Here’s another take on thinking without language excerpted from Keith Devlin’s paper A mathematician reflects on the useful and reliable illusion of reality in mathematics:

**Mathematical thought**

What is the nature of mathematical thought? Although I have been a mathematician for

forty years, I am still not clear exactly what the nature of the mathematical thought

process is.

I am sure it is not linguistic, or at least not totally so, and probably not mostly so.

Mathematicians do not think in sentences; at least not most of the time. The precise

logical prose you find in mathematical books and papers is an attempt to communicate

the results of mathematical thought. It rarely resembles the thought process itself.

I am in remarkably good company in having this view of mathematical thinking. For

instance, in 1945, the distinguished French mathematician Jacques Hadamard

published a book titled *The Psychology of Invention in the Mathematical Field*, in which

he cited the views of many mathematicians on what it feels like to do mathematics. Many

of them insist that they do not use language to think about mathematics. Albert Einstein,

for instance, wrote:

Words and language, whether written or spoken, do not seem to play any part in

my thought processes. The psychological entities that serve as building blocks for

my thought are certain signs or images, more or less clear, that I can reproduce

and recombine at will.

Hadamard himself makes the same point:

I insist that words are totally absent from my mind when I really think . . . even after

reading or hearing a question, every word disappears the very moment that I am

beginning to think it over.

Of particular relevance to my thesis are the mathematicians’ descriptions of the way they

arrived at the solutions to problems they had been working on. Time and again, the

solution came at a quite unexpected moment, when the person was engaged in some

other activity and was not consciously thinking about the problem. Moreover, in that

inspirational moment the whole solution suddenly fell into place, as if the pieces of a

huge jigsaw puzzle had been dropped onto the floor and miraculously landed as a

complete picture. The mathematician “saw” the solution and instinctively knew it was

correct.

No language is involved in this process. Indeed, with a problem for which the solution is

fairly complex, it might take the mathematician weeks or even months to spell out (in

linguistic form) the step-by-step logical argument that constitutes the official solution to

the problem — the proof of the result.

So if mathematical thought is not linguistic, if mathematicians do not think in words (or

algebraic symbols), how exactly does it feel to a mathematician thinking about

mathematics?