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The Allure of Transcendental Numbers

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To quote Marlowe: "all is dross that is not Mathema...."

I can't tell you how I kick myself for not diving into all of this long long ago. I suppose that Logos/Word has been good to me, but mathematics is such an ideal universal language, a world of absolute form....I am humbled before the sight of Math Mountain.

I know this is a philosophy forum, but philosophy and math are quite related. "Let no one ignorant of geometry enter here." Math is arguably the ideal toward which philosophic logos tends. (As philosophy tends toward math, philosophy = yes!!!!)

I just bumped into this strange thing, which is another relation of e and pi. t ties in to probability calculations. Gaussian integral - Wikipedia, the free encyclopedia

And what got me started down this road was researching this: Euler's identity - Wikipedia, the free encyclopedia
At first it just boggled my mind. But a little research into radians, trig, wilder forms of exponentiation got me closer to processing this little gem.

I've spent many an hour reading modern art aesthetic theories & many an hour perusing art books. Much of it seems boring all the sudden. The best math seems like ideal form, a form with unmatched precision and efficiency.

This is a long post, but I hope someone out there will simply share my enthusiasm. I'm no expert, but rather an abid if not rabid student. I'd be grateful for anything the more exposed can share with me. Smile

@Reconstructo,

A transcendental number cannot be expressed as the solution of a finite algebraic equation. Merely irrational numbers can be expressed as roots. With the trannies, it's not so simple. It takes an infinite series.....

@Reconstructo,

YouTube - Pi is an irrational number - Star Trek

@Reconstructo,

That's an illogically action of the computer, such computer should be smart enough to know that Pi has no known end.

@Reconstructo,

Wouldn't the computer have been just as confused if Spock had asked it to compute "to the last digit" the value of 1/3?

@Reconstructo,

Quote:

It is not recorded who it was first realised that not all numbers are fractions. We do know that, whoever he was, he probably lived in the fourth century BC. There is an ancient tradition that says that he was murdered for his pains. Why murder somebody for making a mathematical discovery? The answer is that he was probably a member of (or associated with) the religious cult started by Pythagoras (of the triangle). The Pythagoreans seem to have had a theory of the world that was based on whole numbers and consequently on their ratios. The discovery that whole number ratios were not everything seems to have put the cat amongst the pigeons. Eliminating the discoverer was a predictable reaction, though not nowadays regarded as an acceptable form of mathematical proof.

Beware of Greeks bearing irrational or transcendental numbers.

@Twirlip,

Twirlip;159619 wrote:

Wouldn't the computer have been just as confused if Spock had asked it to compute "to the last digit" the value of 1/3?

I don't think so. Isn't the repeating decimal of one third just a byproduct of our decimal system? Wouldn't 1/3 be "0.1" in ternary notation? But for a number like pi, it doesn't matter which base we use, excepting cheats like base-pi, which would only be to use a "10" glyph instead of the greek letter. (It would be 10 and not 1 because pi^0 is 1, as you probably already know.)

---------- Post added 05-08-2010 at 10:36 PM ----------

HexHammer;159611 wrote:

That's an illogically action of the computer, such computer should be smart enough to know that Pi has no known end.

That's true, I suppose, but the computer is programmed to obey orders, right? And it's a funny piece of screenwriting. It's like War Games, where a computer is forced to play tic-tac-toe against itself.

@Reconstructo,

Reconstructo;161954 wrote:

That's true, I suppose, but the computer is programmed to obey orders, right? And it's a funny piece of screenwriting. It's like War Games, where a computer is forced to play tic-tac-toe against itself.

A computer in far future, should be intelligent enough to not obey unlawful orders, specially in a military installation such as a space craft, with lethal weaponry.

@HexHammer,

HexHammer;162149 wrote:

A computer in far future, should be intelligent enough to not obey unlawful orders, specially in a military installation such as a space craft, with lethal weaponry.

I agree. Of course we could do that now, I think. It just depends on who gets to build and program the computer. I can only assume on Star Trek, they designed their computers to obey.

@Reconstructo,

Recon,

"ternary notation"? Like base 3? To quote GM Moore, "What do you mean?"

Wink

@jack phil,

jack;164798 wrote:

Recon,

"ternary notation"? Like base 3? To quote GM Moore, "What do you mean?"

Yeah, base 3. I should have used the less pretentious phrase. But isn't "ternary" a pleasant word? It's a neat quirk that certain bases will notate certain rational numbers as repeating decimals.

@Reconstructo,

Well, that is ratios; and also why I prefer geometry to calculus; aesthetic reasons, to be sure. Of course, the circle cannot be squared, so the Cartesian system will not do as a substitute.

Nullity: the new number: Philosophy Forums

I've looked into this more; I think I gave you a link a while ago about Fuller and tetrahedrons and there congruence with spheres. I still have much work to do on crafting such a new mathematics... Mathematica Nova or too much vanity?

@Reconstructo,

I wonder if nth decimal place of Pi or e or phi could be approached with probabilities. Could some probabilistic theory predict though not exactly what the nth decimal place is more likely to be? I don't recall how many decimal places out the computers have gotten to as of yet they probably covered a few hundred places as I wrote this. Forget the computers for a moment. But suppose there was a theory that could establish what is most likely to be the billionth place after the decimal point. Still some margin of error. I mean to say there is no definite pattern to the numbers but perhaps some fuzzy probability could be used to place a good bet. Or a theory that would say something like if the preceding number is 1 the next number is more likely to be a 5 and the 4. A fuzzy pattern rather than a definite pattern. I'm guessing with all the empirical data the computers are giving us something like this approach has already been tried...

With any given preceding digit x there are 10 possibilities for what digit comes next. Is the probability for each of these 10 possibility always 1 in 10? As the computers give us more decimal places do we approach this 1 and 10 probability?

If we flip the ideal coin eternally the assumption is that it the empirical data gets ever closer to exactly 50% heads 50% tales. Does the same thing happen for the nth digit of Pi as the computers flip that ideal transcendental coin? 10% 0, 10% 1, 10% 2, 10% 3, 10% 4, 10% 5, 10% 6, 10% 7, 10% 8, 10% 9? If so is that a patter? Can the same be said of e? Can the same be said of Phi? Can the same be said of algebraic irrationals? That last question might be the easiest to answer... but I don't know the answer or where to look for it.

@jack phil,

jack;164798 wrote:

Recon,

"ternary notation"? Like base 3? To quote GM Moore, "What do you mean?"

Who is GM Moore? Are you sure it wasn't Chrysler Moore, his sister?

@Reconstructo,

er, GE Moore. Apologies to GH von Wright Brothers, too.

@jack phil,

jack;164816 wrote:

Well, that is ratios; and also why I prefer geometry to calculus; aesthetic reasons, to be sure. Of course, the circle cannot be squared, so the Cartesian system will not do as a substitute.

Nullity: the new number: Philosophy Forums

I've looked into this more; I think I gave you a link a while ago about Fuller and tetrahedrons and there congruence with spheres. I still have much work to do on crafting such a new mathematics... Mathematica Nova or too much vanity?

You know, I love both passionately. The concept of limits is beautiful. The Fundamental Theorem is strange, lovely. The notion of a derivative. An instantaneous tangent to an exponential curve...damn. Of course integration has its own charms. An infinite sum.

But geometry has its own charms. I like cones. An infinitely sharp point at the top, no? Unbearably straight lines. The perfect circle, as you mentioned, only exist in the visual imagination! Actually, it was the circle and pi that drew me into math...
I looked at the link then, but my memory is rusted...I've tried to stuff so much into this skull lately. It's an interesting theme. I would enjoy reading especially your own words on the subject, for then I could ask you for clarifications if I found it difficult.

@Reconstructo,

Reconstructo;164837 wrote:

You know, I love both passionately.

And love makes the world go round....or is that money? Or is that love of money? I always forget.

@kennethamy,

kennethamy;164840 wrote:

And love makes the world go round....or is that money? Or is that love of money? I always forget.

Perhaps it is the price of love that makes the world go round.

@Deckard,

Deckard;164846 wrote:

Perhaps it is the price of love that makes the world go round.

I hope not!................Anyway, for many, love and passion substitute for knowledge and understanding. In fact, many cannot tell the difference.

@prothero,

prothero;159635 wrote:

Beware of Greeks bearing irrational or transcendental numbers.

Hippasus of Metapontum is the guy in the maths martyr stories. Sometimes he's killed for proving irrationality, sometimes for revealing it to non-Pythagoreans. Usually, he's drowned at sea.

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