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…

Your Name In Lights

New Scientist Magazine are running one of the most curious competitions I’ve ever seen. The prize is having a group named after you!

http://www.newscientist.com/article/dn14106?DCMP=ILC-hmts&nsref=specrt13_pic

Good Luck!

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.