REPLIES TO CRITICISMS. (continued 3)
Reading edition
APPENDIX C. SUMMARY OF RESULTS.
Those who deny a general doctrine enunciated by Mayer as the basis of his reasonings, habitually assumed by Faraday {315} as a guiding principle in drawing his conclusions, distinctly held by Helmholtz, and tacitly implied by Sir John Herschel—those, I say, who deny this general doctrine and even deride it, should be prepared with clear and strong reasons for doing this. Having been attacked, not in the most temperate manner, for enunciating this doctrine and its necessary implications in a specific form, I have demanded such reasons. Observe the responses to the demand.
1. The British Quarterly Reviewer quoted for my instruction the dictum of Professor Tait, that “Natural Philosophy is an experimental, and not an intuitive science. No à priori reasoning can conduct us demonstratively to a single physical truth.” Thereupon I inquired what Professor Tait meant “by speaking of ‘physical axioms,’ and by saying that the cultured are enabled ‘to see at once their necessary truth?’” . . .
No reply.
2. Instead of an answer to the question, how this intuition of necessity can be alleged by Professor Tait consistently with his other doctrine, the Reviewer quotes, as though it disposed of my question, Professor Tait’s statement that “as the properties of matter might have been such as to render a totally different set of laws axiomatic, these laws [of motion] must be considered as resting on convictions drawn from observation and experiment, and not on intuitive perception.” Whereupon I inquired how Professor Tait knows that “the properties of matter might have been” other {316} than they are. I asked how it happened that his intuition concerning things as they are not, is so certain that, by inference from it, he discredits our intuitions concerning things as they are . . .
No reply: Professor Tait told, à propos of my question, a story of which no one could discover the application; but, otherwise, declined to answer. Nor was any answer given by his disciple.
3. Further, I asked how it happened that Professor Tait accepted as bases for Physics, Newton’s Laws of Motion; which were illustrated but not proved by Newton, and of which no proofs are supplied by Professor Tait, in the Treatise on Natural Philosophy. I went on to examine what conceivable a posteriori warrant there can be if there is no warrant a priori; and I pointed out that neither from terrestrial nor from celestial phenomena can the First Law of Motion be deduced without a petitio principii . . .
No reply: the Reviewer characterized my reasoning as “utterly erroneous” (therein differing entirely from two {317} eminent authorities who read it in proof); but beyond so characterizing it he said nothing.
4. To my assertion that Newton gave no proof of the Laws of Motion, the Reviewer rejoined that “the whole of the Principia was the proof.” On which my comment was that Newton called them “axioms,” and that axioms are not commonly supposed to be proved by deductions from them . . .
The Reviewer quotes from one of Newton’s letters a passage showing that though he called the Laws of Motion “axioms,” he regarded them as principles “made general by induction;” and that therefore he could not have regarded them as a priori.
5. In rejoinder, I pointed out that whatever conception Newton may have had of these “axioms,” he explicitly and distinctly excluded them from the class of “hypotheses.” Hence I inferred that he did not regard the whole of the {318} Principia as the proof, which the Reviewer says it is; since an assumption made at the outset, to be afterwards justified by the results of assuming it, is an “hypothesis” . . .
No reply.
6. Authority aside, I examined on its merits the assertion that the Laws of Motion are, or can be, proved true by the ascertained truth of astronomical predictions; and showed that the process of verification itself assumed those Laws.
No reply.
7. To make still clearer the fact that ultimate physical truths are, and must be, accepted as a priori, I pointed out that in every experiment the physicist tacitly assumes a relation between cause and effect, such that, if one unit of cause produces its unit of effect, two units of the cause will produce two units of the effect; and I argued that this general assumption included the special assumption asserted in the Second Law of Motion. . . .
No reply: that is to say, no endeavour to show the untruth of this statement, but a quibble based on my omission of the word “proportionality” in places where it was implied, though not stated.
8. Attention was drawn to a passage {319} from Sir John Herschel’s Discourse on the Study of Natural Philosophy, in which the “proportionality of the effect to its cause in all cases of direct unimpeded action” is included by him among “the characters of that relation which we intend by cause and effect;” and in which this assumption of proportionality is set down as one preceding physical exploration, and not as one to be established by it . . .
No reply.
9. Lastly, a challenge to prove this proportionality. “It is required to establish the truth that there is proportionality between causes and effects, by a process which nowhere assumes that if one unit of force produces a certain unit of effect, two units of such force will produce two units of such effect.” . . .
No reply.
Thus on all these essential points my three mathematical opponents allow judgment to go against them by default. The attention of readers has been drawn off from the main issues by the discussion of side issues. Fundamental questions have been evaded, and new questions of subordinate kinds raised.
What is the implication? One who is able to reach and to carry the central position of his antagonist, does not spend his strength on small outposts. If he declines to assault the stronghold, it must be because he sees it to be impregnable.
The trouble I have thus taken to meet criticisms and dissipate misapprehensions, I have taken because the attack {320} made on the special doctrine defended, is part of an attack on the ultimate doctrine underlying the deductive part of First Principles—the doctrine that the quantity of existence is unchangeable. I agree with Sir W. Hamilton that our consciousness of the necessity of causation, results from the impossibility of conceiving the totality of Being to increase or decrease. The proportionality of cause and effect is an implication: denial of it involves the assertion that some quantity of cause has disappeared without effect, or some quantity of effect has arisen without cause. I have asserted the a priori character of the Second Law of Motion, under the abstract form in which it is expressed, simply because this, too, is an implication, somewhat more remote, of the same ultimate truth. And my sole reason for insisting on the validity of these intuitions, is that, on the hypothesis of Evolution, absolute uniformities in things have produced absolute uniformities in thoughts; and that necessary thoughts represent infinitely-larger accumulations of experiences than are formed by the observations, experiments, and reasonings of any single life.
ENDNOTES TO REPLIES TO CRITICISMS.
“It is hardly a parody of Mr. Collier’s remarks to say:—‘A child discovers that the greater the angle between his legs the greater the distance between his feet, an experience which implicates the notion of proportionality between the angle of a triangle and its opposite side;’ a preconception, as it appears to me, with just as good a basis as that whose formation Mr. Collier illustrates, but one which, as I need hardly add, is soon corrected by a conscious study of geometry or by actual measurement.”
I am indebted to Mr. Hayward for giving this instance. It conveniently serves two purposes. It serves to exemplify the connexion between the crude preconceptions unconsciously formed by earlier experiences, and the conceptions consciously evolved out of them by the help of later experiences, when the requisite powers of analysis and abstraction have been reached. And at the same time it serves to show the failure of my opponents to understand how, in the genesis of intelligence, the scientific conception of exact proportionality develops from the crude, vague, and inaccurate preconception. For while the notion of proportionality acquired by the child in Mr. Hayward’s example, is not true, it is an approximation towards one which is true, and one which is reached when its more developed intelligence is brought critically to bear on the facts. Eventually it is discovered that the angle is not proportional to the subtending side, but to the subtending arc; and this is discovered in the process of disentangling a simple relation from other relations which complicate and disguise it. Between the angle and the arc there is exact proportionality, for the reason that only one set of directly-connected space-relations are concerned: the distance of the subtending arc from the subtended angle, remains constant—there is no change in the relation between the increasing angle and the increasing arc; and therefore the two vary together in direct proportion. But it is otherwise with the subtending side. The parts of this stand in different relations of distance from the subtended angle; and as the line is lengthened, each added part differs from the preceding parts in its distance from the angle. That is to say, one set of simple directly-connected geometrical relations, is here involved with another set; and the relation between the side and the angle is such that the law of relative increase involves the co-operation of two sets of factors. Now the distinguishing the true proportionality (between the angle and the arc) from the relation which simulates proportionality (between the angle and the side) is just that process of final development of exact conceptions, which I assert to be the finishing step of all the preceding development; and to be impossible in its absence. And the truth to which my assailants shut their eyes, is that, just as among these conceptions of space-relations, the conception of exact proportionality can be reached only by evolution from the crude notion of proportionality, formed before reasoning begins; so, among the force-relations, the conception of proportionality finally reached, when simple causes and their effects are disentangled by analytical intelligence, can be reached only by evolution of the crude notion of proportionality, established as a preconception by early experiences which reinforce ancestral experiences.