For those interested in constructive and intuitionistic approaches here Dummett's [0] Elements of Intuitionism is an extremely good read.<p>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.<p>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).<p>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.<p>[0] <a href="https://www.amazon.com/Elements-Intuitionism-Oxford-Logic-Guides/dp/0198505248" rel="nofollow">https://www.amazon.com/Elements-Intuitionism-Oxford-Logic-Gu...</a>
<i>"So, in Borel’s view, most reals, with probability one, are mathematical fantasies, because there is no way to specify them uniquely." (Paraphrasing, because there are only countably many possible math papers that might describe a number.)</i><p>I think Borel has confused names with things. The fact that we can only write down only countably many <i>expressions</i> for numbers doesn't mean that there are numbers that we may never write expressions for - only that in a single symbolic system we can't have expressions for all of them <i>at once</i>.<p>Besides, if you confuse extant with useful you might end up believing that some random large integers aren't "there!"
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.<p>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 whereas the real numbers come about from an abstract "completion" of (say) the rational numbers in a very specific mathematical sense.<p>My point is that deriding "imaginary numbers" (as many do) is total nonsense if you accept real numbers.
>In addition to this mathematical soul-searching regarding real numbers, some physicists are beginning to suspect that the physical universe is actually discrete [Smolin, 2000] and perhaps even a giant computer [Fredkin, 2004, Wolfram, 2002]. It will be interesting to see how far this so-called “digital philosophy,” “digital physics” viewpoint can be taken.<p>Here is how far: Everything written <i>in words</i> about the physical universe is, by necessity, discrete. Thus all information that can be encoded in human languages is discrete. Any non-discrete behavior of the physical universe which causes a change in the discrete information available to us, must, by assumption, have a component which is orthogonal to all of the prior discrete information (otherwise it is fully discrete). Since this component is independent of all previously available information, it looks like randomness.<p>In other words: from the viewpoint of a discrete (linguistic) observer, the behavior of a continuous universe looks identical to that of a discrete universe that contains random fluctuations.<p>What is interesting, then, is that <i>observationally</i>, our discrete observable universe is full of random fluctuations. Speculation as to their true continuous underpinnings is, however, unfalsifiable, unless the randomness itself can be made to disappear. I usually turn the question around: is it inconceivable that there would be a continuous universe with inhabitants that used a discrete language?<p>So:<p>>According to these ideas the amount of information in any physical system is bounded<p>"the amount of <i>observable</i> information in any physical system" -- any unobservable continuous information shows up as unpredictable changes in the observable information.
The author of this paper is Gregory Chaitin of Chaitin's constant fame, among other things (I didn't know that Kolmogorov complexity is also known as Chaitin-Kolmogorov complexity!)
> "the halting probability Ω, which is irreducibly complex (algorithmically random), maximally unknowable, and dramatically illustrates the limits of reason"<p>I really enjoyed the beauty of this statement.
For the parallel historical development of 'the continuum' in physics, I recommend this readable survey:<p>Paper:<p><a href="https://arxiv.org/abs/1609.01421" rel="nofollow">https://arxiv.org/abs/1609.01421</a><p><a href="https://math.ucr.edu/home/baez/continuum.pdf" rel="nofollow">https://math.ucr.edu/home/baez/continuum.pdf</a><p>Blog summary with discussion:<p><a href="https://johncarlosbaez.wordpress.com/2016/09/08/struggles-with-the-continuum-part-1/" rel="nofollow">https://johncarlosbaez.wordpress.com/2016/09/08/struggles-wi...</a><p><a href="https://johncarlosbaez.wordpress.com/2016/09/09/struggles-with-the-continuum-part-2/" rel="nofollow">https://johncarlosbaez.wordpress.com/2016/09/09/struggles-wi...</a>
Lawrence Spector (professor at CUNY, Manhattan) on this topic: <a href="http://www.themathpage.com/acalc/anumber.htm" rel="nofollow">http://www.themathpage.com/acalc/anumber.htm</a>
I like the idea of encoding answers to all questions, or for that matter all books written so far (or both, while we're at it), in one real number between 0 and 1. My favourite number, really.
"According to Pythagoras everything is number, and God is a mathematician. This point of view has worked pretty well throughout the development of modern science. However now a neo-Pythagorian doctrine is emerging, according to which everything is 0/1 bits, ... , God is a computer programmer, not a mathematician, and the world is a ... a giant computer" [p13 of the pdf]<p>If you only have one finger then zero and one <i>are</i> just as real (ahem) as numbers that arise naturally when you have 10 fingers. The 10 toes are a bonus. I doubt that we can really know what Pythagoras really thought but given some of the results attributed to him I think Chaitin does him a disservice.<p>Getting wound up over whether French is a sophisticated enough language to describe numbers and some of the odder consequences of allowing construction to equate existence will probably only lead to a headache.<p>As a civilian wandering on the outskirts of all this philosophical foot stamping, I believe there are a fair few pretty rigorous arguments out there that can't be denied by resorting to "it looks wrong, cos reasons" style illustrations in a 13 page pdf.
Richard's Paradox seems a bit shaky to me (p4): "Since all possible texts in French can be listed or enumerated"<p>Unless I have completely missed the point then he has simply stated a way to generate another member of the set of French texts which of course is part of that set and so on.<p>You can easily squint hard enough to generalize to all texts in all languages, now, earlier and possible then allow that grammar, spelling and so can be pretty slack. Now translate that lot into numbers in some way (a bunch of IT bods should be able to manage that!) To be honest French on it's own is probably more than enough.<p>"How very embarrassing! Here is a real number that is simultaneously nameable yet at the same time it cannot be named using any text in French."<p>The very act of naming the number (in French) constructs the French text that adds to the set of possible French texts.<p>I think that the set of possible French texts is exactly as large as the set of reals. So is the set of all language texts and that the "paradox" is merely trying to use the Cantor argument backwards.
This was a light and interesting read.<p>Here is the direct link: <a href="https://arxiv.org/pdf/math/0411418.pdf" rel="nofollow">https://arxiv.org/pdf/math/0411418.pdf</a><p>"Indeed, the most important thing in understanding a complex system is
to understand how it processes information. This viewpoint regards physical
systems as information processors, as performing computations. This
approach also sheds new light on microscopic quantum systems, as is demonstrated
in the highly developed field of quantum information and quantum
computation. An extreme version of this doctrine would attempt to build
the world entirely out of discrete digital information, out of 0 and 1 bits."<p>It would indeed be quite a blast to discover we are to a high degree of probability in a simulation.
Infinity is a weird thing, isn't it?<p>Now: one of the proofs in the paper relied on an assumption that all possible computer programs are countable, which I think implies that they are finite in length. But it is fairly trivial to generate computer programs that are infinitely long, say by assigning characters or expressions in some language to the digits of transcendental numbers such as pi. It is also possible to generate infinitely many such programs, simply by using pi/2, pi/3... etc.<p>Now, the proof as presented fails, since these programs cannot be ordered by size.<p>Can the proof be modified to take account of this? I don't know... comments invited.
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.<p>And again, even the point about describing the world in 1s and 0s at the end seems to me to be repetitive of Whitehead & Russell's Principia Mathematica which (and please correct me) used logic to construct the integers?
Note that the probability of randomly picking a rational number from [0,1] is <i>exactly</i> 0. That is, with P=1 you will end up with an irrational number, not representable in any modern computer.
Numbers? Real?<p>Neither molecular biology nor the sane part of physics has any of em.<p>Btw, it is heuristic - if there are numbers involved then it is human made. Reality as it is has no such notion. Biology does not count.<p>Numbers require an observer, which is a by-product of the processes in vastly complex brain structures of the cortex, and cannot be the basis of anything in the underlying universe.<p>Any good (which means Eastern) philosophy arrived at these simple conclusions millennia ago.
Richard's Paradox for some reason reminded me of the proof that there is an infinite number of interesting whole numbers. The proof goes like this. Assume instead that the number is in fact finite. Consider the first number higher than any of the interesting numbers, and thus bounding them. Now that's an interesting number! QED.
Just wanted to thank the OP, caustic, for this. I first thought it might be past my available background/resources for a casual read. But I found it accessible and rewarding.
So is there full agreement on what a number is, in the first place?<p>Some would argue that PI is not an actual number; but that it is a concept, like infinity
Are they real? Well, when a human says something about a thing, they can never be completely sure whether they're saying something about an actual thing with an independent existence, or whether they're only saying something about what they say.
i think it's no coincidence Godel's proof of uncountable reals, comes around the same time as Lebesgue integration. As mathematicians started exploring what the serious use of Fourier series
Unrelated, but I read a article a while back which said something similar, but it was based on the fact that our entire mathematical system is designed around "base 10" and as such is only relevant in many constructs to our specific human 'ten fingered', interpretation of math.
1. Given any two real numbers on the real number line, you can find another real number between those two points.<p>2. The Planck length is the smallest unit of distance with any meaning.<p>3. The universe has finite diameter.<p>Discuss.<p>4. For extra credit: Given 2 and 3, above, it follows that both the diameter and circumference of the universe can be expressed in Planck lengths as integers with a finite number of digits. Discuss the concept that Pi is a ratio of two finite integers.
This entire subject is very academic and theoretical, and will never impact anything in the real world. Using Big Fractions instead of floating-point numbers is a far more concrete argument with definite real-world impact!
<a href="https://news.ycombinator.com/item?id=13855198" rel="nofollow">https://news.ycombinator.com/item?id=13855198</a>