My answer to a question in Quora: What is exact definition of mathematics?

I share my small collection of  descriptions of mathematics; some of them sound as attempts to define mathematics.

I. Mathematics is an exact language for description, calculation, deduction, modeling, and prediction — more a systematic way of thinking than a set of rules.

II. Using a legal analogy, mathematics is a language for writing contracts with Nature that Nature accepts as legally binding.

III. The practical importance of mathematics lies in its ability to describe the real world.

The real world consists of what matters. The word “matter” as a noun is used for what the physical world is made of. But if we ask, “What’s the matter with Anne?” we may be asking about a physical ailment, or we may be asking about an idea that is causing Anne to behave strangely. Ideas matter.

The whole point of mathematical education is to make ideas real for students, ideas that were not real for them before. Ideas like fractions, for example. The fact that 2/3 is smaller than 3/4 matters in the real world.

IV. Mathematically educated people are stem cells of a technologically advanced society. Because of the universality of mathematics, mathematicians and well educated users of mathematics are flexible in applying and inventing tools for work in technological environments which never existed before

V. Learning mathematics involves the profound assimilation of intellectual and aesthetic criteria as well as practically orientated ones. The very difficulty in learning mathematics makes it a personality-enhancing experience.

[The above are borrowings from David Corfield, Tony Gardiner, Michael Gromov, Frank Quinn, David Pierce — I do not remember now in which order.]

VI. Mathematics is a part of physics. Physics is an experimental science, a part of natural science. Mathematics is the part of physics where experiments are cheap. — Vladimir Arnold

VII. Mathematics is the music of reason  — James Joseph Sylvester

VIII.  Mathematics is the study of mental objects with reproducible properties. — Philip J Davis and Reuben Hersh

VIII. Finally, my own extension of the thesis by Davis and Hersh:

Mathematics the study of mental constructs with reproducible properties which imitates the causality structures of the physical universe but is expressed in the human language which evolved for social interactions.

My answer to a question on Quora:

Why is math taught differently in school today? What is wrong with the way we learned it twenty years ago?

New technological and economic environment requires a different set of mathematical skills, and taught differently, and — this is the big unmentionable — taught to much smaller number of students. What we see are death throes of the existing system of mathematics education. Politicians are keen to keep the old system of mathematics education for everyone, and for good reason — this is what their electorate expects from them.

This is why there is so much confusion of what, and now, has to be taught at school – in absence of clear economic criteria, anything goes. I heard statements, made by professional mathematics educationalist at meetings at the Department of Education, of the kind: “Why school curriculum should contain fractions? Who of people present here had lately add 3/4 and 7/5? Read more in my papers Mathematics for makers and mathematics for users and Calling a spade a spade: Mathematics in the new pattern of division of labour.

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.

I wish to comment on two specific flaws exhibited by students who encounter proofs first time in their lives.

The first one is

inability to accept the Identity Principle: “$A$ is $A$”, and arguments related to it, as a valid ingredient of proofs.

For many students, a basic observation

For all sets $A$, $A\subseteq A$ ($A$ is a subset of $A$) because every element of $A$ is an element of $A$

is very hard to grasp because of the appearance of the same words about the same set $A$ twice in the sentence: “element of $A$ is an element of $A$”. I have observed that many times and I think that students cannot overcome a mental block created by their

expectation that a proof should yield some new information about objects involved

— and this is the second fundamental flaw.

And, of course, reduction, removal of unnecessary information, is seen by many students as something deeply unnatural.

Every year, I hear from my Year 1 students the same objection:

How can we claim that $2$ is less or equal than $3$, that is, $2⩽3$, if we already know that $2$ is less than $3$, $2<3$?

I think we encounter here a serious methodological (and perhaps philosophical) issue which I have never seen explicitly formulated in the literature on mathematics education:

• a proof of a mathematical statement can illuminate and explain this statement, it may contain new knowledge about mathematics which goes far beyond the statement proved; but
• elementary steps in proofs frequently do not produce any new information, moreover, sometimes they remove unnecessary information from consideration.

A proof can be compared with a living organism built from molecules which can hardly be seen as living entities — and even worse, from atoms which are definitely not living objects.

This is closely related to another issue which many students find difficult to grasp: statements of propositional logic have no meaning, they have only logical values (or truth values, as they are frequently called) TRUE or FALSE. Any two true statements are logically equivalent to each other because they are both true; moreover, the statement

if London is a capital of England then tea is ready

makes perfect sense, and can be true or false, even if constitution of the country has no relation to the physical state of my teapot. [Moreover, the statement is TRUE, because London is NOT a capital of England, it is  a capital of United Kingdom. ]

When my students express their unhappiness about logic which ignores meaning (and I provoke them to express their emotions), I provide an eye-opening analogy: numbers also have no meaning. The statement

The Jupiter has more moons than I have children

compares two numbers, and this arithmetic statement makes perfect sense (and is true) even if Jupiter has no, and cannot have any, connections whatsoever with my family life. Numbers have no meaning; they have only numerical values. Arithmetic, the most ordinary, junior school, sort of arithmetic is already a huge and deep abstraction. We did not notice that because we are conditioned that way.

Learning proofs also involves some degree of cultural conditioning. As a side remark, I suspect (but have no firm evidence) that the role of family — presence of clear rational argumentation in everyday conversations within family — could be important.