straight to the point than economic intelligence misled by a bad method, reasoning without mathematics upon mathematical subjects.
Iv. Let there be an equal number N of equal: natured Xs and Ys; subject to the condition that to every contract made by a Y at least » Xs must be parties, and similarly for an X n’ Ys. First, let n=n’. Contract is as indeterminate as if the field consisted of N n more indeterminate, and the Ys stand to gain. And con- versely.
To appreciate the quantity of indeterminateness likely to result in fact from these imperfections (opera- ting separately and together) would require a knowledge
1 See the remarks in Appendix VII.
Xs and Ys, Let n’ increase. Contract becomes
46 MATHEMATICAL PSYCHICS.
of concrete phenomena to which the writer can make no claim.
The first imperfection applies to Monopolies. It is perhaps chiefly important, as supplying a clue for the solution of the other cases.
The second imperfection may be operative in many cases of contract for personal service. Suppose a market, consisting of an equal number of masters and servants, offering respectively wages and service ; sub- ject to the condition that no man can serve two masters, no master employ more than one man; or suppose equilibrium already established between such parties to be disturbed by any sudden influx of wealth into the hands of the masters. Then there is no determinate, and very generally! unique, arrangement towards which the system tends under the operation of, may we say, a law of Nature, and which would be predictable if we knew beforehand the real requirements of each, or of the average, dealer; but there are an indefinite number of arrangements a priort possible, towards one of which the system is urged not by the concurrence of innume- rable (as it were) neuter atoms eliminating chance, but (abstraction being made of custom) by what has been called the Art of Bargaining—higgling dodges and designing obstinacy, and other incalculable and often disreputable accidents. | |
Now, if managerial work does not admit of being distributed over several establishments, of being sold in bits, it would seem that this species of indeterminateness affects the contract of an entrepreneur with foreman, of a cooperative association of workmen (or a com- bination) with a manager. This view must be modified
1 Exceptions are the multiple intersections of Demand-Curves shown by Mr. Marshall and M. Walras.
COMBINATIONS. 47
in so far as managerial wages are determined by the cost of production (of a manager!), or more exactly by the equation! between managerial wages and the remuneration in other occupations, where the remu- neration is determined by a process of the nature of perfect competition; and by other practical considera- tions. %
The third imperfection may have any degree of importance up to the point where a whole interest (labourers or entrepreneurs) is solidified into a single competitive unit. This varying result may be tolerably well illustrated by the case of a market in which an indefinite number of consumers are supplied by varying numbers of monopolists (a case properly belonging to our first imperfection : namely, limited number of dealers). Starting with complete monopoly, we shall find the price continually diminish as the number of monopolists increases, until the point of complete fluidity is reached. This gradual ‘ extinction’ of the influence of monopoly is well traced by Cournot in a discussion masterly, but limited by a particular condition, which may be called uniformity of price, not (it is submitted) abstractedly neces- sary in cases of imperfect competition. Going beyond Cournot, not without trembling, the present inquiry finds that, where the field of competition is sensibly imperfect, an indefinite number of jinal settlements are possible; that in such a case different final settle ments would be reached if the system should run down from different initial positions or contracts, )The
1 In virtue of permeability between occupations ; postulating (1) free- dom of choice between different occupations, (2) knowledge of cireum- stances determining choice. With the latter sort of knowledge (so warmly impugned by Mr. Cliff Leslie) our free communication about articles of con~ tract (in normal market) is not to be confounded. See p. 18,
2 Cf, Walras’s Elements, 8. 352,
48 MATHEMATICAL PSYCHICS.
sort of difference which exists between! Dutch and English auction, theoretically unimportant in perfect competition, does correspond to different results, different Jinal settlements in imperfect competition. And in general, and in the absence of imposed conditions, the said final settlements are not on the demand-curve, but on the con- tract-curve. That is to say, there does not necessarily exist in the pase of imperfect as there does in the case of perfect competition a certain property (which some even mathematical writers may appear to take for granted), namely, that—in the case all along supposed of Xs and Ys dealing respectively in « and y—if any X X give x in exchange for y,, he gets no less and no more ¥ than he is willing to take at the rate of ex- change fr ;
If, however, this condition, though not spontaneously generated by wmperfect as by perfect competition, should be introduced ab extra, imposed by custom and con- venience, as no doubt would be very generally the case, nevertheless the property of indeterminateness, plurality of final settlements, will abide. Only the final settlements wll now be by way of demand-curve, not contract-curve. Jf, for instance, powerful trades unions did not seek to fix the guid pro quo, the amounts of labour exchanged for wealth (which they would be quite competent to seek), but only the rate of exchange, it being left to each capitalist to purchase as much labour as he might demand at that rate, there would still be that sort of «ndeterminateness favourable to unionists above described. The geometry of this case may be understood from an attentive consideration of
1 As Thornton suggests. Now we believe, but not because that un- mathematical writer has told us.
COOPERATIVE ASSOCIATIONS. 49
the typical illustration at the end of Appendix V., fig. 4.
The fourth imperfection would seem likely to operate in the case of cooperative associahons up to the time when the competitive field shall contain a practically infinite number of such bodies; that is, perhaps for a long time. To fix the ideas, suppose associa- tions of capitalist-workmen, consisting each of 100 members, 50 contributing chiefly capital, and 50 chiefly labour. Let the field of competition consist of 1,000 individuals. The point here indicated is that, not- withstanding the numerical size of the field, contract will not be more determinate (owing to the fact that all the members of the association are tn contract with each other—not, as now usual, each for himself contracting with employer) than if the field consisted of 10 indi- viduals. And w similar result would hold if, with more generality, we suppose members contributing labour and capital in varying amounts, and remunerated for their sacrifices according to a principle of distribution ; in the most, or, at any rate, a sufficiently general case, a function of the sacrifices, the form of the function being a contract-variable, or what comes to much the same thing, there being assumed a function of given form containing any number of constants, which are articles of contract, subject, of course, to the condition that the sum of the portions assigned is equal to the distribuend. And, similarly, if we introduce different kinds of labour and other concrete complications.
The Determinateness will depend not so much upon the number of individuals as upon the number of associations in the field. As cooperative association becomes more prevalent, no doubt, ceteris paribus, the indeterminateness here indicated would decrease.
E
50 MATHEMATICAL PSYCHICS.
Nevertheless, in consequence of the great variety of cooperative experiments, the sundry kinds of contract and divers species of articles, the field of competition being thus broken up, it is submitted that the rise of cooperative association is likely to be accompanied with the prevalence of! indeterminateness, whatever Opinion we may form about the possible regularity in a distant future.
Altogether, if of two great coming institutions, trades- unionism is affected with the third imperfection, and cooperative association with the fourth, and both with the second, it does not seem very rash to infer, if not for the present, at least in the proximate future, a consider- able extent of indeterminateness.
Of this inference what would be the consequence. To impair, it may be conjectured, the reverence paid to competition; in whose results—as if worked out by a play of physical forces, impersonal, impartial—economists have complacently acquiesced. Of justice and huma- nity there was no pretence; but there seemed to com- mand respect the majestic neutrality of Nature. But if it should appear that the field of competition is deficient in that continuity of fluid, that multiety of atoms which constitute ® the foundations of the uniformities of Physics; if competition is found wanting, not only the regularity of law, but even the impartiality of chance—the throw of a die loaded with villainy—economics would be indeed a ‘ dismal science,’ and the reverence for com- petition would be no more.
1 There has been, I believe, observed in cooperative associations, with regard to the comparative remunerations of capital and labour, that dispute without any principle of decision which is the characteristic of contract.
4 Above, pp. 5, 18.
3 Theory of Vortices and Theory of Atoms.
NEED OF ARBITRATION. 51
There would arise a general demand for'a principle of arbitration.
And this aspiration of the commercial world would be but one breath in the universal sigh for articles of peace. For almost every species of social and political contract is affected with an indeterminateness like that which has been described ; an evil which is likely to be much more felt when, with the growth of intelligence and liberty, the principle of contract shall have replaced both the appeal to force and the acquiescence in custom. Throughout the whole region of in a wide sense contract, in the general absence of a mechanism like perfect com- petition, the same essential indeterminateness prevails ; in international, in domestic politics; between nations, classes, sexes.
The whole creation groans and yearns, desiderating a principle of arbitration, an end of strifes.
Corottary.— Where, then, would a world weary of strife seek a principle of arbitration? In justice, replies the moralist ; and a long line of philosophers, from Plato to Herbert Spencer, are ready to expound the principle. But their expositions, however elevating in moral tone, and of great -hortative value for those who already know their duty, are not here of much avail, where the thing sought is a definite, even quantitative, criterion of what is to be done. Equity and ‘ fairness of division’ are charming in the pages’ of Herbert Spencer, and delighted Dugald Stewart with the appearance? of mathe- matical certainty ; but how would they be applicable to the distribution of a joint product between coopera- tors? Nor is the equity so often invoked by a high authority on cooperation much more available; for why is the particular principle of distribution recom-
} Data of Ethics, p. 164, + Besays, Book II. E2
52 MATHEMATICAL PSYCHICS.
mended by Mr. Holyoake (operatives to take net pro- duct, paying therefrom a salary to manager, roughly speaking, and to say nothing of capital) more equitable than an indefinite number of other principles of distri- bution (e.g. operatives to take any fraction which might have been agreed upon, manager the remainder ; either party, or neither, paying wages to the other).
Justice requires to be informed by some more defi- nite principle, as Mill’ and Mr. Sidgwick reason well. The star of justice affords no certain guidance—for those who have loosed from the moorings of custom— unless it reflect the rays of a superior luminary—utili- tarlanism.
But, even admitting a disposition in the purer wills and clearer intellects to accept the just as finis hium, and the useful as the definition of the just; admitting that there exists in the higher parts of human nature a tendency towards and feeling after utilitarian institu- tions; could we seriously suppose that these moral considerations were relevant to war and trade; could eradicate the ‘ controlless core’ of human selfishness, or exercise an appreciable force in comparison with the impulse of self-interest. It would have to be first shown that the interest of all is the interest of each, an illusion to which the ambiguous language of Mill, and perhaps Bentham, may have lent some countenance, but which is for ever dispelled by the masterly analysis of Mr. Sidgwick. Mr. Sidgwiek acknowledges two supreme principles—Egoism and Utilitarianism ; of independent authority, conflicting dictates; irreconcilable, unless indeed by religion.
It is far from the spirit of the philosophy of pleasure to depreciate the importance of religion; but in the
1 See review of Thornton on Labour (as well as Utihtariansem).
PRINCIPLE OF ARBITRATION. 63
present inquiry, and dealing with the lower elements of human nature, we should have to seek a more obvious transition, a more earthy passage, from the principle of self-interest to the principle, or at least the practice, of utilitarianism.
Now, it is a circumstance of momentous interest— visible to common sense when pointed out by mathe- matics—that one of the in general indefinitely nume- rous settlements! between contractors is the utilitarian arrangement of the articles of contract, the contract tending to the greatest possible total utility of the con- tractors. In this direction, it may be conjectured, is to be sought the required principle. For the required basis of arbitration between economical contractors is evidently some settlement; and the utilitarian settle- ment may be selected, in the absence of any other principle of selection, in virtue of its moral peculiarities :
- (a@P\ (dO aqi\ (aP 1 . ae — — oa —emeese = Where the contract-curve is ( 7 ~.) ( d 4) ( 3 4) ( Ty 0, the utilitarian point has co-ordinates determined by the equations
a ad (+5) [P+0] = 0 =) [P+] =0; the roots of which evidently satisfy the contract-equation. The theorem is quite general.
Here may be the place to observe that if we suppose our contractors to be in a sensible degree not ‘economic’ agents, but actuated in effective moments by a sympathy with each other's interests (as even now in domestie, and one day perhaps in political, contracts), we might suppose that the object which X (whose own utility is P), tends—in a calm, effective moment —to maximise, is not P, but P + A II; where A is a coeffictent of effective sympathy. And similarly Y—not of course while rushing to self-gratifica- tion, but in those regnant moments which characterise an ethical ‘ method ’ —may propose to himself as end II + » P. What, then, will be the con- tract-curve of these modified contractors? The old contract curve between narrower limsts. In fig. 1, y. & will have been displaced in a north-westerly and n. 7, in 8 south-easterly direction. As the coefficients of sympathy in- crease, utilitarianism becomes more pure, (cf. pp. 12, 17), the contract-curve narrows down to the uitltartan point.
54 MATHEMATICAL PSYCHICS.
its satisfying the sympathy ! (such as it is) of each with all, the sense of justice and utilitarian equity.”
These considerations might be put clearest in a particular, though still very abstract, case. Let us sup- - pose that in consequence of combinations competition fails to determine the contract between entrepreneur and operatives. The case becomes that described under (a)—deadlock between two contracting parties. One of the parties is indeed here collective; but it is allow- able for the sake of illustration to make abstraction of this circumstance, to abstract also the correlated bar- gains with capitalists, landowners, &c., and to suppose a single entrepreneur in dealing with a single operative.
And, first, let it be attempted to arbitrate upon | some principle of doctrinaire justice—some metaphysical — doyma, for instance, of equality: that the entrepreneur shall have an ‘equal’ share of the produce. Now, there is no presumption that this ‘fair division’ is utilitarianian; in view of the different character of the entrepreneur’s sacrifice, in view also (if one may be allowed to say so) of a possible difference in the entre- preneur’s capacity :® suppose, for instance, that a more highly nervous organisation required on the average a higher minumum of means to get up to the zero of utility. As there is no presumption that the proposed arrangement is utilitarian, so there is no presumption. that it is on the contract-curve. Therefore, the self- interests of the two parties will concur to bulge away from the assumed position ; and, bursting the cobwebs
1 Assuming as economists assume (see Mill, book II. chap. xiv. s. 7, Walker on Wages, &o.), an however slight clinamen from the rectilinearity of the ‘ economic man.’
2 ‘Whereof the unconsciously implicit first principle is: Time-intensity units of pleasure are to be equated irrespective of persons.
5 Seo p. 58.
PRINCIPLE OF ARBITRATION. 55
of doctrinaire justice, to descend with irresistible force to some point upon the contract-curve. Suppose that by repeated experiences of this sort the contract-curve has been roughly ascertained—a considerable number of final settlements statistically tabulated. Now these positions lie in a reverse order of desirability for each party ; and it may seem to each that as he cannot have his own way, in the absence of any definite principle of selection, he has about as good a chance of one of the arrangements as another. But, rather than resort tc some process which may virtually amount to tossing up, both parties may agree to commute their chance of any of the arrangements for the certainty of one of them, which has certain distinguishing features and peculiar attractions as above described—the utilitarian arrangement.
Or perhaps, considering the whole line of possible arrangements, they might agree! to ‘ split the difference,’ and meet each other in the neighbourhood of the cen- tral point—the ‘quantitative mean,’ as it might be called. Well, first, this quantitative mean would likely to be nearer than the extremes to the utilitarian point ; and, further, this very notion of mean appears to be the outcome of a rudimentary ‘implicit’ justice, apt in a dialectical atmosphere to bloom into the ‘ qualitative ? mean’ of utilitarian equity.
1 See p. 185.
* Aristotle’s metaphysical theory that virtue is a mean between two vices is analogous to the mathematical theory that a maximum of pleasure is @ mean between two minima.
So also Aristotle’s notion of two species of excellence (dpern), and more generally all cases in which there seem to be two (or more) best ways of acting (using the superlative in a sense analogous to the proper mathe- matical sense of ‘ maximum’), may be cases of multsple solutions of a problem in the Calculus of Variations, the problem of maximum utility.
It is difficult to allude to Mr. Todhunter’s beautiful and delicate problems
56 MATHEMATICAL PSYCHICS.
Or less specifically may we say that in the neigh- bourhood of the contract-curve the forces of self-interest being neutralised, the tender power of sympathy and right would become appreciable ; as the gentler forces of the magnetic field are made manifest when terrestrial magnetism, by being opposed to itself, is eliminated.
Upon the whole—omitting what it is obvious to understand about the spirit in which very abstract reasonings are to be regarded: a star affording a general direction, not a finger-post to specify a by- path—there may appear, at however great a distance, a general indication that competition requires to be sup- plemented by arlitration, and the basis of arbitration between self-interested contractors 1s the greatest possible sum-total utility.
Thus the economical leads up to the utilitarian cal- culus; the faint outlines of which, sketched in a pre- viously published paper, may be accepted as the second subdivision of our Second Part.
UTILITARIAN CALCULUS,
PROBLEM.—To find (a) the distribution of means and (8) of labour, the (y) quality and (8) number of popu- lation, so that there may be the greatest possible hap- piness.
DerrFinitions.—(1) Pleasure is used for ‘ preferable feeling’ in general (in deference to high authority, though the general term does not appear to call up with equal facility all the particulars which are meant to be
Without once more inviting attention to the versatile features and almost —
human complexion of that species of Calculus which seems most directly applicable to the affairs of men; so different from the brutal rigour ascribed to Mathematics by men who are acquainted only with its elements.
UTILITARIAN CALCULUS. - my
included under it, but rather the grosser’ feelings than for instance the ‘joy and felicity’ of devotion). The term includes absence of pain. Greatest possible hap- puness is the greatest possible integral of the differential ‘Number of enjoyers x duration of enjoyment x degree thereof’ (cf. axiom below).?
(2) Means are the distributable proximate means of pleasure, chiefly wealth as destined for consumption and (what is conceivable if not usual in civilisation) the un- purchased command of unproductive labour.
(3) An individual has greater capacity for happiness than another, when for the same amount whatsoever of. means he obtains a greater amount of pleasure, and also for the same increment (to the same amount) whatsoever of means a greater increment of pleasure.
This ‘definition of a thing’ is doubtless (like Euclid’s) imperfectly realised. One imperfection is that some individuals may enjoy the advantages not for any amount of means, but only for values above a certain amount. This may be the case with the higher orders of evolu- tion. Again, one individual may have the advantages in respect of one kind of means, another of another. But, if one individual has the advantages in respect of most and the greatest pleasures, he may be treated as having more capacity for pleasure in general. Thirdly, the two advantages may not go together. If ‘the higher pleasures, such as those of affection and virtue, can
1 Compare the base associations of ‘ Utilitarianism.’ Surely, as Mr. Arnold says, a pedant invented the term.
® The greatest possible value fff, if dp dn dt (where dp corresponds
to a just perceivable increment of pleasure, dn to a sentient individual, di to an instant of time). The limits of the time-integration are 0 and o, the present and the indefinite future. The other limits are variable, to be deter- mined by the Calculus of Variations,
58 MATHEMATICAL PSYCHICS.
hardly be said to come from pleasure-stuff at all’ (as Mr. Barratt says in his able Note in ‘ Mind X..,’ often. cited below), it is possible (though not probable ?) that the enjoyers of the higher pleasures should derive from the zero, or rather a certain minimum, of means (and @ fortiort for all superior values) an amount of pleasure greater than another class of enjoyers, say the sensual, can obtain for any amount whatsoever of means; while at the same tume the sensual obtain greater incre- ments of pleasure for the same increments of means (above the minimum). In such a case the problem would be complicated, but the solution not compromised. Roughly speaking, the first advantage would dominate the theory of population; the second the distribution of means. A fourth imperfection in the statement of the definition is that the units whose capacities are com- pared are often groups of individuals, as families. With these reservations the reality of the definition may be allowed.
But it may be objected that differences of capacity, though real, are first not precisely ascertainable, and secondly artificial, being due to education. But, first, - even at present we can roughly discriminate capacity — for happiness. If the higher pleasures are on the whole most pleasurable—a fact of which the most scientific statement appears to have been given by Mr. Sully! —then those who are most apt to enjoy those pleasures tend to be most capable of happiness. And, as Mr. Barratt says, it ‘seems (speaking generally) to be the fact that, the higher a being in the scale of evolution, the higher its capacity for pleasure ;’ while greater pre- cision might be attainable by improved examinations and hedonimetry. Further it will be seen that some of the
1 Pessimism, note to chap. Xi.
DEFINITIONS. 59
applications of the problem turn upon supposed, rather than ascertained, differences of capacity. The second objection, William Thompson’s, would hardly now be maintained in face of what is known about heredity. But it is worth observing that his conclusion, equality of distribution, follows from his premiss only in so far as a proposition like our first postulate (below) is true of wealth and labour applied to education, in so far as it is true that improvement is not proportionately increased by the increase of the means of education.
(4) An individual has more capacity for work than another,! when for the same amount whatsoever of work done he incurs a less amount of fatigue, and also for the same increment (to the same amount) whatsoever of work done a less increment of fatigue.
This fourth definition may present the same imper- fections as the third. Indeed the fourth definition is but a case of the third; both stating relation between means and pleasure. The third definition becomes the fourth, if you change the signs of means and pleasure, put means produced for means consumed and the pains of produc- tion for the pleasures of consumption. Or not even the latter change, in so far as labour is sweet (which is very far according to Fourier). It is submitted that this identification confirms the reality of the third definition, since the reality of the fourth is undisputed. Of course, if we identify the definitions, we must bear in mind that they are liable to be separated in virtue of the second imperfection above noticed.
Axiom.—Pleasure is measurable, and all pleasures are commensurable; so much of one sort of pleasure
1 Or this: When the same amount of fatigue corresponds to a greater amount of work done, and the same increment (to the same amount) of fatigue to # greater increment of work.
60 MATHEMATICAL PSYCHICS.
felt by one sentient being equateable to so much of other sorts of pleasure felt by other sentients.
Professor Bain has shown! how one may correct one’s estimate of one’s own pleasures upon much the same principle as the observations made with one’s senses; how one may correctly estimate the pleasures of others upon the principle ‘ Accept identical objective marks as showing identical subjective states,’ notwith- standing personal differences, as of activity or demon- strativeness. This ‘ moral arithmetic’ is perhaps to be supplemented by a moral differential calculus, the Fech- nerian method applied to pleasures in general. For Wundt has shown that sensuous pleasures may thereby be measured, and, as utilitarians hold, all pleasures are commensurable. The first principle of this method might be: Just-perceivable increments of pleasure, of all pleasures for all persons, are equateable.? Impli- cated with this principle and Bain’s is the following: Equimultiples of equal pleasures are equateable ; where the multiple of a pleasure signifies exactly similar plea- sure (integral or differential) enjoyed by a multiple number of persons, or through a multiple time, or — (time and persons being constant) a pleasure whose degree is a multiple of the degree of the given pleasure. The last expression is open to question (though see Delbeeuf ‘Etude psychophysique,’ vii. and elsewhere), and is not here insisted upon. It suffices to postulate the practical proposition that when (agreeably to Fech- nerian conceptions) it requires n times more just-per- ceivable increments to get up to one pleasure from zero than to get up to another, then the former pleasure enjoyed by a given number of persons during a given |
1 Emotions and Wul, 3rd edition. 2 Of. Wundt, Phys. Psych., p. 295; above, p. 8, Appendix ITI,
THE UNIT OF PLEASURE. 61
time is to be sought as much as the latter pleasure en- joyed by 7 times the given number of persons during the given time, or by the given number during the multiple time. Just so one cannot reject the practical conclusions of Probabilities, though one may object with Mr. Venn to speaking of belzef being numerically measured. Indeed these principles of petpytixyn are put forward not as proof against metaphysical subtle- ties, but as practical; self-evident @ priort, or by what- ever ézayeyy or €fiopds is the method of practical axioms.
Let us now approach the Problem, attacking its inquiries, separately and combined, with the aid of appropriate POSTULATES. |
(a)! The first postulate appropriate to the first in- quiry is: The rate cf increase of pleasure decreases as its means increase. The postulate asserts that the second differential of pleasure with regard to means is continually negative. It does not assert that the first differential is continually positive. It is supposable (though not probable) that means increased beyond a certain point increase only pain. It is also supposable that ‘the higher pleasures’ do not ‘ come from pleasure- stuff at all,’ and do not increase with it. Of course there are portions of the utilitarian whole unaffected by our adjustments; at any rate the happiness of the stellar populations. But this does not invalidate the postulate, does not prevent our managing our ‘ small peculiar’ for the best, or asserting that in respect thereof there tends to be the greatest possible happiness. The proposition thus stated is evidenced by every-day experience ; experience well focused by Buffon in his
1 See the cumulative proofs of this postulate adduced by Professor Jevons in Theory of Political Economy.
62 MATHEMATICAL PSYCHICS.
‘Moral Arithmetic,’ Laplace in his ‘Essay on Proba- bilities,’ William Thompson in his ‘Inquiry into the Distribution of Wealth,’ and Mr. Sidgwick in the ‘ Me- thods of Ethics.’
This empirical generalisation may be confirmed by ‘ratiocination ’ from simpler inductions, partly common to the followers of Fechner, and partly peculiar to Professor Delbceuf. All the formule suggested for the relation between quantity of stimulus and intensity of sensation agree in possessing the property under con- sideration; which is true then of what Professor Bain would describe, as pleasures of mere intensity ; coarse pleasures indeed but the objects of much expenditure. Thus pleasure is not proportionately increased by in- creased glitter of furniture, nor generally by increased scale of establishment ; whether in the general case by analogy from the Fechnerian experiments on the senses? or by a more @ priori ‘law of relation’ in the sense of Wundt.
But not only is the function connecting means and pleasure such that the increase of means does not pro- duce a proportionate imcrease of pleasure; but this effect is heightened by the function itself so varying (on repetition of the conditions of pleasure) that the same means produce less pleasure. The very parameter in virtue of which such functional variation occurs is exhibited by Professor Delbceuf in the case of eye-sen- sations ;” that a similar variation holds good of pleasures in general is Bain’s Law of Accommodation. Increase of means then, affording proportionately increased re- petition of the conditions of pleasure, does not afford proportionately increased pleasure. Doubtless there
1 Cf. Fechner, Psychophysik, vol. ix. p. 6. 2 Etude psychophysique, &c.
LAW OF DIMINISHING UTILITY. 63
are compensations for this loss ; echoes of past pleasures, active habits growing up in the decay of passive im- pressions. Indeed the difference of individuals in res- pect to these compensations constitutes a large part of the difference of capacity for pleasure.
It may now be objected: increased means do not operate solely by repeating old pleasures, but also by introducing to new (¢.g. travel); also the ‘compensa- tions’ may more than counterbalance the accommodations. It is generally replied: In so far as a part only of hap- piness increases only proportionately to its means, the second differential of happiness with regard to means does not cease to be negative. That second differential cannot be continually negative. Its being negative for a space may not affect the reasoning. If it does affect the reasoning, one conclusion, the inequality of distri- bution, would probably (if the pleasure-curve is not very complicated) become @ fortiort. Not only would the less capable receive then still less means, but even the equally capable might then not all receive equal means.
This being postulated, let us mark off the degrees of capacity for happiness on an abscissa (supposing that capacity is indicated by the values of a single variable ; if by the values of a function of several variables, the proof differs only in complexity). At each degree erect an ordinate representing the number of individuals of that degree of capacity. On the rectangle correspond- ing to each individual it is required to construct a parallelopiped representing his means. Let us proceed to impart the distribuend means—in the first inquiry a given distribuend to given distributees doing each a given amount of labour—by way of small increments. Let us start with the assumption that each individual has
64 ' MATHEMATICAL PSYCITICS.
and shall retain that minimum of means just sufficient to bring him up to the zero-point of happiness (a con- ception facilitated by, though not quite identical with, the economical ‘natural minimum of wages’). There- after who shall have the first increment of means? By definition an individual of the highest capacity (at least supposing the minimum to be the same in all capa- cities). Who shall have the next increment of means ? Another individual of the highest capacity, in preference to the same individual by the postulate. Thus a first dividend will be assigned to the first section (all the individuals of the highest capacity) exclusively. But they will not continue sole assignees. Their means only, being continually increased, must by the postulate reach a point such that an increment of means can be more felicifically assigned to an individual of the second section (the next highest capacity) than to one of the first. The second section will then be taken into distri- bution. Thus the distribution of means as between the . equally capable of pleasure is equality; and generally is such that the more capable of pleasure shall have more means and more pleasure.
The law of unequal distribution is given by a plane curve, in the plane of the capacities and means, say a megisthedone. To different distribuends correspond megisthedones differing only by a constant. For it is . educible from the postulate that there is only one family of megisthedones. We may have any number of maxima by tacking between different members of the family; but the greatest possible value is afforded by the continuous solution.
If we now remove the condition that each individual shall retain his minimum, what happens? Simply that
1 Compare the reasoning in the ordinary Theory of Rent.
DISTRIBUTION OF MEANS. 65
the megisthedones may now dip below the minimum line. But it is improbable that they should dip very low under the minimum at the lower end while they rise very high above the minimum at the higher end; since excessive physical privations cannot be counter- balanced by any superfluity of refined pleasures. In fact, if we assume that the zero of means corresponds to injinite pain of privation (¢f. Wundt’s curve of pleasure and pain), then by investigating the radius of curvature it is shown that, as the distribuend dimin- ishes, the megisthedone tends to become a horizontal line. In famine the distribution even between unequals is equality—abstracted ulterior considerations, as of posterity.
These conclusions may be affected by the imperfec- tions of the third definition. By the first imperfection, if the ‘minimum’ line were not horizontal. Secondly, suppose that the individuals who have less capacity for pleasures in general have a special capacity for parti- cular pleasures. The bulk of means will be distributed as before, but there will be a residue distributed according to a second megisthedone. The second megis- thedone superimposed upon the first will more or less deform it. Lastly, the unit distributee is often a group (e.g-, a married couple, in respect of their common ménage). The conclusions may be affected, in so far as the most capable groups are made up of individuals not most capable as individuals.
(8) The distribution of labour (to which attention has been called by Mr. Barratt) is deduced by a parity of reason from the parallel second aviom: that the rate of increase of fatigue increases as the work done in- creases, which is proved by common experience and (for muscular work) by the experiments of Professor
F
66 MATHEMATICAL PSYCHICS.
Delbceuf (‘Etude Psychophysique’). As appears in- deed from Professor Delbceuf’s formuls, the first and second postulates are to a certain extent implicated (whereby the first postulate gains strength). Let us now arrange our individuals according to their capacity for work, and proceed as before. Who shall do the first increment of work ? Of course one of the most capable of work. And so on. The distribution of labour as between the equally capable of work 1s equality, and generally 1s such that the most capable of work shall do more work—so much more work,! as to suffer more fatigue.
The inquiry presents the same declensions as the first. In particular, cooperatives are to be compared. not inter se, but with the similar operatives in similar cooperative associations: except, indeed, so far as the work done is a symmetrical function of the effort of fellow-workers. It is deducible that the rowers of a nds étons shall have equal fatigue; but the fatigue of the pilot is not to be equated to that of the oarsman. All the while it is to be recollected that the fatigue or pain of work under consideration may be negative.
(a8) To combine the first and second inquiries, determine by the Differential Calculus the constants of a megisthedone and a brachistopone such that the means distributed by the former may be equal to the work distributed by the latter and that the (algebraical) sum of the pleasures of consumption and the pains of pro- duction may be the greatest possible. - Or, ab initio, by the Calculus of Variations, we may determine the means and fatigue as independent variable functions satisfying those two conditions.
1 This inference requires the second form of the fourth definition, given in the Note. |
DISTRIBUTION OF WORK. 67
Let V=f'n[Rey)— poly Sep)! 42
where z is degree of either capacity, or more elegantly, if possible, a third variable in terms of which both ca- pacities may be expressed ; zx, and x, are the given limits of integration (the number and quality of the distributees being not in the present inquiry variable) ; 7 is the number of each section; F(z y) is a unit’s pleasure of consump- tion, being a function of z his quality (capacity for pleasure) and the independent variable y his means; p is the unit’s pain of work, another independent variable function; ¢ is the constant incidental to problems of relative maximum ; /(xp) is the work done by the unit, being a function of his quality (capacity for work) and fatigue (effort).
Greatest possible happiness = greatest possible value
2 of Jn ey) - plde= | 2, greatest possible value of V, ¢ being taken so that Rs SJ, nly -fep)]d2 = 0. Lo The second term of the variation of V, n[Byo- at 8p Fs) is continually negative by the postulates. Therefore
the greatest possible value of V is when its first term of variation vanishes. The first term of variation,
woo{(E) —] + 9[ 036) -1]
68 MATHEMATICAL PSYCHICS.
vanishes only when both
ad¥ d 1
(7) =c and (35) = * If these equations hold, the two rules (a and £) hold. Q.E.D. The combined solution takes for granted that the means of pleasure and the pain of work are inde- pendent variables. And to a certain extent this may fail to be the case. An individual may want strength or tume to both enjoy the means and do the work which the double rule assigns to him. In that case there will be a compromise between the two rules.
(y) The third postulate simplifying the third inquiry is that capacity for pleasure and capacity for work generally speaking go together; that they both rise with evolution. The quality of population should be the highest possible evolution—provided ? that the first im- perfection of the third definition does not give us pause. To advance the whole population by any the same degree of evolution is then desirable; but it is probably not the most desirable application, given quantity of. a of means of education. For it is probable that the highest in the order of evolution are most capable of education and improvement. In the general advance the most advanced should advance most.
(8) The fourth postulate essential to the fourth in- quiry is that, as population increases, means (the dis- tribuend) increase at a decreasing rate. This is given by the Malthusian theory with regard to the products of extractive labour. And this is® sufficient. For the second differential of the whole means with regard to
1 See New and old Methods of Ethics (by the present writer), p. 72.
2 Ibid. p. 77.
S This is not quite accurate. For a part of the distribuend may increase more than proportionately in virtue of economies effected by increased pro-~
UTILITARIAN SELECTION. 69
population is still negative, even though a part of means increase proportionately to the number of population ; for instance, unproductive labour requiring little or no materials (¢.g., ballet-dancers), or those manufactured articles of which the cost is not appreciably affected by the cost of the raw material. From this Malthusian premiss it is deduced that population should be limited ; but the hedonical conclusion is not necessarily of the same extent as the Malthusian (¢/. below ay8). A simple inquiry under this head is the following, Assuming that all the sections (degrees of capacity or orders of evolu- tion) multiply equally, and that each section reproduces exactly his kind, to find the (utilitarian) rate of increase?
(78) A more important inquiry is: not assuming that all sections multiply equally, to find the average issue for each section, so that the happiness of the next gene- ration may be the greatest possible.
First let us introduce a conception more appropriate than was possible under the preceding head; namely, that each section does not reproduce exactly its kind, but that the issue of each (supposed endogamous) sec- tion ranges on either side of the parental capacity, as thus— @
===) 7 n v= Be x53 where is the capacity of the
7 — 2% parental section, n its number ( = something like Ae a
duction. In the same manner, and for the same reason as a demand-curve may have a plurality of intersections with a vector from the origin (Cf. Mr, Marshall’s theorem) corresponding alternately to maximum and minimum utility, so there may be a plurality of values for the sought number of population, corresponding alternately to utilitarian and pessimistic arrange- ment. The highest value which satisfies the equation to zero of the first term of variation must correspond to a maximum.
‘The imperfection of this postulate does not affect the reasoning based upon the other postulates,
70 MATHEMATICAL PSYCHICS.
since the parental generation is to be conceived as ranging under a curve of possibility; cf. Galton, Que- telet, &c.), y 1s the number of issue of capacity z. Perr haps 5 is constant for all the curves of issue; the variation of 8 alone determines the natural maximum, or artificial limit, of the average issue. But neither the symmetry of the curves of possibility, nor the par- ticulars of this conception, are postulated.
The fifth postulate appropriate to this case is that to substitute in one generation for any number of parents an equal.number each superior in capacity (evolution) is beneficial for the next generation. This being granted, either analytically with the aid of Mr. Todhunter’s ‘Researches,’* or by unaided reason, it is deduced that the average issue shall be as large as possible for all sections above a determinate degree of capacity, but zero for all sections below that degree.
But can we be certain that this method of total selec- tion as it might be termed holds good when we provide not only for the next generation, but for the indefinite future? In the continuous series of generations, wave propagating wave onward through all time, it 1s required to determine what wavelet each section of each wave shall contribute to the proximate propagated ‘wave, so that the whole sum of light of joy which glows in the long lme of waves shall be the greatest possible. If in the distant future, agreeably to the views of Herbert Spencer, population tends inartificially to become nearly stationary ; if to the contemplator of all time genera- tions fade into differentials ; we may conceive formed a differential equation connecting the population of one generation with the population of its successor and in-
1 See Appendix I. p. 93.
UTILITARTAN SELECTION. 71
volving an independent variable function, the average issue for each section. By the Calculus of Variations (if the calculator is not at sea) it is educed that the average issue shall be as large as possible for all sections above a (for each time) determinate degree of capacity, but zero for all sections below that degree. But a further postulate is required for so long as the movement of population is not amenable to infinitesimal calculus; while the present initial irregular disturbances are far from the tranquil waves of the‘ stationary’ state. This sixth postulate might be: To substitute in one generation for any number of parents an equal number each supe- rior in capacity (evolution) is beneficial for all time. This postulate being granted, if possible let the most beneficial selection be not total. Then a total selection can be arranged more beneficial !
If only we have swum through the waves to a terra Jirma, our position need not appear outlandish. For, first, these rules are jvery general, founded on very abstract tendencies, and requiring to be modified in practice. Thus our principle of selection might be modified, in so far as endogamy should not be the rule, if the higher orders of evolution have a greater tendency to reversion (in violation of the fifth and sixth postulates), and so forth. Again, since to exclude some sections from a share of domestic pleasures interferes with the principle of (a), it could not be expedient to sacrifice the present to the future, without the highest scientific certainty and political security. Again to indicate an ideal, though it can only be approached dvOpwrivas, may be useful. What approach is useful in such cases is to be determined by Mr. Todhunter’s principle.t Again, mitigations might be provided for the classes not
» Researches ; below p. 98.
72 MATHEMATICAL PSYCHICS.
selected! In particular, they might have the benefit of rule (8) now almost cut away by the struggle of com- petition. Again, emigration might supplement total selection; emigration from Utopia to some unprogres- sive country where the prospect of happiness might be comparatively zero.
(ayd) In the preceding analysis (5) the distribution of means (and labour) was supposed given. But the reasoning is unaffected, if the distribution of means is _ supposed variable, provided that the later postulates are not affected by that distribution. And this they might be on Mr. Doubleday’s hypothesis. But in Herbert Spencer’s more probable view of the relation of affluence to populousness, the first rule (a) will become a fortiort.
Under this head may be considered the question : What 1s the fortune of the least favoured class in the Utili- tartan community? Let us consider first the case of emigration for the benefit of the present generation. Let us start with the supposition, however inappropriate, that the distribuend does not vary with population; as. in an isolated island where the bounty of nature could not be affected by human exertion.
The happiness of the present generation may be symbolised
fre —cy|jdze+cD
Xo
where D is the given distribuend and the rest of the notation is as above (a8). By the third postulate 2, is given as the highest existing degree of capacity. What ‘ remains variable is 2, the abscissa of emigration. At
1 Cf. Galton, ‘The weak could find a welcome and a refuge in celibate monasteries,’ &c. ; also Sully, Pessimism, p. 392.
THE LEAST FAVOURED CLASS. 13
the limit at Y) — cy = 0. Nowc is positive, for it equals ( 7) the first differential of pleasure with
regard to means, which (presupposed a utilitarian intel- ligence) is probably never negative eens Postulate I.).
But this is not postulated. Only, if (3) is negative,
we are dealing with the external case “of the inquiry, determining what sections shall ¢wmmigrate from our ‘unprogressive country.’ For if the Utopians have such a plethora of means that their happiness would be increased by a diminution of their means, then immi- gration will set in until the point of satiety be at least repassed. Then c¢ is positive, and y is essentially posi- tive. Therefore F(x) ¥,) 18 positive. It cannot be zero, the zero-point of pleasure corresponding to a positive minimum of means.
In this case the condition of the least favoured class is positive happiness. This conception assists us to con- ceive that a similar answer would be obtained if the increase of the distribuend with increasing population were small.
Small in relation to the megisthedonic share of the least favoured class. Write the distribuend
re f ‘nf (epN)dz; where p is the effort of each unit
Lg worker, so far supposed given as a function of 7; N is
the number of population =f. ndzx. Differentiate the Xo
distribuend with regard to 2. Substitute x for 2 and
call.the curve so presented the Malthusian. Then the
condition of the least favoured class is positive, zero, or
74 MATHEMATICAL PSYCHICS.
negative happiness, according as at the limit the ordi- nate of the Malthusian is less than, equal to, or greater than that of the megisthedone.
Our uncertainty as to the condition of the lowest class increases when we consider the case of selection for the benefit of the next generation.
Let n=¢(z) be the curve of prey for the pre- sent generation. Let v= Be/— —(e— oe x5 be the curve of issue for capacity €; where B is the natural maximum of issue. Then n’, the line of possibility for the next
-g2 _ , f % a: generation, 1s 4B € - etz)dz, where by the
Hq fifth postulate 2, is given as the highest existing degree of capacity ; what is variable is 2, the abscissa of total selection. The happiness of the next generation
4+
H!= f [n'(F(z y)—cy)|dz+cD, where « is a con- mm OC
venient designation for the utmost extent of variation— variation in the Darwinian sense. is given by the
equation =(); from which it is by no means clear 0
that the condition of the least favoured in the second generation 1s above zero.
In fact, the happiness of some of the lower classes may be sacrificed to that of the higher classes. And, again, the happiness of part of the second generation may be sacrificed to that of the succeeding generations. Moreover (it is convenient, though out of order, here to add) our uncertainty increases when we suppose the Jaboriousness also of population variable. Nothing indeed appears to be certain from a quite abstract point of
THE LEAST FAVOURED CLASS. 75
view, except that the required limit is above the starving- point ; both because in the neighbourhood of that point there would be no work done, and—before that con- sideration should come into force and above it—because the pleasures of the most favoured could not weigh much against the privations of the least favoured. (Cf. Wundt’s pleasure-curve.)
It may be admitted, however, that a limit below the zero of happiness, even if abstractedly desirable, would not be humanly attainable ; whether because discomfort in the lower classes produces political instability (Aris- totle, &c.), or because only through the comfort of the lower classes can population be checked from sinking to the starving-point (Mill, &c.). Let politics and political economy fix some such limit above zero. If now Hedo- nics indicate a limit still superior (in point of comfort) —well. But if abstract Hedonics point to a limit below that hard and fast line which the consideration of human infirmity imposes, what occurs? Simply that popula- tion shall press up against that line without pressing it back.
(By8) Under this head should be considered whether rule (8) does not interfere with rule (y8). And this upon Mr. Herbert Spencer’s theory of population it would do.! The present then may have to be sacrificed to the future; though in general how much of the present it is expedient to sacrifice to the future must be as nice a question in political, as in personal prud- ence.
(aBy8) Contemplating the combined movements we seem to see the vast composite flexible organism, the play and the work of whose members are continually readjusted, by degrees advancing up the line of evolu-
} Contrast, however, Champagay, Les Antoning, ii. p. 277.
76 MATHEMATICAL PSYCHICS.
tion; the parts about the front advancing most, the members of the other extremity more slowly moving on and largely dying off. The final shape of the great organism, whether its bounding line of possibility shall be ultimately perpendicular, whether thé graduation of (in a Greek sense) aristocracy, or the level of modern revolution, is the ideal of the future, is still perhaps a subject more for prejudice than judgment. Utilitarian- ism, indifferent about the means, with eye undistorted by prepossessions, looks only to the supreme end.
CorotLaries. The application of these inquiries is (I.) to first principles (II.) to subordinate rules of con- duct.
I. The end of conduct is argued to be Utilitarianism, as exactly defined in the ‘ Methods of Ethics,’ by deduc- ing from that general principle maxims of common sense ; perhaps as the constitution of matter is proved by deducing from the theory experimental laws. What inferior accuracy in the moral universe indeed! But before that inferiority should prejudice, let it be settled what degree of accuracy was here to be expected. No one would listen to Professor Clerk Maxwell aBavodo- yourros about the atoms without a mathematical corres- pondence of his theory and the facts. But we have a large experience of the progress of Physics; it is well seen how she goes; but is the movement of Morals so familiar that the true science should be manifest by her method! Whatever the method—for Universal Eude- monism prescribes no dogma about the origin of her supremacy ; affiliated as readily to practical reason as pure passion, the ‘Faith’ of a Green or ‘ Ideals’ of a Grote—whatever our faith, when we descend from faith to works, requiring a criterion for alternative actions, it may be divined that we shall not far err in
PROOF OF UTILITARIANISM. 77
following, however distantly, the procedure of the * Methods of Ethics.’
Consider first then Equality, the right of equals to equal advantages and burdens, that large section of distributive justice, that deep principle which continually upheaves the crust of convention.
moNAdav roMav karéuoe képyva 48 re nad Awe r05 yap pdros Cort peyorov,
All this mighty moral force is deducible from the prac- tical principle of exact Utilitarianism combined with the simple laws of sentience (a and £).
But Equality is not the whole of distributive justice, There may be needed an déia for unequal distribution. Now inequalities of fortune—abstracted the cases of governor and general and every species of trustee for the advantage of others—are generally explained by utilitarians as the consequence of conventions clear and fixed and preventing confusion and encouraging pro- duction, but not otherwise desirable, or rather of which the necessity is regretted. Yet in the minds of many good men among the moderns and the wisest of the ancients, there appears a deeper sentiment in favour of aristocratical privilege—the privilege of man above brute, of civilised above savage, of birth, of talent, and of the male sex. This sentiment of right has a ground of utilitarianism in supposed differences of capacity. Capacity for pleasure is a property of evolution, an essential attribute of civilisation (a). | The grace of life, the charm of courtesy and courage, which once at least distinguished rank, rank not unreasonably received the
1 Pp. 90, 846, 892, &c., 2nd edition. Cf. Buffon, Moral Arithmetic : “Le sentiment n'est en géneral qu'un raisonnement implicite moins clair, mais souvent plus fin et toujours plus sir que le produit direct de la raison,’ (He is proving our first postulate.)
78 MATHEMATICAL PSYCHICS.
means to enjoy and to transmit (a). To lower classes was assigned the work of which they seemed most capable; the work of the higher classes being different in kind was not to be equated in severity.! If we sup- pose that capacity for pleasure is an attribute of skill - and talent (a); if we consider that production is an un- symmetrical function of manual and scientific labour (8) ; we may see a reason deeper than Economics may afford for the larger pay, though often more agreeable work, | of the aristocracy of skill and talent. The aristocracy of sex is similarly grounded upon the supposed superior capacity of the man for happiness, for the &epyetas of action and contemplation; upon the sentiment—
Woman is the lesser man, and her passions unto mine Are as moonlight unto sunlight and as water unto wine.
Her supposed generally inferior capacity is supposed to be compensated by a special capacity for particular emotions, certain kinds of beauty and refinement. Agree- ably to such finer sense of beauty the modern lady has received a larger share of certain means, certain luxuries and attentions (Def. 2; a sub jinem). But gal- lantry, that ‘ mixed sentiment ? which took its rise in the ancient chivalry,’ has many other elements. It is ex- plained by the polite Hume as attention to the weak,® and by the passionate Rousseau ¢voixwrdpws.t Now attention to the weaker sex, and woman’s right not only to certain attentions 1n polite society but to some exemp- tion from the harder work of life, are agreeable to the utilitarian theory: that the stronger should not only do more work, but do so much more work as to suffer § more fatigue where fatigue must be suffered (8). It
1 Cf. Livy, ii., p. 82, B. * Burke. 8 Essay, 14, 4 Emale, iv. 5 See note, p. 66,
PROOF OF UTILITARIANISM.
may be objected : consideration should equally be due from the stronger to the weaker members of the same sex. But in the latter case there is wanting a natural instinct predisposing to the duties of benevolence ; there has been wanting also a fixed criterion of strength to fix the associations of duty; and, lastly, competition has interfered, while competition between man and woman has been much less open (and much less ob- viously useful to the race). Altogether, account being taken of existing, whether true or false, opinions about the nature of woman, there appears a nice consilience between the deductions from the utilitarian principle and the disabilities and privileges which hedge round modern womanhood.
Utilitarian also is the custom of family life, among other reasons, in so far as (contrasted with communistic education) it secures for the better-born better educa- tional influences (y); in particular a larger share of good society in early life, The universal principle of the struggle for life, as Mr. Barratt may suggest, conduces to Utilitarian selection. This being borne in mind, there appears a general correspondence between the popula- tion-theory above deduced (y8) and the current ethics of marriage, which impose! only a precedent condition, success, hereditary or personal, in the struggle for life. Concerning the classification of future society, common sense anticipates no utopia of equality. Physical pri- vations are pitied ; the existence of a subordinate and less fortunate class does not seem to accuse the bounty of Providence.? With the silence of common sense accords the uncertain sound of exact Utilitarianism (ayé).
But, if egoist or intuitivist are not to be altogether
1 In respect to population. * Of. Burke on the ‘labouring poor,’ in Regicide Peace, 3.
80 MATHEMATICAL PSYCHICS.
converted by the deductive process of Mr. Sidgwick, at least the dealing with his exact definition may tend to mark out and reclaim from the indefinite one large common field of conduct, one of the virtues of the in- tuitivist, one of the gratifications of the egoist—rational benevolence. For can there be a rational wish to please without a willingness to estimate the duration of the pleasure, the susceptibility, as well as the number, of the pleased ?
( Exact Utilitarianism may also, as Mr. Barratt thinks plausible, present [the end of Politics; of Politics as based upon self-interest." A political ‘contract’ for the adjustment of conflicting interests should have two qualities. It should be clear and fixed, universally interpretable in the same sense. It should be such that the naturally more powerful class, those who, though fewer, outweigh the more numerous by strength, ability, and capacity to co-operate, should not have reason to think that they would fare better under some other contract. ' Two contracts present these qualities; the rough and ready zsocratical, the exact possibly aristo- cratical, Utilitarianism. The first contract excels in the first quality ; the second in the second.
II. That the same reasonings should lead up to a general principle and down again to its applications— that the theory should be tolerably certain, the practice indefinitely remote—is not more paradoxical than that the demonstrator of the atom-theory should foresee the remote possibility of its application, no less a possibility than to triumph over the second law of Thermodynamics.? The triumphs of Hedonics, if equally conceivable, are equally remote; but they do not so certainly become
1 Compare the Corollary of the Economic Calculus. 2 Clerk-Maxwell, Theory of Heat, p. 308,
APPLICATIONS OF UTILITARIANISM. Sl
more conceivable when considered more remote; for what if in the course of evolution the subtlety of science should never overtake the subtlety of feeling! Faint and vague and abstracting many things which ought not to be abstracted, the Hedonical Calculus supplies less a definite direction than a general bias, here briefly and diffidently indicated.
The end of action being defined as above, the Jacobin ideal ‘ All equal and rude,’ J. 8. Mill’s ideal ‘ All equal and cultivated,’ are not necessarily desirable, not para- ‘mount ends to be sought by revolution or the more tedious method of depopulation. Pending a scientific hedonimetry, the principle ‘Every man, and every woman, to count for one,’ should be very cautiously ap- plied. In communistic association (if such should be) the distribution of produce should be rather upon the principle of Fourier than of Owen. Universal equal - suffrage is less likely to be approved than plural votes é conferred not only (as Mill thought) upon sagacity, but also upon capacity for happiness.
The play of the struggle for life is to be encouraged, in the present state of society, within limits, without prejudice to the supremacy of the supreme principle. Mr. Barratt indeed from the same premisses, the utility of competition, infers a different conclusion: that Utili- . tarianism should resign in favour of Egoism. But surely the inference is, not that the Utilitarian should change his destination from Universal to Egoistic Hedonism (points toto celo apart, as the chart of Sidgwick shows) ; but that, while constant to his life’s star, he should tack (in the present state of storm at least) more considerably than the inexperienced voyager might advise. No one can misunderstand this ‘self-limitation’ of Utilitarianism —for it has been explained by Mr. Sidgwick ; least of all
G
82 MATHEMATICAL PSYCHICS.
the Egoist—for a similar delegation, without abdication, of the supreme command is much more necessary in the case of the supremacy of self-love (Butler, &c.). Lastly, while we calculate the utility of pre-utilitarian institutions, we are impressed with a view of Nature, not, as in the picture left by Mull, all bad, but a first approximation to the best. We are biassed to a more conservative caution in reform. And we may have here not only a direction, but a motive, to our end. For, as Nature is judged more good, so more potent than the great utilitarian has allowed’ are the motives to mo- rality which religion finds in the attributes of God.
1 Mill, Essays on Nature and Religton.
APPENDICES..
ome=6OO =
I. ON UNNUMERICAL MATHEMATICS.
IT seemed undesirable to load our opening pages with a multi- plicity of illustrations which, if the writer’s views are correct, would be superfluous to the mathematician, and, in any case, might be uninteresting to the dyewperpnrds. Indeed, the nature of the subject is such that a single instance—by a sort of ‘mathematical induction,’ as it has been called—a single ‘ representative-particular ’ authenticated instance of mathematical reasoning without numerical data is sufficient to establish the general principle. However, it may be well to add a few words of exposition after first precising the point at issue by citing on our side the father of Mathematical Econo- mics, as the representative of the contrasted view the very able author of a review (on Prof. Jevons’ ‘ Theory’) already re- ferred to.
Cournot says : '—‘ L’une des fonctions les plus importantes de l’analyse consiste précisement & assigner des relations déter- minéesentre des quantités dont les valeurs numériques, et méme les formes algébriques, sont absolument inassignables.
‘D’une part, des fonctions inconnues peuvent cependant jouir de propriétés ou de caractéres généraux qui sont connus, par exemple, d’étre indéfiniment croissantes ou décroissantes, ou d’étre périodiques, ou de n’étre réelles qu’entre de certaines limites. De semblables données quelque imparfaites qu’elles paraissent, peuvent toutefois, en raison de leur généralité méme, et 4l’aide des signes propres 4 l’analyse, conduire & des relations également générales, qu’on aurait difficilement découvertes sans
' Théorie des Richesses, p. 51. See also Preface, p. viii. @ 2
84 APPENDICES.
ce secours. C’est ainsi que, sans connaitre la loi de décroisse- ment des forces capillaires, et en partant du seul principe que ces forces sont insensibles 4 des distances sensibles, les géo- métres ont démontré les lois générales des phénoménes de la capillarité, lois confirmées par l’observation.’ |
The ‘ Saturday Review’ (Nov. 11, 1871):— ... ‘We can tell that one pleasure is greater than another ; but that does not help us. To apply the mathematical methods, pleasure must be in some way capable of numerical expression ; we must be able to say, for example, that the pleasure of eating a beef- steak is to the pleasure of drinking a glass of beer as five to four. The words convey no particular meaning to us; and Mr. Jevons, instead of helping us, seems to shirk the question. We must remind him that, in order to fit a subject for mathematical in- quiry, itis not sufficient to represent some of the quantities concerned by letters. If we say that G represents the confi- dence of Liberals in Mr. Gladstone, and D the confidence of Conservatives in Mr. Disraeli, and y the number of those parties; and infer that Mr. Gladstone’s tenure of office depends upon some equation involving as and on we have merely wrapped up a plain statement in a mysterious collection of letters.’ The reader is referred to the whole article as typical of the literary method of treating our subject. Thus, again, ‘the equations ... ., assuming them to be legitimate, seem to us to be simply useless so long as the functions are obviously indeterminable. They are merely a roundabout way of express- ing what may be better said in words.’ And, again, ‘he wraps up his mysterious conclusions in symbols which are mere ver- biage, as they contain functions which neither are nor can be determined.’
Compare Mill :—‘ Such principles (mathematical) are mani- festly inapplicable where the causes on which any class of phe- nomena depend are so imperfectly accessible to our observation, that we cannot ascertain by a proper induction their numerical laws.’!
Compare also the spirit of his remarks? upon algebra and its exclusive ‘adaptation to the subjects for which it is com-
1 Logie, book iii. chap. xxiv. p. 9. ? Book iv. chap. vi. p. 6.
UNNUMERICAL MATHEMATICS. 85
monly employed, namely, those of which the investigations have been already reduced to the ascertainment of a relation between numbers,’ Compare also the views of Comte to which he refers.
A single instance—that already cited in the text—seems sufficient to oppose to this popular impression about the limits of mathematics. Thomson and Tait, in their ‘Treatise on Natural Philosophy,’ p. 320, discuss the problem of a ball set in motion through a mass of incompressible fluid extending infi- nitely in all directions on one side of an infinite plane, and originally at rest. After constructing the Lagrangian equations from (what may be called in reference to numerical measure- ments) & priori considerations, they go on: ‘ principles suffi- cient for a practical solution of the problem of determining P and Q will be given later. In the meantime, it is obvious that each decreases as « increases. Hence the equations of motion show’ several deductions which are truly ‘ most remarkable and very suggestive,’ ¢.g. (in an analogous problem), that two balls properly projected in a perfect incompressible liquid will seem to attract one another. It is suggested, I think, that a certain hypothesis as to the ultimate constitution of matter corresponds with the observed phenomena of attraction.
Now here is the type of mathematical psychics. The ‘ prac- tical solution of the problem of determining P and Q,’ func- tions denoting quantities of pleasure in terms of external ob- jects (means, &c.), is not yet given. But certain properties of such functions are given. Thus, if P be a person’s pleasure considered as a function of « his means, it is obvious (compare the premises of Thomson and Tait’s reasoning) that P increases
as @ increases, but at a decreasing rate; whence $F con-
tinually positive, a continually negative. And from such
data mathematical reasonings show several interesting results. It has ' been suggested that a certain hypothesis as to the ulti~ mate principle and supreme standard of morals corresponds (to an extent not usually noticed) with the observed phenomena of human action.
' Shove, p. 4.
86 APPENDICES.
One can imagine how facetious the ‘Saturday Reviewer ’ might be in criticising the method employed by Thomson and Tait in the above example, namely, mathematical deduction without numerical measurement. As we are not able to say that P isto Q as 5 to 4, the argument ‘conveys no particular dP dQ dz dx wrapped up a plain statement in a mysterious collection of let- ters.’ Doubtless, I reply, what we know of P and Q might have been stated unmathematically in a roundabout literary fashion ; but that statement, as compared with Thomson’s, would not be a plain statement, nor appropriate nor serviceable. For this same symbol-speech, so harsh and crabbed as compared with literary elegance, is gifted with a magical charm to win coy truth ; the brief and broken language which the love of abstract truth inspires, no doubt foolishness to those who have no sympathy with that passion.!
What need to multiply illustrations of what is self-evident that mathematics, of which the very genius is generalisation, without dipping into particulars, soars from generality to gene- rality! I shall attempt, however, to illustrate a little more fully the method of mathematical physics, hoping that the professed mathematician would pardon in an amateur particular errors, ‘Si modo plura mihi bona sunt,’ if only the general view is correct.
On the theory of sound we obtain an expression for an atmospheric wave involving two (almost) independent arbitrary functions,? ¢ (n dat — 7) + p(nOat+a2). Without suppo- simg the forms of ¢ and w to be known, we may deduce sub-
meaning to us.’ In employing —— ‘we have merely
~ 1 Here may be the place to notice the Saturday Review's criticism upon Professor Jevons’s formule for the ‘law of indifference:’ that his symbols needlessly complicate the plain and simple facts of the market. But the most potent instruments of research are open to similar criticism. The so- ealled ‘equation of continuity’ may no doubt appear to literary common sense a very artificial and complicated statement of some such simple fact, as that matter cannot enter or leave a given space without crossing its boundary. But how fruitful of deductions is this formula in connection with other symbolic statements, needs not to tell to any one, even moderately acquainted with the kinetics of fluids.
2 Airy on Sound, pp. 28, 28,
UNNUMERICAL MATHEMATICS. 87
stantial conclusions; as that, when a tube is stopped at both ends, the forward and backward waves are of identical form.' I would not, however, insist too much on this particular instance, and the very large class of similar physical problems, as in all respects typical of psychical reasoning. For no doubt it may be said that the data from which the expression for wave- disturbance was deduced, the differential equation express-
ing the motion of a particle of airs = f TX “that this
dé da?” premiss is of the nature of numerical precision ; k is made up of factors supposed at least approximatively measurable; whereas (some of) the data of psychics consist of loose general relations, the fact of increase or decrease, positive or negative, possessing not even that degree of grossly approwimative accuracy,’ beyond which even Professor Jevons in his illustrations of mathematical reasoning does not appear to extend his view. At the same time, if we consider as premiss the integral equation for the disturbance, then the method of psychics is fairly well exem- plified by the employment in the theory of sound and elsewhere of arbitrary fumctions; a conception, one might suppose, which had never been entertained by those who object to mathematics’ inability to deal with the complexities of social science ; as if any degree of complexity might not be attributed to an arbitrary function.
But it would exceed the ability and requirements of the present writer to justify the method above postulated (deduc- tion from loose and numerically indefinite relations) by a general review of the uses of arbitrary functions; it will suffice to show the validity of the method in two provinces of mathe- matics least distant from the sphere of psychies—L., the theory of natural forces and energy; and II., the caleulus of varia~ tions. $
I. The hypothesis of natural forces assumes, directly or by implication, as a first or proximate principle, that the attraction or repulsion between two particles is some function of the distance between them, From this loose indefinite relation, without knowledge of the form of the function, the most im-
© Airy, p. 78. 21d, p21. 5 Principles of Science.
88 APPENDICES.
portant conclusions may be deduced. As a very simple example take the motion of a particle round a centre of force. Without knowing the form of the force-function, we deduce that equal areas are swept out by the particle in equal times, that the motion is one plane, that the velocity is inversely proportional to perpendicular from centre upon tangent, and so forth.
No doubt it may be objected that while there is something indefinite and loose in the premisses, the hypothesis of natural forces, there is also something definite and precise, for instance, the very conception of uniform acceleration. But firstly, the
~ hypothesis in question would generally be admitted to hold of
the systems of matter immediately concomitant with mental phenomena, so that the deductions therefrom may well be of great psychophysical interest (especially in view of the analogies to be suggested between energy and pleasure). And again, it is not to be supposed that the data of social science have nothing precise. While there is something in them indefinite and loose, there is also something definite and precise ; for in- stance,' the ‘ law of indifference,’ that there is only one price in a market, a proposition which possesses that degree of at least approximative precision, which is generally, and supposed to be universally, characteristic of applied mathematics. And statis- tical data, as Professor Jevons has pointed out, admit of the same sort of precision. In fine, the objection applies at most to our dynamical illustrations, not to those which will be pre- sented by pure analysis, by the calculus of variations.
The great theories relating to energy present abundantly mathematical reasoning about loose indefinite relations. Con- servation of energy is implicated with such a relation, the mutual attraction of particles according to some function of the distance between them. The principle of conservation of energy affords instances of what is vulgarly supposed a contra- diction in terms, of reasoning at once mathematically and wayvros, obtaining by mathematical deduction a general idea of a state of motion. Suppose a swarm of particles so moving under natural forces that they are now all clustered near each other, now all fly asunder to a distance, then from the principle of the conservation of energy we obtain the general zdea that the
1 As aforesaid, p. 5.
UNNUMERICAL MATHEMATICS. 89
movements of the particles are on an average more rapid, or more correctly their kinetic energy is greater, when they swarm together than when they are widely dispersed.
Peculiarly typical’ of psychics are the great principles of maximum and minimum energy. That a system tends to its least potential energy, this principle affords us in innumerable instances a general idea of the system’s position of rest; as in the very simple case of equilibrium being stable when the centre of gravity is as low as possible. Thus, without knowing the precise shape of a body, we may obtain a general idea of its position of equilibrium.
From the principle of least action we infer that a particle under any (natural) forces constrained to move on an equi-~ potential surface will so move that its path from point to point is of maximum or minimum length. Without knowing the precise law of the forces, the precise shape of the potential surface, we may thus obtain a general idea of the motion.
The great Bertrand-Thompson maximum-minimum prin-
1 The comparison between pleasure and energy may be viewed under two aepects; first (than which not more is asserted here), as not known to be more than a metaphor, yet elegant and convenient, like the hypothesis of flnids in electricity, or the ‘now abandoned but still interesting’ (Thomson & Tait) corpuscular theory of light; secondly, as in the text (pp. 9-15) a deep and real analogy, the maximum of pleasure in psychics being the effect or concomitant of a maximum physical energy.
‘The comparison assists us to conceive what appears to some inconceivable, that equality is not a necessary condition of greatest happiness. Energy is the product of mass and the square of velocity. ‘Therefore the importance of any part of a system, with respect to the total energy, depends not only on its mass, but on its velocity, In the system, consisting of discharged rifle and shot bullets, there lives more energy in the little whiffling bullet than the heavy recalcitrant rifle. And, indeed, the smaller the bullet, the greater ceteris paribus its energy. So, in the social system, we must accus~ tom ourselves to believe that the importance in respect to the utilitarian greatest possible quantity of each class is not necessarily in proportion to its numbers. More energy of pleasure, more évepyeia: in the oracular lan- guage of Aristotle, may exist in one poet than many bors; in Athens than the rest of Hellas, in Hellas than Barbaria; in a century of the age of Phidias, than a thousand years of the declining Roman Empire.
No doubt this property is implicit in the definition of integral pleasure as defined, for instance, in the third Appendix. But the conception of an in- tegral is not, perhaps, so familiar to the unmathematical as not to desiderate illustration,
90 APPENDICES.
ciples and their statical analogues present abundant instances of mathematical reasoning about loose, indefinite relations. We know, in each case, that the energy of a system to which impulses (or finite forces) have been applied is the maximum or minimum consistent with certain data. Without knowing the data precisely, we may obtain certain general ideas of the arrangement of energy in the system under consideration. Thus, if the masses of any part or parts of a material system are diminished, the connections and configuration being un- altered, the resulting kinetic energy under given (however complex and undefined) impulses from rest must be increased.’ If the stiffness in any part or parts of the system be diminished, the connection remaining unchanged, the potential energy of deformation due to given force applied from without will be increased.?, Diminution in the premisses, imcrease in the conclusion, loose, indefinite relations! So again, I think, if certain velocities be imparted by impulses to the bounding surface of an incompressible liquid, we may obtain, without having more than a general idea of the distribution of these given velocities, a general idea of the resulting motion, by reasoning, from the Thomsonian principle, that the motion of the liquid is un-rotatory, that the motion of each particle is perpendicular to a certain velocity-potential surface passing through it, one of the series of such surfaces being the bounding surface, &c. Compare with the last two paragraphs the reasonings in moral science. By first principles the arrangement (of social institutions, &c.) productive of maxi- mum pleasure holds. Without deducing precisely what this best arrangement is, we may obtain mathematically a general idea of it as that one arrangement is better than another.
Upon analogous principles in statical electricity, we know that, if there be a given distribution of electricity over the conductors in a field, the strains throughout the dielectric are such that the potential energy of the whole system is a mini- mum.? We may not know the precise form of the functions which express the distribution of electricity over the conduc- tors; much less, if we had these data, would we be able to
1 Watson & Burbury, Generalised Co-ordinates. * Ibid. 3 Clerk-Maxwell, Electricity, Arts. 98, 9.
UNNUMERICAL MATHEMATICS. 91
calculate the potential, the function whose respective differ- entials shall give the strain in each direction at any point.! Yet it is something both tangible and promising to know mathematically that the potential energy is a minimum, That something is the type of what mathematical psychics have to teach. Analogous remarks are applicable to the somewhat analogous theorem of? minimum energy of electric currents; in a higher dimension, as I think it may be said, and of the. nature of what may be called momentum-~potential rather than force-potential.
II. It is the first principle of the calculus of variations that a varying quantity attains a maximum when the first term of variation vanishes , while the second term is negative (mutatis mutandis, for a minimum). The latter condition is one of those loose, indefinite relations which we have been all along describing. In the simple cases which in the infancy of Mathe- matical Psychics are alone presented in these pages,’ we know by observation not what the second term is, but that it is continually negative. In more complicated cases the re- sources of mathematics are exhausted in calculating, not a. definite numerical, but a loose, indefinite relation, the sign of the second term. The reader should consider Jacobi’s method of discrimination, as stated, for instance, by Mr. Todhunter;4 and Mr. Todhunter’s application of the same to a particular problem,’ and realise how a mathematical reasoning may turn upon the loose, indefinite relations of positive or negative, convex orconcave. Consider also the many of Mr. Todhunter’s ‘ Miscellaneous Observations’ directed to the same relation. All through the calculus of varia- tions the relation is of paramount importance, constituting, indeed, all the difference between a maximum and mini- mum. You find continually, in the statement of a problem,
1 Compare Mill’s or rather Comte’s double objection against Mathematics in Social Science: that the premisses are unattainable, and the reasoning impossible.—Logic, book iv. ch. 24, p. 9.
* Olerk-Maxwell, Art. 283.
> See above, pp. 61-65.
~ 4 Researches in the Calculus of Variations, pp. 21-26.
5 Ibid, pp. 26-30.
92 APPENDICES.
the condition that a required curve shall be, or shall not be, convex ;' so rough and unshaped are the materials with which mathematics is able to build. Now this very relation of con- cavity, not a whit more indefinite in psychics than in physics, constitutes a main pillar of utilitarian calculus; quarried from such data as the law of decreasing utility, of increasing fatigue, of diminished returns to capital and labour; for the exact statement and proof of which the reader is referred to the economical writings of Professor Jevons and Principal Marshall.
It may be said that the former condition of a maximum mentioned lately, the equation of the first term of varia- tion to zero, is of a definite precise rather than a loose indefinite character. But, again, it is to be repeated that all the data of mathematical psychics are not indefinite, but only (as in the case of physics) some. Accordingly, from this equation to zero, combined with an indefinite datum, the increase of one quan- tity with another, of capacity for happiness with evolution, we may deduce another indefinite quantitative relation, namely, in- crease,? or diminution of share of meame in utilitarian distribution.
There are two other leading principles of the calculus of variations which seem calculated to illustrate the method of psychics. First, a consideration of first principles (prior, it may be observed, to any particular measurements or determination of the forms of functions), shows that if the ‘ Haupt Gleichung,’ as Stranch calls it, the leading—in general differential —equa- tion, which must be satisfied in order that the first term of variation should vanish, breaks up into factors, there are, or rather may be,® several solutions, several different functions, each corresponding to a maximum or minimum. (In the simple cases alone presented in these pages, or rather in the - companion paper, in which the expression whose maximum is sought does not involve any differential co-efficients, say
T= ff F (yx) dx between limits, where y is an independent
variable function; then, if ak breaks up into factors, there
dy
1 Researches in the Calculus of Variations, pp. 80, 117, 286. * Above, p. 68. 3 Todhunter’s Researches, p. 262.
UNNUMERICAL MATHEMATICS. ~ _ 93
will in general, I think, be multiple solutions.) A curve between two given points required to fulfil some maximum condition may be discontinuous, may be made up of the different solutions, one step according to one law, and the next step according to another law.! But the different laws or function, though they may thus be employed successively, are not to be mixed and compounded. Any one portion of the required curve must (in general and subject to the exceptions of the following paragraph), obey some one of the laws supplied by the solution of the Hawpt Gleichung. It is submitted that this property has its counterpart in human affairs; the fact that there are sometimes two best ways of attaining an end—if the superlative best may be employed in a technical sense analogous to the superlative maximum. To realise the best, one or other course must be adopted, not a confusion of the two.
The subject of discontinuity leads up to another general remark, It is not universally necessary that the first term of variation should vanish. It suffices for a maximum that the first term of variation should be known to be negative (and obversely for a minimum). Such knowledge is generally the result of imposed conditions ; as in Mr. Todhunter’s problems that a curve must not pass outside a given boundary, must not exclude a given point, must be convex. It is submitted that such complicating imposed conditions have some analogy with the conditions imposed by necessity upon practical politics and applied utilitarianism. For ®povnors has often to be con- tent not with the best course, but the best subject to existing conditions, Compare the subtle spirit of Mr. Todhunter’s calculus of variations with the subtle, and as the ‘plain man’ might almost suppose, sophistical spirit of Mr. Sidgwick’s method of utilitarianism, when it comes to be applied to the actual world in which we live. The abstract maximum, in psychics as well as in physics, is comparatively simple; but the concrete is complicated by imposed conditions; and the complexion of a wise benevolence, in view of each established constitution, custom, church, is affected with a congenital re- semblance to the wily charms of the calculus of variations.
1 Todhunter, pissin.
94 APPENDICES.
I. ON THE IMPORTANCE OF HEDONICAL CALCULUS.
It may be objected that mathematical psychics, though possible, are not valuable ; I say valuable rather than, what “might be understood in a too restricted sense, useful. For no philosophical objector would maintain that the love of the soul for the universal is then only legitimate, when it has been blessed with the production of the useful.
The love of the soul for the universal is undoubtedly capable of extravagance, as in the devotion of Plato to the idea. ‘ Amor ipse ordinate amandus est.’ But the limits are to be traced by a loving hand, and not to be narrowed by a too severe construc- tion of utility. The great generalisations of mathematics have perhaps been pursued and won less for the sake of utility to be produced, than for their own charm. Certainly the superior genius who reduced the general dynamical problem to the discovery of a single action-function was as much affected by the ideal beauty of ‘one central idea,’' as by the practical con- sequences of his discovery. In the example first cited from Thomson and Tait, it might have happened that the generalised co-ordinates employed did not yield that ‘first vindemiation’ of truth above described (p. 85). Yet the Lagrangian conception of considering the energy of the whole system as a function of the position and velocities of the immersed bodies would still have been legitimate, and great, and promising. The Gossenian, the Jevonian thought of referring economics to pleasure as the central idea might be equally splendid, though unfruitful. And so Mr. G. H. Darwin, in his review of Professor Jevons’s ‘ Political Economy,’? appears, not without reason, to prefer the mathematical method on theoretical, abstracted from practical, grounds.
Professor Cairnes* himself admits that the mathematical method might be useful, though not indispensable. If so, the
1 Sir William R. Hamilton, Philosophical Transactions, 1834, 1835. 2 Fortnightly Review, 1875. 3 Preface to Logical Method.
IMPORTANCE OF HEDONICS. * 95
position of the mathematical method in economics might be compared, perhaps, to that of quaternions, which calculus, even if it conduct to no theorem not otherwise deducible, yet, in the opinion of some ' competent judges, deduces theorems already known more elegantly and, as it may be said, naturally and philosophically, than the blind and elephantine formule usually employed for the purpose. At any rate, is it for one who is not conversant with both methods to offer an opinion on their relative value; to declare forbidden, without having himself trodden, the sublimer path ?
But 7s the method unfruitful in social science? The black list in our appendix may show the possibility that mathema- tical ‘reason is here no guide, but still a guard.’ But I go further, and challenge the ayewperpntos to answer the following © examination paper.
SociIAL PROBLEMS TO BE SOLVED WITHOUT MATHEMATICS.
1. A communistic society owns land of varying degrees of fertility, which land it cultivates so as to obtain with a given quantity of labour the maximum of produce. Suppose the quantity of labour at the disposal of the community to be suddenly increased, how will the new labour be distributed ? Will more or less additional labour be employed on any acre according as it is more or less fertile, or otherwise ?
2. When Fanny Kemble visited her husband’s slave planta- tions, she found that the same (equal) tasks were imposed on the men and women, the women accordingly, in consequence of their weakness, suffering much more fatigue. Supposing the husband to insist on a certain quantity of work being done, and to leave the distribution of the burden to the philanthropist, what would be the most beneficent arrangement—that the men should have the same fatigue,? or not only more task, but more fatigue?
3. Commodities being divided into two species, those whose expenses of production (do not diminish or) increase as the
1 Cf. Tait, Edinburgh Phtlosophical Transactions, 1825. 2 Cf. Mill’s Theory of equal sacrifice in taxation.
96 APPENDICES.
amount increases and those whose cost of production diminishes with the amount produced; show that it is abstractedly expe- dient to tax one of these species rather than the other, and even to tax one so as to bounty the other (Marshall’s theorem ).
4. Commodities being divided into two species, according as a slight decrease of price is, or is not, attended with a consider- able increase of demand, which species is it abstractedly pre- ferable to tax ?!
5. The labour market, from an indefinite number of masters and men competing on each side, is transformed by trades- unions and combinations of masters into a small number of competing (corporate) units on each side. Can this transform- ation be advantageous to both sides ?
6. It has been said that the distribution of net produce between cooperators (labourers and capitalists associated) is arbitrary and indeterminate. Discuss this question.
7. Mr. Sidgwick in the ‘ Methods of Ethics’ (iv. chap. i.), having defined the utilitarian end as the greatest possible sum of pleasures, proceeds to observe that with a view to this end equal distribution of happimess, though not necessarily of the means of happiness, is desirable. Assuming what the author’s note seems to imply (cf. ‘Methods of Ethics,’ p. 256, 2nd edition), that individuals have their happiness differently re- lated to means, derive different amounts of happiness from the same means ; show that to attain the end defined happiness and its means must be either both equally or both unequally distributed.
There are those no doubt who see nothing in all this, turn- ing away contemptuously from such questions, as the dog when you try to put him on ascent which nature or discipline haa made to him insignificant. The professed mathematician, it must be owned with regret, is not unlikely to be in this num- ber. But the professed mathematician, however infallible a guide upon the purely mathematical side (and sure to find many errors in these pages should they be so fortunate as to come
' See Notes on Exchange Value, by H. Cunynghame, p. 9.
IMPORTANCE OF HEDONICS. 97
under his notice) is not necessarily an infallible guide over the untrodden pass here supposed to exist between the heights of physics and psychics, supposing that his attention has not been directed to psychological problems. Nevertheless, great is the authority of the masters of the supreme science.
The authority of the mere metaphysician need give us much less pause, The noble Hegelian, from the transcendental heights whence he looks down upon Newton, might smile at the attempt to estimate quantitatively pleasure, A notable authority forsooth, this demolisher of Newton, upon the science of quantity and its limits; and notable authorities and judges of authority are those his followers, whose chosen philosopher and guide is not only blind to truth in her clearest, manifesta- tion, but also, what is even more unphilosophical, is ignorant. of his ignorance and vain of hisinanity. Won ragionam di lor. As the Olympian Zeus, defied by Here and Athena, addresses: his rebuke not to the inveterately obstinate one, but only to the rebellious goddess of wisdom—
“Hp 8 ote récov vepeciferat ov8é yohoorar™ alei ydp of Zabev évxAgy Srre oho"
so a serious argument is addressed not to the incorrigible mystic,
Common sense is addressed and may be persuaded, it is hoped, to forego its prejudices against this sort of calculus. There is the old prejudice still reviving, however often slain, against. the reign of law in psychology, as incompatible withthe higher feelings. But it is too late. The reign of law is estab- lished, and will not, become more oppressive to feeling by be- coming mathematical. And again, common sense, catching sight of such terms as hedonism, is apt to dismiss the whole affair as metaphysical. But, it is to be insisted, the materials with which exact: social science is concerned are no metaphysical shadows, but the very substance of modern civili- sation, destined, doubtless, ere long to become embodied in practical politics and morals. Quantity of labour, quantity of pleasure, equality of sacrifice and enjoyment, greatest average happiness, these are no dreams of German metaphysics, but the leading thoughts of leading Englishmen and corner-stone con-
a
98 APPENDICES.
ceptions, upon which rest whole systems of Adam Smith, of Jeremy Bentham, of John Mill, and of Henry Sidgwick.
Are they not all quantitative conceptions, best treated by means of the science of quantity ?
IT. ON HEDONIMETRY.
IT has been shown that some of the data of physics are as in- definite as some of the data of psychics. And yet it may be admitted that there is a potentiality of precision about even the looser physical demonstrations which gives them a certain prestige. In physics, when we deal with an indefinite P and Q (to revert to an earlier example), there is some understanding that ‘ principles sufficient for a practical solution of the problem of determining P and Q will be given later.’ Whereas in psychics we are so far from expecting, that it seems doubtful whether we can even conceive precise measurement. Yet the conceivability at least may be thought necessary to mathematical reasoning. We must then carefully consider this possibility, or, what is much the same thing, the existence and nature of a unit of pleasure.
There is, no doubt, much difficulty here, and the risen science is still obscured by clouds; and hedonism may still be in the state of heat or electricity before they became exact sciences, as described by Professor Jevons.' Let us, however, following in his footsteps, endeavour to gain as clear a view as may be. At least it is hoped that we may sight an argumen- tum ad hominem, an argument to the man who (with Professor Jevons), admitting mathematical reasoning about self-regarding pleasures, denies the possibility of mathematically comparing different persons’ pleasures. Let us accordingly, with reference to this question of yerpyrcxn and pleasure-unit consider sepa- rately the quantitative estimate which a man can form (I.) of his own pleasure, (II.) of other people’s.
1 Theory of Political Economy, p. 9,
HEDONIMETRY, 99
I. ‘Utility,’ says Professor Jevons (writing exclusively of the first sort of measurement), ‘may be treated as a quantity of two dimensions.’! Now, when it is asked, ‘In virtue of what unit is one intensity said to be greater than another?’ the answer must be, I think, ‘Just perceivable increments of pleasure are equatable,’ which may be shown, perhaps, by that sort of internal experience and handling of ideas which seems to be the method of attaining mathematical axioms.? For if possible let one just perceivable increment be preferred to another. Then it must be preferred in virtue of some differ- ence of pleasurability (non-hedonistic action not existing, or not being pertinent to the present inquiry). But, if one of the increments exceeds the other in pleasurability, then that one is not a just perceivable increment, but consists of at least two such increments. Of course such a way of turning the subject has no pretence to deduction. The stream of thought ‘mean- ders level with its fount.’ Turn the matter as we please, there must, I think, be postulated some such equation as the above, which may be compared, perhaps, to the first principle of probabilities? according to which cases about which we are equally undecided, between which we perceive no material dif- ference, count as equal; a principle on which we are agreed to act, but for which it might be hard to give a reason.
It must be confessed that we are here leaving the terra Jima of physical analogy. It may plausibly be objected, the just perceivable increment, the minimum sensibile, is not treated as a unit in the cases with which physics deal. Let us suppose that for the same objective increase of temperature or weight (as estimated by the approved methods of physics) I have at different times, or with different organs of my body, different subjective estimates. In one sense, certainly more usual, the quantities are the same. In another sense, the minima sensi- bilia being equated, what is felt is. And this latter sense, it is contended, not without hesitation, is appropriate to our sub- ject.
The increments in question are, I think, to be viewed as
1 Theory of Political Economy, p. 51. * Of. Bain on Axioms.
* Laplace, Essai Philosophique sur les Probabitités, 5th edit., p. 7. u2
100 APPENDICES.
finite differences, rather than as genuine differentials (a concep- tion which need not militate with the employment of the nota- tion of the differential calculus).! The conception might be illustrated by that of a force just sufficient to turn a balance overcoming friction. Why, however, each inclination of the will is treated as equal by the rational intelligence, of this, as already intimated, no proof is to be expeeted.
Indeed, the equation, or equatability, in question exists not so much in fact as in the limit of perfect evolution. The im- perfect intelligence does not treat a unit of pleasure in the future as equal to one in the present. Abstracting from the uncertainty of the future, the mere circumstance of futurity affects the estimate of a pleasure; which depreciation the Jevonian factor g? denotes, as I understand. Now it is only in the ideal limit that g becomes equal to unity.
So far about the dimension of intensity. As to the dimen- sion of time a similar line of remark is open. The same ob- jective (say horological) time may correspond to different rates of thought and feeling at different periods, as Locke intimates.?
It is conceivable that two states, presenting to conscious- ness the same number of imtensity-increments above zero, should differ in this rate of flow. And perhaps some states, intellectual exercise in particular, which philosophers have dis- tinguished as more good, though not more pleasurable, than others, may so differ. In dreams, the rate seems high, the intensity low. And so a pleasure would have not only two dimensions, as Professor Jevons says, but three dimensions, namely, objective time, subjective time, and intensity.
And yet the correction may not seem very important, for probably it is more competent to consciousness to combine into a single mark the two considerations of rate and intensity. Suppose one state presents about three pleasure-increments, another about two, above zero, that the rate of the former is double that of the latter, their objective duration being the
1 See the remarks of Clerk-Maxwell, ‘Essay on Atoms,’ Encyclopedia Britannica, p. 38.
2 Theory of Polttical Economy, p. 78.
5 Compare As You Ltke tt, Act iii, ec. 2,and elsewhere. Cf, Mr. Sully’s remarks on Jilusions of Perspective,
HEDONIMETRY. 101
same, is it better to give two marks to each state, say three and two to the former, two and one to the latter, and then to mul- tiply the marks of each; or by a sort of unconscious multiplica- tion to mark at once six and two—about; for the comparison of pleasures as to quantity is here admitted to be vague; not vaguer perhaps than the comparisons made by an examiner as to excellence, where numerical marks are usefully employed.
To precise the ideas, let there be granted to the science of pleasure what is granted to the science of energy ;' to imagine an ideally perfect instrument, a psychophysical machine, con- tinually registering the height of pleasure experienced by an individual, exactly according to the verdict of consciousness, or rather diverging therefrom according to a law of errors. From moment to moment the hedonimeter varies; the delicate index now flickering with the flutter of the passions, now steadied by intellectual activity, low sunk whole hours in the neighbourhood of zero, or momentarily springing up towards infinity. The continually indicated height is registered by photographie or other frictionless apparatus upon a uniformly moving vertical plane. Then the quantity of happiness between two epochs is represented by the area contained between the zero-line, perpendiculars thereto at the points corresponding to the epochs, and the curve traced by the index; or, if the cor- rection suggested in the last paragraph be admitted, another dimension will be required for the representation. The in- tegration must be extended from the present to the infinitely future time to constitute the end of pure egoism.
II. Now it is here contended that there are as many, and the same sort of difficulties, in this estimate of pleasures by the sentient himself (which is yet admitted by Professor Jevons, and substantially by common sense), as in the estimate of other people’s pleasures. We have only to modify our axiom thus: Any just perceivable pleasure-increment experienced by any sentient at any time has the same value. The same primal mystery of an ultimate axiom hangs, no doubt, over this utili- tarian, as over the egoistic, first. principle.
The equation is only true in the limit of perfect evolution.
The variation of subjective time for different individuals,
1 Seo Clerk-Maxwell, Theory of Heat, p, 139,
102 APPENDICES.
presents no greater difficulty than the variation for one in- dividual.
The integration may be equally well illustrated by ideal mechanism. We have only to add another dimension express- ing the number of sentients, and to integrate through all time and over all sentience, to constitute the end of pure utilitarianism.
It may be objected that the just perceivable increment is given by consciousness in the case of one’s own pleasures, only inferred in the case of others.' It may be replied, greater uncertainty of hedonimetry in the case of others’ pleasures may be compensated by the greater number of measurements, a wider average; just as, according to the theory of probabilities, greater accuracy may be attained by more numerous observa- tions with a less perfect instrument. The proposition, ‘ the exercise of higher intellect is accompanied with greater capacity for pleasure,’ is proved By taking a wide average rather than by the self-observation, however accurate, of a single, perhaps exceptional, individual.
IV. ON MIXED MODES OF UTILITARIANISM.
THE distinction between egoism and utilitarianism has been drawn with matchless skill by Mr. Sidgwick. But it has not been observed that between these two extremes, between the frozen pole of egoism and the tropical expanse of utilitarianism, there has been granted to imperfectly-evolved mortals an inter- mediate temperate region; the position of one for whom in a calm moment his neighbour’s happiness as compared with his own neither counts for nothing, nor yet ‘counts for one,’ but counts for a fraction. We must modify the utilitarian integral as defined above (Appendix III.) by multiplying each pleasure, - except the pleasures of the agent himself, by a fraction—a
1 This is a distinction insisted on by Mr. Herbert Spencer, in his remarks op utilitarianism.— Data of Ethics, p. 57.
MIXED UTILITARIANISM. 103
factor doubtless diminishing with what may be called the social distance between the individual agent and those of whose pleasures he takes account.
There is not much more difficulty about this intermediate conception than about the extremes, The chief difficulty is one which is common to the extremes, presented by the phe- nomena which Mr, Sidgwick describes as the self-limitation of a method. For example, in a life ordered according to the method of pure utilitarianism there may be tracts of egoistic action, times when the agent gives full swing to self-interest, leaving out of sight his utilitarian creed. The test whether such an agent is really a pure utilitarian would be, I suppose, whether on having his attention directed to the alternative between methods, having collected himself, in a cool moment, he would or would not calmly and deliberately sacrifice his own greatest happiness to that of others. It seems superfluous to labour a point which has been explained by Mr. Sidgwick.
Yet that there is some difficulty about this rhythm between sovereign and subordinate method may be inferred from the expressions of able thinkers, Thus, Mr. Spencer appears to employ ' as an argument against utilitarianism the utilities of self-indulgence. ‘For his wife he has smiles, and jocose speeches,’ and so forth—the self-indulgent non-utilitarian. But, if self-indulgence and the not taking account of the general good has such an agreeable effect, the intelligent utilitarian will cultivate a temporary relaxation and forgetful- ness of his supreme principle. It never was meant that he should wrap himself up in his utilitarian virtue so as to become a wet blanket to his friends. It never was meant, as Austin says, that the sound utilitarian should have an eye to the general good while kissing his wife. In order that one’s life should be subordinated to the general good, it is not necessary that the general good should be always present to conscious- ness. If I have an hour to prove a theorem at an examination, I shall do well not to keep the quod est demonstrandwm continually before the mind, but to let the mind range among theorems which may serve as premisses, If a man has a day to write an article, though the whole time may be consecrated to
1 Data of Ethics, chap, xi, % See Mr, Spencer's gloomy picture,
104 APPENDICES.
the purpose, it may be expedient to banish the purpose during refreshment or exercise. You cannot disprove the authority of utilitarianism by proving the utility of egoistical, or any other, practice.
To argue, then, that the utilities described by Mr. Spencer could not be grafted upon pure utilitarianism would imply a different conception of a ‘method of ethics’ from that which may be derived from Mr. Sidgwick’s great work. That as a . matter of fact the utilities of egoistic action do not now spring from a root of pure utilitarianism would be freely here admitted ; agreeing with the view suggested that the concrete nineteenth century man is for the most part an impure egoist, a mixed utilitarian.
And the reconciliation between egotism and altruism, gradual process and ideal limit beautifully described by Mr. Spencer, would be upon the view suggested here, the transfor- mation of mixed into pure utilitarianism, the psychical side of a physical change in what may be dimly discerned as a sort of hedonico'-magnetic field.
Y. ON PROFESSOR JEVONS'S FORMULH OF EXCHANGE.
PRoressor JEVons’s formula, oe aD = a? is almost iden- l tical with our a te A = . Almost; for the notation here ¥ 9
employed is slightly more general. The utility is regarded as a function of the two variables, not the sum of two functions of each. The inquiry suggested at p. 34, near foot, could not have been suggested by Professor Jevons’s formula. Our for- mula also is adapted to take account of the labour of produc- tion, the ‘complicated double adjustment’ glanced at by Pro- fessor Jevons.®
Let 2 manufacture the article which he exchanges for y.
1 Above, p. 14, _ * Theory, p. 108. $ Theory, p. 208.
FORMULA OF EXCHANGE. 105
Then (by a violent but not dangerous abstraction) his utility may be written
P=F(f(e)—2,y)— J (¢) where e is the oljectwve measure of labour (e.g. time of work) ; JS (e) is the subjective measure of work, the toilsomeness of fatigue ; f (e) is the produce, corresponding to e. Now, as e is not an article of contract, it appears that (= Te ~) the partial |
differential with regard to e must always be equated to zero. Hence, by eliminating e we come on our old form F (a y), or F (—a, y), as it is convenient here to write.
This ‘ complicated double adjustment’ may be illustrated by a brief reference to that interesting phenomenon pointed out by both Mr. Marshall and Mr. Walras, unstable equilabriwm of trade. From the point of view here adopted the utility of a dealer in # may be written P=F(—za,y). Transformed. to polar co-ordinates P= F ( —p cos 4, p sin 6); when tan @ expresses
the rate of exchange. The demand-curve is (S) =0. For p
this locus expresses the utmost amount of dealing to which the dealer will consent at any given rate of exchange, the amount for which his utility is a maximum at that rate. But the locus also expresses positions for which the utility is a minymum at any given rate. And this part of the locus is not in a genuine demand-curve. Each point represents a position not which the dealer will not consent to change, but which he would by all means wish to change.
By a general property of analysis the maximum and minimum points are arranged alternately along any vector. This property is closely connected with the property of alternately stable and unstable equilibrium of trade. There are, however, I think,
unstable positions where (s) = 0 does not correspond to a
minimum, eg. Mr. Marshall’s figure 8. '
But the most important sort of instability is perhaps that which may be presented in the case of (Mr. Marshall’s) Class II ; of which, as I take it, the definition connects two properties
106 APPENDICES.
(1) diminution of value in exchange upon increase of exports, with (2) diminution in the expense of production upon increase of wares produced for exportation. It is interesting to see from our ndividualistic point of view how these two properties are connected. The analytic condition of the first property is -
(st =-+. For this condition must hold from the point P, Pp where the property in question sets in (see figure) to the point
Fig. 3.
P,, where the property ceases. At each of these points + = 0. The analytic condition of the second property of Mr. Marshall’s definition (the first in the order of his statement) is “Wo. +3 where (as before) ¢ is the objectwe measure! of labour, fle) is the amount of product corresponding to @.
It may be shown, then, that (3) can only be positive when oS is positive. For, agreeably to previous notation, put
1" Other than that which the produce itself presents ; ¢.g., length of time during which a uniform muscular energy is put forth by a workman, —s_ |
FORMULA OF EXCHANGE. 107
P= F(/(e)—pcos 6, p sin 8) — Jf (e). Then we have always the
condition (5) = 0, and we have to find Ee 73 al subject to this
condition. Now, as @ is throughout treated as constant, whereas e is considered as a variable, dependent on p, it will be d,P dp? p’ brackets, denoting by brackets differentiation, which is partial with respect to p, does not take Secount of nd s variation. With
this notation, since (S =) = 0, 7 = (75 +25, Ze e
where oe is to be found from the equation to zero of P
(F)=(92)-2S-. whence $2 = - (gh)
a,F) a,
convenient to denote the object of our inquiry as =? without
Therefore B= (a5 dp? - 2 (7, Adpdes ___ ;)
(32) 2h
Now we may be certain this expression can “say be positive when ol is positive, 4f we are certain of the laws of sentience
which were postulated on a previous! page. For, writing a for f (e) (thea employed i in Professor Jevons’s equation of exchange), and y for p sin 6, we have
sin @ cos 6+ dF ain 8.
cos? 6 + 2 dy dy
(Ta) =F dady
os) =5 44 F d,f dat a? alae) + Ta a ae where it does not seem necessary to bracket the differentials on
the right-hand side. Substituting these values in the expression 1 Page 34,
108 APPENDICES.
for a we see that that expression is certainly negative upon
these conditions :
(1) on fs (both) continually not positive.
d a d y 99 3 (3) Ja continually not negative.
(4) af ; ;
(5) af | not positive.
The first condition is secured by Professor Jevons’s law of diminishing utility, our first postulate (see p. 61).
The second condition is an interesting variety of the same ; that the rate of increase of utility derived from one sort of wealth diminishes with the increase of other sorts of wealth.
The third condition imports that utility at least, does not decrease with increase of wealth; which in a civilised country may be allowed. |
The fourth condition is Professor Jevons’s law of increasing toilsomeness of labour,' our second axiom (see p. 65).
L f then these laws of sentience hold, fe
tive when “Lis positive. It is submitted that this subordina-~ tion—in however abstract and typical a form—of the more complicated phenomena of the market to the simple laws of sentience is not without interest.
But to return to Professor Jevons: the formule here em- ployed, along with a general, and perhaps it ought to be added a filial, resemblance to his, present two points of contrast which deserve especial attention: (1) Graphical clustration has been more largely employed here. Now in some sense pure Analysis may appear to be the mother-tongue of Hedonics ; which soaring above space and number deals with quantities of
1 Theory, p. 1865.
cari only be posi-
FORMULE OF EXCHANGE. 109
pleasure, employing the Calculus of Variations, the most sub- lime branch of analysis,'! as Comte, Caiaphas-like, called the branch most applicable to Sociology. But on the other hand the differential equations which occur in the theory of exchange are of such a peculiar character that it is rather difficult, as may presently appear, to handle them without geometrical apparatus. In this respect at least Mr. Marshall’s preference for geometrical reasoning would seem to be justified.?
(2) It has been prominently put forward in these pages that the Jevonian ‘ Law of Indifference’ has place only where there is competition, and, indeed, perfect competition. Why, indeed, should an isolated couple exchange every portion of their respective commodities at the same rate of exchange? Or what meaning can be attached to such a law in their case? The dealing of an isolated couple would be regulated not by the theory of exchange (stated p. 31), but by the theory of simple contract (stated p. 29).
This consideration has not been brought so prominently forward in Professor Jevons’s theory of exchange, but it does not seem to be lost sight of. His couple of dealers are, I take it, a sort of typical couple, clothed with the property of ‘In- difference,’ whose origin in an ‘open market’ is so lucidly described ;* not naked abstractions like the isolated couples imagined by a De Quincey or Courcelle-Seneuil in some solitary region. Each is in Berkleian phrase a ‘representative parti- cular ;’ an individual dealer only is presented, but there is presupposed a class of competitors in the background. This might safely be left to the intelligence of the reader in the general case of exchange. But in dealing with exceptional cases (pp. 132, 134), a reference to first principles and the pre- supposition of competition would have introduced greater pre- cision, and suggested the distinction submitted in these pages (pp. 19, &c.), namely, that exchange is indeterminate, if either (1) one of the trading bodies (qué individual or qué union) or (2) the commodity supplied by one of the dealers, be indi- visible or not perfectly divisible.
The whole subject of the mathematical theory of exchange
1 Philosophie Positive, Legon 8. . * Foreign Trade, p. 19. 3 Theory, pp. 98, 99.
~110 APPENDICES.
would be put in a clearer light by considering the objections which have been brought against Professor Jevons’s theory by an able critic in the ‘Saturday Review’ (Nov. 11, 1871). The Reviewer says: ‘ When Mr. Jevons proceeds to apply this equa- tion to the solution of his problem, he appears to us to fall into a palpable blunder. Translated into plain English, the equation = ou means, as we see, simply that, however much corn A gives to B, he will receive a proportionate quantity of beef in exchange. If he doubles the amount of corn, that is, he will receive twice as much beef. But the other quantities are obtained on the contrary supposition, namely, that the rate of exchange will vary according to some complex law, determinable, if we could tell precisely what effect will be produced on the mind of the parties to the bargain, by the possession of varying quantities of beef and corn. In fact a is now a function of y, as might easily be foreseen from Mr. Jevons’s statement of the case, in quite a different sense from what it was before. The substitution, therefore, of z for oY is a mistake.’
I submit (1) the following is a significant problem. Given ; ; . dy\ _ dy\ _ two differential equations F, (ay 52) =0, F,(2y Ie ) = 0, find
x and y two quantities such, that if each differential equation be solved, and thereby y for each be found as a function of 2,
and thence for each SY be derived as a function of x; then, if
x be substituted in both (functional) values of y, and both (functional) values of ay, (a) the two (quantitative) values of
y are equal to each other equal to y, and (6) the two (quantita- tive) values of oy are equal to other.
(2) The following is a solution of this problem. Eliminate dy ; dy\ _ dy\_ q. Tx Detween the equations F, (2 ys) =0, F, (x y 5") = 0; the resulting equation in w and y is the locus of the required point. |
(3) The problem and solution correspond to Professor Jevons’s problem and solution.
FORMULA OF EXCHANGE. 111
Let us take these propositions in order.
(1) This proposition by its extreme bumblediness illustrates what was above said about the advantages of graphical illustra- tion. For the geometrical equivalent is simply: Required a point at which two curves each given by a differential equation (of the first order) meet and touch. Or even more briefly: Find the locus of contact between members of two families. |
The conception thus introduced is not only legitimate, but familiarly employed in the Calculus of Variations, in those pro- blems where we have multiple solution subject to the condition that there shall be no abrupt change of direction. The reader will find any number in Mr. Todhunter’s ‘ Researches.’
I am not concerned to show that Mr. Todhunter’s problems are exactly parallel to ours. They could not well be so involv- ing second, where they involve first, differentials. But itis easy to construct an exactly parallel problem with curves presented by maximum analysis, the source of our economical curves. Take the straight line and the cycloid, the shortest line and line
Fia. 4.
of quickest descent. A cycloid is generated by a circle of given diameter rolling on a given horizontal line, the starting-point of the circle—that is where the generating point M is on the horizontal line—being arbitrary. Find (the locus of) a point P on the cycloid such that if a particle starting from rest slide
112 APPENDICES.
down the cycloid from the horizontal line as far as P, and there fly off at a tangent, it will pass through a given point O.
(2) The solution above offered is easily verified. Having eliminated = between F, and F,, take any point x y on the eliminant, and draw through it a curve of each family. Then
F, (x yp,)=0; where p, isthe value of oy for the first curve when x is substituted for z Since the point is on the elimi-
nant ¥, (x yp,)=90. Also F, (x y p,)=0. Therefore p,=p,. Q.E.D.
In the ey case just he let the differential equation 2a—y
of the cycloid' be 2 da ~, and the differential equa-
tion of the line _ hove p and q are the co-ordinates of the given point, Then the required locus is
Vio 2a—y _Y—P, e—q a curve of the third degree passing through the given point, as it evidently ought, if it can; for the given point may be too far from the horizontal line to be reached by generating circle or generated cycloid. In this last case the point is still the scene of contact between a cycloid and line, only the cycloid is imaginary. The mathematician is prepared for such freaks of analysis; the economist should be prepared for somewhat simi- lar freaks? on the part of his similarly obtained ‘demand- curve.’ To avoid misconstruction it may be as well to add that this
solution by elimination of pd would not have been admissible if
1 See Todhunter’s Differential Calculus, p. 342.
2 Thus the origin, though an tntersectton of the demand-curves, is not in any sense a position of equilibrium; not even being on the contract-curve. Again, the alternate intersections of the demand-curves are (as Messrs, Mar- shall & Walras have shown) positions of trade-equilibrium only in name, And we have seen that similar caution is required in handling the analytical expression of the contract-curve (p. 26).
FORMULE OF EXCHANGE. 113
there had been other differentials besides those of the first order. Elimination would in thia case have resulted in that sort of mongrel differential equation, ‘ Mixtumque genus pro- lemque biformem,’ which the Reviewer may be supposed to have had dimly in view.
(3) An attentive consideration of Prof. Jevons’s problem will show that it is a case of the problem here proposed, whether in the language of pure analysis or of geometry. I take the latter for brevity and to illustrate its convenience. Taking for origin the point at which the dealing begins where 2 and y are zero, we see (a) by the law of indifference! that each dealer must move along a straight line given by the differential equation
Had (the Reviewer sees this much). Again under the head-
ing ‘Theory of Exchange’? we may learn (6) that the 2 which expresses the dealer's change of position is at the
oint of equilibrium=?:(2—). But by (a) the 2Y which ex- presses the dealer’s change of position is continually= a
Therefore by the principles just now laid down the locus of the re-
quired point is found by eliminating a between (a) and (b);
whence # (4-2) 2 which is none other than our old friend the . 1 ‘demand-curve.’
We may recognise another old friend in the equation
dy_, (a—2) . . . . . ~%Y=Vi\" “/ eonsidered as an ordinary differential equation. de v1) ny “a
It is the differential equation of our ‘curves of imdifference.’
The problem under consideration may be expressed: Find the locus of the point where lines from the origin touch curves of indifference. If (as before supposed) the curves of indifference consist of a series of circles round a point C, then the locus of the point of contact to any curve of a tangent from O is the locus of vertices of right-angled triangles described on OC ; that is, a semicircle described in OC, a result which of course might
1 Theory, p. 98, et seq. 2 P. 103, sqq. I
114 APPENDICES.
be obtained analytically according to the method here described. Transforming to the point of bisection of OC, and putting c= $ OC, the equation of any indifference-curve is (y—c)?+a?=7°. Whence the differential equation of the family ol =— 7 * , And