Starting From An Early Age

There has been much chatter recently about early life experiencs of mathematics and the effect it has on subsequent learning and understanding.

Hence this research is rather well timed.

It would be interesting to survey how wide spread an early introduction to the subject correlates with later academic achievment in the subject.

(Personally I certainly remeber at comparatively young age forcing my dad to break a sweat struggling to remeber school-boy trigonometry)

Comments, Commendations, Castigations…

Mathematical Methods In Philosophy

A meeting just anounced by the LMS…

A meeting on Mathematical Methods in Philosophy will take place from 19 to 21 September 2008 at the School of Mathematics, University of Bristol. This is the fourth in a series of meetings exploring mathematical methods in epistemology, semantics, theories of truth, and philosophy of mathematics in a British Academy funded research project. This meeting is further supported by the London Mathematical Society and the British Logic Colloquium. Confirmed speakers are:

* Riccardo Bruni (Firenze)
* Martin Fischer (Leuven)
* Harvey Friedman (Ohio State)
* Dan Isaacson (Oxford)
* Peter Koellner (Harvard)
* Ofra Magidor (Oxford)
* Jeff Paris (Manchester)
* Richard Pettigrew (Bristol)
* Gabriel Uzquiano (Oxford)
* Jouko Väänänen (ILLC Amsterdam)
* Andreas Weiermann (Ghent)
* Alan Weir (Glasgow)

There is a registration fee of £20 with a reduced fee of £10 for students and postgraduates. There are some grants for postgraduates to cover the registration fees, travel and accommodation costs from the LMS funding. Apply early by email to Philip Welch (p.welch@bristol.ac.uk) to avoid disappointment.

Further timetabling and titles, etc., will be placed on the meeting webpage at http://users.ox.ac.uk/~sfop0114/rg/meetings/bristol08.html. Contact Helen Craven (tel: +44 117 928 7978, email: helen.craven@bris.ac.uk) with technical and administrative questions or for help concerning the conference. Visitor information including maps can be found at www.maths.bris.ac.uk/events/info/.

Money, money, money…

Like almost any area of human endevour there is of course the critical matter of money – work needs people to do it and people inevitably face the old “food on table/roof over head” problem. Persuits such as science and mathematics rarely see immediate profit though they frequently eventually have uses and there are many examples of this. This timescale makes it almost imposible for private enterprise to play much role in funding these persuits (though, like any rule, there is the odd exception). This naturally forces taxpayers’ money to be used instead.

I don’t know how funding in other countries works, but currently in the UK the principal funding body for mathematics is the Engineering and Physical Science Research Council (becuse pure mathematics is a branch of engineering, right?)

I say this since, over on Mathematics Under the Microscope, Sasha Borovic highlights the fact that a research programme he has been involved in for 25 years and has been extremely fruitious producing many results and yet has never recieved a single panny from EPSRC. He laments:

LMS (the London Mathematical Society) has a different philosophy: it funds people, not projects. The LMS grants are tiny in comparison with EPSRC, but if the outcome is measured in theorems per pound, the LMS grants are likely to be an order of magnitude more cost-effective than ones from EPSRC.

Why is this? Should government bodies change their policies accordingly? Is the situation similar in other countries and disciplines? Could this be taken further (e.g. funding individual papers)?

Comments, Commendations, Castigations…

your science needs YOU!

With all the excitement surrounding the imminant activation of the LHC it is worth remembering that you too can play your own little part in history’s biggest and bravest experiment (can you think of another instance of an entire subject area resting on the outcome of a single test?)

The idea of volunteer computing, getting ordinary people to donate their computing resources over the internet to help perform large calculations using for large science research projects, is not new. It is, however, worth noting that the LHC also uses this form of computing through the project LHC@home. (To run this you need to first download the frequently used BOINC application.) The LHC projects will collectively produce around15 Petabytes (15 million Gigabytes) of data each year and sorting it all out requires a hell of a lot of computing power. I urge as many of you as possible to download the software and get helping: yes kids – particle physics needs YOU!!! (Trying to do this on computors in your department or university network may create problems concerning administrative privilages, but you can at least spare a bit your laptop’s power, surely?)

With more relevence to mathematics, there have been several projects of this kind before, many of which are still active. The GIMPS have broken the record for finding the largest prime number known to man several times over. Other prime number related efforts include prime grid, the rieselsieve project and VTU. Those less interested in prime numbers can also assist with problems such the rectilinear crossing number problem, play with sudoku or even search for generalized binary number systems.

Go forth and calculate…

Marxism in Mathematics

Pretty much everyone has heard of the political theories of the nineteenth century philosopher Karl Marx. Now I am not for a split second suggesting anything about how right or wrong he was or anything about the impact on history of his ideas. I would simply like to observe that his view of how society is/should and will be run appears to be being born out in current trends in mathematics.

For the uninitiated, a little background first. As with most sciences the preservation and continuation of the subject is done through a system of peer reviewed journals. A mathematician wishing to tell the world about his mathematics will submit their paper to a journal for free. The journal will then find other mathematicians in a similar area to read the paper for free. If they believe that paper should be accepted, possibly after a few changes, they will advise the journal to accept the paper. Often the journal will then charge libraries lots of money to have access to it.

Now this has been the case for a very lone time, and before the advent of modern computing it was entirely appropriate for this behaviour to happen, afterall printing and distribution costs have to be met somehow. In recent times however, larger publishing companies have been charging increasing amount of money for their journals. Naturally high prices represent a barrier to access the information within the journals and restriction on academic freedom are bad for mathematics.

Gradually mathematicians have been getting increasingly disatisfied with this situation. This issue became most apparant following the highly publicised resignation of the entire editorial board of the Journal of Topology. This was later followed by the mass resignation of the editorial board of the journal K-theory.

This strikes me as something of a bourgeoisie revolution. This naturally raises the question, what of the proletariat revolution?

The culture of mathematics is changing. Mathematicians are gradually getting used to the idea of looking at papers online, particularly as the arXiv gains popularity. There is even talk on Tim Gower’s blog of “tricks wiki” for the real lifeblood of mathematics, not just the end product, to be transmitted over the web. More agressivly the Banff protocal calls on mathematicians to completely boycott the profiteers. Most significantly of course there already exist several free online journals with new ones being founded all the time. These clearly represent the future.

The prestigue attached to hosting a free journal is sufficiently great that almost any respectable university will fall over themselves to host such a thing. Given how good it looks on a CV up and coming mathematicians are usually more than happy to help edit and referee for a journal. In the age of the internet it almost too easy to publish such a thing.

So when will the proletariat revolution be? Arguably when a profiteering giant of publishing goes under, a small victory will have been won. In the mean time no free online journal has sufficient presteige to lead anyone to victory. In the next couple fo decades some of these journal will hopefully become better established and by in the next couple of decades it may (hopefully) almost become the norm to submit to such a journal. What’s really needed is for someone to submit a major paper such as the Odd Order Theorem to a free journal. Once that has happened the journal in question will almost certainly gain such a level of respectability that the world will no longer be able to deny the future of mathematics journal publishing. Afterall it worked for the Pacific Journal of Mathematics.

Comment, Commendation, Castigations…

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…

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…

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…

I blog therefore I am

The question of existence is perhaps one of the most discussed issues in the history of philosophy. Indeed one of the earliest and perhaps most famous statments of modern philosophy, cogito ergo sum, being precisely that – a statment of existence. I in no way claim to know any sort of answear to any question this kind. The purpose of this post is to pose a similar question. Is the mathematical concept of existence different to the more conventional physical concept of existence?

I’m sure you dear reader are willing to accept that you and many other things such as the very screen you’re reading this on probably physically exist. It is, however, clearly not the case that a mathematical object will exist in any similar sense at all, afterall has anyone ever physically touched the square root of two or seen with their own eyes a 24 dimensional sphere packing?

Now, one can argue that one merely percieves phsically existing and that the existence of an abstract object is as much in the mind as the existence of the chair you’re sitting on, however mathematical concepts differ from phsical ones in a very importance respect that is of course obviouse to any mathematician: PROOF. You take physical existance for granted and don’t particularly require any justification. Many mathematical objects however will only exist once they are proven to do so and in an age when the axiom of choice is widely accepted even this doesn’t necessarily make constructed examples of such a thing immediately obvious. Nor would we necessarily want an actual example in many cases either (do you really want to write down a basis for the real numbers as a rational vector space, or merely use the fact there is one?)

Comments, commendations, castigations…