4

I used the example of π, but this applies to other transcendental numbers as well, such as e

Kant classified statements into 4 epistemic categories based on two criteria: The Analytic/Synthetic distinction (Are statements true by definition or do we need outside information to determine their truth) and the A Priori/A Posteriori distinction (Are they independent of empirical evidence or not).

In particular he arrived at the existence of synthetic a priori truths, in opposition to Hume who believed that all statements were either analytic a priori or synthetic a posteriori.

Neither Kant nor Hume believed that analytic a posteriori truths are possible.

My question is regarding the calculation of π up to an arbitrary number of digits:

  • It is a number, so presumably it contains it's own definition: Is saying "π up to 88 digits = 3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034" is an analytic a priori statement like "Two = 2"?
  • But beyond a certain number of digits, nobody can come with the new digits on their own, they would have to rely on a computer to perform the calculation, so is it a analytic a posteriori (and Kant was wrong to thing analytic a posteriori truths didn't exist)?
  • π isn't really a number, it is a symbol that is shorthand for a complex mathematical relation, and as such "π up to 88 digits = 3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034" is a synthetic a priori truth?
  • But we can't calculate "π up to 2288 digits" without performing a mechanical procedure. So π is actually a empirical fact about the world - i.e. synthetic a posteriori. Is π then an empirical constant, similar to the gravitational constant or the charge of an electron?

How would Kant classify π? How would Hume? If π is empirical, doesn't it make theory laden, per the Quine-Duhem thesis and π would change depending on some changes in the axioms of math or geometry? What is the epistemic status of π? Given that we can never know the "true value" of π completely, is it a thing in itself, part of the noumenon?

  • 2
    Your first three bullet points are incorrect: 1) "π = 3.1415926535897932384626433832795028841971693993751058209749445923078" is merely a falsehood. 2) Every calculation a computer can do, a human can do by hand (more slowly). We only use computers to do these computations faster and more easily. It's not relevant here. 3) You're conflating pi-the-symbol and pi-the-number. Sentences involving pi use the pi symbol, but the propositions themselves are about a number. The number can be defined or represented in a variety of qualitatively different ways. – Era Feb 16 '16 at 19:21
  • @Era thanks for pointing that out. I made some changes. – Alexander S King Feb 16 '16 at 19:27
  • 3
    I am with @Era, the problem is that pi is not a sequence of digits, it is a geometrical relation, the ratio of a circle's circumference to its diameter. All 'synthetic' approximations have nothing to do with the analytic nature of the actual number. That pi is constant is an a posteriori discovery. We might have lived in some hyperbolic manifold, where we could not have discovered that the relation holds. If it exists, though, the rules about it are not synthetic, nor even its value. It is an analytic a posteriori fact, like much of mathematics. – jobermark Feb 16 '16 at 19:57
  • 3
    Numbers aren't facts or things to be discovered. What you're talking about is facts about pi, such as its decimal expansion. Any given fact about pi is analytic given the definitions of all the terms in the statement. More generally, any fact about pi has the same status as virtually any other fact of mathematics. The existence or other metaphysical status of numbers qua objects is a separate question. Perhaps more interesting than transcendentals like pi or e is those which cannot have their digits computed by any mechanical process, which turns out to be almost all numbers. – Era Feb 16 '16 at 20:01
  • 1
    The digits of pi are completely characterized by a finite length computer program. en.wikipedia.org/wiki/Computable_number Computable numbers are of little ontological interest since a high school student could crank out digits all day long, subject to computing resources. The more interesting question is the existence of the noncomputable numbers. – user4894 Feb 16 '16 at 22:26
3

Kant only had three epistemic categories, analytic a posteriori are highly problematic (even Kripke talks only about necessary a posteriori). As for π it was originally defined as a ratio of the circumference to diameter, and only two thousand years later related to numbers and decimal expansions. Still, one could define it as a number using one of many series expansions, continued fractions, etc., known in Kant's time. Regardless, like all arithmetic and geometry all such definitions (or rather constructions implied in them) are synthetic a priori, the difference would only be whether it is a priori synthesis in space (geometry), or in time (arithmetic).

Kant gives a famous example of a similar synthetic a priori in the Critique of Pure Reason, 7+5=12:"The concept of twelve is by no means thought by merely thinking of the combination of seven and five; and analyze this possible sum as we may, we shall not discover twelve in the concept. We must go beyond these concepts, by calling to our aid some concrete image [Anschauung], i.e., either our five fingers, or five points (as Segner has it in his Arithmetic), and we must add successively the units of the five, given in some concrete image [Anschauung], to the concept of seven. Hence our concept is really amplified by the proposition 7 + 5 = 12, and we add to the first a second, not thought in it. Arithmetical judgments are therefore synthetical, and the more plainly according as we take larger numbers...". Obviously, as we take larger numbers a live human may run out of time to synthesize the requisite intuition. But our understanding also has the capacity to project indefinite extensions of our intuitions (as with mathematical induction for instance), so we can intuit that it is possible in principle. This last part was developed systematically by Kant's mathematical descendants, Hilbert and intuitionists, see Was there a Kantian influence on Hilbert's formalist programme?

Hume would equally have no problem, logic and mathematics for him are relations of ideas, and everything in them is analytic a priori, i.e. tautological. Kant was overly optimistic when he wrote "For he would then have recognized that, according to his own argument, pure mathematics, as certainly containing a priori synthetic propositions, would also not be possible; and from such an assertion his good sense would have saved him", he should have read Hume's Treatise, not just his Enquiry. It would be a more interesting question for Frege, for whom arithmetic was a priori analytic, and geometry a priori synthetic, but he'd probably say that equating geometric and analytic π is where it becomes synthetic. Practical considerations generally did not preoccupy traditional epistemologists, be it Plato, Hume, Kant, Frege or Husserl.

However, the question is interesting even from the modern perspective. If all of knowledge is empirical, as Quine's naturalized epistemology holds, and π is another fundamental constant of nature then how come we have to measure the speed of light, while no physical measurements are involved in computing π to any accuracy? See Is geometry mathematical or empirical?

2

Kant does touch upon it obliquely in the first critique; Cavailles expands upon it - he uses circles whereas Kant used triangles.

First pi as comments have already pointed out is not defined by some decimal expression; it's defined geometrically as a ratio.

Hence a priori, but also synthetic as it must take into account the subjects synthetic construction of geometric space - the Cartesian theatre.

Hence, he would judge pi as a synthetic a priori.

I would take this point, as where Gauss took the 'laxity' introduced by Kant to theorise non-Cartesian spaces.

As every school-boy or girl knows, the interior angles of a triangle adds up to 180 degrees; but Kant demurs, and says this is not neccessarily so, on purely a priori grounds; Cavailles following Kant, or rather expanding his line or flight of thought, introduced the example of a circle, and it would seem the same argument would carry over for pi.

  • 3
    Pi does not require geometry in order to be defined, it just happened to be encountered in geometry first. – Era Feb 16 '16 at 21:23
  • @Mozibur Ullah It seems that Kant should go to school, right? – John B Feb 17 '16 at 3:17
  • @Mozibur Ullah I think you do. – John B Feb 18 '16 at 11:33
  • 1
    @MoziburUllah The arcsin and arctan ones are infinite series definitions of those functions, i.e., they do not use the geometrical definitions of those functions. Similarly any definition using those or other transcendental functions (such as the last example on that page) can be restated using the series expansion. You end up with something ugly but it's correct and non-geometrical. – Era Feb 21 '16 at 3:19
  • 1
    (Penrose does that, in the surreal number construction.) Agreed. I was just pointing out that Era's first objection is kind of beside the point. Whatever definition is basic, it is basic, and the given motivation is unrelated to the issue at hand. – jobermark Feb 24 '16 at 18:23

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.