The Status Of Combinatorics

First my apologies for havnig been so quiet as of late – I’ve been busy.

The status of the subject area known as combinatorics within wider mathematics is a bit of a strange one. It seems to exists in a sort of ‘glorious isolation’ very much seperated from other subjects.

Whilst most mathematicians would be happy to attend the British Mathematics Colloquium combinatorialists seem to insist on holding an alternative event of their own, the British Combinatorial Conference, instead. In the UK most mathematics events are funded by the London Mathematical Society whilst combinatorics event tend to be funded by the British Combinatorial Commitee.

So what is the cause of this oasis in mathematics?

Well from the outside there appears to be a certain level of snootyness – my supervisor has certainly aired the view that in any other area of mathematics one frequently encounters problems in combinatorics, but when this happens you simply solve the problem and move on. There is a view that many people doing combinatorics are simply doing the mathematics that remains when the real meat of an impotant problem is removed.

From within there appears to be a very insular attitude. The combinatorialists, certainly within my university, appear to actively discourage their students from taking any sort of interest in anything (lecture courses seminars etc) outside combinatorics.

This cannot be healthy. It is well known that many of the most fruitful pieces of mathematics come from taking the ideas and techniques from one area and applying them to another. Indeed had combinatorialists acted as they do now fifty years ago then the probablistic method, a powerful means of proving results in combinatorics, may never have been discovered.

This is particularly worrying given the applicability of combinatorics to real world situations such as problems of experiment design encountered in statistics or combinatorial optimisation as well its many uses within pure mathematics.

What should be done about this and how?

Comment, Commendation, Castigations…

The recent DECLINE in knife crime

Whilst the older generation like to hark back to more traditional 3Rs teaching the simple fact is that it is simply not appropriate for the modern world (oldstyle teaching itself not necessarily better than current practices). (The same recent “Now we are 50” report has already been discussed on Mathematics under the Microscope, so I won’t go into it here.) The teaching of mathematical ideas such as the rudiments of statistics being a case in point. The current media hyperbole about knife crime – worsened by the stabbing of a relative of a celebrity – is a prime example. A recent piece on the BBC news website, Safety in numbers, reveals official statistics showing that violent crime is actually decreasing! Clearly a good example of why mathematics teaching in the UK needs to be improved!

Australia Takes the Ashes

It has just been reported on Mathematics in Australia that in true Aussie style, they have just won the inaugural Mathematics Ashes.

Unlike the cricketing tradition, however, its interesting to note that history shows the United Kingdom would generally have won most years had the ashes been contested over the past couple of decades.

Language, Mathematics and Probability Theory

Over on FoxMath there has recently been posted an interesting discussion about an Amazonian tribe whose language seems lack any concept of number or counting like our own.

This raises many interesting questions. The one I would particularly like to consider here is the following: what is their perception of likelyhood?

Ancient Greek Mathematics, as advanced as it was for its time, had essentially no conception of probability theory whatsoever. On the other hand the ancient Hebrews, when considering questions of whether a given piece of meat is kosher or not had a highly developed sense of likelyhood, wilst having little/no other mathematics to speak of. A conseption of probability theory seams somewhat disjoint form other mathematical considerations, such as numbers.

Since the modern treatment of uncertainty and likelyhood hinges so critically on the concept of probability, ie a quantative measure of likelyhood, how do these Amazonians deal with chance? What are the advantages/disadvantages of their approach relative to ours? Does anyone know of any “Fuzzy Logic” forms of probability theory using instead of a number in [0,1] mearly the vague notions of “unlikely” and “likely”?

Comments, Commendations, Castigations…

Bertrand Russell and The Big Bang

Bertrand Russel was an early twentieth centiury philosopher and logician responsible for much modern mathematics: it was his famous paradox that stimulated the development of axiomatic set theory (the ZF axioms underlying modern mathematics) and his his book “Principia Mathematica” coauthored with Alfred North Whitehead is widely considered by specialists in the subject to be one of the most important and seminal works in mathematical logic and philosophy since Aristotle’s “Prior Analytics”.

It is for this reason that I find it so strange that Russell’s philosophy seems so at odds with one of the most widely accepted ideas of modern physics: The Big Bang.
In his 1927 work “Why I Am Not a Christian” (published 1957) Russel addresses many aspects of christian dogma and thought. In particular when considering the existence of God he considers several of the arguments found in Christian teachings. The argument that concerns us here is the argument of the First Cause thus:

“Perhaps the simplest and easiest to understand is the argument of the First Cause. (It is maintained that everything…has a cause, and as you go back in the chain of causes…you must come to a First Cause, and to that First Cause you give the name of God.) That argument, I suppose, does not carry very much weight nowadays, because, in the first place, cause is not quite what it used to be. The philosophers and the men of science have got going on cause, and it has not anything like the vitality it used to have; but, apart from that, you can see that the argument that there must be a First Cause is one that cannot have any validity. I may say that when I was a young man and was debating these questions very seriously in my mind, I for a long time accepted the argument of the First Cause, until one day…I read John Stuart Mill’s Autobiography, and I there found this sentence: “My father taught me that the question ‘Who made me?’ cannot be answered, since it immediately suggests the further question `Who made god?’” That very simple sentence showed me…the fallacy in the argument of the First Cause. If everything must have a cause, then God must have a cause. If there can be anything without a cause, it may just as well be the world as God, so that there cannot be any validity in that argument.”

Now, modern physics, and in particular Quantum Mechanics, has certainly changed our view of causality since John Stewart Mill’s time. In particular the idea of a First Cause almost loses meaning. The first Moment Of Creation simply couldn’t happen since the very fabric of spacetime itself was still forming. In short we now have a physical evidence for a `First Cause’ of sorts, but not quite the kind of First Cause Russell would object to, I think. Do we think Russell would revise his views of God Creation or Science on the basis what we now know?

Comment, Commendations, Castigations…

Basic Linear Algebra Problem

Having helped teach linear algebra classes to undergraduates several times I am often faced with a basic lack of understanding of some basic concepts. I have often found that a real “penny dropping” moment comes when understanding the following elementary exercise:

Let be a vector space and let be a linear map. Let denote the eigenspace for the eigenvalue and denote the kernel of . Show that .

If a student knows that this basic fact is true, then many other things become much easier to eaplain. I would seriously recommend more lecturers setting this as an exercises often.

Comments, Commendations, Castigations…