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

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 ).

The formal logic system is represented as a category in this way:
- the objects of the category are the propositions of the language (all provable propositions)
- the arrows of the category are the derivations (all possible derivations A -> B from prop. A to prop.B)

Two categories A and B can represent different formal logic systems but be equivalent (https://en.wikipedia.org/wiki/Equivalence_of_categories). Basically, this means that there are two natural transformations X and Y (https://en.wikipedia.org/wiki/Natural_transformation) that map every derivation in A in a derivation in B and vice-versa.
X and Y are then adjoint functors (https://en.wikipedia.org/wiki/Adjoint_functors)

In this case, the correspondent propositions (objects) in A and B are different in general, but there is an 1-to-1 correspondence between derivations in A and derivations in B. The derivations on formal systems are (exactly) the computations needed to obtain the results of experiments.

In practice, it means that A and B use a different "encodings" (different languages) to describe the same experiment in equivalent ways. From the point of view of the physical predicting capacity of the model, it doesn't make any difference if you use A or B to perform the computations.

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

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 which percentage) is due to the binding energy due to the coupling between quarks and the Higgs field.
Yes, to be precise, physical states are represented by rays in a Hilbert space (infinite-dimensional complex vector space). A ray is a set of normalized vectors (scalar vector X * X = 1 for every vector X), with X and Y belonging to the same ray if X = a * Y, where a is an arbitrary complex number with modulus(a) = 1.
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)

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

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), and only in QED (that is a part of the full standard model - in the full standard model (QCD + Higgs) I don't know. I never understood how QCD renormalization of path-integrals works).

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.

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

For two people trying to end a conversation we're not doing a very good job.

I want to make a semantic point, which is that intuitionism is too vague. It's way more than constructivism. Intuition in Brouwer's formulation has a mystical component that I can never make sense of. I use the term neo-intuitionism to stand for all the contemporary attempts to revive the idea, minus the mysticism: constructive math, homotopy type theory, etc.

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

But there are. It's a theorem that a Cauchy-complete totally ordered field must be uncountable. The constructivists pretend all the noncomputable numbers don't exist. But that's nonsense. Chaitin's constant exists (as a real number) and it's not computable. The Halting problem is not computable. Lots and lots of naturally occurring phenomena are noncomputable. Newtonian gravity is noncomputable. (The jury's still out on QM). You can't close your eyes to things then say they're not there. There's more to mathematical truth than proving theorems, as Gödel demonstrated. You can prove that the constructive real line is "computably complete," but it's still not complete, as in the example of truncations of Chaitin's constant shows.

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

You seem a little off topic here. I asked you what principle of nature, or math for that matter, forces a sequence of coin flips to be computable. Of course I agree that any finite sequence of flips is computable, we can compress it just by writing down its base 10 equivalent. But if you flip infinitely many coins. you will get a computable sequence with probability zero. How can constructivists hope to get so lucky?

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

Nonsense. I can prove it easily. The measure of the unit interval is 1; the measure of the computable reals in the unit interval is zero. Therefore there must be a whole lot of of noncomputable reals in the unit interval.

You're claiming that if I flip infinitely many coins, they must land in a pattern that is computable. That's clearly nonsense. How would the coins know to do that? On the contrary, it's incredibly unlikely that an infinite bitstring is computable and "almost certain," as they say in measure theory, that it's not.

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

Here's a nice contemporary example of exactly that.

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.

The story goes like this. Astronomers can estimate the amount of matter in a given galaxy. We can also measure the galaxy's rotational speed. It turns out that most galaxies are spinning so fast that they don't have enough matter to hold them together gravitationally. By all our known theories of physics, galaxies should have flown apart long ago. Why didn't they?

We have no idea. Being human, a creature with the power of abstraction (you must have not gotten your share) we give it a name even though we have no idea what it is or what it might be.

Dark matter is the name given to some hypothetical "stuff" that interacts with the gravitational field but no other fields. By contrast, a rock falls to earth so it interacts with the gravitational field. And you can see it, so it interacts with the electromagnetic field. That's normal for the stuff we call "stuff."

Dark matter must therefore be something that's matter, in the sense that it interacts gravitationally; but it's dark. It doesn't interact with electromagnetism. You can't see it. In fact "dark" is the wrong name, it should be transparent matter. But dark matter is the name that stuck.

So, is there dark matter? Maybe. We just don't know what it is yet.

Or maybe there is no dark matter at all. Perhaps the law of gravity needs to be tweaked so that at galactic distances, it has a little extra pulling power to hold the galaxies together. This fascinating idea is called Modified Newtonian Dynamics, or MOND.

"Dark matter" is therefore a symbolic phrase to stand for something that causes some effect, but we have no idea what it is, or even if it exists. Yet we reason about it and write scientific papers about it.

That's scientific abstraction. You know nothing of science. I'm impressed. The more I get to know you, the wider your sphere of ignorance seems to become.

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

Wrong theory. If something is computable or not, that's computability theory. If something is computable efficiently or inefficiently, that's complexity theory. You're conflating the two. Minor point but you've done it twice so I thought I'd clarify this point.

You can calculate the magnetic moment of the electron. Period. The efficiency of the calculation is a separate topic and has nothing to do with whether it's computable. I suspect you know that but forgot to make that distinction as you were typing.

But I don't know why you keep mentioning this. You can't measure the prediction with arbitrary accuracy in the real world. We're agreed on this point.

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

Ok. Not exactly sure what you're saying here. I've already stipulated long ago that I know that all electrons are identical. That is in fact a highly strange phenomenon. You pointed out to me that atoms can be identical to other atoms. That's interesting. This last para I didn't quite follow.

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

As it happens, here is how I learned about what you're describing. I never had much physics background. A few years ago I had the opportunity to seriously study some functional analysis. Functional analysis is basically infinite-dimensional linear algebra combined with calculus, if you think of it that way. Normed vector spaces, Banach spaces, Hilbert spaces. For example you can recover the subject of Fourier series as a particular example of an orthonormal basis. One day I discovered that the mysterious bra-ket notation, which was something I thought I'd never be able to understand in this lifetime, turns out to be nothing more than a linear functional operating on a vector, written in inner-product notation. At that moment I realized I understood a lot of QM without having to study physics. So I actually understand all of what you said, from a mathematical point of view.

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.

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.

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?

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

I think what I was getting at is that you made the claim that constructive and classical physics were equivalent categories; and I asked you to clarify how you were categorifying physics. I don't think you answered but it's not an important point. I would certainly take on faith that what you say is true.

Actually what you've convinced me of so far is that constructive math and standard math give the same theory of physics; since in QM we are only doing computable calculations anyway. Is that right?

I was just a little unclear about which category you're using. I know Baez and others use category theory in physics, but I don't know if there are official categories that describe gravity or QM or whatever. Doesn't really matter. Your main point is that constructive math is just as good as classical, since we only use computable calculations. And you are being agnostic about whether the actual universe is constructive or not. Is that a fair summary of your view?

Yes! ( on all points :smile: )

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 described in a constructivist theory of physics? This is not an experiment, because it cannot be performed in reality: it never ends!
But there is even another problem: you cannot define the term "probability" as a mathematical function from a finite sequence of bits (results of partial experiments - the "total" experiment does not have a result, since it does not have an end) to real numbers (the probability) because of the limitation of the language - and this is true even for ZFC set theory: you simply define the probability of a sequence of N bits as the inverse of the number of possible sequences of N bits, such as if there were N results (many-worlds interpretation), but there is only one result. Probability is "a priori" in QM (not explained from other physical principles). Otherwise, if it's not "a priori", the result of the coin flip is derivable from the theory (such as in Newtonian mechanics), and then it is a computable function.

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).

You're claiming that if I flip infinitely many coins, they must land in a pattern that is computable — fishfry

No, in QM the pattern is NOT computable: the pattern is NOT predictable from the theory, so you DON'T NEED any computable function to predict it!

OK, Yes you are right, I used the term "computability" meaning of computational complexity.

So I actually understand all of what you said, from a mathematical point of view. — fishfry

:smile: Super! So we can speak about QM without equivocating the words!

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

:cry: OK, let's just "pretend" that a Hilbert space is complete even in constructivist logic. Or maybe, let's stop arguing about constructivist theory: you said you are not interested, right?

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

Yes, EXACTLY! :grin:

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

:up: :smile:

Yes, EXACTLY! — Mephist

Ok. So whether we use constructive or classical real numbers, we get the same physics. We get the same theorems and we can't measure any difference.

However, we do not necessarily have the same metaphysics. The world may be classical or constructive. It may consist only of computable things or it may contain noncomputable things. Our theories can't tell the difference and our experiments can't tell the difference. 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. I'm talking a hundred years down the road, maybe longer.

No, in QM the pattern is NOT computable: the pattern is NOT predictable from the theory, so you DON'T NEED any computable function to predict it! — Mephist

Confused by this. Constructive physics wouldn't allow a random sequence.

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.

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

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.

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

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 the two logics (constructive or ZFC set theory) will be the final answer (or maybe I should say the "next" answer), because both have the same common "defect": they assume that the wold is deterministic.
(OK, DO NOT ANSWER NOW please! I know the objection: mathematics is not the real world, it doesn't matter if the world is deterministic or not, I am contradicting myself!)

I don't have time now for a quick explanation, I'll get back to this when I came back from work... :pray:

They do represent objects--abstractions, not existents. — aletheist

Therefore the dualism of Platonic realism.

On the contrary, this is Semeiotic 101--in a proposition, the subjects denote objects, and the predicate signifies the interpretant. — aletheist

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.

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

I'm really tired of your unsupported assertions. As I explained It is not a case of imprecision in practise, it is a case of something being logically impossible within the theory. The theory dictates it as impossible, just like a square circle is impossible, by definition. It has nothing to do with our inability to measure with "infinite precision" (whatever that might mean), as it has been demonstrated that no degree of precision can give us that measurement. This is a defect of the theory, it gives us a so-called theoretical "object" which cannot be measured. 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? It's self-defeating.

Only according to your peculiar theory, not the well-known and well-established theory in question. — aletheist

It's not my "peculiar theory", it's the "Pythagorean theorem". This issue has been known for thousands of years, and I'm shocked by the level of denial in this thread. Accusing me of coming up with my own idiosyncratic theory, that's just ridiculous.

Pythagoras demonstrated how we can construct an abstract mathematical object, using accepted mathematical principles, which is impossible to measure. That is the diagonal of a square. Creating "impossible" abstract objects is nothing new, it is easily done through the use of contradiction. The problem here is that it is done through the use of accepted mathematical principles.

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

To me, this is a new and refreshing perspective of mathematics, much more realistic than the Platonic realism defended by many mathematicians. From this perspective, we can see that mathematics is not the primary tool required for an understanding of "nature's internal dialogue". To continue this analogy we could say that nature speaks a different language, and if we want to translate nature's language into mathematics, we need to first recognize that the language is completely different from mathematics, then proceed to determine the differences and similarities to produce some principles for translation. I'm afraid that most physicists would not see things this way though. The prevalent theme here is Platonic realism, according to which, the physical, sensible universe, is a direct representation of the mathematical objects. Under the precept of Platonic realism therefore, learning mathematics directly enables one to understand the universe.

I meant applications where the grain is indeed uniform, such as the atomic structure of certain materials. — simeonz

I don't think that this is a realistic perspective. I don't know how often, if ever, there are individual atoms naturally existing, as they tend to come in molecules. And the "same" atom has a different structure in a different molecule. The first principle we understand about physical reality is the law of identity, and this recognizes that each particular thing is unique. To adopt a principle of grain uniformity would mean dismissing the law of identity, and I don't believe that would be consistent with our experience of the uniqueness of particular individuals.

The particular is unique, with a unique identity. In abstraction, we look beyond the uniqueness, and class unique things together as "the same" according to some principle of categorization. This allows that we might have "5", or "8", apples. Notice that "apples" is the qualifier, the principle of sameness, by which we class the things together as "the same", thus allowing for the abstraction to take place. If we allow that two distinct instances of particular objects are "the same" in an absolute way, like two distinct grains in "grain uniformity" or two distinct occurrences of the number 5, then we violate the law of identity. We would allow that we might have a group of those things classed together without any qualifier (principle of sameness), because we have already assumed that they are the same in an absolute way. There is nothing which makes them the same except the assumption that they are the same. Now we have entered into an extremely confused and contradictory conception within which distinct things are said to be distinct particulars, and they are treated by the application of the theory as distinct particulars, yet they are stipulated by the assumptions of that same theory to be the same in an absolute way. That's the kind of mess which "grain uniformity" might give us.

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

I pretty much agree with what you say here. I think that there is no problem with making approximations in practise. This is common, and as an engineer one would know the acceptable limits of such approximations, established by convention. When I use pi for example I use 3.14, and this is my personal convention. When I use the Pythagorean theorem to lay out right angles, I might round off to about a quarter inch, more or less depending on the length of the perpendicular sides.

But approximation in practise is not the same as approximation in theory. Approximations within theory are employed when the theory cannot provide accuracy due to some deficiency of the theory. The approximations are used, and the theory proceeds from them. However, the approximations are covering over the original deficiencies, and as the theory extends and extrapolates, the effects of the deficiencies compound and magnify. If we refuse to recognize that the approximations are a manifestation of deficiencies in the theories, and address those deficiencies, we will never overcome the problems which inevitably result.

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

That's not an example at all. We know a lot about dark matter, that's why we can name it. It's not at all like naming something which we have not apprehended. If we have apprehended it, it has an appearance to us, and we can describe it. I describe it as a manifestation of the deficiencies of the general theory of relativity. Maybe you recognize it as this as well, but there seems to be a convention amongst physicists and cosmologists making it taboo to mention deficiencies of general relativity.

By all our known theories of physics, galaxies should have flown apart long ago. Why didn't they? — fishfry

Yes! You do recognize it as a deficiency of the theory. Why hide this? Why not call it what we already know it is, rather than the mysterious "dark matter"?

Therefore the dualism of Platonic realism. — Metaphysician Undercover

No, I am not a platonist; I am not claiming that abstractions exist in the ontological sense. Why keep insisting otherwise?

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

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.

I'm really tired of your unsupported assertions. — 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.

This is a defect of the theory, it gives us a so-called theoretical "object" which cannot be measured. — Metaphysician Undercover

It is a feature, not a bug--it reveals a real limitation on our ability to measure things.

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

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.

Creating "impossible" abstract objects is nothing new, it is easily done through the use of contradiction. — 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.

A proof as to why the real numbers are absolutely countable, in spite of all pretences to the contrary:

1. Enumerate the undecidable set of total functions within the entire set of enumerable Turing Machines of one argument {f1(x),f2(x),f3(x),..}, by running every Turing Machine in parallel on each input x=1,x=2,..., and shuffling their enumeration over time as necessary, so as to ensure that fn is defined when run on input x=n.

2) Define the Turing-Computable total function g(n) =fn(n)+1.

Congratulations, you've "proved" that the countable set of Turing machines is "larger" than the countable set of Turing machines.

[continuation of the previous post]
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 same computation (with the same input values) you obtain the same result.
At first, it would seem that this assumption is self-evident and has nothing to do with physics, but in reality there is no prove (nor even physical evidence) that this is true for very long and complex computations.
In fact the computations that are essential for mathematics and logic to exist are all based on the existence on non-linear physical processes. If the world were made only of continuous (non quantized) and linear (constant first derivatives) fields, there would be no way to perform any computation at all, and no intelligent human beings able to create - or discover - mathematical theorems.
Since we leave in a non-uniform universe, we can build objects that encode digital information, but it still seems to be likely that the quantity of information that can be contained in a portion of space-time and the speed at which this information can be digitally processed are both finite, due to some fundamental principles that seem to be unavoidable consequences of quantum mechanics, whatever the fundamental entities of the universe are (https://en.wikipedia.org/wiki/Quantum_indeterminacy, https://en.wikipedia.org/wiki/Quantum_information).
So, likely there exist quantities, or proves of theorems, whose computation (or prove) is physically impossible, even in principle. This is the some kind of question that we have in physics: does it make sense a model of the physical universe (or "multiverse") whose experimental detection is not possible, even in principle? Or, similarly, does it make sense to imagine an infinite non observable Euclidean space-time that contains the physical space (that is NOT incompatible, as a model, with general relativity), or is it better to imagine a finite non Euclidean space-time?

Now, both ZFC and intuitionistic logic (of whatever kind) assume that there exist some symbols that represent functions. And a function (in the model) is whatever "thing" with the following property: every time you give it the same input, it returns the same output.
But what if there is no such thing in nature, because all physical computational processes are in principle limited? There would be a process that is well defined as an experiment, but you cannot count on the fact that you always get the same output for the same input with absolute certainty: you get results that are statistically determined, but not deterministic.
So, you could have for example a theorem that is statistically true at 90%, but impossible to prove with 100% security even IN PRINCIPLE, due to some fundamental laws of physics (that we BELIEVE to be true, but of course have only an experimental - not mathematical - validity).
My question is: should this still be considered a mathematical theorem, or discarded because we do not have an absolutely certain proof of it? I believe that at the end it will be accepted as valid, but only in some kind of "quantum" logic, based on quantum-mechanical "experiments" instead of "real" proofs.

Confused by this. Constructive physics wouldn't allow a random sequence. — fishfry

Constructive physics (constructivist logic) can ASSUME the existence of a function that you can call "random" (whatever it means: it's an axiomatic theory), representing a physical process. Only that you cannot DERIVE or COMPUTE this function. You have to assume it as an axiom of the theory. The point is that this is allowed by the logic because you cannot introduce inconsistencies in this way!

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

Yes, that's very confusing. And if you look at the mathematical literature, the terms "constructive" and "intuitionistic" seem to have changed meaning, and every author uses them with a different meaning even today.
What I mean by "intuitionistic constructive" logic (because this is the one that I know) is Martin-Lof dependent type theory ( https://ncatlab.org/nlab/show/Martin-L%C3%B6f+dependent+type+theory )

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

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 the amplitude of the wave function), but I cannot prove that this is random and has a continuous distribution: it could well be that the experiment gives always the same result!

You say that the probability will be zero for an infinite random experiment. Yes, but there are no infinite experiments!
You can assign a measure to sets of points, but you cannot prove that these sets are uncountabe! The measure does make sense in physics for open sets, but not for every set: not all sets of ZFC are measurable, so there is no corresponding experiment for non measurable sets! ( just to upset you the same old story :razz: - Banach-Tarsky theorem does not describe a physical experiment: all physical experiments are calculated with integrals over open sets). And in constructivist logic the set of all open sets IS COUNTABLE! You cannot test if it is countable or not with physical experiments limited in time (how do you know that you don't get the same results again after a thousand years?)

. . . all physical experiments are calculated with integrals over open sets. — Mephist

Huh? :roll: Really?? all?

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 point. I am pretty sure what I said is correct.

I cannot communicate with someone who doesn't speak my language. My apologies, as I am not inclined to learn yours. It strikes me that you have disregard for the fundamental rules of logic, and that's why I am simply not motivated toward wasting the effort.

That's not an example at all. We know a lot about dark matter, that's why we can name it. — Metaphysician Undercover

You just make this shit up. How do you measure how much we need to know about something before we can name it? On the contrary, the day they discovered that the galaxies are spinning too fast to hold together, they named the cause "dark matter" while having no idea what it is or whether it exists at all.

In fact I already made that point to you. Read my post again.