My answer to a question on Quora: Do physicists or mathematicians actually memorize hundreds of equations?

Some (I think rare) mathematicians have excellent memory and can remember a lot of stuff. Most of them, however, do not memorise every equation / theorem / definition; they keep in their heads generalised — but well structured —images of their fields and can recover a necessary fact or definition frоm “first principles”. Mathematics is not a sum of facts, it is a system of connections between facts and connections between connections, a system of analogies, and, at a higher level of thinking, analogies between analogies.

Added later: Perhaps I have to emphasise one point: “recovery” (as opposed to “remembering”) is fast because it is frequently used in its incomplete form, something like that “ah yes, and here we shall use this and that theorem…” without recalling the exact formulation of the theorem — and then immediately moving further in the argument. Why this is done? Because in most cases a specific argument will fail at later stages anyway; filling in all details in all intermediate steps is waste of time. Details are filled in only when the logical skeleton of a proof starts to look feasible. In many cases the argument / proof fails at the stage of a final write-up, and had to be started again. Mathematical thinking is a chain of failures; the key obstacle to learning mathematics is failure to learn how to manage one’s failures.

My answer to question on Quora: Why aren’t mathematicians more well known?

Well, this applies to every high-level specialism. Do you know many designers of new electronics? Or neurosurgeons?

This is a very uncomfortable social issue: development of technology, and perhaps the survival (or not) of humanity — critically depends on work of a very small (in relation to the total population of the Earth), and increasingly so, number of people. Mathematics is everywhere around us, but it is totally invisible to 99.99% percents of its users. Mathematics hardwired into a mobile phone ( I am not even talking about smartphones) and in mobile phones networks is beyond understanding of 99% of graduates from mathematics departments of British universities — and now there are more mobile phones in the world than toothbrushes.

My answer to a question on Quora: Why isn’t math presented as a language at school?

As usual, an answer is rather frustrating: because mathematics is taught on the cheap, and by teachers who have not been properly educated themselves (and they should not be blamed for that).

Mathematics has many aspects, and most of them remain unknown to many learners — and teachers — of mathematics. Of all metaphorical descriptions of mathematics, one of my favorites is

Mathematics is a language of contracts with Nature which Nature accepts as binding.

Within mathematics, people talk about “language of set theory”, or “language of categories”, or “language of algebraic topology”, or “language of schemes” — the list could be continued almost indefinitely.

To teach mathematics as a language means, first of all, giving students a chance to see that the same problem can formulated, say, in the language of algebra and in the language of geometry, and see the advantages and drawbacks of the two languages, and translate from one language to another, and combine both in a solution. This can be done — but this will be more expensive. Most school systems in the world cannot afford that; richer countries can afford that, but there is no political will to do that. Indeed, what for?

There is another aspect of mathematics which is almost never mentioned: in humans, mathematics is done by the subconscious, and mathematical languages are to certain degree languages of communication with the subconscious. I wrote about that elsewhere. If you want me to continue discussion of that fascinating facet of mathematics, please leave a comment to this post.