## Tuesday, 23 April 2013

### For Mathematicians

Here's a nice shortish article (talk) by Mark Balaguer aiming to explain the basic ideas of philosophy of mathematics to mathematicians:
A Guide for the Perplexed: What Mathematicians Need to Know to Understand Philosophers of Mathematics
From the introductory paragraph:
My hope is to make clear for mathematicians what philosophers of mathematics are really up to and, also, to eliminate some confusions.

1. Do you think this representation of philosophers is accurate? I'd have thought that Balaguer's main claim - that philosophy of mathematics is, at least implicitly, founded on semantics - would be controversial among philosophers. It seems fair enough to say that the theory that '3 is prime' purports to attribute a property of being prime to an object called '3' is a semantic one, but to say that it is a semantic theory that '3 is prime' purports to be about an abstract object, as opposed to a physical or mental one, looks like a bit of a stretch. When you look at his arguments that the purported referent of '3' cannot be physical or mental, his approach resembles good old conceptual analysis more than anything else. (Perhaps he is just using 'semantics' in a novel way for the sake of non-philosophers?)

Also, a possible psychologistic reply: it is possible for names to have merely possible referents; number terms (or uses of them) can refer either to actual mental objects or to merely possible mental objects; a merely possible mental object is mental, not abstract. Though that theory is bonkers, of course (it came to mind because Balaguer mentions the position we call fictionalism, and merely possible objects are not so far from fictional objects; the difference is that fictionalists think that, if '3' refers to a fictional object, then '3 is prime' is true in the fiction but false or meaningless simpliciter, whereas if there are merely possible objects and '3' refers to one, then '3 is prime' might yet be straightforwardly true).

2. I think Balaguer's approach, focusing on semantics and the theory of meaning for mathematical discourse, is reasonably accurate for most philosophers of mathematics, though he concedes that everyone disagrees about everything. Still,

the denotation of a numeral, say "666", is an abstract entity

is not a purely semantic claim that. It's a semantic claim that

each numeral has a denotation

or

some numerals lack a denotation.

But semantics needn't attribute any particular ontological status to these denotations. Semantics puts together a theory of how semantic notions (denotation, reference, truth, meaning) all interact together. It doesn't, though, say what the "nature" of referents are. In semantics, one has compositional principles like:

(i) $Ft$ is true iff the denotation of $t$ belongs to the extension of $F$.
(ii) $\neg \phi$ is true iff $\phi$ is not true.
(iii) $forall x \phi$ is true iff, for all $o$, $\phi(\underline{o})$ is true.

But it doesn't matter, at least to semantics, what these denotations, extensions, etc., are. (On the other hand, semantics will usually need a plentiful supply of syntactical entities - expressions, names and truth bearers. Usually infinitely many syntactical entities. Again, what these "are" doesn't matter.)

It will matter to one's favourite account of mathematics if one attempts to preserve some body of antecedently accepted statements (e.g., all the theorems currently accepted by the mathematical community). For example, if all of a certain class of statements are true, then it may be impossible for it to have, e.g., a finite model. What the domain of that model contains may be of no relevance to semantics, per se. But its size will cause trouble for anyone who also wishes to maintain scepticism about what mathematics is about. So, wishing to maintain the accepted practice, while rejecting this conclusion may force an, e.g., non-compositional or non-classical, theory of meaning.

"it is possible for names to have merely possible referents; number terms (or uses of them) can refer either to actual mental objects or to merely possible mental objects; a merely possible mental object is mental, not abstract."

Seems fine to me, yes - but I think this belongs more to the metaphysical side of things, though. From Quine to Shapiro, philosophers have pointed out that possibilia are epistemologically problematic in way analogous to abstracta are alleged to be - because it may be hard to see how minds can "access" possibilia. On the other hand, one needs a semantic theory to explain what,

"$\phi$ is true in fiction"

means. Or what,

"according to the fiction $F$, $\phi$"

means.

Jeff

3. Oops - I should include though a mention of Balaguer's

"Semantic Platonism: 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)."

This is probably what you had in mind.

I'd say this isn't *purely* a semantic theory. Balaguer goes on to say,

"... the first point I want to make about this theory is that it is not an ontological theory, and it doesn’t imply any ontological theories. In particular, it doesn’t imply that platonism is true; i.e., it doesn’t imply that there actually exist any abstract objects. This is an extremely important point ..."

This is problematic, because it implies that there are *syntactical entities*.
His point is that the claim "whatever "666" denotes is abstract" doesn't imply "there is an abstract object", unless one adds further assumptions. (E.g., that "666" denotes something). However, it is perfectly plausible that the string "666" is an abstract object, so he needs to be more careful.

Any semantic theory needs a sub-theory of its syntactical entities. These syntactical entities are usually closed under concatenation, and therefore there are infinitely many of them. Presumably they are abstract (I.e., finite sequences of letters). So, I would say that Balaguer's Semantic Platonism is both a semantic and ontological theory.

(But this is where we get into the fine-grained disagreements ...)

Jeff

4. Well, Balaguer is asking whether number terms purport to refer to objects of a particular nature. I like your outline of what semantics is, but I think that he is using a broader, less formal understanding of 'semantics' on which some questions of this kind can be settled by semantics - we can tell from its form and use that 'God' purports to refer to an individual, and perhaps that it purports to be a personal name, while 'water' doesn't. Also, he tries to put a semantic slant on e.g. the case of the pupil who argues from physics that there are only finitely many things, and hence finitely many numbers - he says that the pupil isn't just wrong, but that they don't understand the meanings of claims of mathematics.

Perhaps this is instead what you call ‘theory of meaning’, not semantics, because it deals with particular meanings. The methods for determining facts about how we should use ‘3’ seem to me to be pretty similar to those for determining how we should use e.g. ‘¬’, though. And lots of substantive questions can be made to give rise to semantic ones - rather than asking e.g. if numbers are abstract entities, you just ask if 'x is abstract' is true whenever 'x is a number' is true as a matter of semantics (in the same way that you can ask whether P is true whenever ¬¬P is), i.e. if number terms purport to refer to abstract entities. This is dealing with token linguistic objects rather than types (with words like ‘not’ rather than the propositional operator ‘¬’), but if you think of linguistic tokens as abstract too, is there such a big difference?

But, having said all this in Balaguer's favour, I'd worry that that his method bleeds into conceptual analysis, and I don't think we want to allow that to be subsumed into semantics. I also think we want to be cautious about allowing substantive conceptual content to be loaded onto the claims of mathematics (which is what the idea that abstractness is part of the meaning of number terms amounts to). And we likewise want to be cautious about considering ourselves students only of semantics in his sense, because that would require us to give much more weight to experiment (into what terms purport to refer to in actual use), which would change the discipline a great deal. So, while I'm glad Balaguer is explaining philosophy of mathematics to outsiders, I think he has slightly (perhaps excusably) misrepresented it.

On your last point, I think Balaguer accepts something like ‘if there are infinitely many things of type T, then things of type T are not all physical or mental’. If there are infinitely many syntactic entities, then I agree that he seems to be committed to saying that at least most (and in practice, surely all) of them are abstract, by the same arguments as for the abstractness of numbers.

5. I agree with quite a bit here. Abstractness isn't built into the meanings of numerals.
Still, I'm with Balaguer, in explaining Philosophy of Maths (Phi-M), to separate semantic issues from ontological ones; and, in Phi-M, much focuses on semantics (constructivism, intuitionism, predicativism, "hermeneutic" nominalism). Balaguer's overall line of argument is:

(i) stressing the role of semantics/theory of meaning for Phi-M;
(ii) trying to separate semantic issues from ontological ones.

Admittedly, separating them isn't so easy, and controversies are inevitable!

"I think Balaguer accepts something like ‘if there are infinitely many things of type T, then things of type T are not all physical or mental" "

On the example he gives, it seems that the crucial premise is,

(*) If there are finitely many As and infinitely many Bs, then Bs cannot be reduced to As.

So, if there are finitely many physical things and infinitely many numbers, then numbers cannot be reduced to physical things. On the other hand, if there are infinitely many numbers, and numbers are (reducible to) physical entities, then there are infinitely many physical things. Arithmetic itself, at face value, implies that, for each number, there is a larger one, and so that there are infinitely many numbers.

(This can be rephrased as saying that if physical theory T in $L_{Phy}$ implies that there are at most finitely many physical things, then PA cannot be relatively interpreted into T by any relative interpretation $^{\circ} : L_{PA} \to L_{Phys}$ that maps $\forall x \phi$ to $\forall x(Phys(x) \to \phi^{\circ})$.)

Jeff