The eminent geneticist Francis Galton is also worth quoting:

(uh, founder of eugenics + psychometrics = "eminent"? oooook)

It is a serious drawback to me in writing, and still more in explaining myself, that I do not think as easily in words as otherwise. It often happens that after being hard at work, and having arrived at results that are perfectly clear and satisfactory to myself, when I try to express them in language I feel that I must begin by putting myself upon quite another intellectual plane. I have to translate my thoughts into a language that does not run very evenly with them. I therefore waste a vast deal of time in seeking appropriate words and phrases, and am conscious, when required to speak on a sudden, of being often very obscure through mere verbal maladroitness, and not through want of clearness of perception. That is one of the small annoyances of my life.

Also Hadamard himself writes:

I insist that words are totally absent from my mind when I really think and I shall completely align my case with Galton’s in the sense that even after reading or hearing a question, every word disappears the very moment that I am beginning to think it over; and I fully agree with Schopenhauer when he writes, ‘thoughts die the moment they are embodied by words’.

(Maybe that was why Nietzsche was blathering on and on about Schopenhauer.)

I quote these examples because they very much accord with my own thought-modes. Almost all my mathematical thinking is done visually and in terms of non-verbal concepts, although the thoughts are quite often accompanied by inane and almost useless verbal commentary, such as ‘that thing goes with that thing and that thing goes with that thing’. (I might use words sometimes for simple logical inferences.)

— emperors-new-mindch. 10

(emphases on hilarity added)

Contrast this disparagement of verbalisation and language for purposes of human thought and consciousness in Penrose with the lionisation of language by anthropologists. Penrose specifically points out

This is not to say that I do not sometimes think in words, it is just that I find words almost useless for mathematical thinking. Other kinds of thinking, perhaps such as philosophizing, seem to be much better suited to verbal expression. Perhaps this is why so many philosophers seem to be of the opinion that language is essential for intelligent or conscious thought!

— emperors-new-mindch. 10

More hilarity later on:

A common experience, when some colleague would try to explain some piece of mathematics to me, would be that I should listen attentively, but almost totally uncomprehending of the logical connections between one set of words and the next. However, some guessed image would form in my mind as to the ideas that he was trying to convey – formed entirely on my own terms and seemingly with very little connection with the mental images that had been the basis of my colleague’s own understanding – and I would reply. Rather to my astonishment, my own remarks would usually be accepted as appropriate, and the conversation would proceed to and fro in this way. It would be clear, at the end of it, that some genuine and positive communication had taken place. Yet the actual sentences that each one of us would utter seemed only very infrequently to be actually understood! In my subsequent years as a professional mathematician (or mathematical physicist) I have found this phenomenon to be no less true than it had been when I was an undergraduate. Perhaps, as my mathematical experience has increased, I have got a little better at guessing what others are meaning by their explanations, and perhaps I am a little better at making allowances for other modes of thinking when I am explaining things myself. But, in essence, nothing has changed.

[...]

There may seem to be a paradox in this, since mathematics is a subject where precision is paramount. Indeed, in written accounts, much care is taken in order to make sure that the various statements are both precise and complete. However, in order to convey a mathematical idea (usually in verbal descriptions), such precision may sometimes have an inhibiting effect at first, and a more vague and descriptive form of communication may be needed. Once the idea has been grasped in essence, then the details may be examined afterwards.

How is it that mathematical ideas can be communicated in this way? I imagine that whenever the mind perceives a mathematical idea, it makes contact with Plato’s world of mathematical concepts.

— emperors-new-mindch. 10