Mathematics and Constants- Inconsistency with the Universe?

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VMhent
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Mathematics and Constants- Inconsistency with the Universe?

Postby VMhent » Sun Mar 27, 2011 6:49 am UTC

I do not pretend to be any sort of expert in mathematics, nor even well-educated on the subject. However, that doesn't seem to preclude me from asking a fairly basic philosophical question of math:

If math is out there in the universe for us to discover, derive, and manipulate, why are all the major mathematical and scientific constants irrational? If the value of Pi can never be computed with 100% precision using our concepts of numerology, is it possible that we're just doing it wrong in a very fundamental way? Is God-Math operating on a different paradigm from Human-Math? I mean, Pi can't just compute itself out to an infinite number of decimal places. If it's out there, it's just sitting there, like any other number.

As a disclaimer, I'm not impeaching the very real value of mathematics in any way, just wondering why this seeming contradiction occurs in our very orderly union of math, the natural sciences, and the universe. I ask because I think it seems like an interesting question!

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Re: Mathematics and Constants- Inconsistency with the Univer

Postby taemyr » Mon Mar 28, 2011 4:54 am UTC

Not all major mathematical constants irrational. The by far most important mathematical constant is the number 1. Another important constant is 0. It's just that they are so basic that we seldom think about them as constants.

For physical constants they are not so much irrational as not completly determined. When they are irrational it's usually because of unfortunate selection of units. Eg. speed of light, which is in fact not irrational, is exactly 299,792,458 metres per second, however in physics it's usual to use units for distance and time such that the speed of light is equal to 1.

That leaves us with famous mathematical constants such as pi and e, or the square root of 2.
These are irrational, meaning that they are not the ratio of two natural numbers. You state that "the value of Pi can never be computed with 100% precision using our concepts of numerology". What do you actually mean?
We can't write pi as a finite string using decimal expansion. However there are no real reason to expect us to be able to do so.
We can create a wheel with radius 1. Then dip the rim of the wheel in ink and roll it along a paper so that it turns one full turn. The length of that line will be equal to pi. So we have accurate representations of pi, which are meaningful in a way that the mere symbol perhaps fails to be.

There are irrational numbers, why should there not be? These numbers can not be precisly representated in a system designed to represent a subset of the rational numbers, but why should they be?

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Re: Mathematics and Constants- Inconsistency with the Univer

Postby antonfire » Mon Mar 28, 2011 5:00 am UTC

I don't see a "seeming contradiction" in what you said, OP.

Anyway, most interesting mathematical constants are irrational because those are the ones that need names. If pi were exactly 22/7, we wouldn't bother giving it a name, we'd just say 22/7.

On the other hand, talking about measured constants (like the fine structure constant) being irrational is fairly senseless. We can never measure them with enough precision to be sure about something like that.
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Re: Mathematics and Constants- Inconsistency with the Univer

Postby lightvector » Mon Mar 28, 2011 9:06 am UTC

In this sort of philosophical sense there's nothing really special about being rational either. What's so special about being the ratio of two integers? Why not the ratio of two square numbers? Why not the cube root of an integer? Why not a multiple of the circumference of a circle with unit diameter?

Being rational is just a mathematical property of numbers, like being even, or being prime, or being the square root of another integer. From this perspective, there's nothing particularly special if there happen to be numbers that aren't rational. There are also numbers that aren't even or prime, or the square root of another integer.

For that matter, we can't compute 1/7 to an infinite number of decimal places either, in the sense that we can't write it all the way out. Because it goes on forever too. Of course, you might argue that it's just a sequence of repeating digits, 0.142857142857142857..., so you don't have to write it all the way out. We already know how to compute it however far out we like just by repeating these digits. But we also know how to compute π however far we like as well. Sure, it's not as easy as repeating the same digits over and over, and takes some time. But we still know how to do it, so it's not really any different than 1/7.

And in practice, rather than trying to write out all the decimals, we just write "1/7", which is shorthand for "the number you get when you divide 1 by 7". Similarly, rather than trying to write out all of the decimals, we simply write "π", which is shorthand for "the number you get when divide a circle's circumference by its diameter". Again, there's nothing special here. Both "1/7" and "π" are symbols that we write that are shorthand for certain mathematical ideas, and aside from the fact that they are different numbers with different properties, there's nothing magical, or contradictory, or philosophically special about one or the other.

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Re: Mathematics and Constants- Inconsistency with the Univer

Postby mr-mitch » Mon Mar 28, 2011 2:58 pm UTC

You can't describe most real numbers, but all the important ones are, which is what really counts. You can find them, write algorithms to calculate them (and so you can represent the number).

Mathematics comes up with representations for different things. As said before, a number represents a concept. Can you answer this question, "What is two?" There isn't really an answer, but you can invent ways to represent it. Interestingly, the princeton dictionary google define loves defines it as 1 + 1. This is a pretty good definition, as it's true. But what's 'one'? What's 'zero'?

Mathematics (I believe) is useful for abstraction and inference, and I study it because it's beautiful.

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Re: Mathematics and Constants- Inconsistency with the Univer

Postby ConMan » Mon Mar 28, 2011 11:24 pm UTC

The other reason most constants are irrational is because the irrationals hugely outnumber the rationals. It would be more surprising if there weren't any "special" irrational numbers given how many of them there are.
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Re: Mathematics and Constants- Inconsistency with the Univer

Postby doogly » Tue Mar 29, 2011 12:02 am UTC

No no, this is kind of interesting.
Let's take something like the holographic principle seriously, which says the maximum amount of entropy in a given volume is fixed by a multiple of what the area bounding it is. You can't put more information inside of it. Now, how does nature do calculations? How do you store a real number? It requires quite a few bits of information. It is related to an interesting observation that nature doesn't seem to solve NP problems efficiently: http://arxiv.org/abs/quant-ph/0502072
Kronecker remarked that Lindeman's work on pi was useless, since rational numbers are the only ones that exist, and he was not a slouch mathematician. But rather, I'd propose that only irrational numbers exist! Nature never prefers them, the only seem to crop up when we like to talk about a pattern they have. If we thought something was precisely an integer, like the g_s factor which determines spin coupling, lo and behold, further understanding of QED teaches us that no integers are to be found.
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Re: Mathematics and Constants- Inconsistency with the Univer

Postby Qaanol » Tue Mar 29, 2011 1:25 am UTC

taemyr wrote:We can create a wheel with radius 1. Then dip the rim of the wheel in ink and roll it along a paper so that it turns one full turn. The length of that line will be equal to pi.

You sure about that?
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Re: Mathematics and Constants- Inconsistency with the Univer

Postby VMhent » Tue Mar 29, 2011 2:11 am UTC

Maybe I wasn't entirely clear. I understand that we can have useful representations and such of things like Pi. My question is a little more like this- if we can't write down Pi comprehensively (like, "Oh! There it is on my calculator, completely precise!") then it seems like our paradigm for mathematics is failing (in a very slight way) to describe a fundamental property of the universe. I'm not asking why Mother Nature didn't simply pick 3 for that particular ratio, I'm saying she picked a number that we can't describe perfectly. Does that mean we're doing it wrong?

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Re: Mathematics and Constants- Inconsistency with the Univer

Postby antonfire » Tue Mar 29, 2011 2:40 am UTC

What is the difference between "the number which, when multiplied by the diamater of a circle, gives its circumference" and "the number which, when multiplied by 7, gives 1"? Why is one an "imperfect" description and the other a "perfect" one?
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Re: Mathematics and Constants- Inconsistency with the Univer

Postby skeptical scientist » Tue Mar 29, 2011 3:03 am UTC

I think antonfire has the right of it. This isn't expressing a limitation of our knowledge, or of our ability to describe the universe. It's just a limitation of decimal notation, which is simply one particular way of describing numbers. If you think about it, there's really no good reason why we should be concerned with writing numbers as sums of powers of ten. We do it because it seems to be convenient for things like buying groceries and building bridges, but it's not terribly useful at all for doing mathematics.

Given that, the fact that not every number is equal to a finite sum of powers of ten probably shouldn't be particularly surprising, or troubling. One can even show that most numbers are not equal to a finite sum of powers of ten, so pi is actually just like most other numbers in this regard.
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Re: Mathematics and Constants- Inconsistency with the Univer

Postby Qaanol » Tue Mar 29, 2011 3:21 am UTC

I agree with antonfire and skepsci.

lightvector wrote:For that matter, we can't compute 1/7 to an infinite number of decimal places either, in the sense that we can't write it all the way out. Because it goes on forever too. Of course, you might argue that it's just a sequence of repeating digits, 0.142857142857142857..., so you don't have to write it all the way out. We already know how to compute it however far out we like just by repeating these digits. But we also know how to compute π however far we like as well. Sure, it's not as easy as repeating the same digits over and over, and takes some time. But we still know how to do it, so it's not really any different than 1/7.

I’d like to expand on this a little. Not only do we know (many) formulæ that produce the digits of π in order, but we also know a formula to produce just the nth digit of π (in base 16).
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Re: Mathematics and Constants- Inconsistency with the Univer

Postby z4lis » Tue Mar 29, 2011 3:31 am UTC

My approach to this apparent problem is that you need to stop thinking of numbers as the only thing that's man-made. The concept of the circle is, itself, a man-made object. So, it's not surprising at all that we need an "unnatural" constant such as [imath]\pi[/imath] to describe it. Both geometry and numbers are just ideas that we've hatched up to describe perceptions. There's no such thing as a circle in nature as far as I'm concerned. Even something as basic as the concept of a "point", strictly philosophically speaking, has no real significance to me. Why should I think that the universe acts anything like the real line with its distances? After all, isn't it in such a setting as R^3 that we get things like the Banach-Tarski Paradox? This, if anything, is clear evidence to me that "math geometry" and "real geometry" have some serious problems with one another on the small scale. So I suppose I would say that my answer to your question would be "yes, math and reality are at odds with one another on some level".

That, however, does not tarnish mathematic's excellent track record for being a fantastic model of our observations and predictor of natural phenomena.
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Re: Mathematics and Constants- Inconsistency with the Univer

Postby skullturf » Tue Mar 29, 2011 3:57 am UTC

Qaanol wrote:
taemyr wrote:We can create a wheel with radius 1. Then dip the rim of the wheel in ink and roll it along a paper so that it turns one full turn. The length of that line will be equal to pi.

You sure about that?


More argument for tau! :D

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Re: Mathematics and Constants- Inconsistency with the Univer

Postby Ankit1010 » Tue Mar 29, 2011 4:47 am UTC

skullturf wrote:
Qaanol wrote:
taemyr wrote:We can create a wheel with radius 1. Then dip the rim of the wheel in ink and roll it along a paper so that it turns one full turn. The length of that line will be equal to pi.

You sure about that?


More argument for tau! :D


I think he just means that the diameter must be 1 for the full length turn to be pi, not the radius.

Also, my two cents:

We can also do away with a lot of irrational numbers by changing bases from 10 to something else. There's no reason nature must work in base 10, or a specific base at all times anyways.

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Re: Mathematics and Constants- Inconsistency with the Univer

Postby skeptical scientist » Tue Mar 29, 2011 5:00 am UTC

Ankit1010 wrote:We can also do away with a lot of irrational numbers by changing bases from 10 to something else. There's no reason nature must work in base 10, or a specific base at all times anyways.

No. An irrational number is irrational in any base. I think you're confusing "rational" (is a ratio of two integers) with "has a terminating decimal expansion" (which is the same as being a rational number p/q such that the only primes which divide q are 2 and 5).
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Re: Mathematics and Constants- Inconsistency with the Univer

Postby skullturf » Tue Mar 29, 2011 5:09 am UTC

Ankit1010 wrote:
skullturf wrote:
Qaanol wrote:
taemyr wrote:We can create a wheel with radius 1. Then dip the rim of the wheel in ink and roll it along a paper so that it turns one full turn. The length of that line will be equal to pi.

You sure about that?


More argument for tau! :D


I think he just means that the diameter must be 1 for the full length turn to be pi, not the radius.


Yes, I understand that. I just meant that the very fact that somebody would mistype "radius" when they meant "diameter" is an illustration of how it's more natural to talk about radii than diameters (so maybe the ratio circumference/radius is more "natural" than circumference/diameter).

Anyway, I just meant it as a silly digression. Returning to the point of the thread, I agree with several other posters. I think it's misleading to think of irrational numbers as inherently "weird". In fact, rational numbers are quite "special".

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Re: Mathematics and Constants- Inconsistency with the Univer

Postby Qaanol » Tue Mar 29, 2011 5:17 am UTC

Hey guys what’s up with the computable numbers? They getting all uppity thinking they’re special or something. Got to put them back in their place, after all there’s only countably many of them. I mean seriously, right? Density zero? What do they think, being produced by an algorithm gets them extra privileges? Whatever.
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Re: Mathematics and Constants- Inconsistency with the Univer

Postby skeptical scientist » Tue Mar 29, 2011 5:52 am UTC

Qaanol wrote:Hey guys what’s up with the computable numbers? They getting all uppity thinking they’re special or something. Got to put them back in their place, after all there’s only countably many of them. I mean seriously, right? Density zero? What do they think, being produced by an algorithm gets them extra privileges? Whatever.

The same goes for the definable numbers. But then, I bet you can't name a number which isn't.
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Re: Mathematics and Constants- Inconsistency with the Univer

Postby WarDaft » Tue Mar 29, 2011 6:03 am UTC

So, the definable numbers are countably infinite yah? I think ve should put them in a nice list, and then pick one slightly different from each. Then ve could have a very specific undefinable number, just to be sure.
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Re: Mathematics and Constants- Inconsistency with the Univer

Postby Qaanol » Tue Mar 29, 2011 6:06 am UTC

Pshh, definable who needs that‽

WarDaft wrote:So, the definable numbers are countably infinite yah? I think ve should put them in a nice list, and then pick one slightly different from each. Then ve could have a very specific undefinable number, just to be sure.

That list is not definable.
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Re: Mathematics and Constants- Inconsistency with the Univer

Postby WarDaft » Tue Mar 29, 2011 6:29 am UTC

Qaanol wrote:That list is not definable.

And yet, it's still countable!

Actually yes, such a list will be definable, because we have to define what we mean by definable number - then we have everything we need to see what system we must be in to define the list of numbers definable by our definition.


Also, would the definable reals or the computably enumerable reals be larger?

That is, a real r for which there is a TM such that for any e>0, there is some d such that at step d, the approximation a represented on the TM's tape satisfies r-e < a < r+e. The machine need not actually halt. Chaitans constant is CE, oracles are CE, second level oracles are CE, δ-level oracles for an ordinal δ which has a CE representation are CE.
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Re: Mathematics and Constants- Inconsistency with the Univer

Postby Qaanol » Tue Mar 29, 2011 6:55 am UTC

Every computable number is definable, but not vice versa.

Definable has a specific meaning. Its Wikipedia page even has a section on exactly what you describe.
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Re: Mathematics and Constants- Inconsistency with the Univer

Postby WarDaft » Tue Mar 29, 2011 7:02 am UTC

Computable number or computably enumerable number? I know computable numbers are surely all definable, but I don't know if all computably enumerable numbers are. They might not be fully definable in first order logic.
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Re: Mathematics and Constants- Inconsistency with the Univer

Postby kbltd » Tue Mar 29, 2011 8:59 am UTC

VMhent, would you agree with me when I assert that pi is not a physical constant but a mathematical one?
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Re: Mathematics and Constants- Inconsistency with the Univer

Postby doogly » Tue Mar 29, 2011 12:39 pm UTC

Pi is all over physics, like flies on shit. You can't just rationalize all your units and make it drop. It needs to be there.
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Re: Mathematics and Constants- Inconsistency with the Univer

Postby gmalivuk » Tue Mar 29, 2011 7:35 pm UTC

WarDaft wrote:So, the definable numbers are countably infinite yah? I think ve should put them in a nice list, and then pick one slightly different from each. Then ve could have a very specific undefinable number, just to be sure.
Sure, but to be definable requires a finite description, and so that one wouldn't be, as it requires the whole infinite list.
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Re: Mathematics and Constants- Inconsistency with the Univer

Postby skeptical scientist » Tue Mar 29, 2011 9:09 pm UTC

gmalivuk wrote:
WarDaft wrote:So, the definable numbers are countably infinite yah? I think ve should put them in a nice list, and then pick one slightly different from each. Then ve could have a very specific undefinable number, just to be sure.
Sure, but to be definable requires a finite description, and so that one wouldn't be, as it requires the whole infinite list.

Well, unless you can give a finite description of the infinite list.

Given a language L and an L-structure M, we can say that an element a (or subset A) of M is L-definable if there is an L-formula F(x) with one free variable x such that for all y in the universe of M, M models F(y) iff y=a (or y in A). Then, we could happily define a list of all L-definable elements of M. For example, if M is the real numbers, and L is the language of fields, then the set of definable elements of M is just the set of algebraic numbers. We can certainly define a list of algebraic numbers; that's essentially what we do when we prove that the algebraic numbers are countable. And, using the diagonal argument, we could pick a number which is not on that list (or we could just pick e). In fact, we could do the same thing using the language of set theory, replacing M by a model of ZFC.

But that's where things get a bit tricker. It is possible to mechanically check, given a formula in the language of fields, whether it defines a unique real number. But there is no way to do the same for formulas in the language of sets. In fact, it's even worse than that. Given a "Godel numbering" which identifies L-formulas with elements of M in some nice (computable) way, if M is sufficiently complicated (i.e., can represent arithmetic) then the set of elements of M which correspond to L-formulas which define elements of M is not definable. This is related to Tarski's undefinability of truth. So there's no way to define (in a given model M of ZFC) the collection of sets coding formulas which define unique real numbers in M. Similarly, there's no way to define in M a sequence including all real numbers definable in M. This is a bit of a relief, since cantor's diagonal map is definable in M, so if one could define in M a sequence including all real numbers definable in M, one could also define, in M, a number which is not definable in M. If we could do that, M would suddenly poof out of existence, and we wouldn't be left with any models of ZFC. (Rather, we could prove ZFC inconsistent.)

But one can give a perfectly good natural language definition of such a sequence. That's because we can easily define a sequence including all formulas with one free variable in the language of set theory (just list them in lexicographical order, sorted by length). And we can also define a model of set theory (this takes a bit more work, but one example is Godel's constructible universe). So we can, using natural language, define a sequence which takes the first sequence, and omits all formulas which are true of more or fewer than one real number in Godel's constructible universe. Then, we can replace each formula with the real number so defined, and obtain a natural language definition of a sequence containing every real number which is definable in Godel's constructible universe by a formula of set theory. We can even perform Cantor's diagonal construction on this to get a nice natural-language definition of a real number which is not definable in that model of set theory. (Because of how the constructible universe is defined—iterating definability—this whole endeavor is a bit silly, but you could replace it with just about any other model of ZFC, and everything would work the same, modulo taking "M" to be understood, or else defining it.)
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Re: Mathematics and Constants- Inconsistency with the Univer

Postby Qaanol » Tue Mar 29, 2011 11:21 pm UTC

skeptical scientist wrote:<natural language is inconsistent>

To wit:

If this sentence is true, then Santa Claus exists.
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Re: spelling and Constants- Inconsistency with the Univer

Postby skeptical scientist » Wed Mar 30, 2011 2:32 am UTC

Qaanol wrote:
skeptical scientist wrote:<natural language is inconsistent>

To wit:

If this sentence is true, then Santa Claus exists.

I don't believe I was saying that at all. It is, which is well known, but what I was actually doing (more or less) is giving a (consistent!) natural language definition of a real number which is not definable in formal set theory.

Mod holiday has apparently started early, but hopefully you all can figure out what I meant.

Back on the original subject, the cheesegratered version of this thread title really is very appropriate. Decimal notation in many ways has more in common with spelling than it does with mathematics*, and the OP is basically complaining that we can't spell pi correctly. This happened because our spelling system was designed for English, and is only able to approximate sounds made by Chinese speakers, but pi is in Chinese and not English.

*The italicized word denotes the subject of this subforum.
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Re: spelling and Constants- Inconsistency with the Universe?

Postby kbltd » Wed Mar 30, 2011 7:23 am UTC

Doogly said
Pi is all over physics, like flies on shit. You can't just rationalize all your units and make it drop. It needs to be there.

I agree with you. So I need to clarify what I was trying to say when I was asserting that it wasn't a physical constant.

A constant used in physics may be a different thing from a physical constant. Antonfire had already phrased this rather better by pointing out the distinction between interesting mathematical constants (e.g. pi) and interesting measured constants (e.g. the fine structure constant).
The OP referred to the universe, but in my view mathematical constants like pi aren't part of the universe. It's possible to describe some of the features of a universe with different charge on the electron, different gravitational constant etc, but I don't think it's possible to describe a universe where, starting from a "natural" definition of addition and building from that to the real numbers, the usual series for pi generate a different numerical value from the one we know.
So I think it's worth inviting the OP to clarify whether the question is about the ugly numbers that we measure in the physical world, or the ugly numbers that we calculate in the Platonic realm of pure mathematics.
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Re: spelling and Constants- Inconsistency with the Universe?

Postby doogly » Wed Mar 30, 2011 2:00 pm UTC

What are you, a half Platonist? Now neither the real Platonists nor the empiricists are going to be happy with you. Dangerous turf to tread.
LE4dGOLEM: What's a Doug?
Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.

Keep waggling your butt brows Brothers.
Or; Is that your eye butthairs?


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