## Why I don’t like philosophy of mathematics

I recently posted a link to an explanation of the philosophy of mathematics.  While I thought that Balaguer’s explanation was very good, I also remarked that I don’t find the philosophy of mathematics to be useful.  In this post, I’ll say why I don’t find it useful.

Toward the end of his explanation, Balaguer presents the following argument for platonism:

1. Semantic platonism is true–i.e., ordinary mathematical sentences like ‘2 + 2 = 4’ and ‘3 is prime’ are straightforward claims about abstract objects (or at any rate, they purport to be about abstract objects). Therefore,
2. Mathematical sentences like ‘2 + 2 = 4’ and ‘3 is prime’ could be true only if platonism were true–i.e., only if abstract objects existed. But
3. Mathematical sentences like ‘2 + 2 = 4’ and ‘3 is prime’ are true. Therefore,
4. Platonism is true.

Balaguer, who says he is a fictionalist and not a platonist, questions step 3 in that argument.  However, it seems to me that step 2 is already mistaken.  People simply do not use “true” in the way that step 2 supposes.

For example, it seems to me to be entirely true that Sherlock Holmes resided at 221B Baker street, while at the same time it is quite obvious that Sherlock Holmes never actually existed.  Similarly, it seems entirely true that having one horn is what characterizes the unicorn, while it is also clear that there never were any unicorns.

In ordinary speech, existence is not a requirement for assertions of truth.  And, it seems to me, mathematical talk is similar.

Of course mathematicians do say that the number 3 exists.  However, it has always seemed to me that “exist” is used in a special sense within mathematics.  To say that x exists, in the mathematical sense, is just to say that assuming the existence of x does not introduce any contradiction.

For myself, I’m a fictionalist.  Many mathematicians are platonists.  As to how we do mathematics, I do it much the same way that a platonist would.  The issues that platonism are supposed to address do not seem particularly relevant to mathematics and how it is done.

Yes, there are mathematical aspects to platonism.  A mathematical platonist is likely to say that there is a fact of the matter as to whether the continuum hypothesis is true, though we have not yet determined what is that fact.  As a fictionalist, I am skeptical that there is such a “fact of the matter.”

Although platonism has mathematical implications, the type of argument often used within philosophy of mathematics, such as the one discussed above, does not seem at all compelling or even useful.

### 13 Responses to “Why I don’t like philosophy of mathematics”

1. Just a small note, maybe to spark an interest in philosophy of mathematics following your line of thought – it seems like you share the same ontological views as Hilbert did with his formalism: “(…) if the arbitrarily given axioms do not contradict each other with all their consequences, then they are true and the things defined by them exist. This is for me the criterion of truth and existence” (quoted by Shapiro in “Thinking about mathematics”, 2000).

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• Actually, I take axioms to be neither true nor false, and I take the truth of mathematical theorems to be relative to the assumed axioms. I take “exist” to have a special meaning within mathematics — I need to do that to make sense of “exist” as used in the mathematical literature. In particular, I take “exist”, as used in mathematics, to have no implications for existence in the ordinary non-mathematical sense of the word.

I don’t think I am making any ontological commitments in the way that I view mathematics.

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• Ah I see, my bad. On the notion of truth though, do you then view certain theorems as true, but the axioms they’re built on as neither true nor false? Or do you simply view every part of mathematics as having no truth value? If it’s the former, how would you argue against proof of the soundness of first-order logic (deductions preserve truth-value)? If it’s the latter, how would you explain how mathematics is applied with great results in science, if it’s not true?

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• On the notion of truth though, do you then view certain theorems as true, but the axioms they’re built on as neither true nor false?

I take a theorem to be a true consequence of the assumed axioms. That is to say, its truth is relative to the axioms.

I’m not sure how we could look at it in any other way. If I am doing mod 5 arithmetic, then 3+4=2, while for ordinary arithmetic, 3+4=7. If we take mathematical truth to be absolute, rather than to be relative to the assumed axioms, we quickly run into contradiction.

I do find some of the logician’s discussion of soundness to be a bit puzzling, for they seem to be using truth in a way that does not make sense to me.

how would you explain how mathematics is applied with great results in science

Typically, we apply mathematics to measurements (or other observations). Our ordinary arithmetic models this. The successor function in Peano arithmetic models and idealizes the practice of incrementing a counter, and the Dedekind cut construction of the reals models and idealizes the way that measuring systems are constructed in science.

It seems to me that platonists have more of a problem here. If mathematics consists of truths about a platonic reality, then we could not apply it to science unless we could first show that measurements are entities in that platonic realm.

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• I take a theorem to be a true consequence of the assumed axioms. That is to say, its truth is relative to the axioms.

Am I then understanding you correctly if I say that epistemologically you believe that axioms are arbitrary non-contradictory statements, and all deductions from these axioms, using an appropriate set of deduction rules, are true within that system?

Typically, we apply mathematics to measurements (or other observations). Our ordinary arithmetic models this.

But aren’t you then assuming that the axioms of arithmetic are true, due to the results deducible from them being true in those measurements? How could the axioms of arithmetic be neither true nor false, if they accurately describe physical phenomena?

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• Am I then understanding you correctly if I say that epistemologically you believe that axioms are arbitrary non-contradictory statements, and all deductions from these axioms, using an appropriate set of deduction rules, are true within that system?

That’s a fair assessment. As for “arbitrary”, I’ll add that we usually want our axioms to be interesting and useful, so they are not completely arbitrary.

How could the axioms of arithmetic be neither true nor false, if they accurately describe physical phenomena?

Which physical phenomena are accurately described by the Peano axioms?

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• Which physical phenomena are accurately described by the Peano axioms?

Anything involving natural numbers? (Positive) money transactions, counting etc. I would argue that we all believe in the axioms, when we e.g. are confident that putting more money in our wallet won’t cause it to be empty.

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• WordPress requires configuring a finite limit to the depth of replies. There doesn’t seem to be a setting for infinite depth. I guess we hit the currently configured limit.

Anything involving natural numbers? (Positive) money transactions, counting etc.

Natural numbers do not appear in the physical world. There are only numerals written by people (and machines).

Yes, the use of arithmetic works. But whether the axioms are true is not relevant. All that is required is that if we use numbers in a way that is consistent with the axioms, then the conclusions of the theorems will hold.

If you want to argue this is evidence of truth of the axioms, then you have to also accept that one person flubbing his arithmetic is evidence that the axioms are false. And a single counter-example is sufficient to establish falsity.

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• Natural numbers do not appear in the physical world. There are only numerals written by people (and machines).

Sorry, I meant tokens of the natural number type, objects that behave just like our idea of the natural numbers (i.e. satisfies PA).

But whether the axioms are true is not relevant. All that is required is that if we use numbers in a way that is consistent with the axioms, then the conclusions of the theorems will hold.

But the fact that “the conclusions hold” is exactly what we mean by truth. If I have £500 on my bank account and put in £50, I expect there to be £550 on my account afterwards – anything else would not count as a true conclusion from the assumptions of having £500 as well as PA. I assume my assumptions as true, and that my deduction is sound, thereby requiring my conclusion to be true.

If you want to argue this is evidence of truth of the axioms, then you have to also accept that one person flubbing his arithmetic is evidence that the axioms are false. And a single counter-example is sufficient to establish falsity.

If in the above example, I somehow ended up with having £540 on my account, I wouldn’t account this as being true, I would complain to the bank – and assuming they could see that I had £500 and put in £50, they would see that the resulting £540 is in fact a false amount, and make sure to put in the remaining £10. Thus in accordance with PA.

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• Sorry, I meant tokens of the natural number type, objects that behave just like our idea of the natural numbers (i.e. satisfies PA).

That would be evidence that things that satisfy PA happen to satisfy PA. This seems tautological. I don’t see it as a reason for saying that the Peano axioms are true.

I’ve made a new post on the question of whether axioms are true. Perhaps that will better explain my view.

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2. I’m too new to this argument to take a firm position, but I agree with you that talking about unicorns and Sherlock Holmes is not the same as acknowledging their existence. I say this because some people say that for an atheist to talk about God is giving ground to theists, and I disagree, for roughly the same reasons as you think it’s ‘true’ that a unicorn has one horn.

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• Thanks, Jonny. I agree that “God talk” need not have any implications about the existence of God, though of course that might depend on what is said in that “God talk.”

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