I realize this topic is a bit old, but I just discovered it, and thought that others might like to hear directly from one of the neo-logicists who are being discussed.

Anyway, a few points worth making:

In the original post David Corfield writes:

“By contrast, a much more orthodox philosophical approach to mathematics in the English-speaking world, well represented in the UK, is to address the question of whether mathematics is reducible to logic. To gain an idea of the current state of play here, you can take a look at What is Neologicism? by Linsky and Zalta.”

This is misleading at best. Since Russell’s paradox shattered Frege’s own hopes of reducing arithmetic (and real and complex analysis) to logic, very few philosophers have taken seriously the idea that mathematics is reducible to logic. The reason is simple: Logic (as understood today) has no ontological commitments, while mathematics has many.

This brings up a crucial issue: Neo-logicism (at least, the standard variant of it, as espoused by myself, Crispin Wright, Bob Hale, etc,) is not, strictly speaking, a version of logicism at all. There is no claim that mathematics is reducible to logic, since the abstraction principles used (e.g. Hume’s Principle) are not logical truths. Abstraction principles are viewed as being definitions of a certain kind. As a result, their consequences can be known a priori, etc. But the neo-logicist claim is not, and never has been, that mathematics can be reduced to logic alone (misleading nomenclature notwithstanding).

[It should be pointed out that there are alternative views which have also adopted the”logicist” or “neo-logicist” label, such as Neil Tennant’s project and the object-theory version proposed by Zalta and Linsky. In the philosophical literature, however, “neo-logicism” (without any qualification) is generally understood to refer to the views of Wright and Hale (and myself). Linsky and Zalta’s characterization of this version of neo-logicism is, unfortunately, misleading in this respect. Unfortunately, this confusion has persisted recently, necessitating the need to distinguish the ‘Scottish School’ from other variants of neo-logicism as in the Ebert/Rossberg paper.]

In a later post David also notes that he is unclear regarding “whether it matters to them whether their logical reconstructions of portions of mathematics get to the conceptual heart of those portions.” The reason for the unclarity is that there is substantial disagreement between neo-logicists (of the Scottish school) on exactly this issue. The principle in question is this (quoted from C. Wright’s “Neo-Fregean Foundations for Real Analysis”):

“Frege’s Constraint: That a satisfactory foundation for a mathematical theory must somehow build its applications, actual and potential, into its core – into the content it ascribes to the statements of the theory – rather than merely ‘patch them on from outside.’”

Wright rejects Frege’s Constraint as a general requirement on neo-logicist reconstructions of mathematical theories, arguing that whether it applies to a given theory depends on the nature of that theory (and, in particular, on the nature of its applications). In particular, he requires that our reconstruction of arithmetic satisfy it, but not our reconstruction of analysis. Hale, on the other hand, seems to require that Frege’s constraint be met across the board, and he demonstrates in some of his work how an account of analysis can be formulated within the neo-fregean framework that arguably meets the constraint. (I am also sympathetic to Frege’s Constraint, for what it is worth).

Finally, there is a recurring idea (both in David’s book, but also in discussion) that typical philosophers of mathematics are ignorant of of, or are too lazy to learn, ‘real’ mathematics and instead concentrate on three rather artificial areas: arithmetic, analysis, and set theory. The formulation of this idea is particularly offensive in the following quote from John Baez’s discussion of David’s book:

“Alas, too many philosophers seem to regard everything since Goedel’s theorem as a kind of footnote to mathematics, irrelevant to their loftier concerns (read: too difficult to learn).”

I think there are a number of misunderstandings regarding the way philosophy of mathematics has progressed that are at work here.

First off, it is clear that most philosophers of mathematics of any significance know a lot more mathematics than just these three areas. And there is definitely widespread agreement that the more math one knows (including recent mathematics) the better (and not just areas one might classify as ‘mathematical logic’). So why does discussion of these areas not pop up more within philosophy of mathematics? The underlying cause, as Baez noted, has to do with Gödel’s theorem, I think, but not for the reasons Baez suggests.

It is not that mathematics later than this is irrelevant, uninteresting, or too hard. Instead, the problem is that Gödel’s theorem (and various other limitative results) has shown the problem of accounting for the metaphysics and epistemology of mathematics to be much, much more difficult than it had previously appeared to be (and it appeared hard enough as it was).

As a result, philosophers of mathematics have restricted their attention to a small number of conceptually simple ‘test cases’ (arithmetic, analysis, and set theory) The underlying idea would seem to be that if we cannot adequately handle these cases, then there is little point in trying to tackle more complicated mathematical structures and theories. [Note that although ZFC is complicated, the intuitive notion of ‘set’ is rather simple.] In addition, some philosophers (but not all) think there is something particularly special and central about these specific theories (see the comments regarding Frege’s constraint above), and this might also be contributing to this emphasis on a few mathematical domains.

Additionally, I think that many philosophers of mathematics believe that if we can handle these three cases, then we should be able to deal with the rest of mathematics in a roughly similar manner. This might be optimistic (I suspect it is), but I certainly don’t think there is any knock-down argument to the effect that concentrating on these three domains is causing the thinkers in question to miss out on important aspects of the nature of mathematics (after all, pretty much all mainstream mathematics can be modeled in the powerset of the reals).

## Re: Two Cultures in the Philosophy of Mathematics?

It may also be beneficial to have a conference on pure and applied mathematics.

The latter seems to have done more for civiliztion.

Perhaps this over simplifies the relation of pure and applied mathematics:

Archimedes was a mathematician and mechanical engineer.

Newton was a mathematician and mechanical engineer.

Maxwell was a mathematician and electrical engineer.

Steinmetz was a mathematician and electrical engineer.