Wednesday, June 29, 2016

The problem of induction in mathematics

Let's say I have some algorithm that generates the sequence of numbers, and I run ten iterations on the computer and get

  • 1
  • 1.5
  • 1.41666666667
  • 1.41421568627
  • 1.41421356237
  • 1.41421356237
  • 1.41421356237
  • 1.41421356237
  • 1.41421356237
  • 1.41421356237
  • 1.41421356237

I will now be very confident that the sequence of numbers converges, and indeed that it converges to the square root of two. But why? Convergence is a property that the sequence has at infinity. The first 10 items in the sequence are an infinitely short proportion of infinity. Moreover, why do I assume that the sequence converges to the square root of two. Maybe it converges to the square root of two plus e−100. Such possibilities ensure that my credence that the limiting value is the square root of two is strictly less than one. But the credence stays high.

In other words, the standard problems of induction come up not in just in science, but in mathematics. We should, thus, hope that whatever solutions we adopt to the problems as they come up in science will apply in the mathematical cases as well.

My favorite story about induction, the theistic story that God would have good reason--and hence be not unlikely to--create a well-ordered universe does not apply to mathematics, since pace Descartes, God doesn't choose the truths of mathematics. A relative of the story does apply, however. The truths of mathematics are grounded in the necessary nature of the mind of God, as Augustine held, and it is to be expected that the necessary nature of the mind of God will exhibit beauty and elegance. And where there is beauty and elegance, there is pattern and some purchase for induction.


Joshua Reagan said...
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awatkins909 said...

Don't you get the new problem of induction in mathematics too?

This is an interesting phenomenon, because often -- maybe even usually -- mathematicians use inductive evidence and informal reasoning, at least at first. And it usually works! But the rigorous proof only comes much later.

This seems to me good evidence against formalist and syntactic theories of mathematics, and in favor of some sort of realism. Mathematicians are studying real properties, because they discover the truth by thinking of them in the same way other scientists think about their subject matter (e.g., using induction and informal reasoning).

Note that this sort of argument is actually quite different from some of the other arguments in favor of realism -- it's not based on the fact that mathematicians quantify over numbers/functions/whatever, or that postulating mathematical objects makes a simpler theory, or that there are mathematical explanations, etc.

Alexander R Pruss said...

Hmm. My first reaction is that nothing like "is grue" and "is bleen" will be a mathematical predicate, because mathematical predicates cannot involve time or other physical realities. But Goodman's new riddle doesn't need to be taken so narrowly. It's a general problem with non-natural predicates. And we can generate purely mathematical ones (e.g., is even or bigger than 10^googolplex). So, yes, that's right.

The argument for mathematical realism, or at least against formalism, is really interesting. But maybe some form of conceptualism can survive: maybe mathematics is a study of the human mind, and hence induction works? Nah: the inductive data typically doesn't come from brains but from computers. Hmm. Maybe you should try to publish this argument. Or work with me to do so if you don't want to do it alone? Assuming it's new: apart from phil of probability, I don't know much phil of maths.

awatkins909 said...

I'd love to do that with you! Only problem is that I can't really say whether it has been published or not. I tend to assume anything I think of has probably been published already. ("there's nothing so absurd" etc.)

I'm pretty sure there has been discussion of the fact that mathematicians tend to think "semantically" and informally, as well as inductively, at least while they're trying to discover results that they later prove, and also that this works better for students.

(Personal anecdote: I learned logic and philosophy before I took mathematics in college. So I assumed you should learn mathematics the way philosophers tend to portray it: Axiomatically, starting from first principles, proving each step along the way, and so on. Terrible mistake. But when I started to just think of it more intuitively, informally, even "physically," I did much better, even in the upper division proof-based classes.)

I'm not sure whether this has been turned into an argument for realism though. There is a volume called 'The Philosophy of Mathematical Practice', but I haven't read most of the papers in there so I can't say for sure. Probably would be better to ask a legit philosopher of mathematics. (Do you know any?)

Unknown said...

I read something in Alvin Plantinga's lecture notes on this:

(M) The Argument from induction
Hume pointed out that human beings are inclined to accept inductive forms of reasoning and thus to take it for granted, in a way, that the future will relevantly resemble the past. (This may have been known even before Hume.) As Hume also pointed out, however, it is hard to think of a good (noncircular) reason for believing that indeed the future will be relevantly like the past. Theism, however, provides a reason: God has created us and our noetic capacities and has created the world; he has also created the former in such a way as to be adapted to the latter.

It is likely, then, that he has created the world in such a way that in fact the future will indeed resemble the past in the relevant way). (And thus perhaps we do indeed have a priori knowledge of contingent truth: perhaps we know a priori that the future will resemble the past.) (Note here the piece by Aron Edidin: "Language Learning and A Priori Knowledge), APQ October l986 (Vol. 23/ 4); Aron argues that in any case of language learning a priori knowledge is involved.)

This argument and the last argument could be thought of as exploiting the fact that according to theism God has created us in such a way as to be at home in the world (W olterstorff.)

Unknown said...


Are you saying that the efficacy of the application of mathematics to the physical world, as evinced by the successes of mathematical work being done to predict what science would later prove, substantiates a kind of realism? While I agree that the applicability of mathematics to the physical world reveal that it is mathematical in structure, I think that substantiates theism more than it does realism. Wouldn't realism face the problem that mathematical entities cannot cause anything in the physical world, and so it would be implausible for reality to reflect their tendencies?

Michael Gonzalez said...

I don't understand what the problem is supposed to be, awatkins. Familiarity with the workings of a very complex ruleset, and lots of practice in it, could lead to becoming very intuitive about it... no?

A person who is adept at a very complex game, for example, will often intuit the best moves and will give only partial lines of reasoning as to why it was the best move (e.g. Kasparov explaining how he plays chess). Likewise for the mathematician...