[General] Mathletes on steroids

Sigh.

No matter how hard I try, the most controversial things that I can say on my blog seem to be based around whether mathematics is discovered or invented, is it science or art.

Here are the responses to my short piece on UD. (These have been shortened because they were each the size of a normal post. Scott also commented but actually talked about UD and not maths)

Ben:

…You comment on Pi, and refer to it as "the ratio of every circles circumference to its diameter", and then you go on to say, "...in some other universe the ratio of circumference to diameter might be [something else]”

This is far more shocking to hear than, for example, you suggesting that Space Bisons created life through intelligent rock design.

I would have hoped you would have considered Pi to be something more along the lines of "the smallest positive real so that sin(x) = 0." The fact that there may be some relation to circles to be a truly fascinating, yet almost bewildering, coincidence…

…I also feel inspired to dwell on parts of the Mandlebrot set for a while, now there is complex beauty from the simplest of systems.

Mike (in response):

Re pi - it is completely entwined with circles, so it's not really that big a coincidence!

I mean pi's very definition is that it is the ratio of the circumference of a circle to its diameter.

As for sin(w), where w is in radians or degrees or whatever, it's just the height of a unitary circle above (or below) the x-axis starting at the point (0,1) and moving anti-clockwise. Cos(w) is the distance from the y-axis. (Had to look that up, thank you Mathworld, mathematician-geek heaven.) The unit used for w is pretty arbitrary: pi (as in radians) is used because it simplifies calculations. "The smallest possible real so that sin(x)=0" could be 180 (as in 180 degrees), or 200 (as in 200 gradians), or whatever.

And the interesting thing is that if you put a circle into curved space, you can change the value of pi. There are instances where pi even becomes rational. Couple of links: Cosmic Questions and Ask Dr Math.

Mandelbrot sets rock…

Ben (in reresponse):

I always find it rather disappointing when maths gets related to real world things. I much prefer totally abstract idea. I guess a circle could be considered abstract. But geometry just seems a little to "real world" to be taken seriously.

I like prime integers. These are good. I remember having an argument with someone about whether God had any limits one what he could create do with the universe. I argued that God can not change the prime numbers. The whole concept of integers is a basic one that is a completely abstract thought. It matters not what type of universe you are in, the abstract concept of integers and their primes is outside that and exist even if there was no universe or anyone to think about them. God would be limited to looking at the primes; there is no ability to rearrange them.

Gaussian (complex) integer primes are super cool too.

I never looked into quaternions integer primes. That would be fun. That inspires me to go and look at some.

Primes make pretty pictures. I used one for an Album cover once.

Speaking of quaternions, I always wanted to make a 3D Mandelbrot using quartenions (well I would only use the real, i, and j part, and leave k [as] 0 or I would end up with 4 dimensions). But could never think of a good way to actually display it…

The Mandelbrot is soothing…

Mike (in rereresponse):

Ha haarrr! I'm an engineer. It's my job to take all your wishy-washy mathematical "abstractions" and turn them into real world applications!

Bejeebers, I had enough trouble with imaginary numbers without getting into quaternions! I've got to stop writing all my sentences ending in!

Yeah I know; that’s what I was thinking.

Let’s talk about our old friend Pi (π) shall we? Thanks to Mike for finding what I was too busy lazy to find: that Pi changes under different circumstances. In some curved spaces Pi (either as the ratio of circumference to diameter OR as the limit of x as sin(x) → 0) has a different value.

Is mathematics a science or an art?

art¹ [ ärt ] n.

1. Human effort to imitate, supplement, alter, or counteract the work of nature.
2. a. The conscious production or arrangement of sounds, colors, forms, movements, or other elements in a manner that affects the sense of beauty, specifically the production of the beautiful in a graphic or plastic medium. b. The study of these activities. c. The product of these activities; human works of beauty considered as a group.
3. High quality of conception or execution, as found in works of beauty; aesthetic value.

sci·ence [ sns ] n.

1. a. The observation, identification, description, experimental investigation, and theoretical explanation of phenomena. b. Such activities restricted to a class of natural phenomena. c. Such activities applied to an object of inquiry or study.
2. Methodological activity, discipline, or study: I've got packing a suitcase down to a science.
3. Knowledge, especially that gained through experience.

My money’s on art. Sure people use maths every day, but the stuff that people are actually creating is very unlikely to be used in everyday life anytime soon. There is an argument that has any science pegged as a form of art. However, here is a quick litmus test: Maths can be a Science or Arts degree major at Auckland University. As far as I’m aware no other science has that option.

Is science discovered or invented?

Is science innate and immutable in the universe or is a construct of mankind. It is pretty difficult to argue that maths is just a construct of the human mind especially given that our context is IN THE HUMAN MIND. (Any non-humans who read this, your opinion would be greatly valued.).

This is no longer in science or art or mathematics but rather philosophy (which, yes, could be considered art). Substructural logic is a complex topic that is similar to what I’m trying to get at here. Take Universe A, give it only three of the necessary logical structures it might need and then have fun trying to make Universe A into the Universe we all know and love. For example, what about a universe where A=A is not always true. You end up using all kinds of funny symbols with strange names (we would be saying things like “if A hats B then…”).

So let’s look at Ben’s “God’s Limits”:

Does God have any limits on what he can create or do with the universe?

My answer (and I stress my) is not once he has set out the basic rules. So the instant God (who we assume is a force outside of the universe in the first instance) decides that, say, the speed of light is 299,792,458 m/s then everything else that is associated with that is also set. E must equal MC² and all atomic structures are set, electron orbits are set (or as much as they ever are, stupid quantum physics). And prime numbers are there too.

See God couldn’t set up a universe where we have a circle (as we traditionally know them) and also set the value of Pi (as we traditionally know it) to be 5. It wouldn’t work. He would have to construct a whole other universe and start with Pi=5 and go from there.

It’s kind of like taking the materials and blueprints for a boat and trying to build a rocket. You’ll just build a boat.

Prime numbers are funnier still. If there is something then it follows that there is at least one of it (keep your nose out Quantum Physics). If you have a concept of “one*” then you can build from that and create laws of logic (if they exist) and create the entire real system. I can’t remember the amount of times I have created the natural numbers then the Integers and then the Rationals and then the Irrationals (and hence the Reals).

*I should point out that “one” is the nonsense syllable I have assigned to the idea of a single existing entity in our current logical structure.

It gets kind of hard to imagine a universe that doesn’t have the concept of something being itself (and hence there being no concept of “one”). It’s hard to imagine because we are inside the system itself.

I believe that there is no reason that the Prime numbers have to exist. If only because “abstract” is itself abstract and to go all Zen for a moment: Do primes exist if there is no one to comprehend them?

There that ought to get you all fired up. Ha! We are such geeks!

ps. would Madelbrot sets be as cool if they had been discovered by Prof. Lickbuttock?