When I came home with the World of Mathematics, volume 3 had a bookmark (an old yellowed postal giro form) at the article *Mathematics and the Metaphysicians *by Bertrand Russel. The text is from 1901 I think, and it shows. This is the article from where the famous quip comes:

“Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.”

Not a very humanistic point if view!

Russel discusses “pure mathematics” which he thinks of as consisting

“entirely of assertions to the effect that, if such and such a proposition is true of anything, then such and such another proposition is true of that thing. It is essential not to discuss whether the first proposition is really true, and not to mention what the anything is, of which it is supposed to be true.”

This is the view put forward in his book *The Principles of Mathematics* published in 1903 (most of it written in 1900). Mathematics consists of implications P → Q, one proposition P implying another proposition Q.

The article is really fun reading, written as it is by an obviously very enthusiastic Russel (and young – he was 28). The paradoxes of calculus had been solved by Cauchy and Weierstrass through the limit concept and the epsilon-delta approach. The infinitesimals of the pioneers had been abolished (once – but not for all – as it turned out). The paradoxes of Zeno had been solved and the concept of the infinite reined in by Peano, Dedekind and Cantor. The story is told by someone who was there close in time. Russel pokes fun at the philosophers, who haven’t been able to make any progress with the questions of the infinitely small and the infinitely big, since the Greeks. Now the problems are solved by the mathematicians – but have the philosophers noticed it?

But this was before Frege, indeed before Russel’s own discovery of the Russel paradox in the theory of infinite sets and the flaw in Frege’s logic. The Sisyphus work of *Principia Mathematica* was due in ten years time. The work where Russel and Whitehead tried to reduce all of mathematics to logic. This idea, that mathematics is nothing but logic, is spelled out in the article. This is what became the foundational school of *Logicism*. But also *Formalism* is implicit in the article, the idea that mathematics is a pure play with symbols using the laws of logic.

And then came Gödel … and the rest is history, as they say.

Neither logicism, nor formalism are very humanistic in their approaches to mathematics, but the history of why these approaches arose and their successes and failures, that history is truly humanistic. Basically it is the history of understanding the concept of *number*.

Reading and Writing and Calculating are core subjects in all schools all over the world and has been so for a long time. Understanding numbers ought to be a humanistic endeavour, just as understanding how to read and write.

I think too that learning mathematics should be a humanist endeavour. However, it seems to me that when working with mathematics (and when learning mathematics, including number) you at times have to behave in accordance with any of the major philosophies of mathematics. At times, it is effective and worthwhile to treat in in some Aristotelian sense, “extracting” mathematics from nature and surroundings. At times it is effective to trust mathematics like Platonic eternal truths. At times it help to see it as a humanistic, empirical and fallible enterprise and at times you simply have to follow the logic and play the game, ala Russell above. It is not surprising all those variants of a philosophy of mathematics stayed with us for so long since each of them can be “felt” when working mathematically.

“It is not surprising all those variants of a philosophy of mathematics stayed with us for so long since each of them can be “felt” when working mathematically.”

Yes, I think you are right. All the philosophies have their merits, and I think from a pedagogical point of view, the important thing is not to decide which one is the best, but rather discuss them with students when appropriate, to make the subject more humanistic.