Perhaps you can clarify exactly what you mean here. If you mean that you get the same physics, yes of course that would be the point. If I'm understanding you correctly. You want to be able to do standard physics but without depending on the classical real numbers. So if that's what you're saying, it makes sense. — fishfry
Ok. It was only recently that I learned that protons have quarks inside them. Another thing I've learned is that gravitational mass is caused by the binding energy that keeps the quarks from flying away from each other. How that relates to Higgs I don't know. I've also seen some functional analysis so I know about Hilbert space. I have a general but not entirely inaccurate, idea of how QM works. — fishfry
But this can't be, since calculating machines can't calculate ANYTHING with arbitrary precision. Where are you getting these mystical TMs? If the theory gives a result like pi, I'd accept that as a result having arbitrary precision. But if you are saying that even in theory there is a TM that can calculate anything with arbitrary precision, that's wrong. The best a TM can do is approximate a computable real number with arbitrary precision. That's much less than what you are claiming, if I'm understanding you correctly. — fishfry
OK I'll stop arguing about intuitionism. But I think you didn't get my point here, so let me try one last time: — Mephist
Cantor's theorem is valid in intuitionistic logic, but we know that intuitionistic real numbers are countable. In fact the theorem says: forall countable lists, there is an element that is not in the list, and we know that the set of elements missing from the list is countable because the list of all strings is countable.
Now you read the same theorem in ZFC and you interpret it as "there is an uncountable set of elements missing from the list". How do you know that the set of missing elements is uncountable? I mean: the symbolic expression of the theorem is the same, and the interpretation of the symbols is the same. How can you express the term "an uncountable set" in a language containing only the quantifiers "forall" and "there exists one" ?
And if there is no uncountable set of missing real numbers, there are no holes to fill.. — Mephist
For the first part of the question, I guess your question is how do you say "a finite random sequence" in intuitionistic logic. You can't! (at the same way as you can't do it in ZFC: the axiom of choice does not say "random" function). If the sequence is finite it is always computable, so you can say "there exist a finite sequence of numbers" ( the same as in ZFC ).
There is a definition of randomness as "a sequence that is not generated by a program shorter than the sequence itself" (lots of details missing, but you can find it on the web), but this is about the information content and not about the process used to choose the elements of the sequence.
About the bitstrings that aren't computable: all finite bitstrings are computable of course. So probably you mean the bitstreams that contain an infinite amount of information (not obtainable as the output of a finite program). There is no way to prove that such strings exist using a formal logic system (even using ZFC): we can interpret the meaning of Cantor's theorem in that way, and maybe there is such a thing in nature, but you cannot prove it with a finite deterministic formal logic system. — Mephist
So probably you mean the bitstreams that contain an infinite amount of information (not obtainable as the output of a finite program). There is no way to prove that such strings exist using a formal logic system (even using ZFC): — Mephist
It doesn't make sense that you would have assigned the symbol "2" to something and you know absolutely nothing about this thing which you have assigned the symbol to. — Metaphysician Undercover
No, I didn't say you can calculate anything. You can calculate the magnetic moment of the electron in quantum electrodynamic with arbitrary precision, but only in theory (because the number of operations necessary grows exponentially with the number of calculated decimals), — Mephist
But I wanted to point out that there are parts of QM that are in some sense "mathematically perfect". Meaning: there are a finite set of atoms corresponding to all the possible combinations of electrons' orbitals up to a certain number of electrons (82 stable elements? I don't remember). And that ones are "perfect shapes", in the sense that two of them of the same type are exactly the same shape, like two squares. Usually (before QM) physics was made of objects that only corresponded to mathematical objects in an approximate way (orbits of planets for example), but if you looked carefully enough, every object in the physical world was different, and different from the mathematical object that represented it.
Atoms, and particles in QM in general, are different: they are "digital" (quantized) and not "analogical" shapes. So, in some sense, they are "perfect" (mathematical?) objects. — Mephist
The vectors of this Hilbert space are the wave functions (not observable).
Observables are represented by Hermitian operators on the Hilbert space.
And the results of experiments (the numbers corresponding to the measured quantities) are the eigenvalues o these Hermitian operators.
(P.S. it's impossible to understand how it works from this description, but that's the way it is, if you want to be mathematically accurate) — Mephist
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, in the obvious way: an interpretation is a functor from a category representing the language to a category representing the model ( https://en.wikipedia.org/wiki/Categorical_logic ). — Mephist
So I actually understand all of what you said, from a mathematical point of view. — fishfry
And now that you mention it ... that's one of my arguments against constructive physics! A Hilbert space is a complete inner product space. By complete we mean Cauchy-complete. So you can't even have such an object in constructive math, because the constructive real line is not Cauchy-complete. — fishfry
Now if I'm understanding some of your comments correctly, you are saying this doesn't matter because even if we assume the constructive real line, we can still prove the same theorems. Constructive completeness is just as good as completeness, for purposes of calculations in QM. And even if there is ultimately a difference, we couldn't measure that difference anyway!
Perfectly sensible. We could do physics with the rational numbers and a handful of irrational constants if we needed to. No experiment could distinguish that theory from a theory based on real numbers. — fishfry
This is a very interesting point I hadn't considered before. It makes the enterprise of constructive physics seem somewhat more reasonable to me. Am I understanding you correctly? — fishfry
Yes, EXACTLY! — Mephist
Short answer: for a finite experiment, "a priori" probabilities are simply functions that count the total number of possible results, "assuming" that each result has the same "probability" (yes, that's a circular definition: no formal definition of what "probability" is, even using ZFC set theory). — Mephist
But ultimate reality may in fact be one or the other, computable or not. Which supports my belief that noncomputability is the next frontier in physics. If someone ever proves that a noncomputable real is necessary to explain some observable physical phenomenon, it's off to the races to find such a thing in the world. — fishfry
They do represent objects--abstractions, not existents. — aletheist
On the contrary, this is Semeiotic 101--in a proposition, the subjects denote objects, and the predicate signifies the interpretant. — aletheist
Incommensurability does not preclude (mathematical) existence. Our inability to measure two different objects (abstractions) relative to the same arbitrary unit with infinite precision does not entail that one of them is (logically) impossible. — aletheist
Only according to your peculiar theory, not the well-known and well-established theory in question. — aletheist
If you view mathematics as explanative device for natural phenomena, I can certainly understand your concern. However, I see mathematics first and foremost as an approximate number crunching and inference theory. I do not see it as a first-principles theory of the space-time continuum or the world in general. I see physics and natural sciences as taking on that burden and having to decide when and what part of mathematics to promote to that role. If necessary, physics can motivate new axiomatic systems. But whether Euclidean geometry remains in daily use will not depend on how accurately it integrates with a physical first-principles theory. Unless the accuracy of the improved model of space is necessary for our daily operations or has remarkable computational or measurement complexity tradeoff, it will impact only scientific computing and pedagogy. Which, as I said, isn't the primary function of mathematics in my opinion. Mathematics to me is the study of data processing applications, not the study of nature's internal dialogue. The latter is reserved for physics, through the use of appropriate parts of mathematics. — simeonz
I meant applications where the grain is indeed uniform, such as the atomic structure of certain materials. — simeonz
I do agree that the use of mathematics in real applications is frequently naive. And that further analysis of its approximation power for specific use cases is necessary. In particular, we need more rigorous treatment that explains how accuracy of approximation is affected by discrepancies between the idealized assumptions of the theory and the underlying real world conditions. I have been interested in the existence of such theories myself, but it appears that this kind of analysis is mostly relegated to engineering instincts. Even if so - if mathematics already works in practice for some applications, and the mathematical ideals currently in use can be computed efficiently, this is sufficient argument to continue their investigation. Such is the case of square root of 2. Whether this is a physical phenomena or not, anything more accurate will probably require more accurate/more exhaustive measurements, or more processing. Thus its use will remain justified. And whether incommensurability can exist for physical objects at any scale, I consider topic for natural sciences. — simeonz
Do you happen to know what dark matter is? Don't worry if you don't, because nobody knows what dark matter is. It's a name given to something we can not understand but wish to study. — fishfry
By all our known theories of physics, galaxies should have flown apart long ago. Why didn't they? — fishfry
No, I am not a platonist; I am not claiming that abstractions exist in the ontological sense. Why keep insisting otherwise?Therefore the dualism of Platonic realism. — Metaphysician Undercover
No, some symbols are subjects, while others are predicates, although the predicate of a proposition can also be embodied in the syntax rather than the symbols.And symbols represent subjects, so there's a double layer of representation, exactly what Plato warned us against, what he called "narrative", which allows falsity into logic, sophistry. — Metaphysician Undercover
I'm really tired of your willful obtuseness, insisting on your peculiar metaphysical terminology despite the fact that the same word often has different meanings in different contexts. "Existence" in mathematics is not the same as "existence" in ontology. An "object" in mathematics is different from an "object" in semeiotic, and both are different from an "object" in ontology. A "subject" in semeiotic can be an "object" (direct or indirect) in grammar. And so on.I'm really tired of your unsupported assertions. — Metaphysician Undercover
It is a feature, not a bug--it reveals a real limitation on our ability to measure things.This is a defect of the theory, it gives us a so-called theoretical "object" which cannot be measured. — Metaphysician Undercover
What is the basis for the claim that mathematics is created for the purpose of measuring objects? On the contrary, the purpose of mathematics is to draw necessary inferences about hypothetical states of things. One such inference is that in accordance with the postulates of Euclidean geometry, the length of a square's diagonal is incommensurable with the length of its sides.Why would we produce a theory which presents us with an object that cannot be measured, when the theory is created for the purpose of measuring objects? — Metaphysician Undercover
"Impossible to measure" does not entail "impossible," full stop. There is nothing logically impossible about the diagonal of a unit square or the circumference of a unit circle. Again, given how those figures are defined, it is logically necessary for them to be incommensurable.Creating "impossible" abstract objects is nothing new, it is easily done through the use of contradiction. — Metaphysician Undercover
Confused by this. Constructive physics wouldn't allow a random sequence. — fishfry
One way to define "constructive physics" is simply to say, "it uses constructive mathematics". But definitions of the latter sometimes arise principally from avoiding the LEM. Another tack is to avoid non-computable numbers. Or simply to state that experiments must be conclusive in a reasonable finite amount of time. I'm not sure what you two are referring to here. But I haven't read all the thread. — jgill
You can certainly define a measure on the unit interval of reals and assign probabilities to sets of bitstrings. I didn't follow this post. You said you can't define probabilities for bitstrings but you can. — fishfry
That's not an example at all. We know a lot about dark matter, that's why we can name it. — Metaphysician Undercover
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