Here’s a sketch of my present understanding of mathematical practice. First I’ll say what I think is going on with math assuming an understanding of logical consequence, and then I’ll talk about logic.

This may seem like strange order to do things in, placing logical consequence prior to mathematics and using the former to explain the latter, given that there are definitions of logical consequence in terms of mathematical items. If A being a logical consequence of B is a matter of their not existing any models of B and ~A or of there existing a proof, then these mathematical items (proofs, models) are already involved in the notion of logical consequence. Thus an account of math in terms of logic would be uselessly circular. For now I’ll just note that I don’t think this is the right way to think of logic, and that I think there is a prior sense of logical consequence which these definitions are just descriptions of.

So, assuming logic, I think mathematical practice is a combination of a) answering as route training (like learning one’s multiplication tables) requires and b) asserting the logical consequences of certain socially chosen axioms which might be formal or informal (as e.g. talk about the integers is). We stipulate that there are objects with certain formal or informal features and then logically work out what other features they have to have. The choice of stipulation is arbitrary in one sense, and non-arbitrary in another. It’s arbitrary in the sense that study of the implications of any set of axioms, even an inconsistent one, could be mathematically correct even if, since everything follows from a contradiction, it might be very uninteresting. It isn’t arbitrary in the sense that there are lots of good (extra-mathematical) reasons for wanting to study one mathematical structure/set of axioms rather than another. These include aesthetic considerations, practical/scientific considerations and philosophical ones. An aesthetic reason to study a set of axioms or an idea of a kind of mathematical structure would be that it was simple, or novel in structure or had certain other features which a professional mathematician would know more about than I do. A practical or scientific reason to study a certain mathematical structure would be that it, in some sense, shares a structure with some physical system that we are interested in. If you notice that water behaves in a certain regular way you might specifically concoct a mathematical structure which you could then refer to in stating these regularities. Finally, a philosophical reason for studying a certain set of axioms or mathematical structure might be that you want to use it to model some other notion, like the relations between parts and wholes, or the idea of a collection, or logical relations between sentences. The physical and philosophical uses are fairly analogous, in the sense that a mathematical system is found of concocted to match some other system, be it external and physical or linguistic.

This explains how it is that doing math requires logic and doesn’t require any specific observations in the world. In math we are engaging in communal make-believe in supposing that there are objects of a certain kind and then working out logically what such objects would have to be like. The result of this kind of investigation of what else would have to be true if a certain stipulation was true, is to create a kind of store house of various different logically possible structures. These then, provide a useful vocabulary for physics in the sense that we can state an idea of what will physically happen by relating or comparing it to what does happen in one of these made up structures. Thus there’s a kind of two way relation between the made-up structures of mathematics and the real ones of the physical world around us, where on the one hand counting and measuring procedures show the world to have a physical structure which gets us interested in a corresponding mathematical structure, and on the other hand mathematical structures previously investigated for aesthetic reasons alone allow physists to formulate and test new theories.

Now, what about logic? As I said I think that our understanding of logical consequence precedes either of the mathematical definitions which can be used to articulate it. Following Wittgenstein in the Tractatus, I think that logical necessity and consequence arise from the workings of language. If A implies B this is not just a case of formulating in language some general truth of the universe which exists independently of language. Rather it show something about how we apply A and B, that anything which we would call A we would also call B. Thus which logical laws there are completely depends on which language one is speaking. There could even be a language (It would have to be rather small I think) which had no logical laws because there were no interesting relationships between the conditions of application of each of its sentences.

The above is just a bald statement of a position, and it immediately raises two tough questions. The first concerns giving an epistemology for logic: how do we come to learn these facts about how our words apply? The second concerns my claim that logical truths are completely an artifact of language, as opposed to (some of them) being general truths about the world. Some people would wonder how this distinction can be cashed out, and others might assume that it can but wonder how I can show that the world makes no contribution to logical truth. It might be plausible that the fact that should always assent to P whenever we assent to ~~p merely illustrates something about the application of ~ and says nothing substantive about the world, but what about nessicary truths like nothing is green all over and red all over? A external metaphysical fact that nothing ever is such that B but not A can explain our willingness to assent to A whenever we say B, just as much any relationship that’s purely internal to language. So why am I claiming that all necessities are contributed by language and none by the world, and how do I know?

I’ll get to all of these questions later, but for now I have just tried to give a quick sketch or overview of my present ideas about the practices of mathematics and logic.