04/17/19

When exactly is XY not equal to YX ?

My answer to a question on Quora: When exactly is xy not equal to yx?

The expression \(xy\) can be used in a variety of situation for different kinds of mathematical objects (not only to numbers!) and operations on them, and in many (if not in most) situations \(xy\) is not equal \(yx\).

Let \(x\) and \(y\) be two processes or operations and \(xy\) is the outcome of their consecutive application: first \(x\), then \(y\). A kindergarten level “real life” example:

  • \(x\) is putting a sock on the left foot and \(y\) is putting a sock on the right foot; very obviously, the order of operation does not matter, \(xy = yx\).
  • \(x\) is putting a sock on the left foot and \(y\) is putting a boot on the same foot. You would perhaps agree that \(xy \ne yx\).

In geometry, the result of composition (that is, consecutive application) of rotations and other geometric transformations in the space almost always depends on the order in which they are performed. These rotations and their consecutive execution are described as matrices (certain tables of numbers) and their multiplication (defined by some specific rules) – and, as a consequence, for multiplication of matrices, in most cases, the result depends on the order of multiplicands, \(xy \ne yx\).

In real life, time is the principal source of non-permutability (or non-commutativity, in mathematical parlance) of events. By certain age, you start to understand, that there were things that you had to have done 20 years ago, not today or tomorrow.

Another nasty property of time: you can re-use space (say, empty a cupboard and fill it again), but cannot re-use time.

Both of these principles apply to learning mathematics: certain things have to be mastered at a certain age, and in specific order. Learning mathematics is growing neuron connections in one’s brain; like in growing a garden, processes are not freely permutable, and, in many cases, cannot be reversed and done again.

04/7/19

Will math eventually rule everything?

My answer to a question on Quora: Will math eventually rule everything?

Humanity’s dependence on mathematics implemented in software and hardware in all kinds of electronic devices and information technology systems grows with every day. Mathematics hardwired into a smartphone (or even an old-fashioned mobile phone) is beyond understanding of a vast majority of graduates from mathematics departments of British universities. In the world now, there are more mobile phones than toothbrushes. Importantly, mathematics is increasingly invisible: after all, smartphones can be used by innumerate and illiterate people.

My friends working is information technology increasingly complain that software developers (especially the younger generation) more and more often simply copy chunks of code found on Google without any understanding of mathematical algorithms implemented in them.

The number of people who have sufficient mathematical background for understanding how all that works perhaps grows much slower than the human population. Actually, new technology requires smaller number of mathematically educated workers — but with much higher level skills.

It is not a coincidence that in all western democracies the model of mass mathematics education of the kind that existed in 20th century collapses: this reflects the changing role of mathematics and mathematically educated people in the economy.

And this affects people as well: the community of mathematically educated people is undergoing re-crystallization as highly specialised socio-cultural caste.

State schools cannot give all their students mathematical skills needed for the new economy. Moreover, there is no economic need for giving every child mathematics education at that level — and lower-level skills are economically redundant, see my papers Calling a spade a spade: Mathematics in the new pattern of division of labour and Mathematics for makers and mathematics for users. As a corollary, a child can learn proper, real mathematics only if he/she finds support and understanding in the family. What I mean when I say “support and understanding” should be clear from my answer to another question: As a mathematician, how would you mentor your child and help her to learn, do and live mathematics in her free time as she is growing up?

04/6/19

Do mathematicians feel embarrassed when a conjecture they claim is disproved by counter-example?

My answer to a question on Quora: Do mathematicians feel embarrassed when a conjecture they claim is disproved by counter-example?

I do not remember seeing a mathematicians who was embarrassed by their conjectures disproved.

Why? Because making conjectures and refuting them is a normal cycle of mathematics. I think 90% of conjectures die on the same writing desk where they were born, being killed by the same mathematicians who formulated them. In mathematics, it is a daily routine. Refutations are as important as proofs. There is a famous book about the role of refutations in mathematics, Imre Lakatos’ Proofs and Refutations.

And the famous Lewis Carroll’s lines in Through the Looking-Glass capture the spirit:

“I can’t believe that!” said Alice. 
“Can’t you?” the Queen said in a pitying tone. “Try again: draw a long breath, and shut your eyes.” 
Alice laughed. “There’s no use trying,” she said: “one can’t believe impossible things.” 
“I daresay you haven’t had much practice,” said the Queen. “When I was your age, I always did it for half-an-hour a day. Why, sometimes I’ve believed as many as six impossible things before breakfast.”

Proofs and refutations co-exist in the most natural way. Mathematical problems are conjectures. To solve a problem means to prove this conjecture or refute it.

Proofs are frequently done by constructing, in parallel, a counter-example: when a mathematician identifies obstacles for a proof, he/she may wish to try to use them to construct a counterexample; when this attempt at refutation encounters its own difficulties, a mathematician may try to isolate these difficulties and understand their nature – for use in the proof. In this zig-zag movement the aims — to prove a conjecture and refute it — alternate. In a happy outcome , the process converges on a definite answer: either proof or refutation.

But, if you look back at that zig-zag prowl in search of a kill, you may say that half of the time the mathematician believed impossible. Even worse, it is like lions in hunt: ten chases result in one kill; a mathematician normally solves about one problem out of ten that he or she tries.

There is one extreme case of the proof/refutation balance: the original proof of the Classification of finite simple groups. I quote Wikipedia:

The proof consists of tens of thousands of pages in several hundred journal articles written by about 100 authors, published mostly between 1955 and 2004.

As a rule, almost each of these “several hundred journal articles” contains a proof of a particular theorem, a special case or an intermediate step of the “global” statement. Since all that is about finite objects, proofs frequently use mathematical induction in a specific form: proof of non-existence of a minimal counterexample to the theorem. As a result, it makes thousands of pages of arguments about non-existent objects. At a first glance, it gives an answer to another question on Quora: What are some aspects of mathematics that are nonsense? But these arguments about eventually non-existent minimal counterexamples are not nonsense — for example, they can be re-used in proving theorems in other branches of group theory.

04/6/19

What mnemonics have helped you remember math concepts?

My answer to a question on Quora: What mnemonics have helped you remember math concepts?

I never used any kind of mnemonics in my life, neither as a learner nor as a teacher of mathematics.

If you need mnemonics “to remember mathematical concepts” than this is a sure sign that something is wrong with your learning of mathematics: mathematics is not for memorisation, mathematics is for understanding.

I developed a healthy disregard of mnemonics of any kind at age about 3 and 4, in the kindergarten, where my friend explained to me how to put boots on correct feet: “place boots on the floor so that they look at each other”. I did so, and it worked. Next day, however, I tried to recall: should the boots look at each other or other way round? My difficulties continued until a few days later I forgot to put socks on and tried to put boots on bare feet. It was a classical aha! moment, a revelation, an epiphany: a boot should match the shape of the foot!

In this little episode, “look at each other” is artificial mnemonics; “match the shape of the foot” is understanding.

I do not remember my kindergarten days well. I remember a superstition: do not step on cracks in the pavement. And the smell of burnt milk from the kitchen. And boots.

04/4/19

Why is probability a tough Subject, even for students who are very good at math?

An answer that I was trying to answer on Quora, but was prevented from doing so by a technical fault: Why is probability a tough Subject, even for students who are very good at math?

Because probability as it manifests itself in real life is surprisingly deep and difficult concept. As a knowledgeable colleague once explained to me,

There are at least four distinct interpretations of probability:

  • objective Bayesianism
  • subjective Bayesianism,
  • a propensity theory
  • a frequency theory

along with various pluralists positions.

Unless you work in artificial situations with, say, perfect dice, these differences, which I imagine most school teachers are unaware of, will confuse one’s teaching.

Odds in horse races provide a very good illustration of probabilities. Are they

  • The unique propensity of a horse in that precise situation to win.
  • The limiting frequency in some long series of events.
  • A measure of subjective expectation, reflected in betting behaviour.
  • An objective measure of the expectations of a rational agent given certain information.

The question is: what should we tell to children? At what age they become able to distinguish between “subjective” and “objective”?”

04/4/19

I have 12 animals (rabbits and ducks) loose in my barnyard. How many rabbits and ducks are there if I have counted 34 legs?

My answer to a question on Quora: I have 12 animals (rabbits and ducks) loose in my barnyard. How many rabbits and ducks are there if I have counted 34 legs?

There is a classical solution which uses only arithmetic.

  • Observe that rabbits and ducks have different numbers of legs.
  • Make all animals equal in a way that allows counting: at a pet shop, buy sufficient number of boots, \(1\) pair for each duck and \(2\) pairs for each rabbit, ask them put the boots on, and then ask each rabbit to return to you \(2\) of its boots, so that each creature gets exactly \(2\) boots.
  • Ask a question: How many boots are left? It is easy: 12 animals with two boots each have \(12\times 2 = 24\) boots.
  • How many boots were removed? \(34 – 24 = 10\) boots.
  • From how many rabbits boots were removed? 2 boots from a rabbit means \(10 \div 2 = 5\) rabbits.
  • How many ducks are in the barnyard? \(12 – 5 = 7\).
  • As simple as that. But do not forget to return the boots to rabbits.

A comment for a teacher (if by any chance a teacher reads this reply): in this my solution, I am trying to demonstrate Igor Arnold’s characterisation of arithmetic:

The difference between the “arithmetic” approach to solving problems and the algebraic one is, primarily the need to make a concrete and sensible interpretation of all the values which are used and/or which appear at any stage of the discourse.

I was also using the classical old “questions method” for solving word problems; you many find its discussion in my paper A. V. Borovik, Economy of thought: a neglected principle of mathematics education, in Simplicity: Ideals of Practice in Mathematics and the Arts (R. Kossak and Ph. Ording, eds.). Springer, 2017, pp. 241 – 265. DOI 10.1007/978-3-319-53385-8_18. ISBN 978-3-319-53383-4. A pre-publication version (without editorial changes made by publishers): bit.ly/293orpk

After I published the answer, I found a simpler solution:

Give to each animal \(4\) boots, and then ask ducks to return unnecessary (excessive) boots. The calculation now is

\( 4 \times 12 = 48\) boots all together,

\(48–34 = 14\) excessive boots,

\(14 \div 2 =7\) ducks,

\(12 – 7 = 5\) rabbits.

The moral of this story: generosity pays.

04/4/19

Up to what year of mathematics research, does every mathematician need to know all mathematics discovered until that point?

My answer to a question on Quora: Up to what year of mathematics research, does every mathematician need to know all mathematics discovered until that point?

At a practical level: for a beginner mathematician, it is critically important to read classical papers in the field very carefully, line-by-line. With experience, you start scanning papers for key ideas and critical configurations where mistakes are more likely: “aha, the author uses this particular approach; it is interesting to see where and how that particular case is handled in the proof …” This works because an experienced mathematician maintains in his or her mind a mental image of his/her research field and maps the mathematical contents of the paper to that image.

It is impossible to read all publications even in a relatively narrow area of research. You have to be very selective in your reading. Life is short. Mathematics is done by the subconscious; when reading mathematics, you feed your subconscious. It is like feeding a pet — you are responsible for its well-being, and have to maintain a healthy nutritious diet.

03/30/19

Why are the transpose and inverse of an orthogonal matrix equal?

My answer to a question in Quora: Why are the transpose and inverse of an orthogonal matrix equal?

Orthogonal matrices have several equivalent definitions — this is reflected in previous answers to your question. It could happen that in the book that you are using, or in the lectures that you are taking, the equality of the transpose and inverse, \(A^T = A^{-1}\), is chosen as the first definition of an orthogonal matrix and its equivalence to other statements is proven later.

So my answer: depending on the chosen way of exposition of linear algebra in a book or a lecture, it could be just a definition.

03/16/19

What does the given fact that a matrix A times itself equals -A have to say about its eigenvalues?

My answer to a question on Quora: What does the given fact that a matrix A times itself equals -A have to say about its eigenvalues?

There were two answers in this thread, and both correctly stated that the eigenvalues of \(A\) equal \(0\) or \(-1\). However, one of the answers contains the assertion:

They are all either zero or \(-1\), regardless of \(A\)’s being diagonalizable or invertible.

I wish to add that \(A\) is actually diagonalisable. And here is a proof.

Let \(A\) be a \(n\times n\) matrix. To prove that \(A\) is diagonalisable, it suffices to show that the vector space \(V= \mathbb{R}^n\) can be written as a sum of eigenspaces \(V_0\) and \(V_{-1}\) for \(A\) and its eigenvalues \(0\) and \(-1\), correspondingly, or, which is the same, every vector \(\vec{v} \in V\) can be presented as a sum

\(\vec{v} = \vec{v}_0 + \vec{v}_{-1}\)

with \(\vec{v}_0\in V_0\) and \(\vec{v}_{-1}\in V_{-1}\). But this is easy; indeed, obviously

\(\vec{v} = (\vec{v} + A\vec{v}) – A\vec{v}.\)

Observe that

\(A(\vec{v} + A\vec{v}) = A\vec{v} +A^2\vec{v} = A\vec{v} – A\vec{v} = 0\)

and

\(A(-A\vec{v}) = -A^2\vec{v} = -(-A\vec{v}),\)

so we can take

\(\vec{v}_0 = \vec{v} + A\vec{v} \quad \mbox{ and }\quad \vec{v}_{-1} = -A \vec{v}.\)