[This post is an exact replacement for the now deleted “Number and Reality 3.” One should not, I think we can all agree, have “Number and Reality 3” immediately succeed “Number and Reality 1” in a series devoted to number and reality.]

I am hard at work on the post or posts that I think will be the next one or two in the series of posts that I am devoting to the philosophy of mathematics. It or they will bear on the Platonist, realist, and nominalist theories in metaphysics, i.e., on their “solutions” to the so-called “problem of universals,” and on the philosophical theories immediately opposed to them. If all goes according to plan, it or they will be followed by a post or two bearing on the partially analogous Platonist, realist, and nominalist theories in the metaphysics or ontology of mathematics.

An announcement like the foregoing hardly justifies my having led you to read this far. In
recognition that you deserve something of rather more value for your doggedness, I will recommend that you take note of the post that Prof. Peter Smith’s has published on his blog, one that I have recently begun to follow, Logic Matters. Bearing the title, “Philosophy of mathematics — a reading list,” the post lists more than two dozen recent titles in the philosophy of mathematics and offers his expert’s summary evaluations, not all equally kind, of them.

Readers of this blog may recall that, in my post of October 24, 2020, “Number and Reality 0. An Introduction to a Project in the Philosophy of Mathematics,” I announced my intention to make use of Stewart Shapiro’s Thinking about Mathematics* as an “initial guide” as I re-explore and rethink my understanding of the philosophy of mathematics. I noticed, then, with particular interest what Smith has to say about Shapiro’s book.

Let’s begin [the list] with an entry-level book first published twenty years ago but not yet superseded or really improved on:
1. Stewart Shapiro, Thinking About Mathematics (OUP, 2000). After introductory chapters setting out some key problems and sketching some history, there is a group of chapters on what Shapiro calls ‘The Big Three’, meaning the three programmatic ideas that shaped so much philosophical thinking about mathematics for the first half of the twentieth century — i.e. varieties of logicism, formalism, and intuitionism. Then there follows a group of chapters on ‘The Contemporary Scene’, on varieties of realism, fictionalism, and structuralism. This might be said to be a rather conservative menu — but then I think this is just what is needed for a very first introduction to the area, and Shapiro writes with very admirable clarity.

He also had high praise for the major collection of essays** of which Shapiro was the editor, saying:

Stewart Shapiro (ed.) The Oxford Handbook of Philosophy of Mathematics and Logic (OUP, 2005). The editor’s introductory essay is in fact called ‘Philosophy of mathematics and its logic’, which should surely also have been the whole Handbook’s title — for of the twenty-six essays here, twenty are straight philosophy of mathematics, and the logic essays are mostly closely relevant to mathematics too. Large handbooks of this general type can often be very mixed bags, containing essays of decidedly varying quality; but this one really is a triumph. Some of the essays are very substantial, and as I recall it none is a makeweight. There are often pairs of essays taking divergent approaches (e.g. to contemporary logicism, to intuitionism, to structuralism). Of course, there are variations in the accessibility of the individual essays: but Shapiro seems to have done wonderful work in keepinAfg his very well-selected authors under control! So for any serious student now — perhaps beginning graduate student — this must be the place to start explorations of issues in more recent philosophy of mathematics.

Before closing, I’d like to draw your attention to another good deed that Smith has done for those engaged in, if not immediately the philosophy of mathematics, at least logic and thereby all branches of philosophy and indeed all branches of thought. That is, in an earlier post, “Free introductions to formal logic?” he laid out another list, “curated,” one of a higher degree of sophistication than yours truly might put it, of available introductory logic textbooks. The list includes some that are excellent, in his judgment, as well as free.

One of the logic texts, quite modestly entitled An Introduction to Formal Logic,*** is by Smith himself. He says about it:

Peter Smith, An Introduction to Formal Logic (2nd edition, originally CUP, 2020) Webpage here. Available also from Amazon print on demand. Doesn’t cover as much and more expansive than [the previously mentioned] forallx, so perhaps more accessible for self-study

There we go. Until next time.

Richard

*Stewart Shapiro, Thinking about Mathematics. The Philosophy of Mathematics (Oxford and New York: Oxford University Press, 2000). Thinking about Mathematics is readily available for purchase through Amazon.com. You need only click on the following image to be taken to the Amazon site:

Nota bene: As an Amazon Associate, I earn from qualifying purchases.

**Stewart Shapiro (ed.) The Oxford Handbook of Philosophy of Mathematics and Logic (Oxford and New York: Oxford University Press, 2005). The Oxford Handbook of Philosophy of Mathematics and Logic too is readily available for purchase through Amazon.com. Here too you need only click on the following image to be taken to the Amazon site:

Nota bene: As an Amazon Associate, I earn from qualifying purchases. You may also wish to note that I am not sure how long it will be before I will be making systematic use of this latter text in my posts.

***Peter Smith, An Introduction to Formal Logic (2nd edition, originally CUP, 2020).

Nota bene: As an Amazon Associate, I earn from qualifying purchases. You may also wish to note that I doubt that will be making systematic use of this text in my posts, excellent though it is.