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Mathematics An Empirical Science

184. BY a splendid tour de force Kant answered the question, How are Metaphysics possible ? He approached it through the more fundamental question, How are judgments independent of Experience possible ? Since Metaphysics claimed to solve problems which avowedly transcended the reach of Experience, it was indispensable to prove that the human mind was not restricted to experiential judgments, but was capable of forming judgments independently. Kant rightly saw that Metaphysics might be possible if Mathematics were possible ; but he failed, I think, in proving that â priori metempirical judgments were possible in Mathematics, and therefore were possible in Metaphysics. His conclusion is logical enough could we accept the premises ; but as these involve the fallacy of necessary truths having a metempirical character, the premises cannot be accepted.

Our purpose will be to reverse Kant's procedure, and show that mathematical judgments are absolutely and entirely dependent on Experience, and are limited to the range of Experience, sensible and extra-sensible. While, before Kant, the theory of Experience assumed that the Mind was a kind of mirror in which the images of things were reflected ; after Kant, it became the fashion to re-verse this theory, and to assume that the unknown Existence (Ding an sich) was the darkened side of a mirror from whose bright surface were reflected the forms of our minds, the reflected images being the objective phenomena known to us. Both explanations were radically defective, since they both involved the fallacy that a product could be the product of one factor. The proof which Kant offered in support of his position was the existence of certain judgments which must have been anterior to all Experience, because from the nature of the case no Experience could furnish them, since they transcended its range. The proof that no Experience could furnish them was seen in the characters of Necessity and Universality which belonged to their essence ; for as no Experience could be universal, none could exclude contingency.

185. But having seen the characters of Necessity and Universality to belong to all truths, or to none, we cannot accept those characters in proof of the existence of a particular class of truths independent of Experience ; hence the conception of a Mind existing anterior to all sensible experiences, and capable of framing legitimate conceptions respecting supra-sensible existences, must be placed on another foundation, or given up altogether. I am as firmly convinced as Kant himself (and have argued it fully in Chap. II.), that every experience, and every judgment grouping experiences, must be referred back for one of its factors to a prior result, a judgment already organized, and in this sense à priori, since it is prior to and helps to form the latest experience ; but I can see no tittle of evidence for an à priori in Kant's sense, i. e. of antecedence to all Experience ; or that we bring with us at birth a Mind equipped with Forms and Faculties. But, if we do not bring with us this full-statured Mind, if the stature is acquired through growth and development, then the experiential origin and limitation of all knowledge follow irresistibly.

186. Should any of the truths of Mathematics be shown to have an origin lying beyond Experience, or a range lying beyond the logical deductions from Experience, the claims of Metempirics will so far be made good that we shall be compelled to admit the possibility of metempirical knowledge. I purpose to show not only that the science of Mathematics has its origin in Experience, but that it differs from every other science only as each science differs from every other : it differs from Physics as Physics differs from Chemistry, or Chemistry from Biology, in the circumscription of its object, and the nature of its abstractions ; but it has a similar origin, a similar Method, a similar validity, and similar limitations.

187. The majority of mathematicians and philosophers resist the notion of Mathematics being classed among the sciences of observation and experiment ; a classification which is supposed to degrade Mathematics from its supreme position, and to introduce contingency into its results. Because it is with intelligible and not with sensible space that Geometry deals, and because its constructions are purely ideal, - because the line without breadth, and the surface without thickness, the perfect circle, or perfect parallels do not exist as reals, it is concluded that the science of these cannot be classed with sciences of Observation, Experiment, and Induction, which treat of real objects. The chapter on ideal construction in Science will have prepared us for an entire rejection of these positions (§ 110). The properties of Space and Number are assuredly discovered by Observation and Experiment, and their Laws are reached, like all others, through Intuition, Hypothesis, Induction, and Deduction being indeed simply formulae of the conditions specified, and only true under such conditions. The primary facts, the sensible intuitions forming the basis of this great superstructure, are so general and familiar, are so inevitably given in Experience, that we cannot imagine a mind in which they should not be present, — implicitly or explicitly. Hence by an easy transition they have come to be considered innate, antecedent to Experience. But they are no more innate than the primary facts of Chemistry or Biology. Al-though given in most sensible experiences they require to be observed, and reflected on, equally with less familiar facts.

188. A. single body, seen and touched, presents Extension and Form ; several bodies present Plurality,— Number. The bodies thus perceived are groups of sensibles, from which we abstract the qualities of Extension, Form, and Number. The bodies are also perceived in motion, i. e. changing their places without at the same time undergoing any change in their qualities. Place thus becomes detached from the bodies to be considered by itself ; and the abstract of all places is Space. This Space, which is filled by all bodies, — occupied by their Extension, — is only sensible or intelligible as Extension : the characteristic quality of bodies has been transferred to this abstract Space, and as all places are extended, Space is Extension.

189. The science of Geometry may be defined,— the study of the properties of Extension ; as Mathematics in general may be defined, — the study of the indirect measurement of magnitudes. Both Extension and Magnitude are qualities of reals. The properties of Space are observed, classified, reduced to Types and Standards, in precisely the same way as all other properties are observed, classified, idealized. They are first found in sensible intuitions of figures; and although rapidly carried up into the region of Conception, where they seem to depart from the reals of Perception, this is equally the case in all other sciences. The ideal constructions of Biology are never found realized in Nature. It is no more the sheer observations of the biologist than it is those of the mathematician which constitute the material of construction ; nevertheless without the observations no science would be possible. The mind intuites what the eye cannot see. Not — as is generally supposed—because the mind is independent of sense ; it is dependent on sense as Algebra is On Arithmetic ; and we could never intuite the mathematical and biological Types, had we not seen the real objects of which these Types are the ideal forms. So far from the mathematical intuitions being innate, the majority of mankind pass to the grave without a suspicion of them,— without making explicit to their Consciousness what, as elements of the Logic of Feeling, are implicitly present there. No one supposes biological intuitions to be innate ; yet the majority of philosophers hold that mathematical truths carry with them, in the characters of necessity and universality, evidence of their metempirical origin. How comes it, then, that the savage arrives at explicit biological truths long before he arrives at mathematical truths ; knows, and can state his reasons for knowing, that air is necessary to respiration, and food to growth, long before he has any suspicion that two things equal to a third must be equal to each other, or that parallel lines if prolonged would not meet ?

190. The objects of mathematical study are reals, in the same degree as that in which the objects of any other science are reals. Although they are abstractions, we must not suppose them to be imaginary, if by imaginary be meant unreal, not objective. They are intelligibles of sensibles : abstractions which have their concretes in real objects. The line and surface exist, and have real proper-ties, just as the planet, the crystal, the plant, and the animal exist, and have real properties. It is often said, that "the point without length or breadth, the line without breadth, and the surface without thickness, are imaginary : they are fictions ; no such things exist in reality." This is true, but misleading. These things are fictions, but they have a real existence, though not in their insulation of ideal form, for no idea exists out of the mind. These abstractions are the limits of concretes. Every time we look on a pool of water we see a surface without thickness ; every time we look on a party-colored surface we see a line without breadth as the limit of each color. Both surface and line as mathematically defined are unimaginable, for we cannot form images of them, cannot picture them detached ; but that which is unpicturable may be conceivable; and the abstraction which is impossible to Perception and Imagination is easy to Conception. It is thus that sensibles are raised into intelligibles, and, in the constructions of science, conceptions take the place of perceptions. But the hold on Reality is not loosened by this process. When we consider solely the direction of a line, we are dealing with a fact of Nature, just as we are dealing with a fact of Nature when we perform the abstraction of considering the movement of a body, irrespective of any other relations. We no more think that the line is unreal, than that motion is unreal; we no more believe that a surface can exist without an under surface, than we think that a movement can take place without a moving body. M. Delbœuf pertinently remarks that if Mathematics be called imaginary, there would be equal justice in our saying to Newton and Laplace " Your celestial mechanics is false, for there are not in Nature bodies which are only heavy."

191. Not only is it misleading to call the objects of Mathematics imaginary, it is also incorrect to call them generalizations. They are abstractions and intuitions. Any particular line that we can draw, or imagine, has breadth ; any particular circle is imperfect ; consequently generalized lines and circles must have breadth and imperfection. Whereas the line, or circle, which we intuite mathematically is an abstraction, from which breadth or imperfection has been drop ed, and the figures we intuite are these figures under the form of the limit.

192. Unless the objects of Mathematics were real, in the sense just explained, it would be absurd to suppose that the relations intuited could be applied to the discovery of other real relations. A moment's inspection shows that the properties of angles and circles are discovered and demonstrated by the same principles that are applied to the discovery of gases or organic processes. Gauss, whose authority on such a subject weighs against a whole academy, declared Geometry to be the "science of the eye " ; and Prof. Sylvester, also a very considerable authority, declares that most if not all the great ideas of modern Mathematics had their origin in Observation. Among the surprising examples cited by him may be named Sturm's theorems on the roots of equations "which stared him in the face in the midst of some mechanical investigations connected with the motion of compound pendulums," and the discovery of the method of continued fractions by Huyghens, "to which he was led by the construction of his planetary automaton."

Hence it is that most of the difficulties in this science are difficulties rather of Intuition than of Reasoning; and most of the "vexed questions" which have occupied geometers — notably that respecting parallels and that respecting a fourth dimension in space — have arisen from neglect of Intuition. Because analysts are accustomed to operate on symbols they at last begin to assign a sort of talismanic virtue to symbols which will evoke results in defiance of intuitions. But here the words of the illustrious Poinsot deserve attention : " Ce n'est done pas dans le calcul que réside cet art qui nous fait découvrir ; mais dans cette considération attentive des choses où l'esprit cherche avant tout à s'en faire une idée, en. essayant, par l'analyse proprement dite, de les décomposer en d'autres plus simples, afin de les revoir ensuite comme si elles étaient formées par la réunion de ces choses simples dont il a une pleine connaissance."

193. Even in the higher developments of the Calculus, where sensible experiences seem most widely departed from, it is easy to trace a sensible origin for the extra-sensible intuitions ; precisely as in Dynamics and Physics we detect the sensible origin of intuitions which transcend Sense, e. g. uniform rectilinear Motion and Atoms. If there is one conception which might be supposed to justify a metempirical origin, it is that of infinitèsimals. Now we have this conception, it seems that it might have been evolved a priori, and ;that the active intellect of the Greeks might have reached it through their Method of Exhaustions. What is the fact, however ? It is that the ingenious Greeks were arrested in their course by the impossibility of reaching a conclusion now seen to lie so near at hand. Nor was it until Mathematicians had mastered the theory of the composition of motions, by which the path of a projectile was seen to be compounded of two straight lines in different and unceasingly changing directions, that the conception of infinitesimals arose.

194. Enough has been said, some will think more than enough, to establish' the first part of our thesis, that Mathematics is a science of Observation, dealing with reals, precisely as all other sciences deal with reals. It would be easy to show that its Method is the same ; that, like other sciences, having observed or discovered proper-ties, which it classifies, generalizes, co-ordinates, and sub-ordinates, it proceeds to extend' discoveries by means of Hypothesis, Induction, Experiment, and Deduction. On the large use of Hypothesis and Deduction there need be nothing said here, since no one disputes their importance. Induction and Experiment, however, demand consideration.

195. By some minds the very suggestion of mathematical truths being reached by Induction is resisted ; yet it is certain that not only does Induction play a part, but according to some writers that part is very consider-able. "Induction and analogy," says Professor Sylvester, "are the special characteristics of modern Mathematics in which theorems give place to theories, and no truth is regarded Otherwise than as a link in the infinite chain." Some of the divergence on this point must be attributed to the divergent conceptions of what constitutes Induction, much that is even by Mr. Mill included under that head being either Intuition or Description. No one can refuse to recognize it as purely inductive when having calculated a number of terms of a series, and ascertained the law of the series, we fill up the succeeding terms without calculating them ; the induction here consisting in our inference that the succeeding terms will con-form to the law of the calculated terms ; an inference which may be false in special cases. It was assuredly an induction by which Fermat concluded that + 1 is always a prime number when n has the form 2"', i. e. is 2, 4, 8, 16 . . .; but Euler showed that the induction was erroneous when n was 32 ; for 2' + 1 is not prime.

196. If these are pure inductions, the same cannot be said of numerous other examples, also classed under this head. Thus it is no induction by which we conclude that two straight lines having once met do not meet again, but continue divergent ; we do not infer this truth from comparison of instances, we intuite it. Axioms are not inductions, nor can they have been inductively reached ; they are intuitions universalized. I should therefore propose to qualify Mr. Mill's statement " that every induction which suffices to prove one fact proves an indefinite multitude of facts "; the ambiguity which lies in the word "multitude" renders this proposition misleading. An induction cannot prove an indefinite multitude of facts, unless the facts be all repetitions of the one first proved ; if the multitude include any facts having other relations than those proved, the inference is erroneous. On this ground it is misleading to call axioms inductions. Let us take a case selected by Mr. Mill.* He says that when we have to determine whether the angles at the base of an isosceles triangle are equal or unequal, our first consideration is, what are the inductions from which we can infer equality, or inequality ? He specifies eight axioms. Recourse to inductions is necessitated be-cause "the angles cannot be perceived intuitively to have any of the marks " specified in the axioms, although on examination it appears that they have such marks. I agree with him in considering this a case of discursive, and not of intuitive judgment (§ 153), and that the relations of equality are not immediately presented, but have to be sought and compared. ut I cannot consider that the axioms, " Things which being applied to each other coincide are equals," or " The whole and the sum of its parts are equals," have the characters of Induction ; they are identical equations, — propositions which exclude all contingency by excluding all inference. No one has more clearly shown than Mr. Mill the distinction between inductions properly so called, and "generalization in which there is no induction because there is no inference; the conclusion is merely a summing up of what was asserted in the various propositions from which it was drawn" (I. 324). On this ground we must refuse the character of Induction to those axioms which are simply intuitions generalized. With reference to the particular case chosen, instead of the roundabout demonstration of Euclid, or that proposed by Mr. Mill, we might reduce it to two intuitions : 1°, The isosceles triangle has equal legs ; this equality is intuited in the terms defined ; 2. The legs being equal, what is seen of the one is seen of the other, i. e. the angle formed by one leg with the base will be equal to the angle formed by the other.

197. Were Mathematics founded on induction there would be contingency in all its propositions which extend beyond particular cases ; and each conclusion would require experimental verification, direct or :indirect. But this is true only of portions of the science. The* greater part is founded on Intuition, and its conclusions are universal. We are not, however, to suppose that Experiment has little to do here. "A science is experimental," says Mr. Mill, "in proportion as every new case which presents any peculiar features stands in need of a new set of observations and experiments, and a fresh induction." Mathematics is experimental therefore in this, that for every step in advance, as Professor Robertson Smith has well said, everything not the result of calculation or deduction, there is needed a new figuré and a new intuition. The experimentation is easier, less complicated, than in Physics or Chemistry, the elements are more manageable, because more sharply defined, and we are under no misgiving in dealing with them lest they should include any unknown elements which could affect the result. We are perfectly sure that in bringing a right line into relation with another, it is only this relation we have to deal with ; whereas in bringing a gas into relation with a solid we do not know all the co-operant factors ; and our experiment reveals only some of the actual results. But although easier, the procedure is similar. The necessity for a new figure, and a new intuition, is shown at every step. We could not reach the simplest proposition without these. Ask any one, not already instructed, whether it is possible to let fall from a point more than one perpendicular on a straight line ; or whether all parallelograms between the same parallels are equal when their bases are equal. These propositions are so far from being self-evident, or capable of being deduced from the axioms and definitions, that he cannot answer until he has seen the figures and intuited the relations. It is by experiment alone that he can determine the equality of spaces included in figures so unequal as an oblique and a perpendicular parallelogram on similar bases. The chemist has his elements, or what he regards as such, and these he combines and recombines, in various ways, to watch the reactions, and detect the constant results. The geometer has his elements (points, lines, planes) which he combines and re-combines, watching the results. He draws a circle, and divides it by a straight line into equal halves. This straight line he again divides by another, and thus forms four right angles which fill the space circumscribed by the circle. He goes on adding figure by figure and detecting new relations. Like the chemist he gets at constant results, which enable him to foresee what will be the effect of new combinations : he can calculate as well as count. But although Deduction will carry him much further than it will the chemist, because of the greater homogeneity of the elements he deals with, it will not suffice without Experiment, Verification. The man who first discovered that 7 + 5 = 12, did so by a synthesis, which was experimental, not less than that by which the chemist discovered that two volumes of hydrogen and one volume of oxygen constituted water.

198. If in cases so simple Experiment is needed, it may readily be understood how in more complex cases the mathematician essays the demonstration of a problem through a series of tentatives, till he hits upon the construction which discloses the solution, or finds that no solution is possible under the given conditions. Suppose, for example, he asks himself whether there may not be a quadrilateral figure having equal sides, and having two of its angles equal to three right angles : he cannot construct such a figure ; the attempt would at once disclose that such a figure was inconsistent with the properties of quadrilaterals.

Kant has shown that even identical propositions such as a=a, or the axiom " The whole is greater than its part," are admissible only because they can be presented in Intuition ; and we formerly saw that even these are demonstrable, and demonstrable only on the assumption of homogeneity. Whence we conclude that Mathematics must be dependent on and limited by Experience, which furnishes intuitions.

199. There has been much dispute as to whether Mathematics is founded on Axioms or on Definitions. This dispute may be' cleared up by a more rigorous interpretation of the terms. In the sense commonly assigned, it is neither to Axioms nor to Definitions that the foundation can be ascribed, but to Intuitions ; and to Axioms and Definitions only in virtue of their expressing Intuitions. Nor let this be considered idle cavilling. For those who take their stand on the Axioms, hold that the whole science is nothing but the analytical unfolding of the few Axioms placed at the opening of each treatise and this seems to be doubly erroneous ; first, because those Axioms are too few for the purpose, — each step requiring a new intuition, which, when generalized, becomes a new Axiom ; secondly, because they derive their whole validity from Intuitions. Both objections may be condensed in one : the science is synthetical and not analytical.

Take the Axioms of Euclid and try by them alone to deduce the Pythagorean theorem, and it will be found' as idle as the attempt to deduce the action of a poison from the axiom " Every effect has its cause." Dugald Stewart, fully alive to the barrenness of Axioms, sought in Definitions for the real foundation. But the same argument applies here. Unless the Definitions are intuitions of the figures and relations defined, they are also barren.* Definitions, moreover, must not be arbitrary, if they are to lead to other than arbitrary conclusions. We may, if we please, define parallel lines, lines concave to each other or define 5 to be the sum 2 + 2. But in this case all our deductions must be consistent with these assumptions ; and we cannot then say that two parallel lines will never meet when prolonged, nor that 5 is 3 + 2 or 10 = 2. The mathematician does not begin by assuming the properties of figures, and after defining them proceed to ascertain whether such figures exist ; he begins by ascertaining that such figures and such relations do exist, and then defines them as he finds them. In other words, Definitions are the expressions of the figures, not their foundations. With Definitions we can take no step in advance, we can only analyze them.*

200. In the Imaginary Geometry of Lobatschewsky and Beltrami we have indeed a theory of parallels founded on Definitions. Instead of the intuitions really presented to us by the figures, the definitions are made to express relations different from those intuited ; they are arbitrary, and although the deductions from them are consistent with these arbitrary premises, and are therefore logically accurate, they are inapplicable to the real objects given in our Experience. Lobatschewsky's arbitrary' definition of parallelism j- is as wide a departure from the real intuition expressed in Euclid's definition, as the definition of 5 = 2 + 2 is from our definition of that number.

201. Not only do we find Observation, Hypothesis, Induction, and Experiment everywhere underlying the constructions of Mathematics, as in any other science, we also find that in both the abstractions are all raised from sensibles and extra-sensibles by a similar process. The argument that they cannot have been derived from sensible concretes, because our senses never present them under the forms dealt with by mathematicians, may equally be applied to other sciences : the heavens show no elliptical orbits ; our laboratories show no perfect gases ; our islands and continents show no species. And there is good reason why this must be so. Science deals with conceptions, not with perceptions ; with ideal not real figures. Its laboratory is not the outer world of Nature, but the inner sanctuary of Mind. It draws indeed its material from Nature, but fashions this anew according to its own laws ; and having thus constructed a microcosm, half objective, half subjective, it is enabled to enlarge its construction by taking in more and more of the macrocosm.

Science everywhere aims at transforming isolated perceptions into connected conceptions, — facts into laws. Out of the manifold irregularities presented to Sense it abstracts an ideal regularity ; out of the chaos, order. The imperfectly straight real lines give place to lines ideally straight. Having to introduce Likeness (equations) amid a manifold Unlikeness, we begin by reducing to a first Likeness all the diversities of spaces and numbers presented to Sense; and thus get ideal Space everywhere homogeneous, and ideal Number. And so with the rest. But this recognition of the ideality of Mathematics must not cause us to overlook the fundamental fact that only in so far as the ideals are constructed from reals can they have any validity in reference to reals. Kant teaches that objects conform themselves to our modes of Sensibility, and that it is we who invest them with our forms, which is all we know of them ; and he denies that the things themselves determine our forms. I have already stated in what sense I regard this as true, and in what as false ; it is irreconcilable with the ideality of Science ; for were it true that objects received their forms entirely from us, we should find in Nature those very forms which we do not find there, — the perfect circle, the pure gas, the defined species, the histological tissue. These exist, but they exist in our conceptions, not in our perceptions. How they arise in conception, as abstractions from perceptions, we know very well ; whereas, if we only saw in Nature what the Mind brought with it, and reflected on objects, we should see the perfect abstractions, and not the imperfect concretes : and we should see these unaided by Science.

202. Having pointed out the cardinal characters in which Mathematics resembles other sciences, we have finally to inquire if there are any other essential characters which would suffice for a genuine difference. There is one character which will be considered decisive, and that is the apodictic certainty belonging to mathematical conclusions. Kant in the preface to his Practical Reason declares that we might as well attempt to squeeze water from pumice-stone, ex pumice aquam, as to get at necessary and universal truth through experience.* We, on the contrary, have seen that all truth is necessarily true, under the specified conditions ; all truth is universally true if the conditions be universalized; and in these respects Mathematics has no superiority over Biology. But, it may be argued, mathematical truths have an universality denied to all other scientific truths, in that they relate to fundamental aspects under which all things are perceived by us, — thus all things whatever are numerable, and all things are extended. But mathematical truths are not true irrespective of conditions ; and their universality is restricted to our universe. The space of geometers is a space of three dimensions ; and many of their truths would cease to be necessary and universal in a space of two or four dimensions. We must say, therefore, that the truths of Mathematics, like all other truths, have their origin in Experience, and are true only of the universe known through Experience.

203. The superior certainty of Mathematics arises from the superior facility with which certainty is reached and exhibited to others. There can be but one certainty, — that of an identical equation, or identical proposition, - and this admits of no degrees ; it is, or is not. Nor does the equation of condition admit uncertainty, directly the condition which satisfies the equation has been found.

The laws of Motion, Affinity, Life, and Mind, although in successive degrees less general than the laws of Quantity, are not less exact, less certain. The terms in which they are expressed may be less exact, and their application to particular cases may be far more contingent, than is the application of the laws of Quantity ; but when the laws formulate real relations, and are true, their-certainty is unaffected by contingencies of expression and application. A general law is raised by abstracting the constants from the variables, — out of many particular cases we let drop all the special circumstances which individualize each case, and the residuum is the generalized law. When this law has to be applied to some new case, we have to modify it by the reintroduction of such special circumstances as will individualize the case : unless we do this, the law will not hold good, the case will not fall under it. Now it is our very uncertainty respecting these special circumstances which constitutes the contingency of the law. Could we be assured, as in mathematical questions we commonly are, of having all the co-operant factors within our grasp, contingency would vanish. In many scientific propositions this condition is fulfilled ; the abstract truth in Biology is as absolute as the abstract truth in Geometry. If this condition is rarely or never fulfilled in concrete Biology, the same must be said of Applied Mathematics. The proposition : " Water is indispensable to the vitality of a tissue," is not less exact, not less certain, than the proposition : "In a right-angled triangle the square of the hypothenuse is equal to the squares of the two other sides." Neither proposition is self-evident; both have to be shown by experiment ; and when shown, each is seen to be an identical proposition. The inter-mediate steps through which it is shown that Vitality is never found without chemical change, and that water is necessary for such change, may pair off with the steps by which it is shown that in the parallelograms on equal bases, between parallels, the triangles are equal. In both cases a series of identical propositions forms the sub-stance of the conclusions, and the conclusions therefore are equally identical propositions.

But now observe : although it is indisputable that alcohol in sufficient quantity, or concentration, will with- . draw from a tissue in contact with it so much water as to destroy the vitality of the tissue, — and although it is an indisputable corollary that drinking alcohol in such quantity must cause a man's death, — the inference, which to many seems logical, that any quantity of alcohol must, if not destroy, at least diminish, Vitality, is an inference wholly contingent. Every one knows that quantitative differences must have corresponding functional differences. Because a certain quantity of alcohol will destroy a tissue, we are not to conclude that any smaller quantity will do more than disturb its molecular equilibrium, which temporary disturbance may be a positive advantage to the organism. We are here in the midst of indeterminate quantitative relations, and must call in the aid of experiment to determine whether certain quantities are or are not injurious. Could we once ascertain the precise quantities which had precise functional consequences, our treatment of the alcoholic question would be rigorously exact. But at present it is no more capable of a solution than an equation of the sixth degree.

( Originally Published 1874 )

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