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How real are real numbers? (2004)

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Re: How real are real numbers? (2004)

#171

Post by CuriouslyC »

CogitoCogito wrote: post #160
CuriouslyC wrote: post #154

Earlier quoted context omitted.

Infinite values (e.g. the density of a black hole, the size of the universe) and infinitesimals (e.g. continuous space-time) are assumed to be real, rather than inaccurate but useful approximations.

Assumed to be real by whom? Not all physicists believe the same thing. Also why is this unfortunate? Why does it matter if they do or do not "believe" it? Are they able to make useful predictions with their models? Are they able to better understand physics? If so, why worry about their personal beliefs?

If we want to put physics on the faith table, I'm cool with whatever anyone wants to believe, and more power to them. But you can't be objectively right on the faith table - that's the price of admission.

If we want to be "true" and fully rational then we need to try to accurately represent our degree of knowledge about the world. Thus we shouldn't be making strong statements about things with an absence of evidence.

Re: How real are real numbers? (2004)

#172

Post by tejon »

CuriouslyC wrote: post #154
CogitoCogito wrote: post #130

Earlier quoted context omitted.

What do you mean by this statement?

Infinite values (e.g. the density of a black hole, the size of the universe) and infinitesimals (e.g. continuous space-time) are assumed to be real, rather than inaccurate but useful approximations.

Not a physicist but I thought the Planck length specifically denies infinitesimality? And one of the proposed solutions to the black hole information problem is that due to local relativistic effects, they never actually reach singularity in finite time.

Re: How real are real numbers? (2004)

#173

Post by [deleted] »

chriswarbo wrote: post #155
throwaway5752 wrote: post #137

Doesn't this basically rehash stuff covered 100 years ago by Hilbert, Whitehead & Russell, and Godel? If it wasn't such an eminent author, I would give it a pretty solid eye-roll. As some other poster noted in a link they provided, "pi" and "e" - among the uncountably infinite transendentals - are probably reasonable responses to this article in it's entirety. And again, even the point about describing the world in 1…

> "pi" and "e" - among the uncountably infinite transendentals - are probably reasonable responses to this article in it's entirety pi and e are computable ; we even know a bunch of programs which will compute them (to arbitrary precision; Google for "spigot algorithms"). Computable numbers are countable, so we can identify them with the natural numbers, and there's no need for "reals". For example, we might think ab…

[deleted]

Re: How real are real numbers? (2004)

#175

Post by tel »

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 "free choice sequences". These are essentially "unpredictable" sequences of potentially infinite binary choices. As there is no a priori reason to believe that they can be predicted they are able to be much larger than what is computable and thus can be used to give a characterization of the reals. Atop this you build a constructive understanding of free choice reals which behaves very nicely (at least foundationally... it points out all kinds of weirdnesses about what we assume classically to be the structure of reals).

What's very nice about this solution is that it sidesteps the difficulty. Free choice is a weaker thing to ask for than finite/constructive reals, but finite/constructive reals could be transparently encoded into free choice sequences and all the math would just work.

[0] https://www.amazon.com/Elements-Intuitionism-Oxford-Logic-Gu...

Re: How real are real numbers? (2004)

#176

Post by CuriouslyC »

alexbecker wrote: post #167
CuriouslyC wrote: post #120

Earlier quoted context omitted.

Wouldn't it be more accurate to describe epsilon as an infinitesimal?

There are no infinitesimals in the reals. You can define a mathematical structure containing infinitesimals [0], but it's not the reals. [0] Specifically, the dual numbers: https://en.wikipedia.org/wiki/Dual_number

Interesting. How do you describe the difference between two successive real numbers?

Re: How real are real numbers? (2004)

#177

Post by johncolanduoni »

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?

Although our mathematical systems rely on the existence of computable procedures (that halt) to e.g indicate whether a formula is well formed or verify a proof, they have no trouble talking about uncomputable functions. There actually ends up being a hierarchy, as once you allow "solve the halting problem for this Turing machine" as an operation the expanded set of computable functions is now unable to solve its own halting problem: https://en.wikipedia.org/wiki/Arithmetical_hierarchy

> These mathematical objects do not have to "exist" for them to be useful. I think there is a whole lot of new and interesting math that would be unlocked by dispensing with our logical blinders.

These kind of things are actually pretty well studied. Some interesting examples are:

* The hyperreal numbers which include infinitely many distinct numbers which are infinite or infinitesimal: https://en.wikipedia.org/wiki/Hyperreal_number

* The surreal numbers which include all ordered fields as a subfield (reals, complex numbers, hyperreals, etc.): https://en.wikipedia.org/wiki/Surreal_number

* The quaternions and octonions, which along with complex numbers are the only finite-dimensional algebras over the real numbers that can have "division" and "absolute value" in the usual sense: https://en.wikipedia.org/wiki/Hurwitz%27s_theorem_(compositi...

In general, mathematicians are really good at investigating things of the form "Okay we have structure X with properties A, B, C. What if we get rid of C? How about B? How about B and a weaker version of A? ...".

Re: How real are real numbers? (2004)

#178

Post by sjolsen »

nine_k wrote: post #168
sjolsen wrote: post #165

Earlier quoted context omitted.

>The fact that we can only write down only countably many expressions for numbers doesn't mean that there are numbers that we may never write expressions for It depends, as they say, on what the definition of is is. What does it mean for a number to exist if it cannot be described? What does it mean for a construction to exist if it cannot be constructed? >if you confuse extant with useful you might end up believing…

The standard delta-epsilon type of reasoning crucially depends on existence of arbitrary reals. Many geometry proofs / lines of reasoning crucially depend on the ability to position a point on a line at an arbitrary distance from another point. All numbers ever written are rationals and thus countable. But those endless irrational numbers make a lot of ways of reasoning simpler, or possible at all.

>The standard delta-epsilon type of reasoning crucially depends on existence of arbitrary reals.

What do you mean by "arbitrary?" Do you mean uncomputable? How does analysis require the existence of uncomputable reals?

>All numbers ever written are rationals and thus countable

It is also possible to "write" computable irrational numbers, more or less by definition.

Re: How real are real numbers? (2004)

#179

Post by nilkn »

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…

If by real plane you just mean the standard vector space R^2, the answer is algebraic rather than topological. R^2 as a vector space has an addition operation, but no multiplication operation. The complex plane is obtained by simply picking the appropriate multiplication operator.

In general, completing the rationals into the reals is more complex than constructing the complex plane from the real numbers. For the latter, you just need to adjoin a single element (sqrt(-1)), enforce existing arithmetic rules, and the rest falls into place. For the former, you can't just adjoin a single new element like sqrt(2). Doing so will get you the ring (actually field) Q[sqrt(2)], but not R.

If you take R and adjoin two special new elements (sqrt(-1) and the point at infinity), you do obtain a topologically different result: the Riemann sphere. This sphere is in many ways the more natural domain for complex analysis than the complex plane.

Re: How real are real numbers? (2004)

#180

Post by bzbarsky »

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…

They're also isomorphic as abelian groups under addition. At least if you assume the axiom of choice. See eg. http://math.stackexchange.com/questions/925706/is-it-true-th...
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