Reuben Hersh

Some proposals for reviving the philosophy of mathematics

By “philosophy of mathematics” I mean the working philosophy of the professional mathematician, the philosophical attitude toward his work that is assumed by the researcher, teacher, or user of mathematics. What I propose needs reviving is the discussion of philosophical issues by working mathematicians, especially the central issue-the analysis of truth and meaning in mathematical discourse. The purpose of this article is, first, to describe the philosophical plight of the working mathematician; second, to propose an explanation for how this plight has come about; and third, to suggest, though all too briefly, a direction in which escape may be possible. In summary, our argument will go as follows:

The "Origin" of Geometry

The German phenomenologist Edmund Husserl wrote a famous essay, “The Origin of Geometry” that called for a new kind of “historical” research, to recover the “original” meaning of geometry, to the man, whoever he was, who first invented it. It seems to me not that hard to imagine the origin of geometry. Once upon a time, even twice or several times, someone first noticed some simple facts. For example, when one stick lies across another stick, there are four spaces that you can see. You can see that they are equal in pairs, opposite to opposite. It happened something like this, perhaps at some campfire, 20 or 30,000 years ago. In the dead ashes are lying two sticks, one across the other. Ancient #1: Look at that! Do you see that? Ancient #2: What? See what? 1: Those two sticks. How they cross—see, they make four spaces. Two big ones, two little ones. The big ones are across from each other on the sides, and the little ones are across from each other, one the top and bottom. 2: So what? (Kicks one of the stocks.) Now what happened to your four spaces? 1: You changed them around. Now the top and bottom are bigger, and the ones on the side are littler. There are still four spaces. And you still have little facing little, big facing big. 2: Yes, that’s the way it is now. 1: Turn them any way you like, you always get four spaces, and they are equal in opposite pairs. 2: I don’t believe it. 1: How can’t you believe it? Can’t you see it? 2: Just watch now. I turn the top stick, little by little. The top and bottom spaces get smaller, the side spaces get bigger. 1: All right. 2: What if I stop now? Where are your big and little spaces now, Mr. Smart Aleck Wise Guy? 1: You stopped before they could switch around. Before the little spaces became bigger than the big ones and the big spaces became smaller than the little ones. 2: Yes, that’s what I did. That shows you’re way off, you’re screwed up. 1: When the sticks cross now, they make four equal spaces. That’s a special interesting way to make two sticks cross. I like that. You did something good. 2: Let’s go chase a rabbit and eat it. Something like this must have happened more than once. Someone noticed something interesting about a couple of sticks, or bits of straw, or crossed fingers. Something that has to be so, whether you want it or not. An invariant. A geometric fact.

High-Accuracy Stable Difference Schemes for Well-Posed Initial-Value Problems

For

Boundary conditions for equations of evolution

t > 0

Mathematics has a front and a back

, let

A class of “central limit theorems” for convolution products of generalized functions

u(t)

On the invariance principle of scattering theory

satisfy \[ \frac{{du}}{{dt}} = Au, \] where A is a linear operator, the generator of a strongly continuous semigroup. Let

Why the Faulhaber Polynomials Are Sums of Even or Odd Powers of (n + 1/2).

v(t)

Hyperbolic Equations with Coefficients in an Enveloping Algebra

satisfy

Pluralism as Modeling and as Confusion

v(t + h) = r(hA)v(t)