Numbers: Part 1

9 07 2006

In this post I go over why numbers do not exist. Since in a way, we may say that they are pulled out of thin air. People in the olden days liked to lock themselves into certain dogmatic ways of thinking, something which has cursed the future generations into having more trouble with math concepts than need have been. You can tell the close mindedness of the people of old by their insistance to have such names as postive, negative, natural, rational, irrational, real and imaginary for number types.

The task of this post is to show that the imaginary number i is no more or no less real than the number 4. It is often that you will see people say such things as how can you have the square root of a negative number? Some even go as far as to ask how can you have negative money? This is due to a misunderstanding of what a number is.

On the first, how can you have issue with the root of a negative number? This is simply due to centuries old dogma passed traditionally from teacher to student in grade school with the idea that the positive integers are real or exist somewhere and that only they can be numbers. We are taught properties of integers and are told that only those that follow these rules are numbers. How silly. The second on negative money may be considered to be that the positive integers associations are more concrete. 5 -> cats. 1000 -> money. Numbers are no more than an abstract invention or tool we may attach to appropriate situatations which help us reason on certain things we encounter in nature.

We say 5 boxes and attach the concept of 5 to boxes which enable us to envision the multiple boxes. At 45443454 we can no longer even envision the boxes and simply have another concept, a vague notion of many boxes: 454454 -> vague concept -> many concrete objects: boxes. Attatchments are possible with negative integers as well. Consider negative boxes, we denote a lack. -4 -> concept of needing 4 boxes. We can never make our way down to the concrete object box. But the relationship with concrete objects are still obvious. As we go higher and higher we are less able to associate with concrete objects and the relationship is less visible: say with some number q which is a quaternion for example. This is not so easily associated and we have less concrete experience with such a concept and so feel it to be less natural, where infact the positive integers are just as “fake” or “real” as the the quaternions. We are used to handling integers, the path from the positive integer concept to a concrete object is quite simple, creating an illusionary concept of concreteness. The more complex our numbers the less we take them for granted and the more distrustful we are (their path to experience is too complicated), conversly for simple numbers like the positive integers we have little difficulty, in fact so little it may be considered that we cannot dissasociate positive integers from objects. This causes problems on what a number is and creates misconceptions on what a number may be and in the future resulting in us being mistrustful to treat perfectly valid numbers as numbers.

General sketch on how to construct the natural numbers

One way to construct the natural numbers due to Von Neuman is to use set theoretic formulations to define them. I go into it very informally. Consider the empty set ∅ (a set is an aggregration/conglomeration of a collection of objects. A pack is to set as a set’s members is to wolves, a pack is a conglemeration of wolves). The empty set has no members. It is itself a an existing thing but has nothing whatsoever as its contents. Basically a collection of pure nothings. So we take the empty set and from there make a set with that as its member. We then make another set that has the empty set and the set(s) we have made as its members. We do this ad infinitum. This construction has that each stage in our constructing, the current set in question has all the previous sets as its members.

So we have ∅ -> {∅} -> {∅ , {∅}} -> {∅ ,{∅}, {∅ , {∅}}} -> …

The Set (as an implentation of the peano axioms which tell us what it takes to be considered the natural numbers) which contains all these sets is equivalent to isomorphism to (basically the same as) the natural numbers. Yes that is:
∅ = 0
{∅} = 1
{∅ , {∅}} = 2
{∅ ,{∅}, {∅ , {∅}}} = 3

This has the interesting property that the number 52 is inside 100, the number 52 is contained in 100. In fact the number 100 contains all the numbers previous to it inside it (as members). The Natural numbers, in essence, under this method is basically made out of nothing. We just have collections of empty aggregrations. That is our natural numbers that are so real and obvious.

As a final excercise consider the Real Number Line. Try and point out the location of π on it. We may get arbitrarily close to it, zoom in closer and closer but we will never arrive at it or equivallently, we do so at infinity. How is that “real” number any more real than the square root of -1?





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.





The Tangled Web of Existence

20 05 2006

I read somewhere that blogs allow people to think aloud and to think loudly. Begin.

I am a philosopher, one studying math but only because there is no better tool, no better way to exercise the mind than to partake in mathematics. Of course there are many who would state that musing on philosophy is a waste of time. But truly, since on the most objective scale, nothing we do on this world has meaning [without us], has no impact on the rest of the universe that is not us (which is basically to say we have no impact whatsoever), and all that we do may as well not have been since we die and it as if we never were within decades, what I ask, is a waste of time? And for those few who are remembered what good does it do the dead to remember them? Perhaps in forward thought it is good to think that you will be remembered but again, what good is that to you dead? There is an answer to this and I will give it later, in another entry.

The point of philosophy is to ask why? And to try to answer this "why" as objectively as possible by using purely subjective analogies. This is the best that may be done. This is the human, the finite minded sentient condition. All other endeavors – even physics – they do not ask why. Physics says this is how it is but it does not tell why, is this predilection for ignoring why, going about as if it is not a question worthy of pursuit,is it some sort of defense mechanism for those who would rather not ponder the pointlessness of all they do? I would rebut the last with this that is the core of this series of essays: "To each his own". Such lines of thought may ultimately lead nowhere, but then what does? Leads somewhere, that is. Again, what is time not wasted? I will note though, I feel not but scorn for those who would feel superior to others because their job requires more "skill". Pfeh. If it is not evident, I have strong tendencies towards relativism but I am *not* a relativist.

(a) Why does the universe exist? More importantly without someone self aware to perceive it would the universe exist? I am not thinking about quantum mechanics because that one does say yes to my second question. No I mean, without someone self aware to perceive it would the universe exist? Of course you could say, yes, the universe does not need to be perceived to exist – stuff still goes on and has gone on without us. But then, the notion of existence is not possible without perception, it is an invention of the ability to consciously perceive. That we are here now is required for us to be able to say but of course it existed in the past. Where "we" is the set of all beings in the universe who may perceive. "Meaning" is a sentient invention, without us there is no meaning. Certainly there is some inherent structure with a pattern that we may observe and say, yes "this is", but again, it is required that "we" exist to make this observation. If there was no one to ever observe the universe, what point would there be? Can you see what I am driving at? There may be aliens, even if we do not see them that we acknowledge that they may exist is sufficient that they exist. So, it is true that aliens exist (see modal logic, the philosophical part of my RPG is tied to this). But those who make this observation must themselves exist to make this observation. If no one did, if no one could, would anything exist? This question makes me cry. But I feel that the answer *should* be Yes. Things had to exist before us before we got here. But I feel that they had no meaning, meaning is subjective. We bring our own meaning to the universe. Without us (the set of all self aware) the universe might as well not exist. Because it would be meaningless, essentially, pointless. Do you disagree? If yes, please state, it would mean so much to me.

(b) It is often remarked that humans and similar exist because the universe would not go about the business of creating itself if there was no one to admire it. Is that the purpose to our existence? Is that why we exist? I leave that as un-answerable, and un-addressable. It may be or it may not be. This question may as well not have been asked, it may as well not exist. There is not sufficient data to even speculate on this possibility.

(c) It is more acceptable to think that we were created by random non-directed purposes and that our existence is explainable as being a result as such. But if you agree with me then you accept (a) and thus feel that we bring our meaning to the universe. With (a) in mind one cannot help but consider (b) which immediately leads to (c) which must be considered with (a). Nonetheless (a) does not imply (b). We could very well have been a result of random processes and it is only a coincidence that we validated the universe's existence. What a happy thing.

Whatever you believe, its Objective truthiness will never be known. But hopefully, your conclusion is a happy one. Me, I see that we have no impact on the universe, we may as well not exist but then again, we add our own take of meaning to it. And if we are the only life in the universe, we give the universe its meaning. This is for me, a good ending.





The Tool Trap

12 05 2006

Curses, the mass generator doesnt work too well, its too stupid. Or rather, the distribution is highly gay. Generated objects tend to cluster. It seems the Random class that comes with .NET is a piece of crap. I have a couple of choices, I can implement the Mersenne Twister algorithim, whose psuedo random generation is unparralleled in all areas (save predicatabilty) and is as equidistributed as they get. Or create an editor. The latter is at the moment not an option. Although I plan to create an editor I do not wish to do so till after I have completed one episode because I have no wish to get caught in the Tool Trap.

The Tool Trap

The tool trap is the manifestation of feature creep, greed and having eyes larger than your stomach. It is known that children often overstuff their plates with sweets, cakes and pies beyond what they can eat. Their eyes drink in the goodies and they try to eat as much as they can only to be beset by great stomach pains. This same behavourial pattern carries over to adulthood where those working on projects see this and that cool feature which must be had but only serve to bog the project down and cause it to loose direction.

Attempting implementation of advanced features, even the mere act of listing these features, which are not necessary to the essence of the project is an example of a rapid way to lose sight and control of a project. As the features fail to be implemented one becomes sick to their vision and lose hope and feelings of progress. It is necessary that a tight hold be kept on what is destination and the quickest way to it.

Another method is one of the misunderstanding of originality, creativity and creation. It is necessary that one know what it is they wish to do and have a clear idea of their vision. It is not neccessary to dig a hole if you wish to bake a cake. Nor is it neccessary to grow wheat if you wish to eat bread or frosted flakes. So also must all those who set out to create say to themselves: I must realize that if some kind person has helped to wash all my dishes, then I will not wash them again before I set my dinner on them. It makes more sense to get about the more important task of setting the dinner out. It does no one any good to create from "scratch".

Because truly there is no such thing as original, everything is learned, taken and reworked from somewhere. The human mind is not an original one. It cannot create. The laws of conservation apply even to creativity. Certainly there is a Mapping from the set of all people creating to another Set of all those who have written and taught. Each member of the relation essentially tells who the influences on the set of students are. Certainly it feels good to cut a tree, whittle the wood, carve a handle. Find some ore, a smelter, a forge and make a blade to then attatch to the handle so you can cut some bread with the knife but it is also a massive waste of unnecessary time. Ofcourse if the task was to create a knife then no time was wasted. Except ofcourse if you had cast the forge yourself. Thus it must be that you must know your goals and task and know how to use the tools at your disposal to allow for the most rapid development. Overpride in what must be yours and a severe misunderstanding of what is your original creation does no one any good. The essesence of your creativity is in the shaping of your goal not in the laborious side tasks of the creation of tools which facilitate such.

The final tool trap is where one actually creates more tools than they need. Or make the tools much more fancy than they need be. I fell for this once, building an editor, improving this and that and making it more and more friendly and with more features while the game languished. The tools need not do any more than they must.

To do what you wish you must do no more than you must.