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How real are real numbers? (2004)

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Re: How real are real numbers? (2004)

#221

Post by scatters »

prmph wrote: post #151
CogitoCogito wrote: post #40

The continuity of real numbers provides a clean theoretical basis for continuity of functions. I personally view it as more of a theoretical tool that seems to do pretty well and instead steer clear of the philosophical questions. One thing that I think is important to note though is that the jump from real numbers to complex numbers is nothing compared to the jump from integers/rational numbers/etc. to real numbers.…

An interesting idea that follows from this: what other kinds of "numbers" might we come up with if we relax our logical blinders? I have this concept of "materialization" and wonder if there is a formal mathematical term for it. Complex numbers are actual, in the sense that they can be used in calculations that finally would give us a number we can make sense of (materialization), even if we cannot actually imagine a…

> What if we propose the existence an "imaginary" algorithm (call it omega, if you like) that can decide the halting problem?

This is called an oracle (https://en.wikipedia.org/wiki/Oracle_machine). We can posit oracles for solving computable problems in constant time (e.g. factoring the product of two arbitrarily large primes) as well as for solving uncomputable problems (e.g. halting problems).

Oracles are a great tool for studying complexity and computability, since oracle machines have their own complexity and computability limits; an oracle machine for the halting problem can determine whether a simple Turing machine will halt, but cannot determine whether it itself will halt (this is called a Turing jump). Thus oracle machines for halting problems form a class hierarchy, which Post's theorem shows is precisely the arithmetic hierarchy.

Re: How real are real numbers? (2004)

#222

Post by pron »

tel wrote: post #175

For those interested in constructive and intuitionistic approaches here Dummett's [0] Elements of Intuitionism is an extremely good read. Intuitionism is a form of a constructive foundation for mathematics which (a) notes that any attempt to deny the uncountability of reals leads to difficulties and (b) any attempt to internally define them violates constructivity. The resolution proposed is to posit the existence of…

It sidesteps the difficulty by making the math itself (at least for the time being) much more difficult, and that is the reason it was rejected by most mathematicians in the Hilbert/Brouwer debates. Because here's the question: suppose you can't philosophically justify the "existence" of the real numbers, yet they coincide perfectly with observation and result in math that is much simpler than constructive math. Should you reject them? Brouwer -- who was the first to recognize just how much ordinary math relies on non-constructive principles -- said yes, because non-constructive math is philosophically wrong, period. Hilbert -- who was a finitist -- instead suggested that the propositions of math come in two flavors: real propositions, those that are finitary and can be taken to say something about physical reality, and ideal propositions, that are not. He said that as long as the ideal propositions are consistent with the ideal ones, they should not be rejected on a priori philosophical grounds even if no finitary meaning can be assigned to them. I.e. they are philosophically justified after the fact by virtue of their consistency with the real propositions. This philosophical classification of mathematics into "real" and "ideal" is called formalism, because it does not require that the ideal propositions be assigned a finitary meaning beyond their formal statement (as a finite string of characters).

Of course, most mathematicians are not finitist, so they require neither intuitionism nor formalism -- both essentially finitist philosophies -- and are Platonists, believing that even ideal objects that are beyond physical reality and computation have a "real existence" in some Platonic sense.

BTW, I think that after Turing (who used Brouwer's choice sequences in his construction of computable numbers) it is no longer necessary to rely on "free choice" (or lawless) sequences because of the halting theorem, and both lawlike and lawless sequences can be unified, and Brouwer's "creating subject" identified with a Turing machine. But I'm not sure about that. Turing -- who was a mathematical philosopher himself -- rejected any dogmatic a priory philosophy of mathematics, except for common sense, as the one true foundation, and suggested that the value of a formal system be derived not from its a priori philolosphy but from its ad hoc utility.

Re: How real are real numbers? (2004)

#223

Post by bjourne »

jonreem wrote: post #187
hzhou321 wrote: post #181

Earlier quoted context omitted.

I am not a mathematician (physicist). I think the concept of infinity is a con that mathematicians have pulled on us (as there isn't an easy reality to map on to). I can understand arbitrarily big set; however, I never managed to make the jump from arbitrarily finite to infinity. Mathematicians made that jump and glossed over, then continue to show the difference between countable infinity and infinity beyond. Since…

A set being "countably infinite" only means that you can write a function that maps each distinct entry in the set to exactly one natural number (0, 1, 2, etc.) without duplicates. That's it. So for example, the set of natural numbers is countably infinite and we know this because we can write a function that maps each natural number to exactly one natural number: the id function. We can extend this and say that the…

Here is how you do it. Have a function p(r) which evaluates to the previous real number. Then your mapping function is:

    f(r) = if (r == 0) { 0 } { else f(p(r)) + 1 }
If your objection is "You can't determine what the previous real number is." Then my counter-objection is "Please prove that you can't." Which I don't think is possible without first assuming reals are uncountable.

Re: How real are real numbers? (2004)

#224

Post by jacobolus »

esoterica wrote: post #217
hzhou321 wrote: post #181

Earlier quoted context omitted.

I am not a mathematician (physicist). I think the concept of infinity is a con that mathematicians have pulled on us (as there isn't an easy reality to map on to). I can understand arbitrarily big set; however, I never managed to make the jump from arbitrarily finite to infinity. Mathematicians made that jump and glossed over, then continue to show the difference between countable infinity and infinity beyond. Since…

> Since you never can finish counting You seem to have serious trouble wrapping your head around the concept of words having different meanings in a layman context and a technical context. Just because you can't "count" (in the counting-out-loud-on-your-fingers sense) the naturals doesn't mean they aren't countable in the rigorous, set-theoretic sense. Your entire argument basically boils down to "this word means som…

I downvoted and flagged your comment. Please don’t bring gratuitous personal attacks here.

Re: How real are real numbers? (2004)

#225

Post by jacobolus »

j2kun wrote: post #97
CogitoCogito wrote: post #40

The continuity of real numbers provides a clean theoretical basis for continuity of functions. I personally view it as more of a theoretical tool that seems to do pretty well and instead steer clear of the philosophical questions. One thing that I think is important to note though is that the jump from real numbers to complex numbers is nothing compared to the jump from integers/rational numbers/etc. to real numbers.…

> The complex numbers come about by simply adding one dimension Complex numbers are also best thought of (in my opinion) as an abstract completion of the reals under the operation of taking roots of polynomials. Because otherwise, what would be the difference between the real plane and the complex numbers? They are topologically identical, after all. There has to be something more substantial than simply adding a dim…

Complex numbers are (isomorphic to) a certain subset of linear operators on a 2-dimensional Euclidean vector space which rotate and/or scale the vectors in the plane without skewing or anisotropically stretching them. (We usually also include a zero operator here; depending on use case we sometimes omit it (the “punctured plane”), or sometimes add a point at infinity (the “Riemann sphere”).)

If you want you can write them down as matrices acting on vectors in an orthonormal basis by matrix multiplication:

  [a -b]
  [b  a]
Or if you prefer you can consider i to be a unit bivector in the plane, with a complex a + bi acting on vectors by Clifford’s geometric product.

Or if you want you can write them using a length and an angle measure, and use high school trigonometry to figure out how they apply to vectors.

The difference between the real plane and the complex numbers is that for two vectors in the plane, the product is not a vector. (Indeed, if you use Clifford’s geometric product, then the product of two vectors is a complex number (scalar + bivector).)

Re: How real are real numbers? (2004)

#226

Post by FabHK »

kazagistar wrote: post #141
gertef wrote: post #81

Earlier quoted context omitted.

The encoding of all books written so far (and will ever be written in finite time), is a rational number. Don't need Reals

Thats kinda the whole point of the article, in fact. All possible encodings of all possible thoughts, books, formal systems, and whatever, fit into the rationals, and the reals are categorically outside that.

All books written and will be written (in finite time) - yes, rational.

All possible questions (infinitely many) - no, that would be a non-terminating non-periodic binary, right. From the article:

> 2.4 Borel’s know-it-all number

> The idea of being able to list or enumerate all possible texts in a language is an extremely powerful one, and it was exploited by Borel in 1927 [Tasi ́c, 2001, Borel, 1950] in order to define a real number that can answer every possible yes/no question!

> You simply write this real in binary, and use the nth bit of its binary expansion to answer the nth question in French.

> Borel speaks about this real number ironically. He insinuates that it’s illegitimate, unnatural, artificial, and that it’s an “unreal” real number, one that there is no reason to believe in.

Re: How real are real numbers? (2004)

#227

Post by codethief »

Animats wrote: post #218
CogitoCogito wrote: post #40

The continuity of real numbers provides a clean theoretical basis for continuity of functions. I personally view it as more of a theoretical tool that seems to do pretty well and instead steer clear of the philosophical questions. One thing that I think is important to note though is that the jump from real numbers to complex numbers is nothing compared to the jump from integers/rational numbers/etc. to real numbers.…

The continuity of real numbers provides a clean theoretical basis for continuity of functions. Exactly. Real numbers are an invention to make mathematics simpler and cleaner. If you don't have continuous reals, it takes extensive case analysis to prove relatively simple things for, say, a binary floating point representation. It's possible to do that; Boyer and Moore did work like that to formalize floating point and…

> But today, it seems that time, length, and mass are all quantized. There's thus not a physical basis for the existence of reals.

This is actually not the case. Space and time are not regarded as quantized by the majority of physicists and there's also absolutely no evidence suggesting that. To the contrary, such quantization would lead to inconsistencies – e.g. with relativity – pretty quickly.

Fur further reading:

https://physics.stackexchange.com/questions/9076/does-quantu...

https://physics.stackexchange.com/questions/67899/is-time-qu...

Re: How real are real numbers? (2004)

#228

Post by hzhou321 »

drdeca wrote: post #216
hzhou321 wrote: post #197

Earlier quoted context omitted.

Why should you get to choose your real number -- specifying more and more digits -- implies an un-exhaustive process, while I am not allowed to do the same? An unfair game will have unfair winners, how would it mean anything? If we both are allowed an un-exhaustive process of specifying what we have, this goes back to the counting game. As we never can finish, how does it make countable (or not)?

The reason that the real number being defined can be defined in terms of the function is because, that is the situation being considered in the statement. For any way of doing the mapping, there is a real number that the mapping misses. Alternative statement: "there is no such mapping that doesn't miss any of the reals". This is shown because, given a mapping, I can find a real that the mapping misses. When one says…

Given any finitely described real numbers, there will be a finitely described mapping map it to a unique natural number. But if you use real numbers that can never finish describing, we have to use a mapping that cannot be finished in describing. That is not the same as "there is no such mapping that doesn't miss any of the reals". If all mappings that cannot be finitely described are excluded, what will be the reason? And why that reason cannot be applied to exclude some real numbers (that cannot be finitely described) as numbers? I understand that is what it is, but being what it is seems meaningless.

The general halting problem is uncomputable. It can only be simulated. However, any program is still assumed to be finitely described. Any discussion in finite domain (including arbitrarily big finite) cannot lead to conclusions or insight toward infinite.

> I don't know if you are using the word "countable" in the standard way, so I don't know what you mean by that last sentence.

In the context of infinity, words such as countable, bigger, order, etc. all lose its standard meaning. We don't really know what it means if we don't really know what infinity is. Mathematicians simply made a definition to countable here -- a finitely described mapping -- that is fine on its own, but completely useless. Since we can't draw any parallels from infinity to finite (including arbitrarily big), we can't really relate any definitions over the infinity to "standard" meaning of those words to the domain of finite.

Re: How real are real numbers? (2004)

#229

Post by hzhou321 »

zeeboo wrote: post #211
hzhou321 wrote: post #194

Earlier quoted context omitted.

Since it is non-terminating, you didn't really prove every item gets counted after finite time, did you? Refer to my other reply, you asserted a requirement of predefined (describable) counting scheme here. Why that requirement has any relevance here (in the context of infinity)?

If I start at 0 and successively add 1, do you agree that I eventually hit any positive integer you could pick after a finite number of steps? Does that not prove to you that I hit every positive integers? Which one do I not hit?

You will only eventually hit any given integer after finite steps. It does not prove you will hit every positive integer. You'll miss those that one never can finish giving you -- for example, I'll start with digit 1, and I'll infinitely adding 1 behind it (never finishing). It is an infinite natural number that you can't hit within any finite number of steps.

Re: How real are real numbers? (2004)

#230

Post by cvoss »

hzhou321 wrote: post #194
cvoss wrote: post #186

Earlier quoted context omitted.

I find it helpful to think of countability in terms of the following game: You have a set of items in mind. You propose a (non-terminating) scheme for listing all the items in the set. An adversary attempts to name any item X, hoping your scheme misses it. However, you then show your scheme does, in fact, get to X after a _finite_ amount of time. The set is said to be countable if you prove that your adversary cannot…

Since it is non-terminating, you didn't really prove every item gets counted after finite time, did you? Refer to my other reply, you asserted a requirement of predefined (describable) counting scheme here. Why that requirement has any relevance here (in the context of infinity)?

> you didn't really prove every item gets counted in finite time

The proof that the adversary cannot name such an item is logically the same as a proof that every item gets counted. This is a fundamental logical truth, namely de Morgan's law for universal/existential quantifiers: (not exists x such that P(x)) is the same as (forall x, not P(x))

> Why that requirement has any relevance here

'A counting scheme' is an intuitive way of providing a 'one-to-one correspondence to the natural numbers,' which is used in the technical definition of countability.

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