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…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.)

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/.

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!

Comment, Commendations, 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…

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…

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…

I must know, I will know

A colleague and I recently had a bit of argument concerning the style in which mathematics papers are written. Research papers in mathematics are usually written in the third person plural (“we prove the following”; “we need to show”; “we use the following lemma” etc) whilst papers written in most other scientific disciplins are largely written in the first person singular (“I did the following experiment”; “I performed the following test”; “I found that” etc). My colleague thoroughly dislikes this ’stylistic’ point and avoids this ‘convention’ whenever possible.

I would like to argue that this ‘convention’ is not only NOT a convention, but actually an inherently necessary part of writing mathematics. You see, in almost any other scientific discipline, it is not the case that simply reading about an experiment recreates it. If I do an experiment and YOU read the results it is not the case that YOU are redoing the experiment.

Mathematics is different. Its objects are purely abstract, existing only in the mind - its ‘experiments’ are proofs run through only in the mind. If you discover a proof of a result, I try to understand your proof when I read it. I may do this by filling in little arguments to get myself from one line to the next which may not be the same as how you saw it. I may try to apply your argumnt to my favorite example that may be diferent to yours. I may simply have a very different perseption of the objects we are dealing with. Ultimately, however, the basic form of the argument was constructed by you.

In short, if I read YOUR proof, the form it takes in MY mind is somthing that WE constructed together and it is for that reason that it is genuinly the case that WE prove results.