The Benefits of Jabberwocky

17 06 2006

I was looking up some quick info on p-adic numbers and algebraic structures (irrevelant but provides a background) when I happened upon this page on the Monstrous moonshine conjecture. Here is a quote on etymology:

Quote:

The Monster group was investigated in the 1970s by mathematicians Fricke, Andrew Ogg and John G. Thompson; they studied the quotient of the hyperbolic plane by subgroups of SL2(R), particularly, the normalizer Γ0(p)+ of Γ0(p) in SL(2,R). They found that the Riemann surface resulting from taking the quotient of the hyperbolic plane by Γ0(p)+ has genus zero iff p is 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59 or 71 (that is, a supersingular prime), and when Ogg heard about the Monster group later on and noticed that these were precisely the prime factors of the size of M, he wrote up a paper offering a bottle of Jack Daniel's whiskey to anyone who could explain this fact.


… I am not going to explain what this means considering with words like supersingular prime, I only barely vaguely understand what it states (algebraic topology, it looks – studying quotient groups of hyperbolic plane – an example of a structure formed by taking a quotient of a more general structure, say a plane is a torus) but there is a simple point in this.

This explanation for the origin of the name seems like reaching, it is akin to a backronymn as an example of something explained after the fact to match the event. A more likely viable explanation is that offered prior:

Quote:

The term "monstrous moonshine" was coined by Conway, who, when told by John McKay in the late 1970s that the coefficient of q (namely 196884) was precisely the dimension of the Griess algebra (and thus exactly one more than the degree of the smallest faithful complex representation of the Monster group), replied that this was "moonshine" (a word that is used in English as a moniker for crazy or foolish ideas). Thus, the term not only refers to the Monster group M; it also refers to the perceived craziness of the intricate relationship between M and the theory of modular functions.


The point of this post is two-fold. To show how psychological notions affect how we name things and then how subsequent generations interpretate things based on schemas in use when introduced. This in turn shapes the evolution and form an entire subject takes. For example, Conway percieved a coincidence and craziness in what logically follows from a set of premises. The idea of craziness, naturulness, obviousness have no place in the mathematical world. These things are significances which are entirely imaginary, reflecting biases and due to a lack of full understanding (due to the newness only not any lack of skill, Conway is a flipping genius). Unfortunately the stigma introduced into a foreign new concept often colour it in a way which make it unclear, this is then carried to subsequent generations. Adding baggages of understanding which need not exist.

Secondly, by carefuly choosing a name much confusion may be elimanated in the introduction of a subject. Just because it looks like a duck does not mean it is a duck. It may be a sheep in duck's clothing. The name monstrous moonshine group is sufficiently ambigious to be a good mathematical name. By naming it as such, people do not enter into the subject with preconceived notions due to name sharing and thus they do not wrongly or inappropriately transfer old concepts into this new schema which in the future cause dissonance and conflicts, resulting in confusion and disunderstanding. An empty cup is easier to fill with clear water than one partway filled with mud.

It is important to name new concepts carefully and in a way which discourages pre-assumptions.

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Limits from Beyond

16 06 2006

I just began to write this paper on limits which covers them from a non common direction with the aim to have them be more thoroughly and intuitively understood. Ive still got a bit to go since I wish it to be self contained. A snippet:

A set may also contain itself as a member. So we can have D = {a,b,c,D}. Now, consider a set S={Any set which does not have itself as a member}. So we know that D is not a member of the set S since it has itself as a member. But is S itself a member of this set? Is it a member of itself? Can we have {…,S}?

If S is a member of itself then obviously something is wrong since the requirement of membership is a set must not have itself as a member to be included. But then if we say it is not a member of itself then there is still a problem. Why? Because then it would meet our requirement of a set that does not include itself as a member. Therefore it would in fact be a member of itself! This is obviously a paradox

All these though are solutions to problems offered to make the system which we are working with more consistent and free from problems. Mathematics in this sense is like a patchwork quilt. We sew, sew and if we sew in such a way as to tear our quilt we must patch it up lest the poor person who uses our quilt freeze to death. There are many implications to this paradox and the assumptions required to resolve it (assumption of axiom of choice or well ordering for hierarchy) or ability to prove no strong formal mathematical theory may ever be complete but these are outside the scope of this paper.





Mathematical Landscapes

14 06 2006

How will this project turn out?

So I been talking about the game being based on logic and involving theorem proving in a limited sense. But how to do this? Well the general idea was to create a simple formal system and then figure out a way to port the system to a graphical logic representation. The methodology I am thinking of following is that laid out in this paper. This would be for spell creation of which there will be two types. General spells and creation of planes. Spells will be first implemented and it will be my first experimentation of this concept. Spells will be based on my simple formal system and there will be a mapping of theorems to spell aspects and will be a way to get my feet wet for the proper hardcore aspect of the project.

The other type, planes, allow one to map theorems and properties to textures, landscape creatures of an area… You may create a mapping for the system yourself or simply use the default one I provide. The logics backend will be implemented in F# and require F# assemblies, so there will be no Domain specific language. Different logics will allow different directions of exploration, for example, with modal logic we explore concepts of existance and possibility.

The premise itself is very simple, if you map the mathematical structures and object to more farmiliar things, drop down a level an abstraction by entering another, you will be able to enter realms of thought once too difficult to penertrate. In my head i have a dream, The plane, based on your mapping could represent a theorem on the reals. By setting off an explosion or jumping and watching the results you can gain insight into what to do…

Ill have to buck up on my graph theory, combinatorics and graphical logic concpets though.

sys.jpg

Instead of defining behaviour and maps explicity we create in a world whose final output is not known to us. It would be truly like exploring a mathematical landscape, a step towards mathematical visualization..