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### Useless Math

Posted: Mon Dec 06, 2010 1:15 am UTC
As mathematicians, I'm sure many of you have had to justify it's practicality to people outside the discipline. I'm having the opposite problem right now.

I'm writing a paper and trying to argue that mathematical objects (numbers, shapes, sets and so on) exist abstractly regardless of any relationsihp to the real world, and when they describe the real world (like how calculus can be used to describe motion, and we get physics), it is merely accidental. (I'm contrasting Aristotle, who would say that mathematical objects are just real objects, just thoguth of abstractly, as opposed to as objects)

I'm running into trouble finding examples, though, and a totaly lack of them would make my argument very weak. So my question is this: what are some examples of pieces of mathematics that have no known application to the empirical world, or which were discovered long before any practical application was?

(As a side note, maybe my inability to find any examples means I need to rethink my thesis.)

### Re: Useless Math

Posted: Mon Dec 06, 2010 1:21 am UTC
Number theory, I guess? It existed hundreds of years before modern cryptography.

### Re: Useless Math

Posted: Mon Dec 06, 2010 1:32 am UTC
I think you're right only in a very limited sphere. Rules involving things like numbers and shapes seem to have been very clearly created and investigated precisely because they are good models for real-world phenomena like ancient accounting and engineering. But in the last century or two, mathematicians have created axioms of pure mathematical systems without a thought to applications. My favorite college professor would joke that it would take the world 150 years to evolve an engineer clever enough to understand the mathematical theories of today and deduce what they're good for.

I don't know of any such theories at the moment, although I suppose you could talk about the gaps of theories of the past. I've heard it said that the complex plane languished for quite a while in theory until someone decided that it could be used to describe electromagnetic reactions in a single equation (I cannot vouch for this). Non-euclidean geometries might be another example, or the ability to factor nigh-infinite numbers.

### Re: Useless Math

Posted: Mon Dec 06, 2010 1:44 am UTC
Fractals were once believed to have no practical application, I think.

### Re: Useless Math

Posted: Mon Dec 06, 2010 3:10 am UTC
(I'm contrasting Aristotle, who would say that mathematical objects are just real objects, just thoguth of abstractly, as opposed to as objects)

Not sure if this could be important, but the math Aristotle would have been familiar with is(obviously) nothing like the modern stuff you could talk about.

### Re: Useless Math

Posted: Mon Dec 06, 2010 3:19 am UTC
mmmcannibalism wrote:
(I'm contrasting Aristotle, who would say that mathematical objects are just real objects, just thoguth of abstractly, as opposed to as objects)

Not sure if this could be important, but the math Aristotle would have been familiar with is(obviously) nothing like the modern stuff you could talk about.

That's actually part of my point. Most of his familiarity with math was what became Euclidean geometry, which was originally invented to describe the perceptible world, which led to his idea that it is dependent on the world.

### Re: Useless Math

Posted: Mon Dec 06, 2010 3:27 am UTC
The idea of a non-measurable set has absolutely nothing to do with the real world.

Foundational topics like set theory also have very little in the way of applications. (For example, if the continuum hypothesis were provable, what would that change?)

I think it is also pretty safe to say that the digits of pi beyond, say, the trillionth have nothing to do with the real world.

### Re: Useless Math

Posted: Mon Dec 06, 2010 3:54 am UTC
Atmosck wrote:(As a side note, maybe my inability to find any examples means I need to rethink my thesis.)
This. Because this:
Tirian wrote:Rules involving things like numbers and shapes seem to have been very clearly created and investigated precisely because they are good models for real-world phenomena.

You can argue that they exist independent of the real world all you want, but the fact that the mathematical objects we tend to care about are related to some aspect of the world we live in is no accident.

### Re: Useless Math

Posted: Mon Dec 06, 2010 4:20 am UTC
I think Non-Euclidian metrics and Lorentz manifolds and whatnot had been studied in depth long before Relativity stepped in and made them useful for real-world applications...

### Re: Useless Math

Posted: Mon Dec 06, 2010 8:30 am UTC
Regarding fractals, I was under the impression that some of the first applications of fractal geometry were initiated by Mandelbrot himself in trying to model the change in the price of cotton. He also worked hard (as did many others) in trying to explain a lot of real world phenomena through fractal geometry from things like how a tap drips with minimal water flow, how noise is distributed in telephone wires to how stars are distributed in the universe. In general the theory worked pretty well in describing these very complex systems. It really is an incredible theory for describing the emergence of complex dynamics from simple underlying laws.

I'm not really sure on the applications of modern topology (and algebraic topology), I think there are uses in particle physics and string theory, but I'd guess those applications weren't found for a few decades after Poincare et al. started studying abstract homotopy and homology theory.

### Re: Useless Math

Posted: Mon Dec 06, 2010 9:28 am UTC
Yes, there is 'useless' math. Almost all of it in fact, because there are infinitely many possible statements, and they simply cannot all apply to the real world.

Or we could do something intentionally silly, like removing the set of computably enumerable Reals from all the Reals. We are left with a set the same size and density as the Reals, but just about any number you might actually want isn't there. (Where a computably enumerable real is a real number N such that there exists a Turing Machine M, which, for every digit D of N, there is some natural number K such that by step K, M has printed the correct digit of N at position D of the tape, and for all steps after step K, position D does not change again. This is a very descriptive system.)

### Re: Useless Math

Posted: Mon Dec 06, 2010 1:51 pm UTC
Almost all mathematical concepts had some application in mind when they were defined. Sometimes the application was indirect; they were applied only to another branch of pure mathematics, and were two or three (or more) steps removed from practical applications.

I'm thinking in particular of imaginary numbers and complex numbers. They were invented in order to solve cubic equations - a pure maths application. The direct practical applications (e.g. AC circuits and quantum wavefunctions) came about 3 centuries later.

Even then, complex numbers were indirectly "useful" right from the start, because there are practical applications for cubic equations.

If anything, pure mathematics tends to be more "useful" than applied. A typical branch of applied mathematics has only one application; a typical branch of pure mathematics has thousands.

If I had to pick an example myself, I'd choose the Turing machine. It was invented in order to solve the Entscheidungsproblem, a theoretical problem in mathematical logic which - at the time - was the branch of mathematics with no known application, of purely philosophical interest. It led to the invention of the computer.

### Re: Useless Math

Posted: Mon Dec 06, 2010 4:18 pm UTC
WarDaft wrote:Yes, there is 'useless' math. Almost all of it in fact, because there are infinitely many possible statements, and they simply cannot all apply to the real world.

You're assuming:

1: The universe is either finite or periodic
2: There is a finite number of mathematical statements that can describe any given situation.

IMHO, it's not a matter of whether a piece of mathematics is useful, but rather whether we have found a use for it yet.

### Re: Useless Math

Posted: Mon Dec 06, 2010 9:16 pm UTC
undecim wrote:You're assuming:

1: The universe is either finite or periodic
2: There is a finite number of mathematical statements that can describe any given situation.

IMHO, it's not a matter of whether a piece of mathematics is useful, but rather whether we have found a use for it yet.

Actually, I'm not assuming the universe is finite. I'm just assuming that it's not larger than the reals. There's no reason that infinitely large mathematical statements can't be defined. If math can describe infinities of such and such size, then there can be strings that long, and some of those strings can be valid mathematical statements... which can then describe even larger infinities.

That is, there should be more than [imath]\beth_1[/imath] well formed mathematical statements... more than [imath]\beth_2[/imath], [imath]\beth_3[/imath], or [imath]\beth_{\beth_{\beth_{27}}}[/imath] statements... and I'm not giving the universe *that* much credit without at least a little evidence.

The universe would have to be really... really... really big to compete with math in this regard.

### Re: Useless Math

Posted: Tue Dec 07, 2010 7:42 am UTC
Seconded. In that vein, I would say large cardinals are something indisputably useless. Half of these examples seem to be phrased "well people totally thought x was useless . . . until it turned out to be the best thing ever!!" but I can assure you [imath]\kappa[/imath]-hyper-inaccessible cardinals will engender no such defense.

### Re: Useless Math

Posted: Tue Dec 07, 2010 9:43 am UTC
It would be completely awesome if that turned out to be wrong.

### Re: Useless Math

Posted: Tue Dec 07, 2010 11:30 am UTC
On a more philosphical level, you might be interested in Paul Watzlawicks work. Especially his introduction to "Pragmatics of Human Communication" where he does argue your point that mathematical expressions are purely abstract. A psychologist/philosopher may be an umcommon source for a math-related topic but the guy really does give some interesting points on the more philosophical side of math in his work.

-- edit for typos, etc.

### Re: Useless Math

Posted: Tue Dec 07, 2010 8:20 pm UTC
Nic wrote:Seconded. In that vein, I would say large cardinals are something indisputably useless. Half of these examples seem to be phrased "well people totally thought x was useless . . . until it turned out to be the best thing ever!!"; but I can assure you [imath]\kappa[/imath]-hyper-inaccessible cardinals will engender no such defense.

Hey now, we've used those in the Large Number thread down in Forum Games! Also, they could be used in proofs of highly abstract concepts which lead to slightly more mundane things becoming provable, and all in all have a not easily predicted amount of trickle down to seemingly unrelated problems... once we install our brains onto super computers and can actually work with these kinds of things.

However, if we have some ordinal O for which the size of the smallest equivalent definition (where the axioms of the system are considered part of the size of the definition) of the size of the smallest equivalent definition of ordinal O is a [imath]\kappa[/imath]-hyper-inaccessible cardinal, then no, that ordinal will assuredly never have any non-trivial use no matter how smart people get.

And if somehow, just to spite us, someone finds a use for such a silly thing, there's still always the set of non-computably-enumerable reals.

### Re: Useless Math

Posted: Tue Dec 07, 2010 11:02 pm UTC
I was under the impression that all the numbers in the large number game were finite. Has it moved on to a largest cardinal game or something?

### Re: Useless Math

Posted: Wed Dec 08, 2010 4:09 pm UTC
No, but there are things you can do with infinities that result in a finite quantity of some particular size.

Atmosck wrote:I'm running into trouble finding examples, though, and a total lack of them would make my argument very weak.
Only if it's already a poor argument. All the historical examples you're looking for will necessarily be things humans came up with and chose to study. They'll depend much more on what those humans were interested in finding out than it will on the metaphysical status of mathematics, and therefore won't be of much use to you when you're arguing about the metaphysical status of mathematics.

### Re: Useless Math

Posted: Wed Dec 08, 2010 5:54 pm UTC
One issue is that even if math is useless (doesn't correspond to anything "real"), it might be sufficiently close to something that is useful that it can be used as a simplified model of the useful thing.

And the "more simple" one might be much, much easier to work with than the one with all the hairy details.

So you end up with a proof that all numbers have property X, even if you only care about numbers less than 10^100 in practice. Being able to prove it true for all numbers might be much, much easier than checking each number up to 10^100. What more, we might prove it true for all ordinals using, along the way, theorems that require assumptions about inaccessible cardinals to be sufficiently simple. This is massive overkill -- but still easier than checking 10^100 cases in some cases!
MartianInvader wrote:I was under the impression that all the numbers in the large number game were finite. Has it moved on to a largest cardinal game or something?

Ordinals are about ways of ordering things, in some sense.

Ways of ordering things can be turned into ways to recursively call functions on themselves. Even if the ordinal is an infinite ordinal, the mapping can result in a finite number of self-calls (in some sense, infinite components correspond to "x", the input variable, as opposed to a fixed value). Higher orders of ordinals end up corresponding to higher orders of recursion, in some sense.

Recursively calling a function on itself in ridiculously complex and nasty ways can lead to fast growth. And by hooking into the ordinal hierarchy, you avoid having to check that your generated ridiculously complex recursion reaches its terminating case, so long as your mapping from ordinals to recursion is well designed.

Then all you need to do is pull out a large ordinal, use that to define a function, and call it with the value 3.

I did not follow it well enough to determine if inaccessible ordinals result in a merely finite number of recursions or not.

### Re: Useless Math

Posted: Sat Dec 11, 2010 7:59 pm UTC
Non-Euclidean geometry was developed before the applications were known. Gauss, Lobachevsky and Bolyai were concerned with creating a consistent geometry by rejecting the 5th postulate.

### Re: Useless Math

Posted: Tue Dec 14, 2010 5:57 pm UTC
How about the classification of finite simple groups? Some of the larger exceptional groups have no obvious relevance in the real world. Where in nature do you see an action of the Baby Monster? Nowhere that I know.

Topology is very abstract too. In a three dimensional universe (OK, maybe eleven dimensional depending on which physicists you believe) what is the meaning of a 127-dimensional sphere? And where would you see a map of high dimensional spheres?

On simpler level, do the complex numbers exist in reality? The mathematical concept is very useful, and lets us prove things about real numbers, but did you ever see 2-3i cows in a field?

### Re: Useless Math

Posted: Tue Dec 14, 2010 6:38 pm UTC
I did not follow it well enough to determine if inaccessible ordinals result in a merely finite number of recursions or not.
Technically, they did not, because they were being used to make larger versions of smaller ordinals that later would produce a finite number of recursions.

### Re: Useless Math

Posted: Tue Dec 14, 2010 8:27 pm UTC
bobdylan wrote:How about the classification of finite simple groups? Some of the larger exceptional groups have no obvious relevance in the real world. Where in nature do you see an action of the Baby Monster? Nowhere that I know.

Vertex algebras and conformal field theories.

Topology is very abstract too. In a three dimensional universe (OK, maybe eleven dimensional depending on which physicists you believe) what is the meaning of a 127-dimensional sphere? And where would you see a map of high dimensional spheres?

A 127 dimensional sphere is an object, not a mathematical theory. Topology is extremely important; I do gravity and this sort of thing is my bread and butter.

On simpler level, do the complex numbers exist in reality? The mathematical concept is very useful, and lets us prove things about real numbers, but did you ever see 2-3i cows in a field?

The phase in electrodynamics is described with complex number, and that leads to directly observable interference patterns.

### Re: Useless Math

Posted: Tue Dec 14, 2010 9:28 pm UTC
bobdylan wrote:
On simpler level, do the complex numbers exist in reality? The mathematical concept is very useful, and lets us prove things about real numbers, but did you ever see 2-3i cows in a field?

You also never see exactly two-sevenths of a cow. But people don't tend to think of fractions as being as "mysterious" as complex numbers, probably because we're more used to fractions.

Both fractions and complex numbers are useful in some real-world situations and not in others. (And each of those two systems of numbers is as consistent as the other.)

### Re: Useless Math

Posted: Tue Dec 14, 2010 10:37 pm UTC
bobdylan wrote:On simpler level, do the complex numbers exist in reality?
For that matter, do *any* numbers? Now you're just asking a philosophical question (similar to the one the OP wants to address), not a mathematical one.

### Re: Useless Math

Posted: Tue Dec 14, 2010 10:40 pm UTC
cf Kronecker to Lindemann: "What good your beautiful proof on [the transcendence of] π? Why investigate such problems, given that irrational numbers do not even exist?"

### Re: Useless Math

Posted: Wed Dec 15, 2010 10:09 am UTC
gmalivuk wrote:
bobdylan wrote:On simpler level, do the complex numbers exist in reality?
For that matter, do *any* numbers? Now you're just asking a philosophical question (similar to the one the OP wants to address), not a mathematical one.

Exactly. This is a philosophical question that happens to be about math. The search for an example is a part of a larger attack on the ontology of mathematical objects (numbers, sets, shapes etc.). What I'm trying to say is that they they (inclusive of irrational and complex numbers just the same as natural numbers) don't exist in the sense that objects exist, but exist as ideas in the minds of people, independent of whatever it is they describe. (to contrast aAristotle, who says that numbers and shapes exist In the things they describe.)

### Re: Useless Math

Posted: Wed Dec 15, 2010 10:25 am UTC
doogly wrote:
bobdylan wrote:How about the classification of finite simple groups? Some of the larger exceptional groups have no obvious relevance in the real world. Where in nature do you see an action of the Baby Monster? Nowhere that I know.

Vertex algebras and conformal field theories.

Topology is very abstract too. In a three dimensional universe (OK, maybe eleven dimensional depending on which physicists you believe) what is the meaning of a 127-dimensional sphere? And where would you see a map of high dimensional spheres?

A 127 dimensional sphere is an object, not a mathematical theory. Topology is extremely important; I do gravity and this sort of thing is my bread and butter.

On simpler level, do the complex numbers exist in reality? The mathematical concept is very useful, and lets us prove things about real numbers, but did you ever see 2-3i cows in a field?

The phase in electrodynamics is described with complex number, and that leads to directly observable interference patterns.

Yes, but the whole point is that many of these mathematical objects (the OP did say OBJECTS) are useful but are initially defined in abstract. The fact that phase can be described by the mathematical theory of complex numbers doesn't mean that the phase IS a complex number, and it's certainly not true that complex numbers were developed in order to study electromagnetism. I'd also be surprised (and delighted!) if you have much use for the Adams spectral sequence in your study of gravity.

I'm not trying to argue that mathematical constructs aren't useful in physics, of course they are, but I'm trying to give evidence to support the OP's thesis that this is often an accident.

That thesis might not be true of course; most pure mathematics is guided by the desire to solve existing problems, and the original source of these problems is presumably something in the real world.

### Re: Useless Math

Posted: Wed Dec 15, 2010 10:33 am UTC
phlip wrote:I think Non-Euclidian metrics and Lorentz manifolds and whatnot had been studied in depth long before Relativity stepped in and made them useful for real-world applications...

Non-Euclidean geometry is a great example, as it essentially arose from the quest to prove the parallel postulate from the other Euclidean axioms (which, of course, turned out to be impossible as it is essentially the axiom that makes Euclidean geometry Euclidean).

### Re: Useless Math

Posted: Wed Dec 15, 2010 1:49 pm UTC
Atmosck wrote:Exactly. This is a philosophical question that happens to be about math. The search for an example is a part of a larger attack on the ontology of mathematical objects (numbers, sets, shapes etc.). What I'm trying to say is that they they (inclusive of irrational and complex numbers just the same as natural numbers) don't exist in the sense that objects exist, but exist as ideas in the minds of people, independent of whatever it is they describe. (to contrast aAristotle, who says that numbers and shapes exist In the things they describe.)

"When was the last time you stubbed your toe on a seven?" - John Derbyshire.

I think it's worth your time to look at the history of many mathematical advances and see the motivations that were behind them. For example, complex numbers were first invented/discovered to solve cubic equations, but that's where there uses ended. Wessel's complex geometry was immediately used to derive trigonometric identities, most all of which were already discovered, IIRC. But whether either those are really "useful" is up to you.

### Re: Useless Math

Posted: Wed Dec 15, 2010 3:55 pm UTC
bobdylan wrote:. The fact that phase can be described by the mathematical theory of complex numbers doesn't mean that the phase IS a complex number, and it's certainly not true that complex numbers were developed in order to study electromagnetism. I'd also be surprised (and delighted!) if you have much use for the Adams spectral sequence in your study of gravity.

I'm not trying to argue that mathematical constructs aren't useful in physics, of course they are, but I'm trying to give evidence to support the OP's thesis that this is often an accident

What I would argue is that phase is a complex number in the same way that mass is a real number. If you want to say that both are accidental and just happen to be good models then I suppose that's fair. The argument that some math is less realistic than others is the only one I take issue with. I am in the Dirac camp, that beautiful mathematics leads to correct physics. (can also lead to incorrect physics so one has to check, but it's a good way to think of advances).

And I wish I knew what to do with spectral sequences. Tu taught a course out of his Differential Forms and Algebraic Topology that I attended, but I can't claim to be on very good terms with them. Physics will eventually catch up though, I'm sure.

I think an especially interesting case is knot theory. The impetus for its development was an idea of Kelvin's that different knots could model that different kinds of atoms. A proton neutron model turned out to work much better, but we still had this lovely knot theory worked out and well classified. Then next century we find loads of completed unexpected places where knots are essential. I'm sure Kelvin would be shocked that instead of nuclei, knots are actually perfect for some quantum fields.

I think a way to approach this sort of question without bringing up physics at all is to ask whether there are "coincidences" in math. Some properties of the number 4, for example, might be considered to be coincidences due to the law of small numbers. Maybe? But what about pi? Feynman would play a game in every formula with a pi in it, 'find the circle.' Are all instances of pi referring to the same thing? I'd say very much so. I can imagine pi defined the same way geometrically but the fascinating hexadecimal digit extraction formula for it not being true; it seems really odd and people certainly spent a long time not knowing about it, much less thinking it is essential. But now that it is known, of course it is essential! You can't have pi exist and not obey that formula. I can only imagine it being a sort of "side property" because I am good at cognitive dissonance and entertaining incorrect ideas. Too good maybe.

### Re: Useless Math

Posted: Thu Dec 16, 2010 6:26 am UTC
Or we could do something intentionally silly, like removing the set of computably enumerable Reals from all the Reals. We are left with a set the same size and density as the Reals, but just about any number you might actually want isn't there. (Where a computably enumerable real is a real number N such that there exists a Turing Machine M, which, for every digit D of N, there is some natural number K such that by step K, M has printed the correct digit of N at position D of the tape, and for all steps after step K, position D does not change again. This is a very descriptive system.)

Sorry for being slightly off-topic, but what exactly *is* a non-enumerable real? I've read the wikipedia article, but there are no examples of such numbers. It seems like the definition might be saying that there isn't a way to write one normally, but is there a known solution to a known equation that is non-enumerable? Or maybe some strange integral?

More on-topic: It seems to me, finally having some actual math under my belt (if only a little), that any kind of mathematics has to refer to some object. It doesn't have to be real, or have any properties at all related to reality, it could exist only in minds and on paper, but there has to be some object to perform operations on and to describe the qualities of. Otherwise, we are simply manipulating symbols on paper, and our conclusions are meaningless.

Not that we can't have seemingly nonsensical results, like Euler's identity and the Banach-Tarski paradox, but the results still have to refer to some object(s) that we can imagine. Obviously the objects and operations that BT requires are not feasible in the real world, but that's not what is important. What is important is that a "sphere" and "points" and "rotations" are things that can be imagined.
(Sorry if this doesn't seem clear, it's 1:00 AM here. If anyone wants I'll try to explain later).

I don't think it's a coincidence that many branches of mathematics were developed before they became useful outside of math. Someone mentioned number theory and its relation to code formation and cracking. I don't think it's a big surprise that number theory became useful--the integers, a central focus of number theory, relate directly to the real world. Even regarding more esoteric subjects, it makes a certain degree of sense that many of them are relevant to the real world. If mathematicians investigate a number of systems, and find some of them are not internally consistent, than those are probably going to get less study (well, at least it seems that way to me. Maybe I'm wrong). So internally consistent systems, which I'm guessing are more likely to describe real world systems, get more study, and hence generate real world applicable results.

Of course, this could all be nonsense.

### Re: Useless Math

Posted: Thu Dec 16, 2010 6:58 am UTC
pizzazz wrote:Sorry for being slightly off-topic, but what exactly *is* a non-enumerable real? I've read the wikipedia article, but there are no examples of such numbers. It seems like the definition might be saying that there isn't a way to write one normally, but is there a known solution to a known equation that is non-enumerable? Or maybe some strange integral?

Chaitin's constant is an example, for the same sorts of reasons that the halting function is uncomputable.

### Re: Useless Math

Posted: Thu Dec 16, 2010 1:13 pm UTC
Atmosck wrote:What I'm trying to say is that they they (inclusive of irrational and complex numbers just the same as natural numbers) don't exist in the sense that objects exist, but exist as ideas in the minds of people, independent of whatever it is they describe. (to contrast aAristotle, who says that numbers and shapes exist In the things they describe.)

Well, depending on your stance on the Mind-Body-Problem, ideas are "objects" (processes, rather, but physical... things), and they don't exist separately of what they "really" describe, namely, the patterns in your brain.

### Re: Useless Math

Posted: Thu Dec 16, 2010 1:19 pm UTC
doogly wrote:
I think a way to approach this sort of question without bringing up physics at all is to ask whether there are "coincidences" in math. Some properties of the number 4, for example, might be considered to be coincidences due to the law of small numbers. Maybe? But what about pi? Feynman would play a game in every formula with a pi in it, 'find the circle.' Are all instances of pi referring to the same thing? I'd say very much so. I can imagine pi defined the same way geometrically but the fascinating hexadecimal digit extraction formula for it not being true; it seems really odd and people certainly spent a long time not knowing about it, much less thinking it is essential. But now that it is known, of course it is essential! You can't have pi exist and not obey that formula. I can only imagine it being a sort of "side property" because I am good at cognitive dissonance and entertaining incorrect ideas. Too good maybe.

My favourite potential coincidence in math is the Goldberg conjecture. If it's true, is it because of some deep property of the prime numbers that we don't yet understand, or is it a coincidence?

Let's be vague: There's a postulate of someone or other that any "additive" theorem about the asymptotic distribution of the primes is true if and only if it's true for almost all sequences which have the same asymptotic density as the primes. Something like that . In other words, the primes are random when you look at them from a purely additive point of view.

On topic: Maybe what's most amazing is how quickly areas of math that were initially considered "pure", (let's say "not motivated by the study of physics or statistics") have been discovered to have applications. Almost universally. What that says to me is; the chain of problem->solution->theory->problem->solution->theory really doesn't go that deep. A new game for pure mathematicians to play would be: "What real world problem got me here, and how?"

### Re: Useless Math

Posted: Thu Dec 16, 2010 5:12 pm UTC
antonfire wrote:Chaitin's constant is an example, for the same sorts of reasons that the halting function is uncomputable.
Quite to the contrary. Chaitin's constant is ennumerable, but not computable.

There is a Turing machine which does not halt which will gradually contain more and more correct digits of Chaitin's constant, with no limit on the maximum number of digits it will eventually have correct... but there is no halting Turing machine which contains a representation of it.

Computably enumerable numbers are vastly more diverse than computable numbers. You can enumerate numbers computable by an oracle machine... or by an oracle-2 machine... or, if you have a well defined ordinal oracle hierarchy, each answering the halting problem for those at levels less than it, an oracle-δ for any computable ordinal δ.

### Re: Useless Math

Posted: Thu Dec 16, 2010 5:25 pm UTC
pizzazz wrote:Sorry for being slightly off-topic, but what exactly *is* a non-enumerable real? I've read the wikipedia article, but there are no examples of such numbers. It seems like the definition might be saying that there isn't a way to write one normally, but is there a known solution to a known equation that is non-enumerable? Or maybe some strange integral?

By definition there can't be an example of the type you're asking for. If there were a well defined equation (i.e. one that didn't use non-enumerable terms) that resulted in a non-enumerable solution, you could use various methods to approximate the solution to an arbitrary precision, and that would give you an enumeration.

Imagine an infinitely long string of perfectly, independently random digits. Such a string by definition could not be precisely represented by any finite representation, after all, not having any pattern that you can pick out and use to summarize the string is sort of the definition of randomness. And yet, if you were to use that string as the decimal expansion of a number, it would be a perfectly cromulent element of the reals. That's a non-enumerable real number.

### Re: Useless Math

Posted: Thu Dec 16, 2010 5:36 pm UTC
Is it still non-enumerable if you know the distribution rule for choosing each digit?