Mathematics

Are necessary truths ultimately grounded in induction? For example truths of mathematics are said to be necessary, yet don't they make generalizations about an infinite set of numbers that are not verifiable; wouldn't this be considered induction? And if we ground our necessary truths on axioms, aren't these axioms theorems that a community has agreed to as being true and are not objectively true? Thanks for your answer, John

Comments

First, we need to set an issue aside. The word "induction" is sometimes used to refer to a certain sort of mathematical argument in which we prove something for every case by showing it for a "base" case and then showing that if it holds in the first n cases, it holds in the n+1th case. But it's pretty clear that your question is about induction as a matter of reasoning empirically from a limited set of instances to a claim about all cases, and so we'll use the word "inductive" in that way below.

Here's an example of a necessary truth: every star that has planets orbiting around it is a star. Notice that it's universal; it says something about every star with planets. And if it were like "every star with planets orbiting around it is at least 3 billion years old," we could only show that it was true (assuming it is) by empirical means and thus, in a loose sense, inductively. (What we'd actually do is produce an argument from various theoretical and observational premises, but at some point, something like induction would no doubt enter the picture.) In fact, however, our knowledge that all stars with planets are stars is secure without the need for inductive reasoning. That's because any proposition of the form "Every x that is a both P and a Q is a P" is a truth of logic.

Just what logical knowledge amounts to and how best to account for our having it are big questions. But it seems pretty clear that inductive reasoning presupposes deductive reasoning. Imagine some principle of inductive reasoning that you want to invoke in coming to an empirical conclusion. The principle will be a generalization: it will say somethng about all cases that fit a certain pattern. To get to the case at hand, you have to be able to make the deductive move of applying the generalization to the particular case. We could elaborate this point quite a bit further. (For some amusing related reflections, read Lewis Carroll's "What the Tortoise Said to Achilles.")

As for axioms in mathematics, there's a sense in which communities agree to them, but it's not clear that this takes away from their objectivity. Consider a set of axioms for Euclidean geometry. We agree amongst ourselves that these axioms fix what we call Eucliden geomety whether or not any actual spaces are Euclidean. The Pythagorean theorem follows from those axioms whether or not there are any Euclidean right triangles. And so what we know is this: if there are any situations where the axioms of Euclidean geometry are true, then any right triangles in those situations have hypotenuses whose squared lengths are the sums of the squared lengths of the sides. We don't need to pursue any tricky empirical investigations to be confident of this.

None of this is meant to say that all mathematical truth can be reduced to logical truth. And it may even be true that we can have inductive reasons for being confident of some purely mathematical claims. (Some computer scientists talk about "empirical mathematics.") But the mere fact that a mathematical proposition says something about an infinite set of cases doesn't show that our knowledge of it is empirical or inductive.

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