Poincaré's notion of magnitude

Question

This is what Henri Poincaré says in his book Science and Hypothesis (p. 27):

Measurable Magnitude.—So far we have not spoken of the measure of magnitudes; we can tell if any one of them is greater than any other, but we cannot say that it is two or three times as large.

So far, I have only considered the order in which the terms are arranged; but that is not sufficient for most applications. We must learn how to compare the interval which separates any two terms. On this condition alone will the continuum become measurable, and the operations of arithmetic be applicable. This can only be done by the aid of a new and special convention; and this convention is, that in such a case the interval between the terms A and B is equal to the interval which separates C and D. For instance, we started with the integers, and between two consecutive sets we intercalated n intermediary sets; by convention we now assume these new sets to be equidistant. This is one of the ways of defining the addition of two magnitudes; for if the interval AB is by definition equal to the interval CD, the interval AD will by definition be the sum of the intervals AB and AC. This definition is very largely, but not altogether, arbitrary. It must satisfy certain conditions—the commutative and associative laws of addition, for instance; but, provided the definition we choose satisfies these laws, the choice is indifferent, and we need not state it precisely.

It seems there's a bit of confusion here. Is the idea being suggested that we first establish equidistant points (like 0,1,2,3, etc., for instance), and then define how to add two magnitudes based on these points? According to this concept, we have the freedom to define addition in any way we wish, as long as our definition adheres to two primary mathematical properties: associativity and commutativity. Any concrete examples would be very helpful.

Answer 1

I suppose that the thrust of Poincaré's discussion in the passage given cannot be explained fairly enough without going over the general thesis Poincaré elaborates on, though it would only be a crude outline.

Poincaré situates himself in a neo-Kantian perspective and sets out to characterise the synthetic a priori underlying science and mathematics. He takes two notions, hypothesis and convention, as basic to his articulation.

According to the widespread view of naïve realism about science, we draw consequences revealing the knowledge of nature to us out of experiments and observations employing mathematics of which truths originate from self-evident propositions we hold and equally reliable reasoning. Poincaré points out that the crucial role played by hypotheses scientific and mathematical research which is overlooked in this picture of science and mathematics which rationalist and empiricist attitudes share.

Each scientific and mathematical branch introduce its own hypotheses as well it inherits from the ones it should presuppose. Thus, a hierarchy emerges, as Friedman puts down ("Poincare's Conventionalism and the Logical Positivists", p. 305; emphases in the original):

For Newton’s Principia had already shown clearly how we can empirically discover the law of universal gravitation—on the presupposition, that is, of the Newtonian laws of motion and Euclidean geometry. Without these presuppositions, however, we certainly would not have been able to discover the law of gravitation. And the same example also shows clearly how each level in the hierarchy of sciences presupposes all of the preceding levels: we would have no laws of motion if we did not presuppose spatial geometry, no geometry if we did not presuppose the theory of mathematical magnitude, and of course no mathematics at all if we did not presuppose arithmetic.

In accordance with this hypothetical character of science in general, "the aim of science is not things themselves, as the dogmatists in their simplicity imagine, but the relations between things; outside those relations there is no reality knowable."(Science and Hypothesis, p. xxiv)

Not all hypotheses are alike in character; "some are verifiable, and when once confirmed by experiment become truths of great fertility; that others may be useful to us in fixing our ideas; and finally, that others are hypotheses only in appearance, and reduce to definitions or to conventions in disguise" (ibid, p. xxii)." Those hypotheses in appearance (i.e., conventions) are particularly significant for mathematics, hence, the present context.

For Poincaré, conventions are "guided", but not inferred from experience. They are constructs of intuition, but not free to the extent of being arbitrary stipulations. "From them, indeed, the sciences derive their rigour" (ibid. p. xxiii). The following passage may illustrate what Poincaré has in mind when talking about conventions (ibid. pp. 22-23):

It has, for instance, been observed that a weight A of 10 grammes and a weight B of 11 grammes produced identical sensations, that the weight B could no longer be distinguished from a weight C of 12 grammes, but that the weight A was readily distinguished from the weight C. Thus the rough results of the experiments may be expressed by the following relations: A = B, B C, A < C, which may be regarded as the formula of the physical continuum. But here is an intolerable disagreement with the law of contradiction, and the necessity of banishing this disagreement has compelled us to invent the mathematical continuum. We are therefore forced to conclude that this notion has been created entirely by the mind, but it is experiment that has provided the opportunity. We cannot believe that two quantities which are equal to a third are not equal to one another, and we are thus led to suppose that A is different from B, and B from C, and that if we have not been aware of this, it is due to the imperfections of our senses.

Poincaré deems mathematical induction as essential to arithmetic, for it is the sole hypothesis by which we can make generalisations about numbers. Hence, mathematical reasoning, like its physics counterpart, involves inductive reasoning in its own way. For Poincaré, inductive reasoning is a fundamental method of synthetic a priori knowledge.

To extend from arithmetic to geometry, introduction of mathematical magnitude is required in between. We construct mathematical magnitude in order to have the measuring capability (in addition to combinatory capability of arithmetic), basing it on the concepts of mathematical induction, which is basically a indefinite repetition of an operation, beside the conventions of arithmetic operations.

Therefore, the referred passage should be seen in the light of Poincaré's effort to apply arithmetic to bring about a coherent conception of measurable continuum. Let us illustrate this by devising an example following Poincaré: Suppose we have fixed the whole numbers 2 and 3 as "two consecutive steps". Between 2 and 3, we insert 4 intermediary ones. Then, we stipulate the convention that each interval thus defined between 2 and 3 is equidistant. We also stipulate that arithmetic properties (which have been discussed in the preceding part of the book) are applicable.