Readers of the last two posts may well be wondering why I'm going on so much about the "practical helpfulness" of mathematics.

One thing is, I wish I had a better name for it than "practical helpfulness", so maybe someone will suggest one :).

More seriously, I think the fact that our mathematical methods are (in effect) constantly making predictions about themselves, and other kinds of a priori reasoning - not to mention combining with our methods of observation to yield predictions that observation alone would not have yielded (see the computer example) has two important consequences.

Firstly, it shows that our reasoning about math is NOT the kind of thing you are likely to get just by making a series of arbitrary stipulations and sticking to them. All our different kinds of a priori reasoning (methods for counting abstract objects, logical inference, arithmetic, intuitive principles of number theory, set theoretic reasoning that has consequences for number theory) fit together in an incredibly intricate way. Each method of reasoning has myriad opportunities to yield consequences that would lead us to form false expectations about the results of applying some other method. And yet, this almost never happens!

Thus, there's a question about how we could have managed to get methods of armchair reasoning that fit together so beautifully. Some would posit a benevolent god, designing our minds to reason only in ways that are truth-preserving and hence coherent in this sense. But I think a process of free creativity to come up with new methods of a priori reasoning, plus Quinean/Millian revision when these new elements did raise false expectations, can do the job. This brings us to the second point.

Secondly, if we think about all these intended internal and external applications as forming part of our conception of which mathematical objects we mean when we talk about e.g. the numbers, then Qunian/Millian revision when applications go wrong will amount to a kind of reliable feedback mechanism, maintaining and improving the fit between what we say about "the numbers" and what's actually true of those-mathematical objects-whose-structure-mirrors-the-modal-facts-about-how-many-objects-there-are-when-there-are-n-Fs-and-m-(distinct)-Gs etc.

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