"The enrapturing discoveries of our field systematically conceal (...) the analogical train of thought that is the authentic life of mathematics..."

Question

In the preface of the book Discrete Thoughts, Gian-Carlo Rota writes:

Sometime, in a future that is knocking at our door, we shall have to retrain ourselves or our children to properly tell the truth. The exercise will be particularly painful in mathematics. The enrapturing discoveries of our field systematically conceal, like footprints erased in the sand, the analogical train of thought that is the authentic life of mathematics. Shocking as it may be to a conservative logician, the day will come when currently vague concepts such as motivation and purpose will be made formal and accepted as constituents of a revamped logic, where they will at last be allotted the equal status they deserve, side by side with axioms and theorems. Until that day, however, the truths of mathematics will make only fleeting appearances, like shameful confessions whispered to a priest, to a psychiatrist, or to a wife.

I am unable to decipher the poetry here: what could he possibly mean by this?

Answer 1

"Clearly", in his "oracolar" Preface, Rota is alluding to an inner life of mathematical thinking, where meaning lies.

The mainstream "logician" (???) form of mathematics, shaped into "still-life" collections of axioms and theorems is deprived of meaning.

I think that, in order to appreciate Rota's thinking, we need to practice with Edmund Husserl's Phenomenology and its theory of meaning.

According to me, there is clearly a place for meaning, intuition and creativity in mathematical thinking.

Personally I do not think that symbolic thinking in mathematics is deprived of meaning and I see no contradiction between symbolism and formalization, form one side, and intuition and creativity, from the other side.

As a silly example, try to read Sixteenth-century algebraist (like Cardano and Tartaglia) descriptions of method for the solution of third degree equations [see John Fauvel & Jeremy Gray (editors), The History of Mathematics : A Reader (1987), page 257] :

Find me two numbers such that when they are added together, they make as much as the cube of the lesser added to the product of its triple with the square of the greater; and the cube of the greater added to its triple times the square of the lesser makes 64 more than the sum of these two numbers. [i.e., find $a, b$ such that $a + b = b^3 + 3ba^2$ and $a^3 + 3ab^2 = 64 + a + b$.]

Compare now with modern algebra, and - I think - you can appreciate the enormous creativity effort made in the following century by Descartes and Newton and Leibniz to invent a symbolism so powerful which disclosed completely new fields of mathematical inquiry and thinking.

I prefer an approach to the problem of meaning and intuition in mathematics through the so-called Analytical Philosophy tradition; see :

• Michael Dummett, Origins of Analytical Philosophy (1996)

• Claire Ortiz Hill & Guillermo E.Rosado Haddock, Husserl or Frege? Meaning, Objectivity, and Mathematics (2000)

• Richard Tieszen, After Gödel : Platonism and Rationalism in Mathematics and Logic (2011)

• Charles Parsons, From Kant to Husserl : Selected Essays (2012).

Answer 2

The author is describing a problem that I am extremely familiar with, and in fact so are all mathematics students whether they know it or not: mathematicians apparently don't consider it's worth it to think about how their subject was developed.

Well, that's an unfair generalization, I've had several professors that do make the effort, and I could cite several textbooks from my own shelves that contain passages lamenting the very same issue. But I've also seen many textbooks, and certain classes, which start, on page 1 chapter 1, with the words "Definition 1.1" and then proceed as though written by a robot to vomit forth pages and pages of definitions and theorems, while the reader can only scratch their head and wonder the point of any of them are.

This is a problem in education, however, and I'm not sure that's what the author is talking about, but I would think it's something related (given that in the next sentence he immediately mentions "motivation" and "purpose").

Answer 3

I think that Gian-Carlo Rota is making two very specific claims; that these claims are wrong; and that nonetheless, they're still relevant.

The first thing I think he's saying:

• Mathematics does not (currently) have a formalism to encode why we actually care about particular formal systems, and this is a problem.

Why I think its wrong:

• We give semantics for formal systems in order to explain why we care about them.

Why it is still relevant:

• You sometimes read papers that present formal systems, but give no semantics; and, therefore, no motivation. Such a paper is necessarily badly written, and needs revision.

The second thing I think he's saying:

• Mathematics does not have a formalism to encode how we're actually thinking about something.

Why I think its wrong:

• Structuralist mathematics is all about how we're thinking about things; its all about translating between perspectives.

Why it is still relevant:

• Structuralism still hasn't made significant inroads into our ways of talking; e.g. we still talk about Abelian groups as if they're particular kinds of groups, rather than objects of the category $\mathbf{Ab}$ which is canonically equipped with a forgetful functor $\mathbf{Ab} \rightarrow \mathbf{Grp}$. So the "painful retraining of ourselves and our children" that he speaks of is very much going on at the moment.

Answer 4

Rota's ideas on the "analogical train of thought" that is the life of mathematics have been developed (rather satisfactorily in my opinion) by Grattan-Guinness in his essay

Grattan-Guinness, Ivor Solving Wigner's mystery: the reasonable (though perhaps limited) effectiveness of mathematics in the natural sciences. Math. Intelligencer 30 (2008), no. 3, 7–17. See here.

Grattan-Guinness's article challenges the received view that there is something mysterious about the applicability of mathematics, and also describes some mechanisms by which the process of discovery in mathematics operates. A central place is given to reasoning by analogy from related or not-so-related fields. His article is very broad and covers many fields of mathematics and applications. A true gem that only an experienced historian of mathematics and natural science could have written.