## My own (personal) beef with the real numbers

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In general I have found that working over formalisms is one necessary part of developing understanding for a topic; don't just read it, fight it. Follow enough syllogisms allowed by the syntax and you end up with a decent intuition of how to prove things in a structure; what a structure can do and how to visualise it.

This is a good point to bring up. I'm tempted to call it formal intuition. I used it for all the epsilon-delta stuff in basic real analysis. A continuous function was (in that context) something associated roughly with a set of logical moves. It's almost like playing a guitar. A certain know-how kicks in. A person can sit in class and start imagining how something might be proved. This is kind of what I meant by symbol manipulation. In this mode, the stuff of the proof itself is the medium of thought.

Developing such anchors and being able to describe them seems a necessary part of learning mathematics in general; physical or Platonic grounding deflates this idea by replacing our ideas with actuality or actuality with our ideas respectively. In either case, this leaves the stipulated content of the actual to express the conceptual content of mathematics without considering how the practice of mathematics is grounded in people who use mathematics and whether that grounding has any conceptual structure.

I like this. On of the experienced mathematicians I know liked the math is language metaphor. I find this metaphor deeper than might be apparent. What is it to know English? Perhaps to know math is just as complicated, even if epistemologically the situation is simpler. A stupid computer can check a formal proof, but the world of mathematics is certainly not just a set of formal proofs. Squishy Heideggerian insights are valuable even here. Knowing math is like knowing English is like knowing how to ride a bike. That knowledge is largely tacit. When I've taught math, I mumbled to my students something about intuition coming with practice. 'Keep calculating. Faith will come!' I liked Hersh's mathematical humanism: https://www.amazon.com/What-Mathematics-Really-Reuben-Hersh/dp/0195130871 Maybe you've read that. He gets that math is a kind of intersubjective human practice.
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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.

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?
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This is a strange digression, perhaps, but it connects with the intuitive beauty of mathematics. When I fell in love with math, I also fell in love with the idea of an objective knowledge, a knowledge of some transpersonal structure. It wasn't about me-me-me and some rationalized 'theology' justifying my current lifestyle. Math was 'sculpture.' It was cold as ice, austere. It was ancient stone. What is this eroticism of the mathematical?

[The] impossible synthesis of assimilation and an assimilated which maintains its integrity has deep-rooted connections with basic sexual drives. The idea of "carnal possession" offers us the irritating but seductive figure of a body perpetually possessed and perpetually new, on which possession leaves no trace. This is deeply symbolized in the quality of "smooth" or "polished." What is smooth can be taken and felt but remains no less impenetrable, does not give way in the least beneath the appropriative caress -- it is like water. This is the reason why erotic depictions insist on the smooth whiteness of a woman's body. Smooth --it is what reforms itself under the caress, as water reforms itself in its passage over the stone which has pierced it....It is at this point that we encounter the similarity to scientific research: the known object, like the stone in the stomach of the ostrich, is entirely within me, assimilated, transformed into my self, and is entirely me; but at the same time it is impenetrable, untransformable, entirely smooth, with the indifferent nudity of a body that is beloved and caressed in vain. — Sartre

I like to contrast the mathematician and the novelist. The novelist seeks a personal immortality. As a novelist, I want to crystallize my own precious experience of reality. As a mathematician, I lose myself in the inter-subjectively available object. I can't sign or claim this knowledge, or not in the same way. The ostrich swallows a stone that it cannot digest (as Sartre describes it in another passage of 'Existential Psychoanalysis.')

The philosopher is somewhere between the mathematician and the novelist. That Sartre quote tries to capture in words what he takes for a universal experience, a general structure of experience that is otherwise particular. This is basically the eternal 'behind' time, the invariant that is constantly present if we can grasp it.
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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.

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

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.

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.

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.

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.

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.

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.
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Right, my argument is that there is no such thing as an abstract object represented by "2"

You know, that is a very interesting point of view. As I said I find it nihilistic because you must then reject all of the modern world that sprung from that basic act of abstraction. You would be back at the time of the Greeks, who expressed everything as ratios but did not actually have a concept of number as such.

Now that you mention it, that makes perfect sense relative to your neo-Pythagoreanism. By that I mean that you still profess to be "Shocked, shocked, I tell you!" at the fact that the square root of 2 is irrational. The rest of the world got over that a long time ago.

So ok, your mathematical viewpoint is that of an educated person in the time of ancient Greece, say between Pythagoras around 2700 years ago and Euclid 2400.

Question: Is your outlook on the rest of the world similarly situated in the past? Do you shake your fist in the air at the notion of heavier-than-air flight? How far do your convictions go? Have you heard about heliocentrism? It's the latest thing.

I don't want to comment on the specifics of your post at the moment. I dropped in to ask you a question that occurred to me:

Question: I get that you do not believe in the ontological existence, however you personally define that, of $\sqrt 2$. My question is:

Do you believe in the mathematical existence of $\sqrt 2$?

If you say yes, then our disagreement is over whether mathematical existence is sufficient for ontological existence.

If you say no, then our disagreement is whether $\sqrt 2$ has mathematical existence.

So, do you think $\sqrt 2$ has mathematical existence; even though you maintain that's not sufficient justification to grant it ontological existence as you define it?
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It was clear even then that the real numbers had a certain magnificent unreality or ideality.mask

Yes, great insight. The mathematical real numbers are very strange. I always noticed that the more I learned about the real numbers, the more unreal they got. It's very doubtful to me that such a structure has an analog in the physical world. And if it did, it would be quite a surprise.

Mathematicians have a tongue-in-cheek saying: The imaginary numbers are real; and the real numbers aren't!

When I studied some basic theoretical computer science (Sipser level), I saw the 'finitude' of now relatively innocent computable numbers like pi,mask

It's an extremely widely held false belief that pi encodes an infinite amount of information, when it of course does no such thing. Bad teaching of the real numbers in high school is the root cause of this problem. Whether there is a solution that would serve the mathematical kids without totally losing everyone else, I don't know.

It's basically ridiculous to do philosophy of math without training in math: sex advice from virgins, marital advice from bachelors.mask

Another fine point to which I've endeavored to draw @Metaphysician Undercover's attention from time to time.

I always follow your posts. You know much more set theory than me, so I learn something.mask

Thanks. I loved all that stuff in school. I couldn't do calculus integrals for beans, but I took naturally to Zorn's lemma. Did you know that the proposition that every vector space has a basis, is fully equivalent to the axiom of choice? Isn't that wild?
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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.

What a great topic. If you Google "constructive physics," you find a small but nonzero number of paywalled articles on the subject . I believe there's a book about it, too. In fact here it is, the whole book.

https://arxiv.org/pdf/0805.2859.pdf

The table of contents is an awesome read. He has definitely done his homework.
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I couldn't do calculus integrals for beans, but I took naturally to Zorn's lemma. Did you know that the proposition that every vector space has a basis, is fully equivalent to the axiom of choice? Isn't that wild?

I had heard that, but never studied the proof. It is indeed wild.

At my school we only had to learn the set theory that comes with analysis and algebra. I did look into ordinals on my own. I remain impressed by the usual Von Neumann constuction. I used it in visual art and I also think it has a philosophical relevance. It's a nice analogy for consciousness constantly taking a distance from its history. 'This' moment or configuration is all previous moments or configurations grasped as a unity. It works technically but also aesthetically.

It's very doubtful to me that such a structure has an analog in the physical world. And if it did, it would be quite a surprise.

That would be surprising indeed. I think we agree on the gap between math and nature. As you mention, our measurement devices don't live up to our intuition and/or formalism. I have a soft spot for instrumentalism as an interpretation of physical science.

Mathematicians have a tongue-in-cheek saying: The imaginary numbers are real; and the real numbers aren't!

I haven't heard that one. But I know a graph theory guy who thinks the continuum is a fiction and an analyst who believes reality is actually continuous. Another mathematician I know just dislikes philosophy altogether. I like philosophy more than math when I'm not occasionally on fire with mathematically inspired, though I have spent weeks at a time in math books, obsessed. (At one point I was working on different models of computation, alternatives to the Turing machine, etc. Fun stuff, especially with a computer at hand.)

It's an extremely widely held false belief that pi encodes an infinite amount of information, when it of course does no such thing. Bad teaching of the real numbers in high school is the root cause of this problem. Whether there is a solution that would serve the mathematical kids without totally losing everyone else, I don't know.

Right! Because of the infinite decimal expansion. One of my earliest math teachers had lots of digits of pi up on the wall, wrapping around the room. Some of the problem may be in the teaching, but I've wrestled with student apathy. Math tends to be viewed as boring but useful, the kind of thing that must be endured on the path to riches. Its beauty is admittedly cold, while young people tend to want romance, music, fashion, fame, etc.
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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.

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.

f 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.

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.

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 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. If the claim is that "space-time" is not supposed to be an object, then we have nothing being represented except mathematical objects, and no grounding principle, defining what a mathematical object is.

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.

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..

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.

Yes, as I explained, I agree that this is the way to go. Once we recognize that this is what's needed, we can proceed towards a proper analysis of space and time, to establish some principles.

Why should this background of mathematics remain a secret? And is it merely aesthetic in nature (a consideration of mathematical beauty alone)?

The secret background is the intent of the author. So long as the intent remains a secret, pragmaticism remains unacceptable because we do not understand the end. "It works" has no meaning when the end remains a secret.

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?

This is good, a very good start. Suppose there actually is this distinction in "objects". Suppose there are both "quantized objects", and "continuous objects". We would need different principles to apply to each one. Then we would need to establish some principles of application, are we working with a continuous object, or a quantized object. 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.

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

The 3,4,5, triangle is just as faulty as the square, because it is validated by the faulty Pythagorean theorem. The point was that I can make a 3,4,5 triangle without any knowledge of the Pythagorean theorem. So let's say I am in the habit of doing this, and I know absolutely nothing about the theory. I am producing right angles at will, and I believe that the right angle is a perfectly natural thing. Then I learn about the square, and realize that there is a problem with the right angle, and therefore it is not a natural thing. Likewise, I could tie a string to a stake, and make a simple compass, and go around creating circles at will, thinking that the circle is a natural object, until I realize the irrationality of pi. This tells me that these are not natural objects, they cannot exist.

So now I want principles to explain why I can make a figure which is theoretically impossible to make. I make excuses, I rationalize that I am not making a "perfect" square, or a "perfect" circle, and the theory says that such "perfect" figures are impossible. But there's something fundamentally wrong with this rationalizing. The theory is supposed to give me the "ideal", the perfect geometrical figure, and my inability to construct it ought to be due to my imperfect procedure. But here, what is indicated by the mathematics is that this supposed "ideal" is actually less than perfect, so that the more perfect the procedure is, all it does is demonstrate how much less than ideal the ideal is. Therefore I can conclude that the "ideal" is not the ideal at all, and there is a fundamental contradiction here which tells me that we need is a better, more ideal "ideal".

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?

Aristotle established and used the law of identity as a fundamental tool against the deception of sophism. So let's assume as you say, that logic is the manipulation of symbols, and it doesn't say anything about any real things. To say something about a thing is an act of description, and this is distinct from logic which works with symbols. Therefore if a logician is claiming to say something about real things, we can charge that person with sophism, deception. This is what we do with "identity" then, use it to demonstrate that someone is falsely claiming identity. So how do we approach set theory, doesn't it look like sophism, a false claim of identity, to you?

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

Good question, I think the jury is out still on that decision.

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

No, it does not prove that continuous change does not exist. But it proves that numbers, which represent change in finite steps, are the wrong tool for representing continuous change. This is where mathematicians demonstrate their stubbornness. They want numbers to be capable of representing everything, so they twist and turn the systems, adding layer upon layer of sophistication, mixed with deceptive sophism, and voila, numbers represent the fundamental reality, or even more extreme, numbers are the fundamental reality. It's like physicalism, or scientism, but it's more properly called mathematicism. and it appears to be the root of the other two. People think that mathematics is all we need to describe reality, when in reality mathematics cannot describe anything.

So let's revisit this root problem. Numbers, which demonstrate finite increments of difference, cannot properly represent continuous change. Either we assume that continuity is not something real, therefore numbers can be applied to all of reality, or we assume that continuity is real, and numbers cannot be applied to all aspects of reality. I suggest the latter is what is really the case. But then mathematicism looms menacingly in front of me, and I feel it my duty to demonstrate the sophistry of the mathemagician.

As I said I find it nihilistic because you must then reject all of the modern world that sprung from that basic act of abstraction.

This is not the case. To reject that "the abstraction" exists as an object does not require that I reject abstraction. What I reject is any instance where an abstraction is presented as an object.

Now that you mention it, that makes perfect sense relative to your neo-Pythagoreanism. By that I mean that you still profess to be "Shocked, shocked, I tell you!" at the fact that the square root of 2 is irrational. The rest of the world got over that a long time ago.

I'm not "shocked" at the fact that the square root of two is irrational, what shocks me is that the rest of the world got over this.

Question: I get that you do not believe in the ontological existence, however you personally define that, of 2–√2. My question is:

Do you believe in the mathematical existence of 2–√2?

If you say yes, then our disagreement is over whether mathematical existence is sufficient for ontological existence.

If you say no, then our disagreement is whether 2–√2 has mathematical existence.

So, do you think 2–√2 has mathematical existence; even though you maintain that's not sufficient justification to grant it ontological existence as you define it?

I'd answer this, but I really don't know what you would mean by "mathematical existence". Many things can be expressed mathematically, but what type of existence is that? I suppose the short answer is no. The symbol √2 does not stand for anything with real "existence". I agree that is has a large amount of mathematical significance, and it is quite important mathematically, so the symbol definitely has meaning, but I don't think I'd agree that the symbol stands for anything which has "existence", in any proper sense of the word. All existence is "ontological existence" so it makes no sense to try and separate "mathematical existence" from "ontological existence".
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All existence is "ontological existence" so it makes no sense to try and separate "mathematical existence" from "ontological existence".
Mathematicians and philosophers of mathematics, with the presumed exception of platonists, reject the premiss that all "existence" is ontological existence. Specifically, they acknowledge that mathematical existence does not entail ontological existence.
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In this mode, the stuff of the proof itself is the medium of thoughtmask

In my experience, formal intuition works more like an open neighbourhood around a proof than of proofs themselves. There are essential details and inessential details for the understanding of a structure. The essential details are what enable you to generate expectations of how the structure behaves; envisage theorems and ask questions about it.

EG: "We can label set elements however we like, so functions can be interpreted as ways of permuting the labels of the elements on a set... I wonder if every collection of functions on a set behaves just like a set of permutations on that set?"

It seems you can write something like mathematical pseudo-code to suggest an intuition and play about with it, an example for the above:

Let's take the function like f(x)=x+1 on the natural numbers, envisage it as a list of pairs:

(1,2)
(2,3)
(3,4)

and so on.

And you can read that as "relabel 1 with 2, relabel 2 with 3, relabel 3 with 4" and so on. If you have a familiarity with disjoint cycle notation for permutation groups, you might think both "those look a lot like permutations" and that whole thing looks like the permutation (123456...) in the cycle notation, which is the original function in another representation. The procedure to generate this intuition didn't look to depend on much besides the choice of set.

There are a lot of syntactic details that facilitate every step of this "pseudo-code", sometimes (often) I can get them wrong and that blocks the intuition from working. Furthermore, the "essential details" as interpreted by me might not (usually don't) generate all the behaviour of the structure- IE enable me to expect every provable theorem as provable. (They also often make me expect things are provable when they are not.)

So the essentials of a structure look like "necessary highlights", in a sense they cover the structure in question for the purpose with all relevant detail. These "highlights" I think, ideally, map onto your anchor points for formal intuition. In such a case it seems to me that someone would understand the conceptual content of a mathematical structure (insofar as it is relevant to the concerns) well. If you have mastery over the domain, I imagine the essential details allow you to generate expectations for many provable theorems from the structure, and allow you to easily see if something is inconsistent with it.
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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.
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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.

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.

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.

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..

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.

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..]
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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've never looked at the book and have no interest in constructive physics. You'll have to forgive me. Personally I think it's a fool's errand. I have disagreements with constructivists. I'm well aware that various neo-intuitionistic foundations are in vogue, homotopy type theory and so forth. What I'm saying is that I'm totally unequipped to respond in detail to your points about constructive physics; not only by knowledge, but also by interest.

I also wanted to mention that I'm falling a little behind in my mentions and a lot of interesting points are being made lately. In fact you wrote me a great reply that I wanted to get to. You know a lot more physics than I do and I didn't realize atoms can be identical. I hope to respond to that post at some point.

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.

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. I used to read John Baez back in his loop quantum gravity days, and I didn't understand how he was applying category theory to physics either.

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.

But as I say, I am a little skeptical about the rejection of noncomputability. My own belief is that the next revolution in physics will involve going beyond our current notions of computability. So I don't think the constructivists are going to win. And from what little I've seen, every flavor of constructive math these days at some point has to sneak in at least a weak form of the axiom of choice; and I believe that would have to be true in physics as well. Even the constructivists will allow a certain amount of nonconstructability; because it turns out to be necessary to get a decent mathematical theory.

And probably, after Voevodsky, it doesn't make much difference even from a mathematical point o view.

I'm not sure exactly what you mean by that but I'm not sure I want to know. Voevodsky did a lot of things. But generally I don't like to jump down the constructive rabbit hole (having spent enough time learning about it to satisfy my own curiosity) so please don't feel obligated to write more than a few words here, if any.
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As I said I find it nihilistic because you must then reject all of the modern world that sprung from that basic act of abstraction.
— fishfry

This is not the case. To reject that "the abstraction" exists as an object does not require that I reject abstraction. What I reject is any instance where an abstraction is presented as an object.

That's what abstraction is! It's giving a name to something immaterial in order to manipulate it. We see 2 cows and 2 pigs and 2 chickens and 2 barnyard metaphors. So we abstract a thing, called 2. You may object and say that you only mean 2-ness. But I say that's no different than declaring a thing called 2. You either believe in abstraction or not. I don't buy the distinction you're making here.

Now that you mention it, that makes perfect sense relative to your neo-Pythagoreanism. By that I mean that you still profess to be "Shocked, shocked, I tell you!" at the fact that the square root of 2 is irrational. The rest of the world got over that a long time ago.
— fishfry

I'm not "shocked" at the fact that the square root of two is irrational, what shocks me is that the rest of the world got over this.

That's very funny. I do see your point of view. Like I say it's nihilistic because you must therefore reject the entirety of the modern world. You say you have found a loophole that allows you to accept the world yet reject ... something. The square root of 2. What on earth have you got against the square root of two? Even the Babylonians chiseled a few digits of the square root of 2 into a rock. It is in some sense inevitable that people will crawl out of caves and figure out how to use fire, and invent the wheel, and build cities and do commerce, and discover the square root of 2. It's not something some set theorist made up. It's out there in the world. It has an existence independent of us if anything does. Somehow. It's mysterious, I agree. But you seem to just reject it on grounds that you haven't explained to me yet.

I'd answer this, but I really don't know what you would mean by "mathematical existence".

Since you don't know any math, perhaps you will take my word for it. $\sqrt 2$ has mathematical existence.

Now to describe to you what mathematical existence is, I would have to do some math. Which you don't like. But $\sqrt 2$ has mathematical existence because:

* If we believe in the rationals, we can build a totally ordered field containing the rationals in which there is a square root of 2.

* We can then construct such a field within set theory. If you prefer we can do it in category theory. In fact we can do set theory within category theory, so if you don't like set theory there's a more fundamental theory we can build it out of. There are plenty of alternative foundations about these days. They're very trendy in fact. Since you hate set theory you'll be glad to know that in some circles, nobody cares about it any more.

But see now you've got me ranting math again and it's a waste of time because you don't like it.

So forget everything I wrote that you don't want to read, and just know this: $\sqrt 2$ has mathematical existence because I say it does; and if you were willing to follow a symbolic argument I'd prove it to you, in fact I already have several different ways.

Many things can be expressed mathematically, but what type of existence is that?

Mathematical existence. Something you know nothing about because you don't know any math.

I suppose the short answer is no. The symbol √2 does not stand for anything with real "existence".

Yes I know you already believe that. The question is whether you're willing to believe that it has mathematical existence. You ask me what that is but I've given you many demonstrations of the mathematical existence of $\sqrt 2$; as the limit of a sequence, as an extension field of the rationals, as a formal symbol adjoined to the rational numbers. All these things are part of mathematical existence. You will either have to take my word for it, or work with me to work through a proof of the mathematical existence of $\sqrt 2$ .

I agree that is has a large amount of mathematical significance, and it is quite important mathematically, so the symbol definitely has meaning, but I don't think I'd agree that the symbol stands for anything which has "existence", in any proper sense of the word.

How can you tell, personally, whether that's your deep philosophical mind talking, or just your mathematical ignorance? From where I sit ... well let me tell you that you didn't score any points with me when you totally ignored my beautiful demonstration that 2 + 2 = 4 assuming only the Peano axioms. You showed me that you have a psychological block in dealing with symbology; leading to a massive area of ignorance of math; leading to making large errors in your philosophy. That's my diagnosis.

All existence is "ontological existence" so it makes no sense to try and separate "mathematical existence" from "ontological existence".

Well see THAT is something we can agree to talk about. Whether there is any difference. I think a rock exists in a way that $\sqrt 2$ doesn't. The latter only has abstract mathematical existence.

Like Captain Ahab, who only has fictional existence in a novel. Nevertheless statements about him can can truth values, such as whether he's the captain of the Pequod or the cabin boy. So there's fictional existence.

Do you think Ahab has fictional existence? I've been meaning to ask you that, actually. If you say yes, then you accept at least one "lesser type" of existence, so why not accept mathematical existence? If you say no, then how can statements about him have definite truth values?
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I had heard that, but never studied the proof. It is indeed wild.mask

The context here is the fact that the statement "Every vector space has a basis," is fully equivalent to the axiom of choice. Each implies the other. The proof is a simple application of Zorn's lemma, an equivalent of choice. Given a vector space $V$, you consider the partially ordered set of all linearly independent subsets of $V$, ordered by set inclusion. You then convince yourself that if you have an upward chain of set inclusions, the union of the chain is in fact also linearly independent, and is an upper bound for the chain. Then you apply Zorn's lemma to conclude that there must be a maximal linearly independent set; which must therefore be a basis; because if it weren't, you could add an element to it so it wouldn't have been maximal. QED.

I started writing a much longer and more elementary explanation of the jargon to make this more accessible, but it quickly got way too long. It's all on Wiki or just ask :-)

That's one direction, that choice implies basis. The other direction, that the statement that every vector space has a basis implies the axiom of choice, was actually proven only as recently as 1980 I think. That surprised me when I looked that up a while back.

At my school we only had to learn the set theory that comes with analysis and algebra.mask

Right, that makes sense. Everyone needs the basics of unions and intersections and so forth but even most math majors don't ever take a course in set theory. I did because I was interested. I always had some kind of affinity for that stuff. Like I say I can rattle off a Zorn proof like that but when I see those crazy freshman calc integrals my brain freezes.

I did look into ordinals on my own. I remain impressed by the usual Von Neumann constuction. I used it in visual art and I also think it has a philosophical relevance. It's a nice analogy for consciousness constantly taking a distance from its history. 'This' moment or configuration is all previous moments or configurations grasped as a unity. It works technically but also aesthetically.mask

That's a cool idea. I get it. In von Neumann you go up one step at a time, by successors and by limits. You're suggesting a continuous analog of that. Every moment in time is the union of all that's come before. That's good

That would be surprising indeed. I think we agree on the gap between math and nature.mask

Yes. It seems obvious to me. But then again there's that pesky "unreasonable effectiveness." And so often, the physicists discover the math before the mathematicians do. So the relation between nature and math is different than the relation between nature and, say, chess. Math and nature are intertwined, but they're not the same.

As you mention, our measurement devices don't live up to our intuition and/or formalism. I have a soft spot for instrumentalism as an interpretation of physical science.mask

I had to look that up. You mean science is valid insofar as it's useful. I'd disagree. I like math for the sake of math and science for the sake of science. In fact a lot of the uses of science are far more evil than the intent of the scientists. Atom bomb and all that, arising from beautiful theoretical work on the nature of the universe. The pacifist Einstein inventing the physics that led to the bomb. One of history's ironies I'd say.

I haven't heard that one. But I know a graph theory guy who thinks the continuum is a fiction and an analyst who believes reality is actually continuous. Another mathematician I know just dislikes philosophy altogether.mask

Exactly. Some working practitioners of math and physics have philosophical opinions and most don't care at all. And among those with opinions, they're all over the place. Like everyone else I suppose.

I like philosophy more than math when I'm not occasionally on fire with mathematically inspired, though I have spent weeks at a time in math books, obsessed. (At one point I was working on different models of computation, alternatives to the Turing machine, etc. Fun stuff, especially with a computer at hand.)mask

I've heard a little about that. Continuous TMs and the like.

Right! Because of the infinite decimal expansion. One of my earliest math teachers had lots of digits of pi up on the wall, wrapping around the room. Some of the problem may be in the teaching, but I've wrestled with student apathy. Math tends to be viewed as boring but useful, the kind of thing that must be endured on the path to riches. Its beauty is admittedly cold, while young people tend to want romance, music, fashion, fame, etc.mask

I'm afraid that even if you totally reformed math education in such a way that everyone loved it, they'd still prefer music, fashion, fame, etc.
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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

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 ).
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Nevertheless, how you stipulate or construct the object lends a particular perspective on what it means; even when all the stipulations or constructions are formally equivalent.[/quotre]

Yes definitely. Each perspective adds to your intuition and understanding of what's going on. A little like being baffled by trig and then baffled by "the square root of -1" in the context of solving quadratics; and then at some point in the future, maybe, you find out that trig and complex numbers are two ways of talking about the same thing; and everything becomes so much more clear.

Learning math is sort of about learning more and more abstract and general viewpoints for the same thing.
I remember studying abstract algebra at university, and seeing the isomorphism theorems for groups, rings and rules for quotient spaces in linear algebra and thinking "this is much the same thing going on, but the structures involved differ quite a lot", one of my friends who had studied some universal algebra informed me that from a certain perspective, they were the same theorem; sub-cases of the isomorphism theorems between the objects in universal algebra. The proofs looked very similar too; and they all resembled the universal algebra version if the memory serves.

That's interesting. I never looked at universal algebra but what you describe is a lot like category theory. There's a construction called a "product" which particularizes to the Cartesian product of sets, the direct product of groups or rings, etc. There are also some surprises. The coproduct, which is what you get when you take the definition of the product and simply reverse all the arrows, is the disjoint union in the category of sets; and the direct sum in the category of Abelian groups. Without the abstract point of view you wouldn't necessarily realize that disjoint unions and direct sums are essentially the same thing in different contexts.

Regarding that "nevertheless", despite being "the same thing", the understandings consistent with each of them can be quite different. For example, if you "quotient off" the null space of the kernel of a linear transformation from a vector space, you end up with something isomorphic to the image of the linear transformation. It makes sense to visualise this as collapsing every vector in the kernel down to the 0 vector in the space and leaving every other vector (in the space) unchanged. But when you imagine cosets for groups, you don't have recourse to any 0s of another operation to collapse everything down to (the "0" in a group, the identity, can't zero off other elements); so the exercise of visualisation produces a good intuition for quotient vector spaces, the universal algebra theorem works for both cases, but the visualisation does not produce a good intuition for quotient groups.

I've always felt that the symbology is how we communicate, but the intuitions are private. Some people gravitate to one visualization or another.

If you want to restore the intuition, you need to move to the more general context of homomorphisms between algebraic structures; in which case the linear maps play the role in vector spaces, and the group homomorphisms play the role in group theory. "mapping to the identity" in the vector space becomes "collapsing to zero" in both contexts.

I think you have to see the particular examples before you're ready to absorb the abstract viewpoint that integrates and clarifies things. There's no easy way through it.

There's a peculiar transformation of intuition that occurs when analogising two structures, and it appears distinct from approaching it from a much more general setting that subsumes them both.

I guess the original example of all this is Descartes's great invention of analytic geometry. A problem in geometry can be turned into an equation, solved algebraically, and the result transferred back to the geometry. That was a great leap forward.

Perhaps the same can be said for thinking of real numbers in terms of Dedekind cuts (holes removed in the rationals by describing the holes) or as Cauchy sequences (holes removed in the rationals by describing the gap fillers), or as the unique complete ordered field up to isomorphism.

With the reals, I think most people think of them via their axiomatic definition as a complete totally ordered field. The constructions are incidental, serving only as a proof that if we were called upon, we could cook up the reals within set theory. But in practice we don't care about the construction; only the properties. We add, subtract, multiply, divide, take limits, etc. It doesn't matter what set of Tinker Toys we use to build a model of them. It's their behavior that matters; and that's the start of modern categorial or universal thinking.
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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).

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.

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

Yes I used to read his articles on Usenet about how he uses n-categories in physics. I was quite amazed, having only seen category theory years before in math classes and not realizing it had escaped into the wild. Now it's a big deal in computer science too. It's taking over.

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 ).

I think Vovoedsky's name gets used way too much in vain in these types of discussions. It's a perfectly commonplace observation that isomorphism can be taken as identity in most contexts. The univalence axiom formalizes it but informally it's part of the folklore or unwritten understandings of math.

But really, that's not my point about constructive math. I don't care if it gives the right theory. The constructive real line is full of holes. The intermediate value theorem is false. It is not a continuum. Doesn't that bother anyone?

And since physics is supposed to be about the world, this is the kind of thing that should matter a lot! That's my thesis, based on a my admittedly limited understanding of these matters.
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Ok this is the post I wanted to get to.

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).

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.

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.

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.

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.

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.

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?

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).

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.

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.

Well of course. I agree with that. It's like saying that if I have a circle with radius 1 its circumference will be 2pi but of course in the real world we can't measure pi.

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.

I think we're in violent agreement here but I'm not sure that you're fully appreciating the point. You can solve an equation using the real numbers. But you can't measure it to be correct to infinite precision; and you can't know that your model is true.

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.

I'm not following this.

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.

Ok.
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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.

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 ]
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This is the way I look at mathematical objects in general, and real numbers in particular. They can be physically represented, if they happen to be. But generally, they are specifications more so then anything. As all specifications, they express our epistemic stance towards some object, not the properties of the object per se. Real numbers signify a process that we know how to continue indefinitely, and which we understand converges in the Cauchy sense. Does the limit exist (physically)? Maybe. But even if it doesn't, it still can be reasoned about conceptually.

This paragraph seems to bear on my conversation with @Metaphysician Undercover.

I can indeed specify $\sqrt 2$; but when I do so I am merely "expressing my epistemic stance" toward $\sqrt 2$; yet not necessarily saying anything about $\sqrt 2$ itself, whatever that means. I think this is @Metaphysician Undercover's point perhaps.

And then "expressing my epistemic stance" towards a mathematical object is what I mean by endowing that object with mathematical existence. Perhaps this is the distinction being made.
• 459
I interpret the Axiom of choice to be a 'prayer to nature' to send me the desired object already made. The axiom refers to my opponents choices that are not modelled in the formalism i am using.

If I have no internal strategy for constructing a basis from my formalism of a vector space, then I am reliant upon nature sending me a basis, which I have no control over. But suppose nature never sends me a basis?

Arguments between constructivists and classical logicians are caused by a fundamental disagreement about the nature of proof. The former equates proofs with fully-determined algorithms under the control of the mathematician, whereas the latter allows proofs to interact with nature in an empirically-contigent and indeterminate fashion.

Unfortunately, classical logicians are usually in denial about what they are actually doing. Instead of admitting that their notion of proof is empirically contigent and not internal, they insist their notion of proof is internally constructive in a transcendental platonic realm.
• 459
An important question is the relationship of the Axiom of Choice (AC) to the Law of Excluded Middle (LEM), for Classical Logic is normally distinguished from Intuitionistic Logic on the basis of the latter axiom rather than the former axiom. Furthermore, intuitionists often claim that AC is constructively acceptable by interpreting AC to refer to the very construction of a function, for intuitionists do not accept the existence of non-constructive functions. This is very confusing, because AC's natural role is to refer to an unspecified function for which we do not possess a constructive description. This situation arises all the time in computing when a program points to an externally provided input data-stream that the programmer cannot further describe.

Now according to the SEP's article on the Axiom of Choice , AC implies LEM in the presence of two further axioms, namely Predicative Comprehension (PC) and Extensionality of Functions (EF). The former says that the image of every predicate applied to individuals is a set, whereas the latter says that every extensionally equivalent pair of sets has the same image under every Set Function.

https://plato.stanford.edu/entries/axiom-choice/#AxiChoLog

The author of the article proves that PC & EF => ( AC => LEM), but he then argues that whereas PC is constructively valid, EF isn't. See the argument, but I don't find his argument about EF persuasive, at least as I understand it.

In my current view, EF is also constructively admissable, implying that the precise difference between classical logic and intuitionistic logic is AC as much as it is LEM, which then cements the view that classical logic describes a game between two players, whereas intuitionisitic logic describes solitaire.

A further motive for my view (and indeed the most natural motive), is that classical logic involves sequents of the form (a AND b) => x OR y, where it isn't known which of x or y is true, in which the negation of one implies the existence of the other. On the other hand intuitionistic logic only involves sequents with a single conclusion, of the form (a AND b) => x. Thus there is indeterminism in the case of classical reasoning, but not in the case of intuitionistic reasoning.
• 108
I think that people develop intuitions, depending on whether they consider mathematical objects as rigorous philosophical metaphors for some physical counterparts. As I said, when I think about mathematics, I reason from an information processing perspective, where knowledge itself is the object of investigation. For me, the diagonal of a square is "square root nothing", because perfect squares rarely exist in our daily experience. At the atomic and subatomic level, perfect geometries may actually make sense (at least probabilistically), but that doesn't explain how we managed to exploit mathematical ideals for so long. I think that the information revolution pushes the emphasis gradually from natural sciences, towards a more "means to an end" perspective of logical studies.

For me, when we construct mathematical models, we are factoring in uncertainty, efficiency, capacity (for control, measurement, computation, etc), and we concoct a useful approach to solving a practical problem. This does not mean that our solution is completely disconnected from the underlying reality, but it is not literal representation of the natural phenomena either. It is mostly practical and has many interesting properties on its own. Those properties determine how it behaves when we use it and even if they are not necessarily properties of the original system (although they could be), they are worth investigating if we plan on applying the solution many times and want to have understanding of its behavior.

So, to me, algebraic structures, real numbers in particular, are "ways" of dealing with problems. They are inspired by nature, but are not necessarily literally representative of natural objects. The specifications of those algebraic structures, the solutions based on those specifications, etc, have interesting, in some cases paradoxical properties, that are worth investigating simply because of our usage of those structures. If they happen to coincide with nature's geometries on some fine-grained level, this is particularly interesting, and bodes further investigation, but it doesn't affect the originally intended utility.
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I think Vovoedsky's name gets used way too much in vain in these types of discussions. It's a perfectly commonplace observation that isomorphism can be taken as identity in most contexts. The univalence axiom formalizes it but informally it's part of the folklore or unwritten understandings of math.

Bit about the univalence axiom was in one of @Mephist's posts, no idea what happened there.
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Ok this is the post I wanted to get to.

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.

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.

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.

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?

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.

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.

Yes, exactly. That's what I mean.

I'm not following this.

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)
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Mathematicians and philosophers of mathematics, with the presumed exception of platonists, reject the premiss that all "existence" is ontological existence. Specifically, they acknowledge that mathematical existence does not entail ontological existence.

Ontology is the study of existence. Isn't it? How could there be a form of existence which isn't ontological existence? That sounds very contradictory to me.

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.

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.

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.

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.

So, we create a grid of points according to a geometry of space-time, like we might position points according to Euclidian geometry. There is an assumed figure produced by this positioning like a square etc., and we might assume that the figure could exist as a real physical object. If the geometry we use to position those points, is not consistent with what is allowed for by the real positioning of objects, because the medium is not continuous, or if the nature of the continuity is completely misunderstood, then I would say that this is a problem. This is what we see in Euclidian geometry. The geometry allows that we can place a point virtually anywhere. But when we create the figures which connect the points, like a square or a circle, we see that there is a problem with this assumption, that we can put the point anywhere we want. 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".

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.

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.

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.

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...

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.

That's what abstraction is! It's giving a name to something immaterial in order to manipulate it.

I believe you do not have a very thorough education in philosophy, or you would not characterize "abstraction" in this way. Abstraction is a process. That process is sometimes described as producing a thing which might be called "a concept", or "an abstraction". There might be a further process of manipulating that thing called "an abstraction", but notice the separation between the process which is abstraction, creating the immaterial thing called an abstraction, and the process which is fixing a name to the supposed "immaterial thing" (an abstraction) and manipulating it.

To begin with, we need to analyze that process of abstraction, and justify the claim that an immaterial object is produced from this process. If there is no immaterial object produced, then the name which is supposedly given to an immaterial object, simply has meaning, and there is nothing being manipulated except meaning. But if you are manipulating meaning you stand open to the charge of creating ambiguity and equivocation. This is why we separate logic, which is manipulating symbols, from the process of abstraction which is giving meaning to those symbols. So it is very good to uphold this principle. In logic we manipulate symbols, we do not manipulate "something immaterial" (meaning) which the symbols represent. What the symbols represent is determined by the premises. The "something immaterial" (meaning) precedes the logic as premises, and extensions to this, as new understanding, may be produced from the logical conclusions, but what is manipulated is the symbols, not the immaterial thing (meaning).

You say you have found a loophole that allows you to accept the world yet reject ... something. The square root of 2.

I don't say that I've found a "loophole", I say that there is weakness. And, it's not me who found this weakness, which is a deficiency, it's been known about for ages. You look at this deficiency as if it is a loophole, and insist that the loophole has been satisfactorily covered up. But covering a loophole is not satisfactory to me, I think that the law which has that deficiency, that weakness, must be changed so that the loophole no longer exists.

* If we believe in the rationals, we can build a totally ordered field containing the rationals in which there is a square root of 2.

Until you provide me with a definition of "field" for this premise, your efforts are futile. If a field requires set theory, I'll reject it for the same reason I rejected your other demonstration. If you can construct a field with square root two, without set theory, then I'm ready for your demonstration. If you produce it I'll make the effort to try and understand, because I already believe that you would need to smuggle in some other invalid action, because that's what's occurred in all your other attempts.

Yes I know you already believe that. The question is whether you're willing to believe that it has mathematical existence. You ask me what that is but I've given you many demonstrations of the mathematical existence of 2–√2; as the limit of a sequence, as an extension field of the rationals, as a formal symbol adjoined to the rational numbers. All these things are part of mathematical existence. You will either have to take my word for it, or work with me to work through a proof of the mathematical existence of 2–√2 .

You never explained to me what you mean by "mathematical existence" that remains an undefined expression.

You showed me that you have a psychological block in dealing with symbology; leading to a massive area of ignorance of math; leading to making large errors in your philosophy. That's my diagnosis.

It's not the case that I have a block in dealing with symbology, but what I need is to know what the symbol represents. Until it is explained to me what the symbol represents I will not follow the process which that symbol is involved in. I believe that whatever it is that is represented by the symbol, places restrictions on the logical processes which the symbol might be involved in. Supposedly, you could have a symbol which represents nothing (though I consider this contradiction, as a symbol must represent something to be a symbol), and that symbol might be involved in absolutely any logical process. However, once the symbol is given meaning, the logical processes which it might be involved in are limited. So if you start with the premise that a symbol might represent nothing, I'll reject your argument as contradictory.

Like Captain Ahab, who only has fictional existence in a novel. Nevertheless statements about him can can truth values, such as whether he's the captain of the Pequod or the cabin boy. So there's fictional existence.

"Fictional existence" is contradiction plain and simple. To be fictional is to be imaginary, and to exist is to be a part of a reality independent of the imagination. If you are handing to "existence" a definition which allows that an imaginary, fictional thing, exists, then it's not the rigorous philosophical definition which I am used to. I think that if you cross this line, you have put yourself onto a very slippery slope, denying the principles whereby we distinguish truth from falsity.

I can indeed specify 2–√2; but when I do so I am merely "expressing my epistemic stance" toward 2–√2; yet not necessarily saying anything about 2–√2 itself, whatever that means. I think this is Metaphysician Undercover's point perhaps.

And then "expressing my epistemic stance" towards a mathematical object is what I mean by endowing that object with mathematical existence. Perhaps this is the distinction being made.

I interpret this as your "epistemic stance" requires Platonic realism as a support, a foundation. I deny Platonic realism, so I think your epistemic stance is ungrounded, unsound.
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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.

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!
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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.

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...
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