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I am wondering about this and have wondered about it for a short while. Usually physics is modeled using things based off of the Real Number Line $\mathbb{R}$, which is uncountable. (E.g. we may use powers of $\mathbb{R}$, we may use $\mathbb{C}$, we may use "manifolds" which essentially glue together "warped pieces" of $\mathbb{R}^n$, etc.) But is it possible to dispense with this uncountable set, and uncountable sets in general, and still do ALL of the mathematics used in ALL physical theories which are supported by empirical evidence?

In particular, there is a type of mathematics called "constructive analysis" (a subfield of more general "constructive mathematics") and one approach to this is to jettison $\mathbb{R}$ and replace it with the sub-field of "computable real numbers", which are real numbers for which we can write a (possibly computationally intractable but not impossible) computer program to approximate to any desired $\epsilon$ of accuracy (where $\epsilon$ is a rational number). This reduces our set of real numbers to a countable set, and if we keep all our other objects computable as well, we can get a good deal of analysis done. There are some caveats -- e.g. equality is not decidable, we can only tell if two computable reals are within a given $\epsilon$, however this may not be a problem since it lines up neatly with the way empirical science works -- we can never actually prove two physical quantities are equal by empirical measurement, only that they are equal to within some $\epsilon$, namely, our measurement error. E.g. in all true honesty, we cannot say the mass of a photon is 0 -- we can only say that it is $<10^{-18}\ \mathrm{eV}$, at least, according to Wikipedia as of now. Another caveat is that bounded monotone convergence fails: we can find a bounded monotone computable sequence of computable reals which has no supremum. But despite these, like I said, you can do differentiation, integration, etc. . It also has some other interesting properties, e.g. all functions are continuous.

So with that in mind, is what I am suggesting possible? Or is there some type of math in physics that for some reason requires uncountable sets? Can we do every mathematical proof needed for physics' math to work out in these constructible, countable sets and maybe even better, with no proofs by contradiction, i.e. intuitionistic logic and no law of excluded middle?

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First of all, a small historical note. As far as I know, in constructive mathematics you can construct uncountable sets, e.g. the reals, since the usual way of introducing them (by Dedekind cuts) is constructible, i.e. involves a countable process. Of course it is not done in a finite number of steps method, but indeed constructible. Most of the people that are unhappy with uncountable mathematics are unhappy with "uncountable/unconstructive" logical reasoning: the fundamental point being the use of the non-constructive full axiom of choice, opposed to some weaker version (e.g. countable choice or dependable choice). Of course intuitionism has a more nonstandard approach than just restricting the axiom of choice, but apart from the book by Bishop (that is not purely intuitionistic), I don't know any other systematic and rigorous approach to reproduce known mathematics in a constructivist fashion. However in the following I adhere to the OP question.

The problem with finitist/constructivist methods in mathematical physics (in my opinion also with not assuming the full axiom of choice, but in that case eventual problems are more difficult to spot) is not that you have experimental evidence of unconstructible sets. Obviously in experiments and measurements there are errors that prevent you from saying with certainty that a quantity has this precise value, and also a supposed real number would never be calculated with complete precision.

The problems you have are of two types:

  • There is no experimental evidence of a constructive universe. I mean that as you can't do any experiment to prove with absolute certainty that a quantity is measured by a real number, you cannot also do any experiment to prove with absolute certainty that a quantity is a constructible one. On that point, intuitionistic/constructible and "standard" mathematics are on the same ground.

  • The predictive power of constructivist mathematics is inferior compared to the one of standard mathematics. This is, in my opinion, the most important argument. The importance of theoretical and mathematical physics is to make predictions. A mathematical result is important for physics as long as it can be interpreted as the model of a concrete system, and makes predictions that agree with experiments. In that context, since with standard mathematics much more results can be proved than with constructivist methods, you can efficiently predict much more interesting physics that can confirm observations, or be tested in new ones. There are no constructive models, and I suspect it would be extremely difficult or impossible to have some, that describe as well as standard mathematics classical mechanics, quantum mechanics or relativity. Modern mathematical physics modelling makes predictions using highly non-trivial mathematics that is not constructivist at all. To reproduce that with constructivist math would be highly non trivial if not impossible, and even if successful it would employ a very difficult mathematical apparatus (because a constructive proof is often more difficult than a standard proof, since you have less powerful tools to work with).

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  • $\begingroup$ Thumbs up for a very good answer, but truthfully, it's not about physics, at all. Even where you are trying to connect to experiments you are missing the point of what experiments are: the explanation of complex natural phenomena using less complex ones, mathematics is merely an intermediate deductive tool in that chain. $\endgroup$ – CuriousOne Jul 3 '15 at 7:29
  • $\begingroup$ What sort of prediction cannot be made from a constructively-formulated physics theory that CAN be made from one formulated in nonconstructive mathematics AND is wholly physical? $\endgroup$ – The_Sympathizer Jul 3 '15 at 7:32
  • $\begingroup$ @mike4ty4: Whether a prediction is physical or not can not be decided with mathematics, at all. It can only be decided with an experiment. $\endgroup$ – CuriousOne Jul 3 '15 at 7:35
  • $\begingroup$ Right. So then I'd say show me an example of a testable prediction (the only kind that matters in science) that can be made by, say, "classical" classical mechanics and NOT by Computable-Intuitionistic classical mechanics. Something involving things along the lines of creating objects like Banach-Tarski-like sets doesn't count because that is not testable since we cannot make anything like that. $\endgroup$ – The_Sympathizer Jul 3 '15 at 7:37
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    $\begingroup$ @mike4ty4 to any lie symmetry of the action it corresponds a conserved quantity. This is a very important mathematical theorem with huge applications in physics (noether's theorem). Do you think you would be able to prove it by constructivist methods? And not just in a particular case, but in a fair generality, since it is omnipresent in physics, from systems with just one to infinite degrees of freedom. $\endgroup$ – yuggib Jul 3 '15 at 7:52
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Math for physics (and for all other fields too) can only be expressed without any uncountable sets!

What does it mean to express something X, for instance to communicate X in discourse, dialogue or at least monologue? "X", the name of X, must be realized by physical means, i.e., in writing, or speaking, or thinking, or what other tools might be available. This will happen in a not vanishing domain of space-time. Every domain of space-time contains infinitely many rational spatio-temporal coordinates, at least one them must be reserved for realizing "X".

As Cantor has shown, the set of rational spatio-temporal coordinates even in an infinite and eternal universe is countable. That proves that all expressions (by the way including all "diagonal numbers" ever created) belong to a countable set. Uncountable sets can never play a role in expressed mathematics other than as names of imagined but, with respect to their individual elements, not completely imaginable sets.

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  • $\begingroup$ The problem is to reconcile this fact with the other fact that calculus passes through the continuum and all physical laws are formulated using calculus. $\endgroup$ – Ryan Thorngren May 9 '18 at 12:17
  • $\begingroup$ Calculus is based on potential infinity and has nothing to do with uncountability. On the contrary, transfinite set theory contradicts calculus. See, e.g., philosophy.stackexchange.com/questions/51038/… $\endgroup$ – Otto May 9 '18 at 12:24
  • $\begingroup$ That's wrong. There are no countable dense linear orderings that are closed under taking limits. $\endgroup$ – Ryan Thorngren May 9 '18 at 12:39
  • $\begingroup$ Of course. "Countable" is as meaningless as "uncountable". Limits in classical mathematics are defined according to Cauchy. Nothing uncountable there. $\endgroup$ – Otto May 9 '18 at 12:42
  • $\begingroup$ @RyanThorngren The reader might be interested to know that Wolfgang Mückenheim's mathematical theories (e.g. concerning quantification), "proofs", and "contradictions" are considered as utter nonsense by mathematicians. (There are good reasons why he publishes on vixra.) The reader might also wonder why Mückenheim recently started to participate in Physics.SE using three different identities "Heinrich", "Wilhelm", "Otto". $\endgroup$ – Uwe May 9 '18 at 13:37

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