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:

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:

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

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, 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).

Yes! ( on all points :smile: )

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. 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)

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.

Cantor's theorem. |X|<|P(X)||X|<|P(X)|. This is a theorem of ZF, so it applies even in a countable model of the reals. You mentioned Skolem the other day so maybe that's what you mean. Such a model is countable from the outside but uncountable from the inside. — 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:
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..

On a different topic, let me ask you this question.

You flip countably many fair coins; or one fair coin countably many times. You note the results and let H stand for 1 and T for 0. To a constructivist, there is some mysterious law of nature that requires the resulting bitstring to be computable; the output of a TM. But that's absurd. What about all the bitstrings that aren't computable? In fact the measure, in the sense of measure theory, of the set of computable bitstrings is zero in the space of all possible bitstrings. How does a constructivist reject all of these possibilities? There is nothing to "guide" the coin flips to a computable pattern. In fact this reminds me a little of the idea of "free choice sequences," which is part of intuitionism. Brouwer's intuitionism as you know is a little woo-woo in places; and frankly I don't find modern constructivism much better insofar as it denies the possibility of random bitstrings. — fishfry

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.

I do not see how this can be. The constructive real line is not Cauchy-complete. It's only countably infinite. It does not contain any of the noncomputable numbers. It can not possibly be an intuitively satisfying model of a continuum. I'm troubled by this and I'm troubled that the constructivists never seem to be troubled. — fishfry

[continuation of the previous post]
So, you can see constructivist logic as an algebra of propositions built with computable functions (https://en.wikipedia.org/wiki/Heyting_algebra). You cannot build non-computable functions using only the operations of this algebra, but you can add elements that are not part of the algebra ("external" non-computable functions) to obtain a new algebra that uses all computable functions plus the function that you just added.
That's exactly the same thing that you do adding square root of 2 to the rationals: you obtain a new closed field that contains all the rational numbers plus all that can be obtained by combining the rational numbers with the new element by using the operations defined on rational numbers.

But I see that the main problem for you is not about the soundness of logic, but about the cardinality of the set of real numbers.
So, my question is: how do you know that the cardinality of the set of real numbers is uncountable?
- answer (let's speed up the interaction :smile: - you can add additional answers in the next post if you want): because of Cantor's diagonal argument (https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument).
Well, Cantor's diagonal argument is still valid in constructivist logic: it says that any function that takes as an argument an integer and returns a function from integers to integers cannot be surjective (cannot generate all the functions). The proof is exactly the same: take F(1,1) and change it into F(1,1) + 1, then take F(2,2) and change it into F(2,2) + 1, etc... This is a computable function if all the F(n,n) were computable, ( n -> F(n,n) - very simple algorithm to implement ) but it cannot be in the list: if it were in the list ( call it X(n,n) ), let "m" be the position of X ( X is the m-th function ). What's the value of X(m,m) ?
X(m,m) cannot be computable.
The problem is well known: you cannot enumerate all computable functions because there is no way to decide if a given generic algorithm stops.
So, computable functions are as uncountable as real numbers are. Where's the difference?

Prediction is not a good indicator of understanding. Remember, Thales predicted a solar eclipse without an understanding of the solar system. All that is required for prediction is an underlying continuity, and perhaps some basic math. I can predict that the sun will rise tomorrow morning without even any mathematics, so the math is not even prerequisite, it just adds complexity, and the "wow' factor to the mathemagician's prediction. So, continuity and induction is all that is required for prediction. Mathematics facilitates the induction, but it doesn't deal with the continuity. Real understanding is produced from analyzing the continuity. This is an activity based in description, and as I mentioned, is beyond the scope of mathematics.

Again, we encounter the problem of pragmatism. If prediction is all that is required, then we gear our epistemology toward giving us just that, predictability. If this is easiest done using false premises like Platonic realism, then so be it. But we do this at the expense of a real understanding. — Metaphysician Undercover

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 equations of quantum mechanics: physicists learned how to use them to predict the results of experiments. Maybe that is a problem, but it is a problem of physics since the beginning: Newton didn't know how to make sense of a "force" that acts from thousands of kilometers of distance.
I heard somebody say that now it's clear: everything is filled with a "field", and it's the exchange of particles of that field that transports the force. So, can you try to imagine how to generate an attractive force by exchanging an object?
The reality is that there are equations that work, and you can apply a mathematical theory made of imaginary things with imaginary rules that happen to give the right results. The real "ontological" reason why this system is able to "emulate" the experiments of the real world, nobody is able to explain. And it's not only about the use the square root of 2.

Notice that the problem is with the conception itself, it has nothing to do with "the real". The idea that we can conceive a point anywhere is false, as demonstrated by the square root problem. Conceiving of continuity in this way, such that it allows us to put a point anywhere is self-defeating. Therefore we need to change our concept of the continuity of "space". — Metaphysician Undercover

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 equations are partial differential equations defined on a continuous 4-dimensional space.

Now. let's add time to the mix. We already have a faulty conception of space assumed as continuous in a strange way which allows us to create irrational figures. Special relativity allows us to break up time, and represent it as discontinuous, layering the discontinuous thing, time, on top of the continuous, space. Doesn't this seem backward to you? Time is what we experience as continuous, an object has temporal continuity, while space is discontinuous, broken up by the variety of different objects. — Metaphysician Undercover

Special relativity allows us to represent time as discontinuous?? Why? On the contrary, in special (and general) relativity space and time have to be "of the same kind", because you can transform the one into the other with a geometrical "rotation" ( https://en.wikipedia.org/wiki/Lorentz_transformation ) simply changing the point of view of the observer.

Your analogy is faulty, because what you have presented is incidents of something representing what is meant by the symbol "5". So what you have done is replaced the numeral "5" with all sorts of other things which might have the same meaning as that symbol, but you do not really get to the meaning of that symbol, which is what we call "the number 5". The point being, that for simplicity sake, we say that the symbol "5" represents the number 5. But this is only supported by Platonic realism. If we accept that Platonic realism is an over simplification, and that the symbol "5" doesn't really represent a Platonic object called "five", we see instead, that the symbol "5" has meaning. Then we can look closely at all the different things, in all those different contexts, which you said could replace the symbol "5", and see that those different things have differences of meaning, dependent on the context. Furthermore, we can also learn that even the symbol "5" has differences of meaning dependent on the context, different systems for example. Then the whole concept of "a number" falls apart as a faulty concept, irrational and illogical. That's why you can easily say, anything can be a number, because there is no logical concept of what a number is. — Metaphysician Undercover

Well, my idea was much simpler, I guess: just treat a number like an attribute of an object, like the color of the object, or it's volume. Do you agree that the volume is an attribute of an object? A lot of objects may have the same volume, or the same color. Well, an object can have even a number, if I consider the object as made of several distinct parts.
How would you teach a child what is 5? You show him a picture with 5 flowers and you say: this is 5. The child understands that there is some attribute of that picture that is called 5. Then you show him a picture with 5 trees and you say: that is 5 too! Then the child should understand what's the attribute (the characteristic) that the two pictures have in common.
You can do the same to show him what is a color: show two pictures of different objects with the same color.
I know, this explanation is not very "philosophical"... and, to say the truth, I don't really understand why is this such a philosophical problem :yikes: But what's the problem with this interpretation?

I don't quite get this. "Space-time" here is conceptual only, like "the square" we've been talking about, or, "the circle". Therefore, the positioning of the points is what creates the "object" called space-time, just like we could position points in a Euclidian system, to outline a line, circle, or square. What is at issue, is the nature of the medium which is supposed to be between the points, which accounts for continuity. The continuity might be "the real", what exists independently of our creations of points according to some geometrical principles. — Metaphysician Undercover

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 that they are 4-dimensional "cubes": 3 dimensions of space and 1 dimension of time. These "cubes" are not all of the same size, they don't have straight angles and each edge in general can have a different length. The measures of these little "cubes" are described by the metric tensor. The information of which little cube is attached to which other on which side is described by the ordering of the "indexes" that I assigned to each cube:
for example: cube (1,1,1,1) is attached to cube (2,1,1,1) in this way: the right side of cube (1,1,1,1) is the left side of cube (2,1,1,1). Cube (1,1,1,1) is attached to cube (1,1,2,1) in this way: the up side of cube (1,1,1,1) is the down side of cube (1,1,2,1). I don't know if that gives the idea...

Actually, if you analyze this situation closely, "wave functions" are produced from observations, so they are still mathematical representations of the movements of objects. The wave function is a use of mathematics to represent observable objects. There is no such separation between the representation of a physical particle and the wave functions, the wave functions represent the particles. They are of the same category, and I think the physicist treats the particle as a feature of the wave functions. Wave functions are used because such "particles" are known to have imprecise locations which they can only represent as wave functions. With observed occurrences (interactions) the particles are given precise locations. Wave functions represent the existence of particles when they are not being observed. — Metaphysician Undercover

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 are spread all over the space, or on a space much larger than the observed spot. Nobody has never seen an elementary particle that looks like a wave function!

Ok this is the post I wanted to get to. — fishfry

Yes, that's one of the most interesting subjects even for me :grin:

Ok that's beyond my pay grade, but maybe I can tell you what I know about it. Say you have a hydrogen atom, one proton and one electron, is that right? The electron can be in any one of a finite number of states (is that right?) so if you take two hydrogen atoms with their electrons in the same shell (is that still the right term?) or energy level, they'd be exactly the same. — fishfry

Yes, everything right until now.

But you know I don't believe that. Because the quarks inside the proton are bouncing around differently in the other atom. Clearly I don't know enough physics. I'll take your word on this stuff. — fishfry

No, because even the motion of the quarks inside the proton is quantized, at the same way as the motion of the electrons is. If the proton is in it's base state (and that's always the case, if you are not talking about high-energy nuclear collisions), ALL that happens inside of it is described ONLY by an eigenfunction of the Hamiltonian operator with the lowest eigenvalue: it's a well-defined mathematical object. And all protons in their base state are described by the same function. No other information is required to describe COMPLETELY it's state (even if quarks were made of "strings" and strings were made of "who knows what"). What would change in case quarks were made of strings is that the Hamiltonian operator would have a different form, probably EXTREMELY complex, but the wave-function would be the same for all protons anyway.

I don't believe you. I do believe that you know a lot more physics than I do. But I don't believe that there is an exact length that can be measured with infinite precision. I'm sorry. I can't follow your argument and it's clearly more sophisticated than my understanding of physics but I can't believe your conclusion. — fishfry

No, there isn't an exact length that can be measured with infinite precision. But you don't need to be able to measure an atom with infinite precision to check if two atoms are exactly identical: identical particles in QM have a very special behavior: the wave-function of a system composed of two identical particles is symmetric (if they are bosons) or anti-symmetric (if they are fermions) (https://en.wikipedia.org/wiki/Identical_particles). Because of this fact, the experimental result conducted with two identical particles is usually dramatically different from their behavior even if they differ from an apparently irrelevant detail.
For example you can take a look at this: https://arxiv.org/abs/1706.04231

Wait, what? You just agreed with me. The "real physical predictions" are only good to a bunch of decimal places. There are no exact measurements. The theory gives an exact answer of course but you can never measure it. You agree, right? — fishfry

Yes, I agree. There are no exact measurements, but there are exact predictions in QM. For example, the shapes of hydrogen atom's orbitals are regular mathematical functions that you can compute with arbitrary precision: http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hydwf.html
Of course, you cannot verify the theory with arbitrary precision, but the theory can produce results with arbitrary precision (at least in this case).

All of this is quite irrelevant to whether we can measure any exact length in the world. Since it's perfectly well known that we can't, it doesn't matter that you have this interesting exposition. There's some QED calculation that's good to 12 decimal digits and that's the best physics prediction that's ever been made, and it's NOT EXACT, it's only 12 decimal digits. Surely you appreciate this point. — fishfry

Yes, that's because physical experiments are more and more difficult to realize if you want more and more precision, and in this case ( the exact measurement of electron's magnetic moment ) even the computational complexity of the theoretical computation grows exponentially with the precision of the result. But in theory the result can be calculated with arbitrary precision.

If that's all you mean, you have gone a long way for a small point. Of course if we have a theory we can solve the equations and get some real number. But we can never measure it exactly; nor can we ever know whether our theory is true of the world or just a better approximation than the last theory we thought was true before we discovered this new one. — fishfry

Yes, exactly. That's what I mean.

I'm not following this. — fishfry

Well, let's abandon the discussion about this "theory" ... :smile:

P.S. In my opinion, that's one of the most interesting aspects of QM: the information required to describe exactly an atom is limited: it's a list of quantum numbers, each of them is an integer in a limited range. So, just a bunch of bits.
If the equations of QM were the same as in classical mechanics (orbits of planets depending on their initial conditions without any limit to the precision of measurement), chemistry would be a complete mess: every atom would be different from all the others (as every planetary system is different from all the others)

I do not see how this can be. The constructive real line is not Cauchy-complete. It's only countably infinite. It does not contain any of the noncomputable numbers. It can not possibly be an intuitively satisfying model of a continuum. I'm troubled by this and I'm troubled that the constructivists never seem to be troubled. — fishfry

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 the language. You cannot say anything about them. There cannot be disagreement about sentences that do not exist in one of the two languages: you can only disagree about what you can say in both languages.
So, if you don't need to speak about the things that you cannot speak about, there's no problem.
If you need to speak about those things (for example the incomputable real numbers) you can add their existence to constructive logic as an axiom, and that axiom is independent from the other axioms. So it cannot cause inconsistencies, or alter the results that you just deduced using only the constructivist part.

[ sorry, I have to go now: I'll continue this another time ]

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 "constructivist" theory. For example, from page 55:
"As in the classical logic, we can add to intuitionism the axioms of arithmetic or of the set theory, which gives the constructive versions of these logical theories"

All the results are exactly the same, and all theorems are equivalent, only reformulated in a different way (encoding the rules of logic in a different, but equivalent way)

For physics, if the formulas are the same and the method to calculate the results is the same, there's no difference: the difference is only in non-essential mathematical "details" (from a physicist point of view).

Are you saying that classical and constructive physics are equivalent as categories? I'm afraid I don't know exactly how you are categorifying physics — fishfry

Well, basically category theory can be used as a foundational theory for physics. It's rather
"fashionable" today, here's an example: https://arxiv.org/abs/0908.2469
One of the advantages is that equivalent formulations of a given theory can be seen as the same theory: pretty much the same of what Vladimir Voevodsky did with homotopy type theory and his univalence axiom ( https://ncatlab.org/nlab/show/univalence+axiom ).

The mathematical system being employed premises that a symbol represents an object, and that each time the symbol appears within an expression, like an equation, it represents the very same object. Any conclusions produced must uphold this premise. — Metaphysician Undercover

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), and then there are other mathematical "objects" (such as wave functions- https://en.wikipedia.org/wiki/Wave_function) that are not meant to represent something that normally we could call "objects" in physics.

Right, clearly there are "problems" if we represent mathematical figures as real objects. Notice I removed your qualifier, "physical" objects. If we begin with a statement as to the nature of an "object", a definition, such as the law of identity, then we must uphold this definition. If the claim is that a "mathematical object" is fundamentally different from a "physical object", such that the same definition of "object" cannot apply to both, then we need to lay out the principles of this difference so that equivocation can be avoided. — Metaphysician Undercover

Yes, exactly!

The problem is that any such "grid of points" is laid out on a spatial model. If a square is an invalid spatial model, then so is the Cartesian coordinate system Then "space-time" itself is improperly represented. — Metaphysician Undercover

Well, the grid of points represents only the topology of space-time, not the metric.
Meaning: there are 4 integer indexes for each point (the grid is 4-dimensional), and then the ordering of the points, only defines which are the pieces of space-time that are adjacent (attached to each-other), not their size or the orientation of the edges.

Then you have to associate to each point two 4-dimensional tensors (a tensor, basically, is a 4x4 matrix of real numbers - approximated as floating-point numbers with limited precision) (for picky mathematicians reading this: these are the components of the tensor relative to an arbitrarily chosen base, not the tensor itself): one is the metric tensor (https://en.wikipedia.org/wiki/Metric_tensor_(general_relativity), defining basically the size of this piece of space-time in each dimension, and the other is the stress-energy tensor (https://en.wikipedia.org/wiki/Stress%E2%80%93energy_tensor), describing the matter that is contained in this piece of space-time.

But this is still a very over-simplified description: for example, to perform the calculations you have to calculate the equivalent of derivatives of these tensors (that are even larger sets of numbers - https://en.wikipedia.org/wiki/Levi-Civita_connection).

And there is the tacit assumption that the topology of space-time is the same as flat space-time (that is true for relatively "small" systems such as for example the solar system).

Pragmatism is not the answer, it is the road to deception. Human objectives often stray from the objective of truth. When we replace "the truth" with "they simply have to work", we allow the deception of sophism, because "what it works for" may be something other than leading us toward the truth.. — Metaphysician Undercover

Well, I wouldn't call this "pragmatism". That has more that only a practical meaning.
I mean: you can call "natural number" anything that "works" in a similar way as a given system of symbols and rules (there are a lot of equivalent systems: for example Peano arithmetics (https://en.wikipedia.org/wiki/Peano_axioms, or the usual decimal symbols with arithmetic rules for operations as they teach in primary school).
So, the fingers of my hand are a number (the number 5), if I specify how to perform the arithmetical operations with fingers. The electrical charges inside a computer are numbers too, because the computer has encoded the rules of arithmetic as logical circuits.
So, basically, every "object" that is composed of well-identifiable parts can be considered to be a natural number, if you specify how to perform the arithmetical operations with the parts.

Thirdly, we'd need some principles to relate the continuous to the quantized. For example, to me time appears to be continuous, and space appears to be quantized. If this is the case, then we need different principles for modeling time than we do for space, and some principles to relate these two systems to each other. — Metaphysician Undercover

Well, the problem is that what you say doesn't seem to be in any way "compatible" with current physical theories. And current theories are VERY good at predicting the results of a lot of experiments.
To me, it seems VERY VERY unlikely that a simple physical theory based on a simple mathematical model can be compatible with current physics at least in a first approximation. The physical world seems to be much more complex than we are able to imagine...

[ TO BE CONTINUED ANOTHER DAY..]

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 the difference?

I mean: OK, you can reformulate all current physics theories in a constructivist logical framework, but is the result really different from the normal formulation?
I like constructivist logical frameworks based on dependent type theory because of the simplicity of formal proofs: that's the thing that makes the difference! But form the point of view of a physical theory, the equations and the computations, and of course the results, are exactly the same! ( I didn't read the whole book, maybe I missed something, but at first sight, that's the way it looks like ).

From my point of view, that's only another way of "encoding" physical formulas and procedures using a different logical framework. Of course encoding is important, if it "refactors" the same concepts in a simpler way: that is basically what I would call a better understanding of the theory. You can even "encode" these theories using only natural numbers using Godel's encoding if you want (https://en.wikipedia.org/wiki/G%C3%B6del_numbering) to obtain an absolutely incomprehensible theory that gives the same results: that's NOT a good formulation of the theory :smile:

But in this case, the two formulations are completely equivalent (in the sense of equivalence of categories, if you see theories as functors from formal systems to models), and from the point of view of physics the choice between two equivalent representations doesn't make any difference. And probably, after Voevodsky, it doesn't make much difference even from a mathematical point o view.

On paper you produce "a representation" of the Euclidean ideals. That representation is something completely different from the square root, which is part of the formula behind the representation which you draw on paper. When I want to lay out a square corner, a right angle, on the ground, I might use a 3,4,5, triangle. In this exercise I am not using a square root at all. I could make this square corner without even knowing the Pythagorean theorem, just knowing the lengths of 3,4,5. But if one side of the right angle is to be 5, and the other side 6, I'll need to know the Pythagorean theorem, and then figure the diagonal as the square root of 61 if I am going to make my right angle. — Metaphysician Undercover

So is the 3,4,5 triangle really straight or not? I don't understand...

That's not quite right. We, as human beings, cannot necessarily distinguish two distinct things, due to our limited capacities of perception and apprehension. So it's not quite right to say that you can always distinguish a thing from all other things. A thing is distinct from other things, but we cannot necessarily distinguish it as such. And that difference may be a factor in quantum mechanics — Metaphysician Undercover

OK.

Right, but to perceive a thing, name it "X", and then claim that it has the "identity" of X, is to use "identity" in a way inconsistent with the law of identity. You are saying that the thing's identity is X, when the law of identity says that a thing's identity is itself, not the name we give it. The law says a thing is the same as itself, not that it is the same as its name. — Metaphysician Undercover

OK, the identity cannot be identified with the name.

Consider that human beings are sometimes mistaken, so it is incorrect to say "the name is a reference that identifies always the same concrete object". The meaning of the name is dependent on the use, so when someone mistakenly identifies an object as "X", when it isn't the same object which was originally named "X", then the name doesn't always identify the same concrete object. And, there are numerous other types of mistakes and acts of deception which human beings do, which demonstrate that the name really doesn't always identify the same concrete object, even when we believe that it does. — Metaphysician Undercover

OK, so what can I do with identities?

If I cannot refer to them with names, I would say that it's impossible to speak about identities. So, they surely cannot be used in logic arguments. Logic is basically manipulation (operations) of language, isn't it?

Do you recognize that Einstein's relativity is inconsistent with Euclidian geometry? Parallel lines, and right angles do not provide us with spatial representations that are consistent with what we now know about space, when understood as coexisting with time. My claim is that the fact that the square root of two is irrational is an indication that the way we apply numbers toward measuring space is fundamentally flawed. I think we need to start from the bottom and refigure the whole mathematical structure. — Metaphysician Undercover

But Einstein's relativity is based on differential calculus and real numbers. How can it be correct, if the whole system is wrong?

Consider that any number represents a discrete unit, value, or some such thing, and it's discrete because a different number represents a different value. On the other hand, we always wanted to represent space as continuous, so this presents us with infinite numbers between any two (rational) numbers. This is the same problem Aristotle demonstrated as the difference between being and becoming. If we represent "what is" as a described state, and later "what is" is something different, changed, then we need to account for the change (becoming), which happened between these two states. If we describe another, different state, between these original two, then we have to account for what happens between those states, and so on. If we try to describe change in this way we have an infinite regress, in the very same way that there is an infinite number of numbers between two numbers. — Metaphysician Undercover

OK, continuous change cannot be identified by a finite number of steps. But does this prove that continuous change cannot exist?

If modern (quantum) physics demonstrates to us that spatial existence consists of discrete units, then we ought to rid ourselves of the continuous spatial representations. This will allow compatibility between the number system and the spatial representation. Then we can proceed to analyze the further problem, the change, becoming, which happens between the discrete units of spatial existence; this is the continuity which appears to be incompatible with the numerical system. — Metaphysician Undercover

Quantum physics is based on Hilbert vector spaces, that are infinite-dimensional continuous vector spaces (even "more" infinite than the infinite 3-dimensional euclidean space). I don't know if there is a way to express the same theory with similar results on first approximation making use only of mathematics based on integral fields. But even if there is a way, I suspect that it would become an extremely complex theory, impossible to use in practice. Would it then be more "real" then the current theory making use of real numbers? At the end, the only way to decide which theory is more "real" in physics is only agreement with experiments.

No, division and multiplication are not at all symmetrical, because you never have a remainder in multiplication. In multiplication, you take a designated number as the "base unit", a designated number of times, and you never end up with a remainder. You have no such "base unit" in division, you have a large unit which you are trying to divide down to determine the base unit, but you often end up with a remainder.

Evidence of this difference is the existence of prime numbers. These are numbers which we cannot produce through multiplication. We can still divide them, knowing there will be a remainder, but that doesn't matter, because there's often a remainder when we divide, even if the dividend is not prime. — Metaphysician Undercover

OK, division and multiplication are not symmetrical for integers, because integers are "quantized": you can't give one candy to three children, because candies are "quantized". But physical space is not quantized, or is it? The mathematical description of continuous measures is not inconsistent: there are several ways to make them at least as consistent as natural numbers are.
So, if integers (quantized) objects exist in nature, why shouldn't continuous objects exist?

The problem of irrational numbers arose from the construction of spatial figures. That indicates a problem with our understanding of the nature of spatial extension. So I suggested a more "real" way of looking at spatial extension, one which incorporates activity, therefore time, into spatial representations. Consider that Einsteinian relativity is already inconsistent with Euclidian geometry. If parallel lines are not really "parallel lines", then a right angle is not really a "right angle", and the square root of two is simply a faulty concept. — Metaphysician Undercover

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 1-dimensional, if you don't consider the internal surface), and all real physical objects are 3-dimensional. The usual "trick" to make some sense of this kind of model is that they are so small that are not directly observable. Einstenian general relativity is the same as Euclidean geometry in this respect: world lines are just a mathematical abstraction to represent trajectories in space-time. They are not real objects, and there is no problem with the distinction between finite or infinitesimal distances: it works even if you consider space-time as discrete. In fact, in practice it's very common in GR simulations to approximate space-time as a 4-dimensional discrete grid of points.

The main point to keep in mind with physical models is that they don't have to be considered the real thing: they simply have to WORK as the real thing. So you have to choose which characteristics (or properties, or attributes) of the model correspond to characteristics of physical real objects and which ones are only mathematical approximations. For GR, the trajectories are only abstract 1-dimensional "lines": what's important (measurable) is only their length, and the angle between them, but only up to a certain approximation. Physical bodies can be represented as "points", or "spheres", just to make calculations simpler: the small-scale details are not important for the model, so they don't correspond to anything "real".

Now, if you think that the distinction between measures expressed with rational or with real numbers is essential in your theory (represents some important characteristics of the real physical space), I don't see any other way other than making lengths become discrete at the microscopical level.
If this is not your idea, in which way the use of rational numbers instead of real numbers could make a difference? I know that you think that real numbers do not exist, but what's the difference if they exist or not, if your model doesn't care of what happens at the smaller scales?

If you mean electron microscope photos of a lattice of atoms, those are still subject to the quantum and classical measurement problems. To clarify what I said earlier:

* In quantum theory, nothing has an exact position at all. Before it's measured, it doesn't have a position. Sometimes that's expressed by saying that it's in a "superposition" of all possible positions. Then when you measure the particle, it (somehow -- nobody understands this part) acquires a position drawn randomly from a probability distribution.

This applies to all objects, large and small, though the effect is much more pronounced when an object is small.

For example you yourself are where you are in space right now because that's the most likely place for you to be. It is statistically possible that you might suddenly find yourself in a statistically improbable place. For example all the air molecules in your room could move to the corner of the room and you'd have no air. That is extremely unlikely, but it has a nonzero probability. It could happen.

So even if all instances of a given particle are the same, you still have no idea exactly where it is, or exactly how long a line made up of these particles is.

Atoms, by the way, are way too large and they're all different. I don't even know if two hydrogen atoms are exactly the same.

However it's interesting that every electron in the universe is (as far as we know) exactly the same. Why is that? It's another thing nobody understands. — fishfry

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 atoms (of any element), and even for entire molecules, and this has been verified experimentally. Two atoms in the ground state (https://en.wikipedia.org/wiki/Ground_state) are EXACTLY IDENTICAL (as mathematical objects in the mathematical model of QM) if the ground state is not degenerate (https://en.wikipedia.org/wiki/Degenerate_energy_levels).
The tricky thing to realize experimentally is to obtain a non-degenerate ground state for a complex object as an atom: very low temperature, external magnetic field, confined position in a very little "box" (usually a laser-generated periodic electromagnetic field). But this is possible, and in this state the whole atom is COMPLETELY DESCRIBED from by one integer number: the energy level.
In this state you can put a bunch of atoms one over the other, if they are bosons (https://en.wikipedia.org/wiki/Bose%E2%80%93Einstein_condensate) and the theory says that you can have N IDENTICAL objects all in the same IDENTICAL place.

The result is obtained by purely mathematical considerations on objects made of complex number functions (the states are the eigenvalues of the system's wave function), but the effects predicted using a purely mathematical abstract model generate real physical predictions in the form of measurable quantities. That seems very strange if mathematical objects are only symbols subject to arbitrary rules. In some way, the rules that we invented for the symbols correspond exactly to some of the "rules" of the physical (real) world.

* And even in classical physics, a measurement is only an approximation.

So now I'd like to re-ask your question but pertaining to electrons, which are all exactly the same. But electons are very small and extremely subject to quantum effects. You simply can't say exactly where an electron is at any time. Only where it's statistically likely to be. One, because nothing is exactly anywhere at all in quantum physics; and even when it is, after a measurement, the measurement itself is subject to classical approximation error. You made the measurement in a particular lab with a particular apparatus, built and operated by humans. It's imperfect and approximate from the getgo. — fishfry

Yes, but the indeterminacy is only for the product position * momentum, and not the position alone (for example an electron emitted from the nucleus of an atom has an indeterminacy of initial position of the size of the nucleus from which it was emitted). And the curious thing is that the wave function, if you want the path-integral over the trajectories to be accurate enough, must be described with a much finer granularity of space than the size of the atom. The wave equation works the best if it's defined on the (mathematically imaginary) real numbers (at least for QED). The renormalization of electron's self-energy (https://en.wikipedia.org/wiki/Renormalization) is a mathematical theorem based on a mathematical model where space is the real euclidean space (real in the mathematical sense: vector space defined on real numbers) (I know the objection: it works even on a fine-enough lattice of space-time points, if you make statistics in the right way, but the lattice of positions have to be much smaller of the wavelength of the electron - that for "normal" energies is comparable with the size of an atom).

Well there are no computers with arbitrary precision. That's the problem with the computational theory of the universe. There's too much it can't account for.

It's those pesky noncomputable numbers again, one of my favorite topics. If the universe is "continuous", in the sense that it's modeled by something like the real numbers; then it is most definitely not a computer or an algorithm. Because algorithms can't generate noncomputable numbers. — fishfry

Yes, however in same cases, the system is symmetric enough that you can use analysis to compute the results instead of making simulations, so you can get infinitely precise answers, (such as for example in the case of hydrogen atom's electronic
orbitals) that however you'll be able to verify experimentally only with finite precision.

So in your hypothetical world there would be squares and if you want to go from (0,0) to (1,1) you simply have to move 2 units, one unit right and one unit up. You can't travel along the diagonal because at the finest level of the lattice, you can't move diagonally. I have no idea what that means physically but I think you are overthinking this or underthinking it. It's kind of tricky, which is a problem for the theory. — fishfry

Well, that was a simple example that doesn't have much sense as a real theory of physics (and I absolutely don't believe that it can be a good model of physical space), but it's still a mathematical model suitable to be used to make predictions on the physical space (well, you should say how big are the sticks: surely there are a lot of missing details). However, as a model, you can decide to make it work as you want: in our case, the squares made with sides of one stick can't have a diagonal (so, let's say, nothing can travel along the diagonal trajectory, as in the Manhattan's metrics), and big "squares" can have diagonals but can't have right edges, or straight angles.

Some people do! There are some discrete or quantized theories of reality around, like loop quantum gravity. From the article: "The structure of space prefers an extremely fine fabric or network woven of finite loops." — fishfry

Yes, but in loop quantum gravity loops are only "topological" loops: they are used to build the metric of space-time, not defined over a given metric space.

But I don't speculate about the physical world. Math is so much simpler because it doesn't have to conform to experiment! In math if you want a square root of 2, you have your choice of mathematically rigorous ways of cooking up such a thing. — fishfry

I agree with you on the square root of 2, of course! But I am not so convinced that mathematical objects are only cooked-up fictions not related to physical reality.

A wave is active, so it requires the passing of time, for its activity. So let's assume "space" is an active medium. Now suppose we try to make something static, like a circle or a square, within this medium which is active. The shape won't actually be the way it is supposed to be, because the medium is actively changing from one moment to the next. So if we want to make our shape, (circle or square), maintain its proper shape while it exists in an active medium, we need to determine the activity of the medium, so that we can adjust the shape accordingly. Understanding this activity would establish a true relationship between space and time, because defining this activity of space would provide us with a true measure of time. — Metaphysician Undercover

OK, but I don't understand how all this can be related to irrational numbers.

What I am talking about specifically, is dividing numbers. Divide ten by three, and you have a remainder of one. It is the remainder which is a problem. When we multiply numbers we never get remainders, yet we tend to treat division as the inversion of multiplication. It's actually quite different from multiplication because multiplication starts from premises of fundamental base units, whereas division presupposes no such base units.. So I think we need to pay close attention to this fact, that constructing a magnitude through multiplication is really a completely different process from destroying a magnitude through division. — Metaphysician Undercover

Division between integers is repeated subtraction ( A/B you count how many times you have to subtract
B from A to reach 0 ); multiplication between integers is repeated addition ( A*B you add A B times starting from 0 ).
The definitions are quite symmetric between each-other. What do you mean by "division presupposes no such base units"? OK, A/B is not an integer ( there is a reminder ) if A is not a multiple of B. Again: what does this have to do with physical space-time?

I don't understand how you would build an irrational length segment. — Metaphysician Undercover

By using compass and straightedge (as described by Euclides) you can build all the lengths that can be obtained from the integers with a finite sequence of operations of addition, subtraction, multiplication, division, and taking square roots (https://en.wikipedia.org/wiki/Straightedge_and_compass_construction). Square roots are not so special from this point of view.

What the law of identity says is that a thing is the same as itself. This puts the identity of the thing within the thing itself, not as what we say about the thing, or even the name we give it — Metaphysician Undercover

Hmmm... :worry: maybe...

First, to be a thing is to have an identity — Metaphysician Undercover

OK

Second, a thing is unique, and no two things are alike, and this is the principle Leibniz draws on — Metaphysician Undercover

OK, I translate this as: you can always distinguish a thing (meaning: physical entity) from all the other things. Not quite true in quantum mechanics, but let's assume it is.

So the law of identity is not concerned with how we refer to objects, it is a statement concerning the real existence of objects, as the objects that they are, independent of what we say about them — Metaphysician Undercover

OK, but when you give a name to a concrete object, the name is a reference that identifies always the same concrete object, isn't it?

Anyway, my main objection to what you say is that you don't explain how to use the fact that square roots are irrational (some of them) to deduce something about physical space-time. A physical theory in my opinion (even if limited) should be falsifiable in some way (meaning: should be usable to predict that something should happen, or that something else can't happen). And if it's not physics but only mathematics, then there should be some kind of logical "proof". Don't you agree?

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 be EXACTLY the same (indistinguishable, even in principle, with absolute precision), right?

You are right, we will never be able to check if this theory is correct with absolute precision, not even in principle, because all physical measurements must necessarily have a limited precision.
Nevertheless, in principle (if you have enough computing power and the model is complete and consistent - I know, that's a big if) you can use the mathematical model to make predictions about the result of experiments with arbitrary precision.

So, in a model of the physical world where all distances have to be multiple of a given fixed length (I don't know if such a model exists, but let's assume this as an hypothesis), there cannot be squares
made of unit lengths. I don't know what these unit lengths are made of: they are simply the building blocks of my model, the same as the "strings" of string theory or the "material points" of Newtonian mechanics!

By the way, to be clear, I don't believe in this theory! :smile:

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 one after the other, along the side of a square. N is a natural number (abstract mathematical object), but the process of putting N sticks in line is a real, physical experiment.

2. Call Build_Diagonal(M) the physical process of putting M sticks in line one after the other, along the diagonal of the same square.

Try to find M and N such that the sticks arrive at the same point. Since M/N is irrational, you can't do it, and the physical process of trying to build the square withe the sticks cannot be realized. ( well, OK, you have to build two sides of the square and the diagonal at the same time, but you get the what's the point! )

To state the problem succinctly, set theory allows that two distinct things have the same identity, in the same way that we might say two distinct things are equal. The faulty premise is that things with the same value "2" for example, are the very same thing. In other words set theory premises that, "2" refers to an object, rather than a value assigned to an object. It is a category mistake to treat what "2" refers to, as a particular object, rather than as a universal principle. — Metaphysician Undercover

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 understand why this "law of identity" is so important...).
However, that's my question: how do you refer to an object instead of to it's value? I mean: if every symbol refers to a different object, even if the symbol is the same as the one that you used before, you can never refer to the same object twice, can you?

I think there are two issues becoming evident. One is that we do not know how to properly represent space. The irrational nature of the "square", and the "circle", as well as the incompatibility between the "point" and the "line" indicate deficiencies in our spatial representations. — Metaphysician Undercover

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 enters both in the side and in the diagonal of the square an integer number of times (no matter how little you take your unit length).

So there cannot exist any fundamental minimal length of physical space (kind of a microscopic indivisible stick) that can be oriented in any direction. If there is such a thing, every physical object at the microscopic scale should be made of tetrahedrons, or something similar. So circles and squares are really only approximations of the real "physical" shapes. Is your idea something of this kind? If not, in what other way can you make all the lengths be rational numbers?

If this is the idea, I think the problem with this kind of physical theory is that all laws of physics are expressed in terms of differential equations (even the ones that describe "quantized" entities), and if quantum mechanics is right, it doesn't even make much sense to speak about an exactly determined physical length: physical space appears to be much more weird than a simple 3-dimensional geometric structure.

The other is that we do not know how to properly divide something. There is no satisfactory, overall "law of division", which can be consistently, and successfully used to divide a magnitude. We tend to look at division as the inversion of multiplication, "how many times" the divisor goes into the dividend. Because there is often a remainder, division really cannot be done in this way. The "square root of two" is a more complex example of this simple problem of division, the issue of the remainder. — Metaphysician Undercover

You mean that there is no defined physical procedure to divide a generic geometrical segment by another? If you take two generic segments of whatever length, you can always build a third segment that is proportional to their ratio (whatever it is, even irrational). That's in Euclides' elements. Can't be this counted as division? If not, what do you mean by "law of division"?

Why not move on, and inquiry what this principle tells us about numbers and spatial relations, instead of trying to disprove it. — Metaphysician Undercover

I am curious to know: do you have an answer to this question?

However, re-reading that thread, I see that I threw even harder (and even less comprehensible) stuff, like this one: "A formal proof makes only use of the computational (or topological) part of the model. The part that remains not expressed in formal logic is usually expressed in words, and is often related to less fundamental parts of physics, such as, for example, the geometry of space.".
I guess nobody replied to this one because everybody thought that it doesn't make sense at all :joke:

Well, I am surprised. I didn't expect somebody to agree with that kind of categorical assertions! :razz:
I mean: it's clear that finding the right definitions it's not all. And it's extremely conceited to say "I'll tell you what the whole mathematics is about!". But there's something true in what I wrote, and I wanted to see if somebody agrees without spending too much time to explain what I mean :smile:

Yes, ok. — fishfry

:smile: :smile: :smile:

Sure. Agreed. All open sets are measurable. — fishfry

Yes!! I was starting to despair that there is a way to make me understand...

My objection was only this one: BT doesn't make integration inconsistent. You can reason about infinitesimal parts and be confident of the fact that integration works, if you decompose the object in open sets. That's all I wanted to say!!!

Like f(x) = 2x stretching (0,1) to (0,2)? Yes. Open sets are easy if you don't require the maps to be isometries.

I don't know if the rotations or translations carry open sets to open sets. I'd tend to doubt it. — fishfry

Yes of course they have to be isometries. I meant: there is no way of decomposing an object in an infinite set of open sets and then recomposing them in a different way so that each peace has the same measure but the sum of the measures of all the pieces is different. If this were possible, the theory of integration would be inconsistent.
I know your objection: if there is an infinite number of pieces the measure of each peace cannot be finite. OK, but you can build the limit of a sequence of decompositions, like you do with regular integration.
I am not arguing that BT theorem is false, I am arguing that it works only because you perform the transformation on pieces that are not measurable. If the pieces were made using the decomposition in open sets, as with regular integration, it couldn't work. I know that you can even define a Lebesgue integral that is working on sets that are not open: this is not a necessary condition, but is a sufficient condition to preserve additivity.

If you're having intuitions of topologies or continuity, those are the wrong intuitions to be having. What's interesting though is that each orbit is dense — fishfry

If topology has nothing to do with it why all the proofs of decomposition of objects that don't preserve volume are decomposing the objects in pieces that are not open sets?

Can you find an example of decomposing an object and then recomposing it with different volume where the pieces are open sets?

By "models' factorizations" I mean finding the right definitions that allow you to describe some complex (containing a lot of information) models in a simple way, or that allow you to prove something that was too complex to prove without these definitions.
In a sense, this is a form of compression of information: understanding something means compressing the information contained in something in a new simpler way (by using a different point of view, or definitions). That's mainly what mathematicians are doing today.

I'm not sure what you mean. Can you please link your earlier post on B-T? — fishfry

I linked it in the post just before this one. Here's the link:
https://thephilosophyforum.com/discussion/comment/302364

However, with ubiquitous and incredibly powerful computing and no need for physicists to believe in a physical continuum, I would argue the average student is much better served by focusing on "what can the computer do for me", viewing constants algorithmically with arbitrary (to a physical limit of computation) precision potential determined in practice by one's problem, and building up intuitions around machine calculation (and analytical work including error bounds, computational complexity, along with analytical proofs of convergence when available, just in the "arbitrarily close to the limit" finitist framework); rather than, what we seem to all agree here, building up wrong intuitions about the real number system. — boethius

Yes, I see your point. Maybe you are right.

But you did give a wrong and misleading definition of an open set. I do have to say that. Open sets are really important. An open set in the reals is just like an interval without its endpoints. What matters about it is that "all its points are interior points." It doesn't include any points of its boundary. That's what makes open sets have the interesting properties that they do.

They're not really infinitesimal. They can be arbitrarily small. But they aren't "infinitely" small. In fact that is the great "arithmetization of analysis," the great founding of the continuous world of calculus on the discrete world of set theory. Instead of saying things are infinitely small, from now on say they're arbitrarily small. For every epsilon you can go even smaller. But in any individual instance, still nonzero. That's the essence of open sets. — fishfry

OK, I think I should give some explanation on this point:

I wrote you have to take "open sets" as infinitesimal pieces
What I meant is you should impose the restriction that the infinitesimal pieces are also "open sets"

The definition of open sets is of course what you wrote: "all its points are interior points", or "there are not isolated or border points in the set", or "each point of the set is surrounded by other points"

Now, if I wanted to explain under what assumptions additivity of volumes (or surfaces, or segments) works without using a formal logic system, I would say that it works even if you consider infinitesimals as really existing entities (with the appropriate rules of calculus: for example: integrating over a line, dx squared is zero), but you cannot take as dx isolated points: you have to take pieces that don't contain points that are isolated from each-other, because otherwise the topology of the object is not preserved (the functions are not continuous), and you can build a sphere using the points of a line, or two equal spheres using the points of one sphere. As I wrote in my explanation about Banach-Tarski mounts ago, the theorem works because it uses isometric transformations, but applied to set of points that are isolated from each other (not on open sets). If you impose the restriction that your isometric transformations should be even continuous (going from open sets to open sets), you can't do it any more.

I really didn't want to enter in the discussion about Banach-Tarsky theorem again :worry:
I found what I wrote about six months ago:

What's wrong with the Banach-Tarsky paradox — Mephist

. It's still valid!