Khinchin's Constant
mathworld.wolfram.com
mathworld.wolfram.com
Or as Wikipedia puts it:
> Although almost all numbers satisfy this property, it has not been proven for any real number not specifically constructed for the purpose.
There is no uniform probability distribution over all real numbers.
The familiar real numbers, those that people (including mathematicians) actually use, and that our machines use, are atypical of the set of real numbers as a whole in many important respects. The real numbers are a formal abstraction to an extent that few people realize.
Our computers are designed to compute finite values using instruction sets that are explicitly designed to be precise, which has the side effect of these instruction sets and encodings being highly brittle and un-evolvable. This is what makes things like neural nets and genetic programming hard-- making these systems work involves creating virtual anti-computers within our anti-biological existing ones that are fuzzy, mutable, evolvable, etc.
I speculate that this is because we've built computers explicitly to do things we are not good at: crunching massive amounts of data very precisely. We are pretty good at more "organic" forms of cognition so there was little evolutionary "market need" for us to augment ourselves in that way.
Analog computers can be indeed be used for these types of problems, and their usage in the field predates their digital siblings. Sampling of the result of an analog calculation is of course limited by analog noise (e.g. thermal noise), which we must contend with in this universe. On the other hand, processing in the analog domain gets us further than quantization noise and heat dissipation (due to clock signal & switching) would in the digital domain.
Edit: If you are interested further, here is a discussion of the application of analog computing in solving differential equations: http://chalkdustmagazine.com/features/analogue-computing-fun...
Also, someone on here once posted a link to a recent dissertation comparing the efficiency of solution finding for a problem tackled analog vs. digital, wherein given the problem's nature, the analog computer outperformed its digital counterpart both in terms of performance and energy consumption.
Edit2: Here is the dissertation http://www.cisl.columbia.edu/grads/gcowan/vlsianalog.pdf
And the comment it was from: https://news.ycombinator.com/item?id=11728186
There is a common pattern in AI where a problem that humans do easily seems intractable, right up until we find a cheap trick that gets the job done and seems to have similar quirks to the biological equivalent. If we're lucky, we find an even better way than what evolution has stumbled upon. For example, computers are way better than humans at landmark recognition (using SIFT-inspired algorithms), but it took us a while to find the tricks to do that efficiently.
This is what makes me dislike the name "real" so much; I long ago grew tired of mathematicians making non-intuitive claims about the world which turn out, in fact, to be claims about the real numbers.
As Kronecker said, "God made the natural numbers; all else is the work of man".
According to whom? Our measurements aren't arbitrarily accurate, but as far as we know, physics itself doesn't suffer from accuracy loss at some point.
There are a few ways we can avoid the reals in physics.
Abstractly, there are results such as Skolem's "paradox" ( https://en.wikipedia.org/wiki/Skolem%27s_paradox ) which tell us that theories involving uncountable sets (such as the reals) can be satisfied by countable models.
More concretely, physicists never manipulate real numbers directly; only symbols which may represent real numbers. Such symbolic manipulation could, in principle, be performed by a turing machine, and hence any value obtained from such theories (e.g. experimental predictions) must be in the set of computable numbers. Hence the reals are just an 'implementation detail' of any theory, and their purported presence cannot have any influence on the theory's predictions.
As I said previously, I have nothing against the reals as a useful mathematical tool; I just disagree with the name, and claims that they are physically real. Others go further though, arguing that discrete mathematics is more general and useful than continuous, e.g. http://www.math.rutgers.edu/~zeilberg/OPINIONS.html
Your entire premise (that we might replace reals in current physical theory with some other structure and get the same result) is trivially debunked. Pi shows up all over the place, and e.g. rationals are not closed under exponentiation by e.
The schroedinger equation applied to physical hilbert spaces is inextricably real-valued.
Sorry, I was taking it for granted that the "reals" are problematic and unphysical (based on the sentiment of the grandparent). For clarification, the real numbers are a superset of the "computable" numbers (those numbers which can be approximated, to any given precision, by a universal turing machine).
The computable numbers are countable (since there are countably-many turing machine tapes), but the reals are uncountable. That means "almost all" of the reals are uncomputable.
Since all the laws of physics are computable, all physical processes must be based on computable numbers; even if some initial condition of the universe resulted in an uncomputable number, e.g. if the digits of the muon's magnetic moment are the same as the digits of the 100th busy beaver number, we can never know since we'll never compute what that busy beaver number is (we could test such predictions empirically, but it could always just be a coincidence).
> Pi shows up all over the place, and e.g. rationals are not closed under exponentiation by e.
Pi and e are computable numbers (quite trivially so, there are loads of known algorithms to do it). Exponentiation is a computable operation (again, the algorithm is pretty trivial; it's just repeated multiplication, multiplication is repeated addition and addition is repeated increment, so you only need 3 loops)
https://en.wikipedia.org/wiki/Computable_number#Can_computab...
> The schroedinger equation applied to physical hilbert spaces is inextricably real-valued.
It's been a while since my physics master's, so I wouldn't want to refute this face-on; however, it's trivial to notice that (by definition) we can only use computable numbers in our calculations, and hence even if there are any non-computable reals in a theory, they cannot affect the predictions of that theory. If some prediction depended on a value outside of the computable numbers, e.g. the 100th digit of Chaitin's constant, or the 100th busy beaver number modulo 7, then that prediction would not be calculable and could only be given in symbolic form, e.g. 1.8 * Chaitin(100) ^ 3.45. This is not the case for the schroedinger equation, which gives results that are analytically and/or numerically computable.
> Your entire premise (that we might replace reals in current physical theory with some other structure and get the same result) is trivially debunked.
It's not my premise to "replace reals in current physical theory" (although it seems that Doron Zeilberger might be pleased if that happened).
I was pointing out that physicists, mathematicians and many others claim to use the reals, when in fact they're only using the computable numbers. This leads to confusion when unintuitive results about the reals are applied to the world.
For example, when choosing a real number randomly from the unit interval, the probability of each point is 0, but the sum of their probabilities is 1. That's an example of the reals being unintuitive, and that's fine. However, trying to apply such statements to the world (e.g. "I can prove that the chance you've measured exactly 1 litre of water is zero") leads to confusion. Even worse, the confused person (the one making the claim, confusing uncomputable numbers with reality) may then "prove wrong" some perfectly justified refutations.
In case you can't see the problem, the "proof" that an amount other than 1 litre was poured relies on checking more and more decimal places until a difference is found between the measured amount and 1 litre. However, we have no way of knowing that the "keep checking decimals" algorithm will halt, so the claim might be an unsound counterfactual based on the "result" of an infinite computation. Any claim that the algorithm will halt is pre-supposing that there is a difference, and hence is not a proof. At any finite time, we can only ever compare the amounts to within some epsilon > 0, hence we can only ever claim the probability is arbitrarily small.
Dubious non-constructive, uncomputable results like this appear all over the place when dealing with reals. This makes the reals an interesting mathematical curiosity worthy of study, like the surreals, the hyperreals, transfinite cardinals, etc. and their properties can certainly be useful when required.
However, their status as the "default" set of (non-whole) numbers should have been removed after Turing's "On Computable Numbers" paper.
> It sounds like you just have some sort of aesthetic agenda.
When it comes to the reals I'm very serious. When it comes to renaming the "negatives" to the "other imaginaries", replacing spurious use of "integer" with "natural" and outright banning the awful phrase "unsigned integer", then that's an aesthetic agenda :)
In this sense, I agree with Robert Harper (e.g. Practical Foundations for Programming Languages, section 22.2) that unprovable claims regarding computability may be merely "theses" to logicians, but to scientists they should be considered "laws" just as much as Newton's law of universal gravitation.
In fact, other than purported claims of "hypercomputation" ( https://en.wikipedia.org/wiki/Hypercomputation#Criticism ) or the anthropocentric "microtubule"-type theories popularised by Penrose ( https://en.wikipedia.org/wiki/Orchestrated_objective_reducti... ), it seems that the computability of the universe is commonly accepted 'background knowledge' by most researchers in the field.
Even those making extreme claims in this area, like those refuted in http://www.scottaaronson.com/papers/npcomplete.pdf are presumably assuming that the universe is computable; there doesn't seem to be much point debating computational efficiency of soap bubbles if I thought (for example) that halting oracles existed!
Still, if there are reasons to allow uncomputable numbers as elements of a quantum theory, that still doesn't invalidate the larger point that any predictions of those theories are computable; it's just a constraint on which axiom scheme we follow when manipulating our equations.
Again, even if a theory involves uncountable sets, it may still have countable models (following either Skolem's "paradox", or a similar argument; since QM operates at much higher level than formal systems, I'm not sure if it's first-order, second-order, higher-order, etc.).
My best example: epsilon delta proofs...well, usually when I run physics simulations, I have an explicit error I can't overcome, so I don't really need to know whether a limit converges for all epsilon, just some larger than the largest error in my problem.
The first thing on that page are the computable real numbers. They have this annoying property that the set itself is not computable enumerable: there a real numbers that, if we only knew which ones they were, we could know all their digits; but there is no algorithm that will list all such numbers. https://en.m.wikipedia.org/wiki/Computable_number
It can be calculated to n digits in O~(n^2) time, which isn't really that bad. Something like Brun's constant or the area of the Mandelbrot set is probably a better example of a real number that is "notoriously difficult" to compute.
Have I misunderstood something?
Mathematics is abstract. That means something. It means when I say "I have $50", you can think of those $50 as peices of paper, portions of goats, amounts of hot dogs, or investment potential. $50 is not a thing, it's a model for things that have arithmetical properties and value.
That's how numbers work: they have properties, and you look at the world and map those properties to things that are relevant to you.
So you read something about Khinchin's constant, you understand what the terms are, understand the structural implications of it's properties, and then you go and try to find things that behave similarly in the world. Sometimes it fits well, sometimes you have to force it, sometimes it's just overly complex. The point is that the behavior of something like this is a model, which means it will never map perfectly to reality, and also that it may apply with varying degrees of success to any problem.
So where can you use it? Everywhere. You just have to think in terms of its properties. Do that enough and maybe you'll find out why it's stupid to use for some problems and smart for others.
But there's no point in asking someone else to do all the thinking for you.
Simple Example: Did you know that the ratio of a circle's circumference to its diameter is the same regardless of the size of the circle? I suppose you could use that to figure out how much stuff you need to make a wheel. I don't. Never have. Would you believe I use it to figure out what to add to an electric circuit to counteract the power factor of the inductive load of an electric motor the size of a pickup truck?
Math is funny like that. Once you understand a concept, you might see it in nature in a place you didn't expect it, and it may be suddenly quite useful.