Don’t worry, this post is not about factors irrelevant to mathematics such as gender, ethnicity and sexual orientation.

Nor is it about important factors, highlighted in the 2011 International Review of UK Mathematics although largely ignored by the research council that commissioned it: research area, size of research group, and size of institution.

But I discovered recently a diversity in the way we do mathematics, which I found surprising and potentially significant.

A few weeks ago, I was at dinner with a visiting colloquium speaker. The conversation turned to whether mathematical thought is done in words, or is “pre-linguistic”.

This is a topic about which Jacques Hadamard, in his book originally called *The Psychology of Invention in the Mathematical Field* but re-published as *The Mathematician’s Mind*, had a lot to say. Some linguists and linguistic philosophers, notably Max Müller, insist that language is essential to thought, and that no thoughts can be pre-linguistic. Hadamard, from his own intuition and from the writings of others from Poincaré to Einstein, is convinved that this is not the case, and is bewildered that Müller can hold this view with such vehemence. In a footnote, he says,

I have also seen the following topic (a deplorable subject, as far as I can judge) proposed for an examination—an elementary one, the “baccalauréat”—in philosophy in Paris: “To show that language is as necessary for us to think as it is to communicate our thoughts.”

For me, I know for sure that my best insights (those which are not just routine calculations) are pre-linguistic, and I struggle to put them into words: similarly, if the insight is a conjecture, I struggle to see how the conjecture might be proved. I assumed that most mathematicians would be like me, and would agree with Hadamard rather than Müller.

So it was a bit of a surprise when, of the five research mathematicians at the table, we were split 3 to 2 in Hadamard’s favour.

This is of course an anecdote, and not survey data. But we noticed a curious thing. The two who said they did mathematics in words had something probably significant in common: their cradle tongue (in both cases, a Slavic language) was not the language in which they do mathematics (in both cases, English); and both of them had learnt English at a comparatively advanced age. The other three of us were all native English speakers.

Not sure what to make of this. But I am glad that it drove me back to Hadamard’s book. I had completely forgotten that, at a certain point, he admits to his failure to be able to think creatively about group theory!

Expanding the sample beyond mathematicians, I find that people who think primarily in ordinary language forms have the greatest difficulty grasping the simplest mathematical facts, even such as those we encounter in combinatorics and discrete mathematics.

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Why is “pre-linguistic” the alternative to thinking in words? Could mathematical concepts, their relationships and manipulations not be treated by our brains as another language?

What is a language? If it is something with words, then my experience (and Hadamard’s) is clearly that I don’t do mathematics in language. If you say that a language may have a symbolic representation more general than conventional words, then why can’t I just write the symbols down? If it is more general than that, then I fear it is so general that the statement doesn’t have much meaning.

I think that’s a very early point at which to give up looking for a useful description of language š

The machine learning/natural language processing crowd like to think of words as points in a high dimensional vector space; the popular example of this is “king – man + woman = queen”. Language acquisition can then be seen as generating some useful internal representation of such a space: useful in the sense that a person who had never heard the word “monarch” could still conceive of a word that might mean something like “king – man”.

I tend to think about mathematical research in the same sort of way – refining my internal representation of the space of mathematics (i.e. improving my understanding of objects, their properties, and their relationships to each other). Considering axioms gives straightforward examples (group – invertibility = monoid), but there are more interesting cases: what is “dual group – abelian”? Or “groupoid – group + ring”?

I think we have two type of dialogues: 1) internal dialogue (ID) (is not equal to something which is spiritual) 2) external dialogue (ED). These two type of dialogues have a lot of things in common, but are not identical. For ED we use contractual languages, but the language of ID is not so clear. ID is constructed by our genes, our body, our life and many other things which affect to our unconsciousness. So, ID live in our unconsciousness and its mechanism of work (language) is not so clear (therefore we do not know the origin of innovation or creativity). But, ED lives in our consciousness. I think mathematics use both ID and ED, but it is not one of them exactly. I think, Richard Guy is an instance which he used ID many times to solve some problems (he obtained the solutions of some problems when he was slept and unconsciousness has fortune to appear as lucid). Anyway, since we use ED much more than ID in the society, we see mathematics as a purified contractual language.

Sasha Borovik quotes a paper by Amalric and Dehaene, “Origins of the brain networks for advanced mathematics in expert mathematicians”, which you can find at

http://www.pnas.org/content/113/18/4909.abstract

They say “Our work addresses the long-standing issue of the relationship between mathematics and language. By scanning professional mathematicians, we show that high-level mathematical reasoning rests on a set of brain areas that do not overlap with the classical left-hemisphere regions involved in language processing or verbal semantics. Instead, all domains of mathematics we tested (algebra, analysis, geometry, and topology) recruit a bilateral network, of prefrontal, parietal, and inferior temporal regions, which is also activated when mathematicians or nonmathematicians recognize and manipulate numbers mentally. Our results suggest that high-level mathematical thinking makes minimal use of language areas and instead recruits circuits initially involved in space and number. This result may explain why knowledge of number and space, during early childhood, predicts mathematical achievement.”

Stanislaw Lem, in

His Master’s Voice, is much more absolute than I have been above. He was a writer, not a mathematician, but this novel is written in the persona of a great, but uncompromising and iconoclastic, mathematician. At one point he writes:I had to laugh, for instance, at the assurance of those who determined that all thought was linguistic. Those philosophers did not know that they were creating a subset of the species, i.e., the group of those not gifted mathematically. How many times in my life, after the revelation of a new discovery, having formulated it so solidly that it was quite indelible, unforgettable, was I obliged to wrestle for hours to find for it some verbal suit of clothes, because the thing had been born, in me, beyond the pale of all language, natural or formal?

I call this phenomenon “surfacing”. It defies description, because what emerges from the unconscious with difficulty, slowly, finds nests of words for itself; it exists as an entity before it settles inside those nests; yet I can give no indication, no hint, to explain in precisely what form that non- and preverbalness appears; it is heralded only by a keen presentiment that the expectation of it will not be in vain. The philosopher who does not know such states from introspection is, with respect to the quality of certain mechanisms of the brain, a man unlike me; we may belong to the same species, but we differ far more than such thinkers could wish.

So many modes of mathematical thought,

So many are learned, so few are taught.

There are streams that flow beneath the sea,

There are waves that crash upon the strand,

Lateral thoughts that spread and meander ā

Who knows what springs run under the sand?

ā Notes Found In A Mathtub • 2016-09-09