Reuben Hersh was a mathematician, unorthodox philosopher of mathematics, and a very kind and friendly man.
Some of you may already know that there is a threat to the abolish the combinatorics group at the University of Strathclyde: http://combinatorics.cis.strath.ac.uk
Peter Cameron is helping to consolidate national and international efforts to prevent this from happening. His comments can be found here: https://cameroncounts.wordpress.com, and the petition in support of the combinatorics group here: https://britishcombinatorial.wordpress.com/2019/06/20/combinatorics-at-strathclyde-2/
Tim Gowers has also written briefly on this: https://gowers.wordpress.com
If you would like to show support for this combinatorics group, Peter Cameron is collecting signatures for the above petition, which only requires emailing Peter at pjc20 >>at<< st-andrews.ac.uk, simply giving your name and University affiliation (if you have one) and the statement “I support this petition”. Peter will do the rest.
Also, feel free to distribute this email to your colleagues/friends who might be interested in supporting the combinatorics group. Your help is much appreciated.
Thanks in advance for your help!
David Bevan, Sergey Kitaev and Einar Steingrimsson [from an e-mail message]
My answer to a question on Quora: Why are so many mathematicians platonists?
Because mathematics is consistent (or, at least, so far has been consistent). Even more so, it is more consistent than the “reality” as we see it around us, or hear other people talking about. At an ordinary, everyday level, our vision of the “real” world is full of contradictions, ambiguities, uncertainties, and — the last but not least — direct and intentional lies. In comparison with that mess, our vision of mathematics is crystal clear. Subjectively, mathematics can be perceived as something more real than the world around us.
Please notice that I do not claim that the Platonic world on mathematical objects exists. I am talking only about perception of mathematics by working mathematicians. Almost everyone who I know is, to some degree, a Platonist.
My answer to a question on Quora: Why are mathematicians so obsessed with proofs? Many theorems are just intuitive.
Your belief that “many theorems are just intuitive” indicates that you are at very early stages of learning mathematics. 99.99% of mathematics is beyond “immediate intuition”. If you do not agree – then please give an example of an “intuitive” theorem about, say, Hilbert spaces, or etale cohomology.
Mathematical intuition is something that has to be trained and controlled — and proofs is the only known tool for achieving that.
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 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.
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?
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.
“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.
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.
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”?”