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.

My answer to a question on Quora: Why is school 8 hours long?

Admittedly it was in primordial times, but, in my country, at my time at primary school (7 to 11 years old), school day was 4 lessons of 45 minutes long, with two breaks of 10 minutes and one break of 25 minutes in between, from 8:30 to 12:30 in the morning. There was some homework, but not very taxing. A plenty of time was free for whatever children wished to occupy themselves with. Parents were at work until 17:00.

A short school day is actually a physiological norm. Why in the UK, say, school day is abnormally long? Because it is an offence to leave children alone; the law is vague — The law on leaving your child on their own, but it applies with unnecessary, in my opinion, rigour. Schools are forced to act as storage rooms for children while parents are at work.

Of course, in old times there were risks involved; legs and arms could occasionally be broken while skiing (unsupervised), or playing ice hockey (also unsupervised), etc., but these were very rare events, and were seen as unavoidable and normal risks. There were no modern culture of over-protection which would, of course, cut accidents — but at expense of loss of child’s precious independence. Analysing now my and my friends’ behaviour of that time, I see that we were quite risk aware and knew how to avoid danger — it was a normal part of growing up.

My answer to a question on Quora: How many years could it take me to study and understand all the mathematics fields that exist so far?

If you mean understanding at the level of ability to do research work in every field of mathematics, then, I am afraid, there is no hope to achieve this goal. Mathematics expands, and the cutting edge of mathematical research moves further and further away from any fixed reference point, say, undergraduate mathematics. From the point of view of an aspiring PhD student, mathematics looks like New York in the Capek Brothers’ book  A Long Cat Tale:

And New York – well, houses there are so tall that they can’t even finish building them. Before the bricklayers and tilers climb up them on their ladders, it is noon, so they eat their lunches and start climbing down again to be in their beds by bedtime. And so it goes on day after day.

It was written in the first half of the 20th century, but Joseph and Karel Capek understood thing or two about futurology (although the term “futurology”, most likely, did not exist in their time): they were the people who coined the word “robot”. We live in the world where, in almost every field of human endeavour, no-one can understand everything. The human civilization that we transform and build is immensely complex, and mathematics is perhaps its most complex part.

[For this post, I cannibalized some bits of my paper Mathematics for makers and mathematics for users; it discusses some relevant themes.]

The famous geneticist James Watson, of the double helix fame, about his relations with mathematics:

All through my undergraduate days I worried that my limited mathematical talents might keep me from being more than a naturalist.  In deciding to go for the gene, whose essence was surely in its molecular properties, there seemed no choice but to tackle my weakness head-on.  Not only was math at the heart of virtually all physics, but the forces at work in three-dimensional ;molecular structures could not be described except with math. Only by taking  higher math courses would I develop sufficient comfort to work at the leading edge of my field, even if I never got near the leading edge of math.  And so my Bs in two genuinely tough math courses were worth far more in confidence capital than any   I would likely have received in a biology course, no matter ;how demanding.  Though I would never use the full extent of the analytical methods I had learned, the Poisson distribution analyses needed to do most phage experiments soon became satisfying instead of a source of crippling anxiety. [From J. D. Watson, Avoid Boring People , Vintage Books, New York, 2010, p. 51]

My answer to a Quora question:

What’s something about math that still amazes you, even after knowing it for a long time?

That mathematics is consistent: regardless of how long and complicated are proofs, everything miraculously gets worked out without contradiction.

Mathematics is an ideal world; what strikes is its stability. You may revisit some its corner after being away for 30 years, and discover that everything there is the same as it was when you left it.

I am obsessed with stories of how people learn, and of their motivation for learning.

This is Dr Brian May, and his personal story that appears to be unbelievable: the interesting bit is  2”07 – 3”32 of the BBC film. Aged 7, Brian May got obsessed with stereo photography and very soon started to produce his own stereopictures.

By the time he joined Queen, he was doing PhD in Astrophysics (he formally defended his PhD only years later).

Well, the story is quite believable to me. Once upon a time I knew a boy who, at age 14, was repairing TV sets (primordial by modern standards, black and white, vacuum tube) for all his neighbours in a small provincial town. This job required an oscilloscope; he made one from his family’s TV set by adding an additional circuit and a switch between the two modes of operation: as a normal TV set and as an oscilloscope. In later life, he became a guru and wizard of the black art of fine-tuning of accelerators of elementary particles and was in charge of one of the biggest one in the world.

And, of course, there was Richard Feynman who, as a boy, famously “Fixed radios by thinking“.

Back to Brian May: his PhD thesis is published, and the preface contains this passage:

“I inherited a Fabry-Perot spectrometer and pulse-counting equipment from Prof. Ring, and spent 18 months entirely rebuilding and updating both the optics and electronics, in preparation for obtaining essentially first viable set of radial velocity measuremnents, all around the elcliptic, of the Zodiac Light. The writing of my thesis was virtually complete in 2006, but the submission was deferred due to various pressures.”

It is easy to believe that May, as the lead guitarist of Queen, did not have the same issues with scales of measurement as Nigel Tufnel of Spinal Tap famously had:

This goes to 11…  [watch from 1”16].

My answer to a question in Quora:

Who was a notable person that was originally evil, but eventually regretted their evil and became good later on?

One of more obvious answers is St Paul the Apostle (or Saul, how he was known prior to his inversion on the road to Damascus).  Acts 9:1–6 KJV say:

[1] And Saul, yet breathing out threatenings and slaughter against the disciples of the Lord, went unto the high priest,
[2] And desired of him letters to Damascus to the synagogues, that if he found any of this way, whether they were men or women, he might bring them bound unto Jerusalem.
[3] And as he journeyed, he came near Damascus: and suddenly there shined round about him a light from heaven:
[4] And he fell to the earth, and heard a voice saying unto him, Saul, Saul, why persecutest thou me?
[5] And he said, Who art thou, Lord? And the Lord said, I am Jesus whom thou persecutest: it is hard for thee to kick against the pricks.
[6] And he trembling and astonished said, Lord, what wilt thou have me to do? And the Lord said unto him, Arise, and go into the city, and it shall be told thee what thou must do.

There are conflicting interpretations of this episode, but, in any case, Paul was a changed person since then. An evil man would not write in 1 Corinthians 13:4-7 KJV :

[4] Charity suffereth long, and is kind; charity envieth not; charity vaunteth not itself, is not puffed up,
[5] Doth not behave itself unseemly, seeketh not her own, is not easily provoked, thinketh no evil;
[6] Rejoiceth not in iniquity, but rejoiceth in the truth;
[7] Beareth all things, believeth all things, hopeth all things, endureth all things.

This is the abstract of a talk given by me at the Meeting “Mathematical Academic Malpractice in the Modern Age“, Manchester, Monday 21st May 2018, with the title

Confident students do not cheat: how to build mathematical confidence in our students

I think it could be useful to address the question which, in my experience, is almost never asked: what pushes problem students to cheat by plagiarising work from their peers and, increasingly, from the Internet? Some answer can be found in Denizhan (2014):

“These students exhibit an inability to evaluate their own performances independent of external measurements.”

Plagiarism is one of the psychological defences of a student who does not otherwise know whether his/her solution / answer is correct. Mathematics provides a simple remedy: systematically teach students how they can check their solutions. This will boost their confidence in their answers – and in themselves. I teach linear algebra; I have at least two dozen undergraduate linear algebra textbooks in my office — none of them provides systematic advice on these matters. The same applies, I think, to any other undergraduate subject. In my view, the most efficient methods for checking answers in a particular class of problems usually provided by a more advanced point of view. For example,

• all these elementary problems about systems of linear equations can be effectively checked if the concepts of the rank of a matrix is used;
• the correctness of eigenvalues of a matrix can be checked by using the fact that the sum of eigenvalues is the trace of the matrix, and the product is its determinant, etc.

This retrospective reassessment of previous material can give students a chance to see how actually simple it is — and boost their mathematical confidence. In my talk, I’ll discuss how to incorporate error-correcting aspects of mathematics into course design.