i — Mephist

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. — Mephist

“No man ever steps in the same river twice, for it’s not the same river and he’s not the same man”, said Heraclitus in 544bc.

But you asked, "What if there exists no such thing in nature?" But that is not a problem for math, because math is not physics. In math there is certainty about the repeatability of functions.

I like the idea of a Boltzmann brain. The universe is completely random. Like static on a tv set. Once in a while, by perfectly random chance, a region of the universe suddenly coheres into a conscious mind. That's me. Or all of us. We'll be blinking out in an instant or two.

The idea that the universe is repeatable, or orderly, or understandable, etc., is a philosophical assumption. You can do science without making that assumption but most scientists prefer to believe their work is "about" something. I'm sure the bloodletters and proponents of the phlogiston theory of heat felt the same way in their time.

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! — Mephist

My understanding is that the constructivists allow in nonconstructivity whenever they paint themselves into a corner. For example every type of constructivism includes some form of the axiom of choice, because you can't do math without it. Arguably you can't even do physics without it. There's a weak form of choice needed to prove the Hahn-Banach theorem, a key theorem in functional analysis; which, as I've mentioned, is the mathematical framework of QM. Whether you can do QM without any form of choice I do not know. But the constructivists aren't as pure as they claim to be. And in the end they'll be proven wrong about the world. That's my opinion, but it may take centuries for me to be right.

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 — Mephist

Why are you trying to convince me that math isn't physics? I'm talking about the measure of computable bitstrings in the space of all bitstrings. The measure of the computable bitstrings is zero.

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

Right. Constructive physics is physics based on constructive math. That's all it is. Instead of using the real numbers, you only use computable numbers. Or something.

How do you measure how much we need to know about something before we can name it? — fishfry

You missed the point. You still do not seem to see the difference between describing something and measuring something. I said that if you're at the point of naming something, you ought to be able to say something about that thing.

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. — fishfry

You said: "By all our known theories of physics, galaxies should have flown apart long ago. Why didn't they?"

I think the answer is very obvious, the theories are wrong. If the galaxies don't behave the way that the theories say they should behave, then the theories are wrong. We know what "dark matter" means, it means that the theories are wrong. But instead of facing this fact, that the theories are wrong, someone has dreamed up a name "dark matter", and they attribute the fact that the theories are wrong to this mysterious thing, "dark matter". Why not just call it like it is, "the theories are wrong", dump the theories, and the "dark matter" which the theories necessitate because they're wrong, and get on with producing a new theory which doesn't make this mistake?

I cannot communicate with someone who doesn't speak my language. — Metaphysician Undercover

That explains a lot. Why should I (or anyone else) accept the constraints of your peculiar language?

It strikes me that you have disregard for the fundamental rules of logic ... — Metaphysician Undercover

How could you ever make such a determination, given your admission that you are unwilling even to try to understand my (or others') usage of the terms, simply because it is different from yours?

But instead of facing this fact, that the theories are wrong, someone has dreamed up a name "dark matter", and they attribute the fact that the theories are wrong to this mysterious thing, "dark matter". Why not just call it like it is, "the theories are wrong", dump the theories, and the "dark matter" which the theories necessitate because they're wrong, and get on with producing a new theory which doesn't make this mistake? — Metaphysician Undercover

I hope you'll forgive me but I prefer not to engage with your scientific nihilism, which itself is driven by scientific ignorance. If we threw out everything we don't understand, we'd have never understood gravity or electromagnetism. You simply know and understand nothing about the things you're talking about.

That explains a lot. Why should I (or anyone else) accept the constraints of your peculiar language? — aletheist

The constraints of my language are the fundamental laws of logic, identity, non-contradiction, excluded middle. These are what facilitate discourse.

How could you ever make such a determination, given your admission that you are unwilling even to try to understand my (or others') usage of the terms, simply because it is different from yours? — aletheist

You have demonstrated that you change the meanings of the words that you use, at will, as you use them, which results in the appearance of contradiction. You justify this with the claim that the same word has a different meaning in a different context. Yet there is only one context here, discussion between you and I. So all this does is lessen my charge against you from contradiction to equivocation, if it happens to be that the apparent contradictions are actually just ambiguous use, and not intentional deception.

Since you seem incapable of getting beyond this way of speaking, I see no point in continuing. I'm sorry, but I truly tried to understand your usage of terms, but I am incapable because your usage is extremely undisciplined. Let me make it clear though, it is just your usage, not others on this thread which I have this difficulty with. I disagree with others, but I understand their usage of language, not yours though.


You made the following statement:

By all our known theories of physics, galaxies should have flown apart long ago.

If you cannot see that the truth of this statement indicates that the theories have been falsified, then I'm afraid your denial is beyond hope.

If you cannot see that the truth of this statement indicates that the theories have been falsified, then I'm afraid your denial is beyond hope. — Metaphysician Undercover

Name me a theory of science that hasn't been falsified or will not someday be falsified. You're a scientific nihilist. You deny the entire enterprise because it doesn't serve up absolute truth on a platter.

'Will someday be falsified', is not the same as 'has been falsified'. The falsification is what determines the faults, demonstrating the weaknesses of the theory, showing us where improvement is needed. There is no point in dismissing theories which have not yet been falsified, because we would not know what needs to be improved. That's the scientific method, observations which are inconsistent with what the theory predicts reveal the faults in the theory. But until those inconsistent observations come about, we don't know where the weaknesses of the theory lie.

Will someday be falsified', is not the same as 'has been falsified'. The falsification is what determines the faults, demonstrating the weaknesses of the theory, showing us where improvement is needed. There is no point in dismissing theories which have not yet been falsified, because we would not know what needs to be improved. That's the scientific method, observations which are inconsistent with what the theory predicts reveal the faults in the theory. But until those inconsistent observations come about, we don't know where the weaknesses of the theory lie. — Metaphysician Undercover

Which has what to do with what we were talking about, the use of symbols to represent things we don't understand?

The constraints of my language are the fundamental laws of logic, identity, non-contradiction, excluded middle. — Metaphysician Undercover

None of those dictate the peculiar metaphysical definitions that you insist on imposing for terms like "existence" and "object," even in the context of non-platonist mathematics where they entail nothing ontological whatsoever.

... I see no point in continuing. — Metaphysician Undercover

On this we agree.

But you asked, "What if there exists no such thing in nature?" But that is not a problem for math, because math is not physics. In math there is certainty about the repeatability of functions. — fishfry

Yes but mathematics needs computations for proofs, and computations are physical processes. You may think that computations are not physics if you make them by mind, but if they are too long to be made by mind you need a computer, right? If a computation is too long to be performed by any computer even in principle, is it still valid? This sounds a little like the existence of parallel lines that never meet: they may not exist if the geometry of the universe is not Euclidean. OK, Euclidean geometry exists as a mathematical object, but can you use it to make deductions in general about any geometry? You can't, because there are sound geometries where that assumption is false.
At the same way, you can assume that unfeasible computations exist as a mathematical object, but can you use their existence to make deductions in general about any physical theory? You can't, because there are sound physical theories where that assumption is false.

All right, as I said, I just gave up arguing about constructivism.. :meh:

Why are you trying to convince me that math isn't physics? I'm talking about the measure of computable bitstrings in the space of all bitstrings. The measure of the computable bitstrings is zero. — fishfry

How do you define this measure in pure mathematical terms? You cannot use probability, because probability is physics (unless you find a sound mathematical definition of probability)

How do you define this measure in pure mathematical terms? You cannot use probability, because probability is physics (unless you find a sound mathematical definition of probability) — Mephist

https://en.wikipedia.org/wiki/Measure_(mathematics)

Measure theory is an abstraction of the concept of length, area, volume, etc; and also of probability. In fact probability theory is based on measure theory.

In fact if you know the Kolmogorov axioms of probability, that's basically measure theory.

A measure is a function from the collection of subsets of a given set, to the nonnegative real numbers satisfying some conditions. If in addition the measure of the entire set is 1, that's a probability measure.

The measure of the rationals in the reals is zero. And in fact the measure of any countable set must be zero. This follows from countable additivity. The measure of all computable bitstrings has measure zero for the same reason. That means if you pick a real number "at random," the probability is zero that you'll pick a computable real.

. . . but mathematics need computations for proofs. — Mephist

I have conjectured and proven lots of theorems - some quite challenging - that do not require computations beyond basic inequalities and a little arithmetic of complex numbers. However, creating examples and imagery in the complex plane usually requires programming skills and a computer. So, even if not for proofs, computations are necessary in many areas of mathematics. :cool:

If a computation is too long to be performed by any computer even in principle, is it still valid? — Mephist

A TM with a program too long to write down in the age of the universe is still a TM. Practical resource limitations do not apply to the theory of computability.

No computation is too long to be performed "in principle." In principle a TM is a finite sequence of instructions. No matter how long it is, as long as it's finite it's computable in principle, if not necessarily in practice.

We know that Cantor's Theorem concerning the cardinality of the power-set of integers isn't a constructive proof, for we cannot enumerate and diagonalise only the Turing machines representing the recursively enumerable sets due to the Halting Problem, so we must enumerate the larger set of TMs.

And yet, the entire set of countable TMs can be diagonalised to prove that the set of countable TMs are "uncountable", by dynamically enumerating the halting TMs so as to ensure the termination of the diagonal TM for each of it's inputs; but in fact all that my proof of "uncountability" amounts to with respect to Turing Machines, is the construction of an enumeration of Turing Machines in such a way that the diagonal Turing machine cannot be part of the enumeration. This is analogous to enumerating the odd numbers and then diagonalising them to construct an even number.

To spell out the difference, in the case of Cantor's Theorem the constructed enumeration of the sets of natural numbers is considered to be prior to the construction of the diagonal set, but in the case of my method, the enumeration of TMs was constructed via the construction of the diagonal function. In other words, selectively constructing a non-exhaustive infinite enumeration via a diagonal procedure isn't a proof that a bijection with the natural numbers doesn't exist under a different enumeration. And in the case of Turing Machines we know that such a bijection does exist.

But this raises doubts about Cantor's original diagonal argument, for I might have been lucky enough with my original enumeration of TMs to produce a diagonal function without requiring any shuffling of the enumeration. Therefore Cantor's original argument isn't proof enough that the power-set of N is literally larger than N.

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. — Metaphysician Undercover

I think that you are ascribing to mathematics the kind of role that I don't think it has. At least, not directly. Or maybe I did walk into this when elaborating over my example. Its aim wasn't to model the structure of physical objects, but to illustrate how coarse structures not literally represented by mathematical ideals, can still be usefully approximated by those ideals. It was designed to have some similarity with the atomic structure of materials. But it is not a theoretical model for physics. The idea was, that actual physical structures approximate the mathematical ideals, and our numerical algorithms approximate those same ideals, and thus, under certain assumptions of the magnitudes of the involved deviations, our numerical algorithms match the physical structures within the required precision.

About the grain uniformity. let's say that we are talking about Voronoi partitioning of space, and not triangular tessellation, because it simplifies the definition. When I say uniform grain, what I mean is that the distances between the generator points of the partitioning follow some known distribution with finite variance. This is a simple example, and not a physical model. However, it does illustrate, I think, that the diagonal will usefully approximate some quantity over a rough spacial structure, despite being itself defined as a continuous object.

But approximation in practise is not the same as approximation in theory. — Metaphysician Undercover

Which theory do you mean? For me, real numbers theorize some characteristics of computation. And do so imperfectly. The algebraic structure is defined over some converging computational sequences. It does allow for imaginary objects that do not correspond to actual computational processes, because the latter can not be specified procedurally. And thus it works with incomplete specification. But it is a best effort theory. I am not necessarily subscribed to the idea that real numbers correspond to the points of physical lines, whatever that may mean. It may turn out that this conceptualization works, but I am not convinced. I do think that it works for approximations however.

P.S. I want to clarify that I do actually think that our computational and logical ideals are naturally inspired. They are not literally representative of any particular physical structure, but they are "seeded" as concepts by nature, whether our sentience existed or not.

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 bitstrings: there is an infinite number of computable bitstrings
(2). - The set of all possible bitstrings: there is an infinite (not countable) number of possible bitstrings

So, to calculate the probability of choosing a computable bitstring, I have to divide (1) by (2), and the result is zero. What kind of computation (limit, theorem, or whatever) should I use to divide (1) by (2)? How is this computation defined? That was my question.

Second question: supposing that the division of (1) by (2) is zero, what sense can I make of that measure?
- I cannot perform the experiments because they are infinitely long.
- I cannot even choose one bit at a time from each of the sequences, because both are computable at every step, until they are finite. But they will always remain finite forever: the experiment never ends.

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 computations can be considered to be proofs, right? You reduce an expression in a normal form following some rules (if it's a multiplication between integers the "proof" can be made automatically with a calculator).

The point is that a lot of interesting physical problems (for example computing the exact shape of biological proteins by following the lows of quantum mechanics) require an amount of computation that is beyond the computing power of any (classical) calculator, and probably even if you could use a computer big as the entire universe, if you want to be sure of the result at 100%. Maybe there are cases where the computation is easy because of particular symmetries of the system, but there are rather exceptional cases.

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 can be deduced from the axioms of ZFC set theory.
But what I am saying is that you can equally well assume as an axiom (that would be incompatible with ZFC) that Turing machines DO NOT exist!
So, what is the reason to assume the existence instead of non existence? It is possible (and even probable in my opinion) that the theory that assumes the non existence of Turing machines is even more interesting (both mathematically and for physical theories) than the theory assuming their existence.
And the alternative to TM's existence is not univocal: there are a lot of possible "intuitionistic" and "constructivist" theories that are not equivalent between each-other. But, at the same way, there are even a lot of non-Euclidean geometries. Constructivist theories correspond to elegant constructions in topology, represented as internal languages of certain categories. In comparison, ZFC axioms seem to be much more arbitrary, from my point of view.

Well, my argument was simpler than this:
- We know that we cannot enumerate all halting Turing machines, so for every supposedly complete list of halting Turing machines I can find another halting Turing machine that is not in the list.
Does that mean that the set of halting Turing machines in uncountable? No! It only means that there is no way to enumerate that list!
Now I have a second list: the list of all functions - even not computable - from N to N, and I have a theorem that says exactly the same thing: for every supposedly complete list of functions from N to N, I can find another function from N to N that is not in the list. But in this case I conclude that the set of functions from N to N is uncountable! Why? What guarantees that in this case I can build that list? The theorem only says that there have to be some more elements not present in the original list (the list cannot be complete), but not that there has to be an uncountable number of missing elements.

But what I am saying is that you can equally well assume as an axiom (that would be incompatible with ZFC) that Turing machines DO NOT exist! — Mephist

I've never heard of this idea, that TM's don't exist. I see no problem expressing TMs in set theory. An unbounded tape of cells is modeled as the integers. You have some rules that let you move right or left. It's pretty straightforward.

Can you say more about this? I have never heard this idea at all. It seems VERY restrictive. Perhaps it's like denying the axiom of infinity. Logically consistent but too restrictive to do math with.

I'm not even sure what that means, that TMs don't exist. The Euclidean algorithm to find the greatest common factor of two integers is 2400 years old. That's the world's first algorithm. It's a Turing machine. It exists. I think "TMs don't exist" introduces a contradiction.

Constructivist theories correspond to elegant constructions in topology, represented as internal languages of certain categories. In comparison, ZFC axioms seem to be much more arbitrary, from my point of view. — Mephist

I'll stipulate that a lot of very clever people are on the neo-intuitionist bandwagon these days and that their viewpoint represent the future, while mine represents the past; at least for now. My unease with constructivism is psychological. I love the catechism of standard classical math. It's a matter of faith, not science. Or if faith is too strong a word, familiarity and preference.

I've never heard of this idea, that TM's don't exist. I see no problem expressing TMs in set theory. An unbounded tape of cells is modeled as the integers. You have some rules that let you move right or left. It's pretty straightforward.

Can you say more about this? I have never heard this idea at all. It seems VERY restrictive. Perhaps it's like denying the axiom of infinity. Logically consistent but too restrictive to do math with. — fishfry

Well, OK. Turing machines are a model of computation equivalent to Church's untyped lambda calculus.
Limiting Turing machines to have finite dimensions would in fact be too restrictive to do mathematics (including the infamous square root of two :wink: ). But you can limit Church's untiped lambda calculus by introducing a typed lambda calculus, and with dependent types this is a formal system powerful enough to do mathematics - and corresponding to some form of intuitionist theory (omitting details).
I don't know what would be the equivalent limitation to Turing machines that corresponds to dependently typed lambda calculus (if there is one). So, I should have said that we can assume that the "original" (non limited) Turing machine does not exist.

I don't know what would be the equivalent limitation to Turing machines that corresponds to dependently typed lambda calculus (if there is one). So, I should have said that we can assume that the "original" (non limited) Turing machine does not exist — Mephist

You're saying TMs don't exist but finite state machines do? Maybe so, but then you'll make your physics a lot harder if you can't even run an algorithm to approximate a constant of nature. I never studied lambda calculus so I can't comment on the rest. But since lambda calculus and TMs are equivalent, they either both exist or neither do. As far as I understand.

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.wikipedia.org/wiki/Lambda_calculus) and Turing machines are equivalent as computing power.
But there are various other less powerful (as computing power) forms of lambda calculus, that are called in general "typed" lambda calculi (https://en.wikipedia.org/wiki/Typed_lambda_calculus) that do not correspond to Turing machines as computing power, but are still more powerful than finite state machines.
One of the most powerful (as computing power and expressiveness of the resulting language) lambda calculi is called Martin-Lof dependent type theory (https://ncatlab.org/nlab/show/Martin-L%C3%B6f+dependent+type+theory). Yes, it's not called "dependently typed lambda calculus", but it is!
Meaning: it's still a version of lambda calculus using types with a very powerful type system. However, this is still less powerful than the original lambda calculus invented by Church, but powerful enough to do mathematics. The interesting thing is that it's not true that this limitation produces a logic that is difficult to use. On the contrary, the logic is easier to use than ZFC, and that's the reason why most of currently developed proof assistants (Lean, Coq, Agda) make use of this language (https://en.wikipedia.org/wiki/Category:Dependently_typed_languages).
Now, about Turing machines: a Turing machine equivalent to this kind of lambda calculus is basically one of these proof assistants (without hardware limitations of the real machines, of course), so probably it's not impossible to define. But I think that nobody described exactly these kind of logic in term of Turing machines. However, they are equivalent to a kind of Turing machine that is less powerful than the original Turing machine, but powerful enough to do all mathematics.

P.S. Here's a citation taken from wikipedia: (https://en.wikipedia.org/wiki/Typed_lambda_calculus)
All the systems mentioned so far, with the exception of the untyped lambda calculus, are strongly normalizing: all computations terminate. Therefore, they cannot describe all Turing-computable functions.[1] As another consequence they are consistent as a logic, i.e. there are uninhabited types. There exist, however, typed lambda calculi that are not strongly normalizing.

P.S. Here's a citation taken from wikipedia: — Mephist

So the lambda formulation is more granular, able to support more nuanced theories? Something like that?

Anyway I know about Coq and the proof assistants and such, but my eyes glaze over every time I read the phrase, Martin-Löf type theory. So I know all about the existence of all this stuff, without necessarily knowing much about it. On the other hand, on something like homotopy type theory, I don't know anything about types. But I do happen to know what homotopies are, and that gives me a little insight into what they're doing.

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, right? :smile: )
But I think that now the picture is becoming quite clear (even thanks to Voevodsky's work): there is a very strict correspondence between topology and logic. But you have to "extend" the notion of topology to the one of topoi (a category with some additional properties). ZFC is the logic corresponding to the standard topology (where lines are made of uncountable sets of points). But ZFC and the "standard" topology are not at all the only logically sound possibility! (that in a VERY short summary)