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On an Evolutionist Theory of Axioms

On an Evolutionist Theory

of Axioms

Inaugural Lecture

Delivered October 15, 1889


J. Cook Wilson, M.A.

Wykeham Professor of Logic in the University of Oxford
Fellow of Oriel College, Oxford

B. H. Blackwell, 50 & 51 Broad Street
London: Simpkin, Marshall & Co.


I could have wished, in an introductory lecture of this kind, to say something of my predecessors; but I have not the knowledge necessary to do them justice, and I am afraid also that it would seem unbecoming in me to offer to pronounce a judgment on them or their work.

The names of the two professors of philosophy who have so recently vacated their office will occur to every one's mind. The one is my immediate predecessor, who is fortunately still with us. His work is well known, and forms a part of the studies of the University: it would be unnecessary and presumptuous for me to praise it. The other, who was Waynflete Professor of Moral and Metaphysical philosophy is, alas, dead. I would I were able to pay a fitting tribute to his memory: but it requires far more learning than I can claim to appreciate fully his loss to the scientific world, and convey a true sense of it to others. And there is another difficulty—in him I have lost a kind and dear friend. This mention of the dead reminds me that it has also been my privilege to enjoy not only the teaching but the friendship of two of the most distinguished philosophers of our day in England and Germany—Professor Green and Professor Lotze. The feeling of those of us who came into more immediate contact with Professor Green is too deep for words. Professor Lotze was a striking figure even in Germany. He united diverse gifts in a remarkable way. He was an acute metaphysician and logician, a man of great artistic taste, a classical scholar, one of the foremost psychologists, and one of the foremost biologists in Germany.

To enter into the work of any one of these three dead teachers would be a great ambition. I hardly feel worthy to entertain it, much less, do I hope to realise it.

I must not forget that it was also my singular fortune to be the pupil of one of the most distinguished mathematicians in Europe, Professor Henry Smith. Would that my abilities had allowed me to profit more by his instruction.

I do not forget others distinguished in mathematics, classics, or philosophy, to whom I am indebted for instruction, but I do not name them, for I am thankful to say they are still alive.

Whatever a man's own unworthiness may be, he is allowed to praise his teachers. For me, however, the task is altogether too difficult—I have had such great teachers.

Leaving these personal matters, I propose, after giving a very short account of what I imagine, rightly or wrongly, to be the chief tendencies of English Logic, to offer a criticism of what has been thought the crowning achievement of a school of philosophy which is rather popular in England.

What is called Formal Logic, does not seem to have had many attractions for the greater number of the principal English philosophers. Something like a paradox is true of it. While its subject-matter has perhaps seemed neither to demand nor to reward any great effort of thought, the question as to how it should be defined, and what constitutes the difference between the Form and the Matter of Thought has been an object of laborious and subtle investigation to thinkers of high repute in Metaphysics.

Within Formal Logic itself, however, there has been a particular development, which, as far as I know, has been more seriously undertaken in England than in any other country—and that is the reduction of Formal Logic to a sort of mathematical calculus. It is, however, confined to a few specialists. I do not here propose to offer any consideration of its merits.

The idea of Logic as an Organon, that is, as an instrument of scientific discovery, or as subservient to scientific discovery, has been one of the strongest influences in English Logic. It was hoped that herein a trustworthy method might be supplied to those sciences which had not reached the precision of mathematics. This hope so eloquently expressed in the Novum Organum, led ultimately to the preponderance of Induction in our home-grown Logic.

Quite different to this, not home-grown, but due rather to the influence of German Metaphysics, is a treatment of the subject which has established itself of late among us. In this the abstractions of the old Formal Logic have gained new interest. Studied for themselves, and not as elements in an Organon, they are found to suggest important questions about the inner nature of the thought which expresses itself in them. It seems characteristic of this logic that it subjects to a thorough criticism, the distinctions commonly accepted as fixed by Formal Logic and used without question in other departments of English Logic and Philosophy. Its results are at least stimulating.

But it concerns me more at present to point out within the more properly English Logic, another subject which has been well represented. Whether it belongs really to Logic or not—for there is much dispute about the province of Logic—it is most intimately connected with even the narrowest conception of the subject, and has interested students of Logic from the first beginnings of this part of philosophy till now. I mean the inquiry into the origin of what are called axiomatic truths, or first principles.

The question, as is well known, is one of those fundamental ones which divide philosophers into opposing schools.

I propose to consider the contribution made to it by the Evolutionist school of philosophy.

May I be allowed to say that in the discussion which will follow, I use the words 'evolution' and 'evolutionist' in reference to evolutionist philosophy, and not to evolutionist science. Why I am anxious to make this distinction will appear hereafter.

The Evolutionist claims to have solved the vexed problem, and to have reconciled all previous schools of philosophy.

The principal representative of this philosophy says—'Already I have pointed out that the hypothesis of Evolution "supplies a reconciliation between the experience-hypothesis as commonly interpreted and the hypothesis which the transcendentalists oppose to it;" and here we see how complete the reconciliation is.'

This claim is a high one and made, as we see, with some conviction. The theory, though distinctly empirical, recognises that induction is insufficient for axioms. It is well known, but its essential features must be re-called.

The sufficient and the only guarantee, it is said, of a universal proposition is that we cannot conceive its contradictory. This is the ground of our belief in axioms; and thus they are à priori for the individual, being due to the constitution of his mind and not to his experience. But this constitution of his mind has been itself produced by experience, not his own, but that of the race.

The effect of experience is to modify the nervous structures: the modifications have been accumulated by inheritance from generation to generation, so that the mind is at last, in the case of axioms, completely adapted to the object Thus, in the well-known formula, axiomatic beliefs are said to be à priori for the individual, but à posteriori for that entire series of individuals of which he forms the last term.'

I will read a passage which describes how all this happens:—

'Little more need be said to make it clear how certain primary space-relations are presented to consciousness under the form of necessary relations. If a segment of a circle be looked at, the image of it cast on the retina is necessarily such that the arc covers a greater number of retinal elements than the chord; and since each of these retinal elements yields its separate impression to consciousness, the series of impressions produced by the arc is felt as larger than the series produced by the chord. This continues to hold however much the arc is flattened: so long as it has any perceptible curvature at all, it is felt to be longer than the chord uniting its extremes. Parallel experiences are derived from the ocular muscles. Carrying the eye along the line of the curve, yields to consciousness a greater quantity of sensation than carrying the eye along the chord does. As the curve is flattened this difference diminishes; but some of it continues as long as the curve continues appreciable. Thus the truth that a straight line is the shortest line between two points, lies latent in the structures of the eyes and the nervous centres which receive and co-ordinate visual impressions. We cannot think otherwise because, during that adjustment between the organism and the environment which evolution has established, the inner relations have been so moulded upon the outer relations that they cannot by any effort be made not to fit them. Just in the same way that an infant's hand, constructed so as to grasp by bending the fingers inwards, implies ancestral hands which have thus grasped, and implies objects in the environment to be thus grasped by this infantine hand when it is developed; so the various structures fitting the infant for apprehensions of space-relations, imply such apprehensions in the past by its ancestors and in the future by itself. And just as it has become impossible for the hand to grasp by bending the fingers outwards instead of inwards; so has it become impossible for those nervous actions by which we apprehend primary space-relations to be reversed so as to enable us to think of these relations otherwise than we do.'

This theory it may be remarked does not necessarily imply that the mind is identical with physical organs, but it does imply that the mind not only perceives by them, but can only think and imagine by their means, and is therefore restricted by the functions of which they are capable.

Divested of technicalities unnecessary for our present purpose, it comes to this. In its early experiences the mind was not obliged to think the arc greater than its chord. But every arc presented to the experience of every generation has been greater than its chord: and the constant perception of this relation gradually modified the perceiving and thinking organs, so that it became more and more difficult to think arc and chord in any other relation. The difficulty transmitted by inheritance became stronger by accumulation in successive generations, till it became altogether impossible to think the contradictory of the axiom. This is how the individual's belief in the axiom is formed, and the process secures that the subjective belief shall correspond to an objective fact.

This has an appearance of clearness and simplicity which recommends it to many.

Before asking if it is true, we may ask whether we should like it to be true. Such a question may seem unscientific, but in philosophy at least we should clear our reasonings a good deal if we confessed to ourselves what we wanted to believe and what we hoped was untrue.

Now at first sight the theory may well have attractions. Many will feel that induction does not account for the conviction they have had as far back as they can remember in the same strength about axioms: and yet they will feel on the other hand the force of Mill's objection—'I must protest,' he says, 'against adducing, as evidence of the truth of a fact in external nature, the disposition, however strong or however general, of the human mind to believe it'

The evolutionist view seems to bring a welcome help and to shew how the desired correspondence between subjective belief and objective fact has been brought about by the operation of known causes.

No doubt then many may think, it would at least be a good thing if it were true. The evolutionist philosopher probably believes that it is a beneficent result of the great law of Evolution; and there may be scientific evolutionists who feel at least a prejudice in its favour.

But let us look a little nearer to see whether we should have cause to congratulate ourselves if the evolutionist philosopher were right.

The theory that the mind has become modified by its environment, and that its functions have become fixed, implies that at first it was not modified and its functions not thus fixed.

If 'Just as it has become impossible for the hand to grasp by bending the fingers outwards instead of inwards; so has it become impossible for those nervous actions by which we apprehend primary space-relations to be reversed so as to enable us to think of these relations otherwise than we do,' then it follows that once it was possible that these nervous actions should be reversed. And what does this mean? It means that the mind and its organs were so constituted that they were capable of thinking the contradictory of axioms, for instance of conceiving a chord equal to or greater than its arc, or two straight lines which enclosed a space, only that this capacity was never developed.

To some this will appear a sufficient reductio ad absurdum, but I do not propose to take that line of argument.

Let us rather imagine that such a condition of the mind is possible.

It will follow that our minds have been deprived of half their powers beyond recovery. The geometrician, for instance, is cut off from a field of thought as large as the present science—a field in which among many other things he would have developed the properties of those straight lines which are not the shortest distance between any two points upon themselves. In fact, in respect of all that is axiomatic, that is, of all that appears self-evident, the whole human race is in the position of that part of it which has lost or never had the use of a particular sense—the deaf, the blind, or the colour-blind, or those who have no ear for music. We are like the Proteus which in dark caves has gradually lost the eyes which would be useless to it: except that we are worse off, for though an object for our lost faculties may never come into our experience, we should obviously have plenty of employment for them.

The same process has woven a falsehood into our nervous tissues and so into our minds: for the confidence we have in axioms is made to come from our conviction that their contradictories are not even thinkable, and yet it is shewn at the same time that this is an utter mistake, inasmuch as thought in itself, though not as developed in us, is capable of thinking these contradictories. The individual then at best has got a true belief at the expense of entertaining a false belief, and at the expense of the loss of half his mental powers.

Is this a result to be contemplated with satisfaction?

May one not look for sympathy if one confesses to the hope that it is not true? Surely it would be one of those things for which we could not honestly say we were thankful. At best it would be a matter for resignation and not for gratitude. We may bow to it as an inevitable decree of nature; we deceive ourselves if we say we can see it is good and desirable. The pessimist would find another instance of Nature's unkindness. Even where she seems to confer a benefit, she has done us a deep injury. She has mutilated our minds with a refinement of cruelty. For instead of mercifully concealing a loss which could only be deplored and never repaired, she has evolved the philosophy which has betrayed the secret.

And is there really any compensation?

If we have gained a belief which happens to be true, can we be said to understand it if we ground it on a falsehood? Can we be said to 'know' at all in the proper sense of the word?

Can a belief be even intelligent which is simply caused by the want of a power to think otherwise, a power which might have been developed, and is lost to us by a process as unconscious as that which deprives us of taste or smell when we have a cold?

In ordinary life, when we are not philosophising we have a great contempt for beliefs which are merely the result of imperfect mental development; we expect them among savage tribes, the prejudiced, and the uneducated.

An attempt has been made to remove a bias in favour of the evolution theory. If the attempt were successful, it must be admitted that it would create a bias against the theory: and there is all the more reason to try to take a purely scientific attitude, and ask whether we have to believe whether we like it or not.

In the first place attention must be expressly called to the fact that this is a theory of knowledge. It does not explain merely how we come by a belief, but how that belief is true; that is how we have knowledge.

I will read a passage to make this clear:—

'Hence the inconceivableness of its negation is that which shows a cognition to possess the highest rank—is the criterion by which its unsurpassable validity is known. If the negation of a cognition is conceivable, the discovery of this amounts to the discovery that we may or may not accept it. If its negation is inconceivable, the discovery of this is the discovery that we are obliged to accept it. And a cognition which we are thus obliged to accept, is one which we class as having the highest possible certainty. To assert the inconceivableness of its negation, is at the same time to assert the psychological necessity we are under of thinking it, and to give our logical justification for holding it to be unquestionable.'

Other passages to the same effect might be quoted.

I begin with an objection, relating to a matter already referred to, which perhaps concerns the form of the theory rather than its essence. The evolutionist does not seem sufficiently to notice that, while seeming to make inconceivability an ultimate test or criterion, he is really throwing entire discredit on the mind's power to conceive or not conceive as such.

Men do indeed believe firmly when they think the contradictory of their belief inconceivable; not as being merely inconceivable for them, but as being inconceivable for thought as such. For the moment a man is convinced that some one else can conceive what he cannot, he ceases to suppose that his own state of mind can determine the truth in the particular case. Thus a man without a musical ear nevertheless may believe that there is such a thing as a musical order. But the supposed history of the mind's modification implies that though we cannot conceive the contradictory of an axiom we might have been able, and our ancestors perhaps were able: and thus the mind's limitation in respect of the conceivable becomes as such entirely indifferent.

Any reliance placed on the inconceivableness of the contradictory as a criterion must lie, and indeed according to this theory does lie, in the way in which it has been produced, and this is the essential thing to be considered.

The criterion has its value because it is the effect, as supposed, of an experience infinitely greater than that of one individual: and thus when Mill objected that if the test got its value from experience, we had better appeal directly to experience, it was replied that the large experience represented by the test was inaccessible to the individual.

It is easy to see that the maximum logical worth of the test would be that of an induction drawn from all the instances which have occurred in the experience of the race, supposing these could be presented to one mind which would draw the inference. The induction has registered its conclusion automatically in the physical organism, and, through that, in the mind. The race has come to believe that all A must be B, because the constant experience through its history of instances of A which were B has so modified the organs that they cannot be used for thinking A except as B.

It has been inadvertently assumed that in axioms such a belief has been produced in one way only.

From the very assumptions about biological processes which this theory makes, it is clear that the sufficient and necessary condition that the belief 'all A must be B' should be produced, is not that every A should be B in the nature of things, but only that those instances of A which have acted on the organs of perception should have been B. And thus the conviction that A must be B could be perfectly well produced in the automatic way supposed, in a case where A was not necessarily B. The limitation of our experience to a certain species of A which would be necessary for this, would correspond to known facts. For instance, generation after generation of a particular race had never seen swans which were not white. Countless generations living in certain parts of the earth have never seen water in a solid form.

Thus from the very manner of its formation our belief that the arc must be greater than its chord is compatible with the existence of arcs which are not greater than their chords. There are ways enough in which our limitation to one kind of arc may have come about. For instance, naturalists expect to find new forms of flora and fauna in regions which are being explored for the first time. How do we know—not to speak of what may be found in other planets and stars—that the polar regions have not an abundance of that other kind of arc and chord? And was it not as important scientifically that the naturalists in the Challenger should have dredged the deep seas for them as for those objects in which they were immediately interested?

Or again, allowing, what could not be known if the theory we are examining were true, that all the arcs in the world now are greater than their chords, how can the evolutionist know that in early geological periods there were even any arcs at all which were greater than their chords } For aught he can say the arcs not greater than their chords may have been contemporary with the 'dragons of the prime' and have perished with them. Why should he expect us to allow arbitrary assumptions here, which are allowed in no other empirical science?

It is quite remarkable to find the evolutionist confidently making such statements as the following, without even asking himself how he could be entitled to them.

'Space-relations have been the same not only for all ancestral men, all ancestral primates, all ancestral orders of mammalia, but for all simpler orders of creatures.'

But there is another and more serious form of the difficulty.

It is not even necessary that the experience of the race should have been uniform. Arcs may have been seen equal to or less than their chords, but if they only came seldom, or if in the course of time their number much diminished, like that of an expiring race of animals, then according to the biological principles presupposed, the infinitely greater accumulation of contrary experiences would in time wipe away all trace of them from our organism.

Worse than this—owing to the way in which Nature has fixed our functions we could not perceive arcs which were not greater than their chords, even if there were plenty of them about us. It is therefore useless to dredge the deep seas or to go to the Poles, and indeed we do not know whether in our own parts of the world these arcs have not begun to appear. We could no more see them than a blind man could.

And thus the evolutionist, professing to have established the validity of the test of inconceivableness, has unwittingly shewn it to be consistent with the contradictory of what he supposes it to absolutely guarantee. It follows irresistibly that all that seems to us simplest, clearest, most self-evident, and more certain, according to the evolutionist himself, than anything else we can believe, may be an illusion.

The theory which was to have reconciled great philosophies has destroyed itself, and has ended in a scepticism which has not even the merit of being self-conscious.

The contradiction cannot be avoided by dogmatically affirming that the experience which has fixed our functions does, as a matter of fact, correspond to a universal truth which has no exceptions: for this involves, according as we look at it, a new inconsistency or an argument in a circle.

(i) We are told that the ultimate and only criterion for the individual who forms the last term of the series is the test of inconceivableness, and yet on the other hand that the criterion is trustworthy because an invariable experience has produced it. To vindicate this position the evolutionist must know the latter proposition to be true. But if he does he has a higher knowledge than the criterion itself, because it is knowledge from which the criterion derives its value, and thus the criterion is not the ultimate criterion for him. The evolutionist has fallen into this contradiction apparently because he, has forgotten that he himself is one of the last terms of the series.

(ii) Or the difficulty may be put thus. The inconceivableness of the contradictory of an axiom has been produced, it is said, by a constant experience in the race, which again corresponds to a universal truth without exception. How does the evolutionist know that the experience of the race has been such, for he cannot have had it; and à fortiori how can he know the universality of the corresponding fact in nature? Apparently from what has been quoted he thinks he knows. Being the last term of the series he can only know the fact by the inconceivability which he supposes it has produced in himself: and thus the invariable experience and the corresponding fact give the criterion its value, and they themselves are only known by the criterion: which is arguing in a circle.

This may be shortly put by saying that the evolutionist philosopher has cut himself off from the possibility of giving the necessary evidence in favour of his own theory.

But it may be answered that though satisfactory evidence for the theory cannot be given, it is a possible hypothesis, and though it may have to surrender all claims to establish the validity of our beliefs, it may be true, for constant experience might produce such an effect on our organism.

This of course would be giving up a great part of the evolutionist position.

In the first place, it must be answered that according to biological laws the loss of a function or of an organ may not only be caused by the want of a use for them, but also by the operation of various processes, among which, for instance, are diseases, and how can it be known that some such process has not destroyed our capacity for thinking the contradictory of axioms?

But the most important difficulty is this.

The objector can hardly have reflected on the real consequences of his hypothesis. It would throw a doubt, as we have seen, on every principle however simple, self-evident, and certain it may appear, and therefore it would leave nothing even to found itself upon.

It would throw doubt also upon experience which is supposed to be our most direct access to reality, for owing to the limitation of the nervous structures to one kind of function, our minds may be misrepresenting the object.

How deep this unconscious scepticism has gone will appear still better from another aspect of the theory.

'For logical intuitions' it is said 'there is no warrant assignable other than that assignable for all intuitions accepted as certain: namely, the impossibility of thinking the opposite. Unless it be alleged that the consciousness of logical necessity has a different origin and a higher origin, it must be admitted that the consciousness of logical necessity is just as much a product of past experiences as is every other consciousness of necessity.'

It follows then, from what we have seen, that these logical intuitions must share the uncertainty of axioms. But among them we find included the principle of the syllogism, the law of excluded middle, the principle of contradiction. They are in fact those simple laws or forms of thought to which thought must conform to be thought at all. Thought therefore cannot throw any doubt on them without committing suicide.

As Aristotle has said:—

ὁ δ' ἀναιρῶν ταύτην τὴν πίστιν οὐ πάνυ πιστότερα λέγει.

But there is one short general criticism which seems enough, and would have explained beforehand without such an examination of details that the evolutionist theory must end in self-contradiction.

There is an elementary principle on which we should expect all philosophies to agree, which is that thought cannot question the validity of its own presuppositions or even try to establish them without self-contradiction.

Now it is evident that the evolutionist theory violates » this principle, whether we consider the account given in it of the supposed criterion of all truth, inconceivableness of the opposite, or the account of those primary laws of thought which have just been spoken of.

The mistake is of the more elementary character when the attempt to establish the laws of thought, or a general criterion, is made by help of the reasonings of a special science, for that of course must presuppose the general laws and the criterion.

But this is precisely the use here made of biology.

The reasonings of biology would collapse if it did not assume for instance the principle of contradiction, it is futile therefore to prove its objective validity by biology.

As to the criterion itself, which is to shew that a cognition is of the highest rank (its 'unsurpassable validity,' etc.), the principles of biology cannot depend on it since biology has to establish it. Hence either they have not the highest rank, which involves the absurdity that the criterion of cognitions of the highest rank is shown to be a valid criterion by appealing to cognitions which are not of the highest rank: or else the principles of biology have the highest rank, and then it turns out that there are some absolute truths which are not derived from the criterion, and, what is still worse, that evolution has altogether forgotten to explain their origin, thus leaving the old problem so far from being solved that it is not even attempted.

This work was published before January 1, 1924, and is in the public domain worldwide because the author died at least 100 years ago.