Posted by: Alexandre Borovik | August 30, 2018

Trigger reflex

When I see a statement like that, I just cannot stop myself from pulling the trigger:

Why does Wittgenstein want surveyability? He seems to think that to be capable of the specific use of a theorem which a new proof makes possible we must be able  to reproduce its proof. This is just false, indeed perversely so — without understanding anything about Wiles’ proof of Fermat’s Last Theorem you can use it to rule out the truth of \(a^17 + b^17 = c^17 \) where \(a\), \(b\) and \(c\) are any three integers, even hundreds of digits long — for example I know that \(123456789^17 + 12233445566778899^17\) can’t be equal to \(12345678901234567890^17\) without needing to calculate any of the three powers. [Edwin Coleman, The surveyability of long proofs, Foundations of Science, 14, Issue 1–2pp 27–43.]
Indeed, I believe most mathematicians will make an instant observation that \(17\) is a odd natural number, and therefore the last digit of \(9^17\) is \(9\), and therefore the last digit of  \(123456789^17 + 12233445566778899^17\) is \(8\) and does not equal to the last digit of  \(12345678901234567890^17\) , which is, of course, \(0\). One does not need Fermat’s Last Theorem for that (and, for the sake of historical integrity of the narrative, the case \(n = 17\) had been settled by Kummer in 1847).
Posted by: Alexandre Borovik | August 22, 2018

Do mathematicians need funding?

My answer to a question on Quora: Do mathematicians need funding?

There was a number of answers in this thread regurgitating the old joke about mathematicians and philosophers – apparently the latter require even less funding. What really matters is that mathematicians really need (very modest, admittedly) funding for meeting regularly other mathematicians – either at conferences, or, which is much more important, in direct visits to their universities, or in various programs like “Research in Pairs” run by the famous Mathematisches Forschungsinstitut Oberwolfach. There are serious psychological and neurophysiological reasons why doing serious mathematics in isolation from the community of fellow mathematicians is very difficult, next to impossible. I served, for a number of years, on a committee of a learned society which was awarding quick, easy to apply, small grants for international collaborative research in mathematics (basically, short visits, for about a week) . Quality of papers arising from this research is astonishingly high, on the value for money count it beats any state-funded bureaucratic schemes.

A fascinating paper by Simona Cafazzo, Sarah Marshall-Pescini, Martina Lazzaroni, Zsófia Virányi, Friederike Range in the Royal Society Open Science

Highly cooperative social species are expected to engage in frequent reconciliation following conflicts in order to maintain pack cohesiveness and preserve future cooperation. By contrast, in social species with low reliance on cooperation, reconciliation is expected to be less frequent. Here, we investigate the pattern of reconciliation in four captive wolf packs and four captive dog packs. We provide evidence for reconciliation in captive wolves, which are highly dependent on cooperation between pack members, while domestic dogs, which rely on conspecific cooperation less than wolves, avoided interacting with their partners after conflicts. Occurrence, intensity, latency, duration and initiation of wolf reconciliations appeared to vary as a consequence of a compromise between the costs (e.g. risk of further aggression) and the benefits (e.g. restoring relationship with opponents) of such interactions. Our results are in line with previous findings on various wolf packs living under different social and ecological conditions, suggesting that reconciliation is an important strategy for maintaining functional relationships and pack cohesiveness. However, current results on dogs are in contrast to the only other study showing that reconciliation can occur also in this species. Therefore, the occurrence of reconciliation in dogs may be influenced by social and environmental conditions more than in wolves. Which factors promote and modulate reconciliation in dogs needs to be further investigated.

Posted by: Alexandre Borovik | August 14, 2018

Understanding Mathematics

Question on a LinkedIn discussion group:

Which way do you learn and understand Mathematics in general? -> From the “forest” to the “tree” (-> first the Big Picture and then going into the details) or vice versa (-> seeing the Big Picture after learning and understanding some details) and explain why.

My answer:

In learning mathematics, it definitely starts from a tree to a forest. I think only professional research mathematicians, or exceptional teachers of mathematics, can really think from a “forest” to a “tree”. I recall a recent conversation with my former MSc and PhD student who now works, in a senior position, in software development for a serious, and widely known, Internet company. I asked her: “Does it help you that you have a PhD in mathematics”. Her response: “When they have a project too big to handle, they come to me and I cut it in smaller more manageable parts”. This is thinking from top down. It remains a rare skill, hard to teach, and hard to learn. And the issue is of critical importance not only for mathematics, but for other walks of life, too. Just a few weeks ago I had a conversation about that with a lecturer of architecture (and she was also a very successful practicing architect).

Posted by: Alexandre Borovik | July 10, 2018

Why it matters that so many people became innumerate

BBC Capital published this post, Why it matters if we become innumerate. The article says:

“What we all need in daily life is quite simple maths,” says Mike Ellicock, chief executive of the charity National Numeracy. “But we also need a conceptual understanding applied to complex situations.” In essence, this understanding applies to a broad range of mathematical information that may be intricate, abstract, or embedded in unfamiliar contexts.

For instance, you might need to calculate the true cost of buying versus hiring a car; whether to use award points or money to buy an airline ticket; or how to adjust a recipe to feed six people instead of four.

In the first two examples, you deal with businesses which are not much interested in you being numerate. I have already said on a number of occasions that

If banks and insurance companies were interested in having numerate customers – as they occasionally claim – we would witness the golden age of school mathematics: fully funded, enjoying cross-party political support, promoted and popularised by the best advertising companies in all forms of mass and social media. But they are not; banks and insurance companies need numerate workforce – but even more so they need innumerate customers. 25 years ago in the West, the benchmark of arithmetic competence at a consumer level was the ability to balance a chequebook. Nowadays, bank customers can instantly get full information about the state of their accounts from an app on a mobile phone together with a timely and tailored to individual circumstances advice on the range of recommended financial products. This kind service can be described in a logically equivalent form: a bank can instantly exploit the customer’s vulnerability.

Growing innumeracy is a quite alarming socio-economic phenomenon, and it has very deep roots. Read mor on that in my papers:

Mathematics for makers and mathematics for users,

Calling a spade a spade: Mathematics in the new pattern of division of labour,

Posted by: Alexandre Borovik | July 8, 2018

Why does algebra have letters in sums?

My answer to a question on Quora:

Why does algebra have letters in sums?

One should not underestimate the influence of François Viète who was the first to use algebraic notation (letters) not only for unknowns but also for parameters (knowns) in a problem. He also used, as Wiki states, “simplification of equations by the substitution of new quantities having a certain connection with the primitive unknown quantities”. Importantly, Viète is the first cryptographer and cryptanalist know to us by name. His decryption of intercepted diplomatic correspondence had direct effect on European politics of his time. A really juicy bit from the Wiki:

In 1590, Viète discovered the key to a Spanish cipher, consisting of more than 500 characters, and this meant that all dispatches in that language which fell into the hands of the French could be easily read.

Henry IV published a letter from Commander Moreo to the king of Spain. The contents of this letter, read by Viète, revealed that the head of the League in France, the Duke of Mayenne, planned to become king in place of Henry IV. This publication led to the settlement of the Wars of Religion. The king of Spain accused Viète of having used magical powers.

At that time, encryption of texts mostly used substitution ciphers, and the idea of substitution of letters for numbers should be very natural for Vieta.

I modestly suggest that teachers could perhaps use this idea: teaching primary school children some basic substitution ciphers: it is fun, it is a natural spelling exercise, and, I believe, a good propaedeutic for later study of algebra and computer coding.

Alyssa J. Kersey, Emily J. Braham, Kelsey D. Csumitta, Melissa E. Libertus & Jessica F. Cantlon. No intrinsic gender differences in children’s earliest numerical abilities. npj Science of Learning  3, Article number: 12 (2018).


Recent public discussions have suggested that the underepresentation of women in science and mathematics careers can be traced back to intrinsic differences in aptitude. However, true gender differences are difficult to assess because sociocultural influences enter at an early point in childhood. If these claims of intrinsic differences are true, then gender differences in quantitative and mathematical abilities should emerge early in human development. We examined cross-sectional gender differences in mathematical cognition from over 500 children aged 6 months to 8 years by compiling data from five published studies with unpublished data from longitudinal records. We targeted three key milestones of numerical development: numerosity perception, culturally trained counting, and formal and informal elementary mathematics concepts. In addition to testing for statistical differences between boys’ and girls’ mean performance and variability, we also tested for statistical equivalence between boys’ and girls’ performance. Across all stages of numerical development, analyses consistently revealed that boys and girls do not differ in early quantitative and mathematical ability. These findings indicate that boys and girls are equally equipped to reason about mathematics during early childhood.

Posted by: Alexandre Borovik | July 8, 2018

What are some real-world uses of the determinant of a matrix?

My answer to a question on Quora:

What are some real-world uses of the determinant of a matrix?

At the time of writing, I am engaged in a small debate with a colleague on one of the LinkedIn discussion groups: he teaches students to solve systems of 2 linear equations with 2 variables using Cramer’s rule (that is, via determinants), without giving any justification or proof for it, but I personally prefer self-justified solutions: for systems of 2 linear equations with 2 variables, the honest Gaussian elimination is quick, and it is easy to explain to students why it gives the right solution. Moreover, every intermediate step of Gaussian elimination can be naturally interpreted in terms of the original system of equations.

And this goes to the heart of the matter: in life, determinants are almost never used in computation. Someone said that

mathematics is the art of avoiding calculations;

in that sense,

linear algebra is the art of avoiding calculations with matrices,

and the rule of thumb is

avoid calculations with determinants!

For example, you can invert a matrix in essentially the same time as compute its determinant; after that the use of the cofactor formula for the inverse of a matrix and Cramer’s rule for solving systems of linear equations becomes waste of time.

However, determinants provide extremely efficient tools for thinking about problems of linear algebra, including those in practical applications. Linear algebra in its development or exposition goes through more and more compressed expression of relevant mathematical meaning, and the value of the determinant: zero or not zero is perhaps the most compressed form of expression of linear dependence / independence of \(n\) vectors in \(\mathbbR^n\).

Determinants have wonderful algebraic properties and occupied their proud place in linear algebra because of their role in higher level algebraic thinking.

In this thread on Quora some uses of determinants were mentioned, for example, computation of eigenvalues of a matrix; I am not an expert in numerical linear algebra, but I have a feeling that most methods for computation of eigenvalues do not even mention the word “determinant”. Even at a theoretical level, determinants can be excluded from the standard treatment of linear algebra, see Sheldon Axler’s paper Down with Determinants!

So, let me summarise:

  • If you need more that just application of existing computer programs for solving practical problems of linear algebra and have to think about the process of solution, you may find determinants very useful indeed.
  • Determinants can be meaningfully used for compact formulation of mathematical models of physical phenomena (perhaps this applies not only to physics). This thread in Quora contains some nice examples.
  • But is is best avoid calculation of determinants.
Posted by: Alexandre Borovik | July 6, 2018

My answer to a question on Quora:
Why is mathematical illiteracy socially acceptable?

Because what is known as “mathematical literacy” is economically redundant: having it, or not having it does not affect earnings of 95% of people. An ever decreasing pool of jobs which require “mathematical literacy” is filled (at least in Britain) by recruiting university graduates from mathematics-intensive disciplines, like mathematics, or physics, or electronic engineering. These jobs require no more than basic school level mathematics skills which are supposed to be given to every school leaver. However, big employers do not trust school marks, and rightly so.

However, a small number of professional occupations require knowledge of mathematics far beyond “mathematical literacy”. Critically, form nay nations, this is the defense and security sector.

The summary: everyone is taught mathematics at school not for his/her personal advancement and enjoyment, but for alleged future employment – which is a fiction for majority of learners. This is misselling of “educational product” on a grandiose scale. Wide acceptance of “mathematical illiteracy” is a natural, and healthy, reaction of the society to this scam. I support the idea that mathematics has to be taught as music: for learner’s enjoyment and personal development, without any promise of future employment in “music-intensive industries”. This will make mathematics more popular — and much more expensive to teach, which, of course, kills this idea at its roots.

I apologise fro plugging my papers, but they contain more on that:

I apologise fro plugging my papers, but they contain more on that:

Mathematics for makers and mathematics for users,

Calling a spade a spade: Mathematics in the new pattern of division of labour,

Perhaps I have to add adisclaimer:

Views expressed are my own and do not necessarily represent position of my employer, or any other person, corporation, organisation, or institution.

My answer to a question in Quora, slightly edited:

What is your favorite visual mathematical proof?

Perhaps it is Tennenbaum’s proof of the irrationality of the square root of 2. What follows is reposted from a blogpost by David Richeson at his wonderful blog Division by Zero.

Suppose \(\sqrt2 = a/b\) for for some positive integers a and b. Then \(a^2=2b^2=b^2+b^2\). Geometrically this means that there is an integer-by-integer square (the pink  \(a\times a\) square below) whose area is twice the area of another integer-by-integer square (the blue \(b \times b\) squares).

Assume that our \(a\times\) square is the smallest such integer-by-integer square. Now put the two blue squares inside the pink square as shown below. They overlap in a dark blue square.

By assumption, the sum of the areas of the two blue squares is the area of the large pink square. That means that in the picture above, the dark blue square in the center must have the same area as the two uncovered pink squares. But the dark blue square and the small pink squares have integer sides. This contradicts our assumption that our original pink square was the smallest such square. It must be the case that \(\sqrt2\) is irrational.

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