The Four Cultures of Mathematics

A well-known point of agreement amoung mathematicians is an often sigted discussion by Tim Gowers, namely “Two cultures of mathematics“.

It has, however, been argued that there are other, perhaps more accurate, divisions of mathematicians. I sight Peter Cameron’s view that “those who delve
deep and those who range more widely.” This is, of course, not entirely incompatable, but it does rais several questions. If these two cultures both eist ae they entirely independent, ie is it more sensible to classify mathematicians as points on aplane, rather like the well known political compass? Or it simply the case that one is right and the other not, or worse that neither is right? If neither are right, then what is right? Is this case division a help or a hinderece? (I suspect the latter.)

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Upload an Avatar

…goes the imortal cry, but do wordpress actually know what they’re talking about? Acording to WordPress’s help pages:

An avatar is a small image that appears when you post to the forums and when you comment on any posts anywhere. It’s your profile picture. The avatar should be 128 * 128 pixels square. It will be smaller when used in comments so do check how good it looks when it shrinks.

The actualy concept of an avatar, however, goes back much further. In Hindu philosophy an avatar is the ‘descent’ or incarnation of a divine being (deva) or the supreme being (God) onto planet Earth. The Sanskrit word avatāra literally means “descent” (avatarati) and usually implies a deliberate descent into lower realms of existence for special purposes. There have been many people over the years claiming to be avatars, usually quite harmlessly.

The concept of an avatar was most recently re-invoked by the author Scott Adams in his wonderful books “God’s Debris” (which is freely available online, if you click on the link) and its sequeal “The Religion War” dicussing the philosophy of the mysterious, powerful and subtle control probabilities exert upon the world and our daily lives (taking a view largely justified through a sophisticated ammendment to Occam’s Razor). (And yes that’s Scott Adams as in the infamous Dilbert Comics.)

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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)

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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/.

Beyond Reason…

It is not often that you see the views of people such as the the famous Oxford mathematician and physicist Roger Penrose presented alongside those of the famous MIT linguist and philosopher Noam Chomksy and the Archbishop of Canterbury Rowan Williams. Recently the popular science magazine New Scientist recently published an opinion piece doing precisely that. The article discusses the growing influence of logic and rational thought on our dailey lives and why people are becoming increasingly unhappy with it.

Thankfully for us, all of the written articles along with a large amount of additional content can still be found online!

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Deep Mathematics I: “Typical” Theorems

This is the first of what I hope will be a series of (not chronolgically consecutive) posts discussing what makes “deep” mathematics deep. This is, of course, not a new issue and has been dicussed many times before. I suppose this is more an issue of semantics than philosophy, but this is certainly one place the fact that mathematics is done by human beings rather just being being the “cold and austere” * discipline that it is so often viewed as being becomes apparant.

In this post I would like to argue that one sufficient condition for a piece of mathematics to be described as ‘deep’ is that it indicates the “typical behaviour” of somthing. Now any mathematical theorem is, of course, a precise statment, usually of the form “Every X is a Y” or “No X is a Y”, but the sort of theorems that I am talking about are the ones that can often be summed up as “Xs tend to be Ys”, “your average X on the street is a Y”, “Every X looks a lot like a Y” etc. The ability to paraphrase a theorem in this way indicates that it somhow gives us a better understanding of the ‘true nature’ of a particular kind of object. A few examples will hopefully make what I mean a bit clearer.

• Coxeter Groups: The concept of a reflection (a function from to fixing a hyperplane and mapping everything perpendicular to it to minus itself). Since isometries can always be expressed as products of reflections its natural to ask what groups are generated by them. In the 1930′s Coxeter showed that finite groups of this kind are built up from certain “irreducible” reflection groups (a bit like prime numbers) and successfully classified these. What does the classification state? That reflection groups are trpically built from symmetric groups. In what sense does it state this? Of the three infinite families, one of them is the set of all symmetric groups and the other two consist of groups made by sitting asymmetric group ontop of a set of much smaller ones.
• Kuratowski’s theorem: thinking of a graph as simply a bunch of points joined up by a bunch of lines it is natural to try and draw these minimizing the number of edges that cross to make the structure of the graph clear. If we can do this we say the graph is planer. Kuratowski’s theorem chracterizes the graphs that are non-planaer: there are essentially only two non-planer graphs, usually written and . Every other non-planer graph is consists of copies of graphs made from these glued together. (I’m aware that this generalizes to the deep and jaw-dropping Robertson-Seymore theorem, but that’s a little harder to describe and not quite the sort of theorem I’m talking about.)
• Hurwitz’s theorem: a Riemann surface is beautiful kind of surface that looks a bit like contorted version of the complex plain. Hurwitz’s theorem is a precise statment about the number of automorphism (“symmetries”) of a Riemann surface that preserve angles. What does the theorem state? That Riemann surfaces tend to have few automorphisms.

This is by no means an exhaustive or even particularly representative list. Further examples would be appreciated.

Do people agree/disagree with this view? If you agree can you provide further examples of “typical” theorems? I you disagree can you give counter examples?

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(*=”Mathematics, rightly viewed, possesses not only truth, but supreme beauty – a beauty cold and austere” Bertrand Russell)

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)?

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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.

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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?

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