Probably you think that I completely missed the "meaning" of what a mathematical proof. But that's the way computers (formal logic systems) see proofs...
OK, let's start from examples of normal forms: "245 * 10" is a number that is not in normal form. "2450" is the same number in normal form (one of the...
OK, I'll try. But I think it will be a long post anyway. So, I'll do it after answering the other posts before. But I can give you the reference to a ...
OK, I'll try to explain this point. The fact that there is a relation between topology and logic (mediated by category theory) was well known even bef...
Yes, exactly! (well, I don't understand what energy has to do with this, but more or less...) However, that's only MY idea (and it's more about philos...
Yes, there are a lot of formal logics based on type theory - and even Martin-Löf type theory has a lot of variants (too many to be something important...
No, a finite state machine is equivalent to a finite Turing machine, I guess. Yes, lambda calculus (the original one invented by Church - https://en.w...
Well, OK. Turing machines are a model of computation equivalent to Church's untyped lambda calculus. Limiting Turing machines to have finite dimension...
Well, my argument was simpler than this: - We know that we cannot enumerate all halting Turing machines, so for every supposedly complete list of halt...
Yes, but the existence of Turing machines cannot be proven from other principles of logic: it has to be assumed as an axiom. Maybe (I don't know) it c...
Yes, of course practically all "normal" proofs are short and all the computing power needed is a pen and a peace of paper. But in reality all computat...
OK, so let's try to follow this definition of measure to calculate the measure of the two sets that you mentioned: (1). - The set of computable bitstr...
How do you define this measure in pure mathematical terms? You cannot use probability, because probability is physics (unless you find a sound mathema...
Yes but mathematics needs computations for proofs, and computations are physical processes. You may think that computations are not physics if you mak...
Any counter-example? P.S. I see there are several persons that studied physics visiting this site: maybe we could create a post especially on this poi...
I can assign a number to an experiment (calculated with some well-defined algorithm) and call it "probability" (for example defined as the square of t...
Yes, that's very confusing. And if you look at the mathematical literature, the terms "constructive" and "intuitionistic" seem to have changed meaning...
Constructive physics (constructivist logic) can ASSUME the existence of a function that you can call "random" (whatever it means: it's an axiomatic th...
There is in fact a physical assumption that is necessary for both mathematics and logic to make sense: the fact that each time that you repeat the sam...
Yes, exactly! If there were an experiment that could tell the difference, then it's no more metaphysics! But my guess about the future is that none of...
:smile: Super! So we can speak about QM without equivocating the words! :cry: OK, let's just "pretend" that a Hilbert space is complete even in constr...
OK, I'll not insist going ahead on the first part. Only about this part. Short answer: this is not a computable sequence. - So how is this experiment ...
No, I didn't say you can calculate anything. You can calculate the magnetic moment of the electron in quantum electrodynamic with arbitrary precision,...
The binding energy due to the coupling between quarks and gluons is responsible for the most part of the mass, the rest of it (I don't remember now in...
I didn't answer to this yet, so I'll do it now. In general, category theory can be used to represent formal logic systems and their interpretations, i...
OK I'll stop arguing about intuitionism. But I think you didn't get my point here, so let me try one last time: Cantor's theorem is valid in intuition...
So, you can see constructivist logic as an algebra of propositions built with computable functions (https://en.wikipedia.org/wiki/Heyting_algebra). Yo...
But I am afraid that's all what physics (at least contemporary physics) does: prediction. Nothing else! Nobody knows how to make sense of the equation...
Yea, but I was speaking about a way to approximate space-time with discrete pieces to make computer simulations, not of the real equations. The real e...
No, maybe I shouldn't have talked about "points": you simply split space and time in a lot of little "cubes" that are attached one to the other. Only ...
Yes, but the problem is that (for example) particles are always detected as little spots (such as a point on a photographic plate) and wave functions ...
Yes, that's one of the most interesting subjects even for me :grin: Yes, everything right until now. No, because even the motion of the quarks inside ...
Let's put it in this way: what you call "noncomputable" in boolean logic should be called "nonspeakable" in constructivist logic: they are not part of...
OK, I'll avoid to get into trouble with constructivism again :smile: Basically, what I wanted to say is that there is a "trick" in his kind of "constr...
Yeah well, let's say so... The way QM is formulated is: there are "observables" that represent the objects (or better: the results of experiments), an...
Thanks for the reference! I took a quick look at the book (just a quick look at the equations, really) and the first think that I thought is: what's t...
So is the 3,4,5 triangle really straight or not? I don't understand... OK. OK, the identity cannot be identified with the name. OK, so what can I do w...
OK, division and multiplication are not symmetrical for integers, because integers are "quantized": you can't give one candy to three children, becaus...
If you consider geometric spatial figures as real physical objects, there are a lot of "problems" with them: first of all, they are 2-dimensional (or ...
The even more interesting thing (that's why I talked about atoms) is that this is true not only for elementary particles as electrons, but even for at...
OK, but I don't understand how all this can be related to irrational numbers. Division between integers is repeated subtraction ( A/B you count how ma...
Well, in the current theory of the physical world (standard model, or whatever variant of it you prefer) all atoms of the same element are supposed to...
But I can use numbers to describe (or model) physical processes (experiments): 1. Call Build_Side(N) the physical process of putting N sticks in line ...
Sorry for the intrusion, but I am curious of this issue (only one premise: I didn't study philosophy :yikes:, so, for example, I don't really understa...
Well, the "issue" of the irrationality of the diagonal of the square is the one that ancient greeks recognized: you cannot find any unit length that e...
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