@Metaphysician Undercover, Let me just put this remark here because it's the core of the problem.

In order to figure out things that we don't understand, we give them names. We write down the properties we want the names to have. We apply logical and mathematical reasoning to the names and the properties to learn more about the things. That's how science works. That's how everything works.

So we are ALWAYS writing down and using symbols without much if any understanding of the things we're representing. It's exactly through the process of reasoning about the symbols and properties that we LEARN about the things we're interested in. That's Newton writing that force = mass times acceleration. At the time nobody knew what force, mass, and acceleration were. Newton defined those things, which may or may not "really" exist; then he applied mathematical reasoning to his made-up symbols and terms; and he thereby learned how the universe works to a fine degree of approximation.

Of course no scientific theory is "true" in an absolute sense. But that's the point. That's how rationality works. We make up symbols for things we don't understand; and we come to understand the world we live in by means of reasoning about our symbols.

You reject all of this. If Newton had said F = ma to you, you'd have said, what's force? And Newton would describe it to you, and you'd say, well that's not real, it's only a symbol. You reject all science, all human progress, rationality itself. If I tell you all x's are y's and all y's are z's, and you REFUSE TO CONCLUDE that all x's are z's because I haven't told you what x, y, and z are, you are an absolute nihilist. You believe in nothing that can help you get out of the cave of your mind.

I believe you do not have a very thorough education in philosophy, or you would not characterize "abstraction" in this way. — Metaphysician Undercover

This is surely no criticism of me since; for one thing, I have already admitted my ignorance of much of classical philosophy. And you are abysmally ignorant of mathematics, yet you insist on your right to make lofty pronouncements on mathematical topics. Why shouldn't I have the same right?

But in fact you are wrong that I don't know what abstraction is. You're right that I haven't read all the deep thinkers on how to define abstraction. But as someone who's had some small bit of exposure to higher math, I've seen a heck of a lot of absraction in the wild. I know what abstraction is. It's giving a name to something you can't kick with your foot. You don't like that definition but it seems quite serviceable to me. We can't see justice but we have a word for it and we can write down some of its properties and over time, we come to learn what justice is.

I suggest, based on our conversation, that you may be highly expert on what the great philosophers say about abstraction; but your actual experience and knowledge about how abstraction works is virtually nil. At least when it comes to mathematical abstractions. And what's more abstract than mathematical abstractions?

In high school algebra they use numbers, like 47. That's abstract. Then they ask you to find 'x'. A lot of students never get past that trauma. In college you look at derivatives and integrals, more abstractions. If you're a math major they tell you about sets, groups, rings, metric and topological spaces. Past that there's category theory, which is so abstract that mathematicians jokingly refer to it as, "abstract nonsense."

You may know what people have SAID about abstraction. But you know nothing of abstraction. You've proved that to me over and over.

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. — Metaphysician Undercover

I see no reason to abandon my casual definition, paired with my experience of grappling with mathematical abstractions.

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). — Metaphysician Undercover

This is all bullshit. It has nothing to do with the subject at hand. If you don't credit me with having a deep personal connection with the process of abstraction, that seems like more of a personal issue than anything else.

Aren't you conflating book learnin' with actual experience? How can you tell a math person they don't know abstraction? That's like telling a pizza chef he doesn't know marinara sauce.

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. — Metaphysician Undercover

If you think the mathematical existence of the square root of two is a "weakness" or defect in mathematics, it is because you are so ignorant of mathematics, that you haven't got enough good data to reason soundly about mathematics. I would think someone in your position would be desirous of expanding their mathematical understanding. Think of it as "opposition research." Learn more so you can find more sophisticated ways to poke holes.

Until you provide me with a definition of "field" for this premise, your efforts are futile. — Metaphysician Undercover

Meta my friend sometimes you are a very funny guy. I mean that sincerely. I have in fact defined field once or twice, but the reason I haven't burdened you with the details is because you have expressly asked me not to burden you with mathematical details.

But I will be glad to walk you through this, a little later. Next post, after I've gotten your clear permission to walk you through a little math.

If a field requires set theory, I'll reject it for the same reason I rejected your other demonstration. — Metaphysician Undercover

Well, a field is typically defined as a type of set; but the definition really has nothing to do with set theory. It's about what algebraic operations are allowed. In fact I will be happy to build you a field, which briefly is a collection of numbers that you can add, subtract, multiply, and divide (except by zero) just like you can with the rational, that contains a square root of 2.

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, — Metaphysician Undercover

I take this as an honest and brave statement on your part. If I'm understanding correctly, you are asking me to walk you through a mathematical argument, and that you will make a good faith effort to understand me. Is that correct? I have your permission to do this?

Because I can in fact, and without much difficulty at all, show you the square root of 2 without using any set theory; not even by a different name. In other words I won't just sneak in set theory without using the words.

If you will grant me the existence of the rational numbers; I'll build you a square root of 2.

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. — Metaphysician Undercover

No actually I don't. All I need is for you to believe in the existence of the rational numbers.

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

I thought I'd defined it several times; or at least given many examples of it. But perhaps you're right. Let me give some thought to what a formal definition of mathematical existence might look like.

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. — Metaphysician Undercover

Symbols don't necessarily need to represent anything. If I have a symbol that behaves a certain way; that's just as good as a thing that behaves that way. At some level one can take the symbol for the thing.

That's abstraction.

Until it is explained to me what the symbol represents I will not follow the process which that symbol is involved in. — Metaphysician Undercover

Why can't I write down a symbol.

$x$

Why, pray tell, may I not type that symbol on the page? And say that it stands for a green thing? Why can't I do that? It's the foundation of civilization.

You know, I bet you failed the kinds of questions like:

If there are three fraggles in a snaggle; and four snaggles in a boodle; then how many fraggles are in a boodle?

Are you telling me that you would not be able to determine that it's 12 if I didn't tell you what a fraggle, a snaggle, and a boodle are?

If you assert that to me then there's no hope of communication here.

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. — Metaphysician Undercover

You categorically reject abstraction. I can't work with you anymore. I'm not going to bother to show you the square root of 2 because you can't solve the fraggle problem.

"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. — Metaphysician Undercover

Ok at least you're consistent. You deny mathematical existence but you also deny fictional existence.

So I ask you? Is Ahab the captain of the Pequod? Or its cabin boy? Do you really claim to be unable to answer on the grounds that Ahab's a fictional character? Nihilism.

Ok to sum up:

1) You asked me to walk you through a construction of the square root of 2 that does not require set theory. I want to make sure I have your permission to do some math and that you'll engage with my exposition in good faith. Do I have that agreement from you?

2) In order to do (1) I require only one premise from you. You must grant me the existence, in whatever way you define it, of the rational numbers. If you'll do that I'll whip up sqrt 2 in no time flat, no set theory needed, and no cheating on that point. I will use no set theoretic principles.

3) We have a sticking point, which is that you don't accept a symbol unless it comes with a meaning attached. If you truly believe that then you can't solve the snaggle problem, which requires you to reason logically about symbols whose meaning is not defined.

If you are unwilling to do that, I can't give you a proof and frankly I can't continue our conversation. You reject rationality entirely with such a stance.

4) We have stumbled on an interesting point. You say that a symbol can never be conflated with the thing it's supposed to symbolize. But in math, we often do exactly that. We don't know "really" what the number 2 is. Instead, we write down the rules for syntactically manipulating a collection of symbols; we use those rules to artificially construct a symbol that acts like the number 2. Then we just use that as a proxy for the number 2.

There is even some philosophy behind this. When we talk about the number 2, we don't care about what the object is; we only care about how it behaves.

This is the modern viewpoint of math. We don't care what numbers are; as long as we have a symbol system that behaves exactly as numbers should.

I think we've just stumbled into category theory and structural thinking in math. A thing is not what it's made of; a thing is what it does. And even more, a thing is entirely characterized by its relationships to all other things. If you know the relationships you know the thing. That's category theory.

So anyway this got long but the last four numbered paragraphs enumerate the points that need reply. We needn't squabble about the definition of abstraction.

But really, reading over this, we're done. You won't allow a symbol that behaves like a thing, to be used as a proxy for that thing in a chain of reasoning, in order to get more understanding of the thing. It's like solipsism. I can't refute it but I don't waste my time arguing with solipsists.

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. — Mephist

Ok. It was only recently that I learned that protons have quarks inside them. Another thing I've learned is that gravitational mass is caused by the binding energy that keeps the quarks from flying away from each other. How that relates to Higgs I don't know. I've also seen some functional analysis so I know about Hilbert space. I have a general but not entirely inaccurate, idea of how QM works.

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 — Mephist

If you agree that there's no way to measure an exact length to infinite precision, then you accept my point that the idea that measuring the square root of 2 is purely a mathematical exercise and not a physical one; as would be measuring a length of 1. But if you accept this point we're in agreement.

Yes, I agree. There are no exact measurements, — Mephist

So we're in agreement and there isn't actually any disagreement.

but there are exact predictions in QM. — Mephist

And no way to exactly verify them. We're in complete agreement.

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). — Mephist

Yes the mathematical theory gives an exact prediction. But that makes sense. And of course the relation of QM to reality has been the subject of physical and metaphysical argument for a century now.

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 ) — Mephist

Yes that's the one, thanks.

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. — Mephist

But this can't be, since calculating machines can't calculate ANYTHING with arbitrary precision. Where are you getting these mystical TMs? If the theory gives a result like pi, I'd accept that as a result having arbitrary precision. But if you are saying that even in theory there is a TM that can calculate anything with arbitrary precision, that's wrong. The best a TM can do is approximate a computable real number with arbitrary precision. That's much less than what you are claiming, if I'm understanding you correctly.

Yes, exactly. That's what I mean. — Mephist

We are in agreement. Though I'd like you to clarify your belief in magic Turing machines that can calculate "anything" with arbitrary precision. TMs can only approximate computable numbers. If you believe the universe is computable that's a proposition I disagree with.

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

Second the motion.

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) — Mephist

I've heard that without QM, atoms would collapse. So QM seems to be a good thing. But it can't be the ultimate answer. Something more is out there.

[continuation of the previous post]
So, you can see constructivist logic as an algebra of propositions built with computable functions (https://en.wikipedia.org/wiki/Heyting_algebra). You cannot build non-computable functions using only the operations of this algebra, but you can add elements that are not part of the algebra ("external" non-computable functions) to obtain a new algebra that uses all computable functions plus the function that you just added. — Mephist

I had a long convo about all this with @alcontali I believe, a while back. At that time I felt that I'd satisfied my curiosity about the subject of constructive math. I hope you'll forgive me but I prefer not to spend much time talking about this. It just doesn't hold my interest, trendy as it all is these days.

That's exactly the same thing that you do adding square root of 2 to the rationals: you obtain a new closed field that contains all the rational numbers plus all that can be obtained by combining the rational numbers with the new element by using the operations defined on rational numbers. — Mephist

You miss all the noncomputable numbers. You have holes in your real line.

But I see that the main problem for you is not about the soundness of logic, but about the cardinality of the set of real numbers. — Mephist

No, the cardinality argument is secondary. The primary argument is the lack of Cauchy-completeness of the constructive line. But it turns out that you can prove that Cauchy-completeness implies uncountability, so in a sense they're the same question.

So, my question is: how do you know that the cardinality of the set of real numbers is uncountable? — Mephist

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

Well, Cantor's diagonal argument is still valid in constructivist logic: it says that any function that takes as an argument an integer and returns a function from integers to integers cannot be surjective (cannot generate all the functions). The proof is exactly the same: take F(1,1) and change it into F(1,1) + 1, then take F(2,2) and change it into F(2,2) + 1, etc... This is a computable function if all the F(n,n) were computable, ( n -> F(n,n) - very simple algorithm to implement ) but it cannot be in the list: if it were in the list ( call it X(n,n) ), let "m" be the position of X ( X is the m-th function ). What's the value of X(m,m) ?
X(m,m) cannot be computable. — Mephist

It makes sense that a constructive version of Cantor's theorem is true. But as I said I'm not primarily concerned with cardinality arguments.

The constructive line is not Cauchy-complete. As an example consider the sequence made up of the successive finite truncations of the binary digits of Chaitin's Omega. This is a Cauchy sequence of computable numbers that fails to converge to a computable number.

Like I say, I am perfectly well aware thatl the constructivists have a million ways to wave their hands at this. I truly don't understand why they aren't bothered by a continuum with holes in it.

** EDIT ** I just realized what you'll say here. I cannot computably form the sequence of successive finite truncations of Omega because I can't computably determine the bits. The sequence I gave is noncomputable so you don't see it and it causes no problem for you. You can prove some version of "all computable Cauchy sequences converge," and that satisfies a constructivist. I'm learning to think like a constructivist! I don't know if that's good or bad.



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

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

The problem is well known: you cannot enumerate all computable functions because there is no way to decide if a given generic algorithm stops.
So, computable functions are as uncountable as real numbers are. Where's the difference? — Mephist

Well they're internally uncountable but actually countable. Analogous to the fact that Cantor's theorem still holds in a countable model of ZF. But the computable numbers are in fact countable. There is no computable enumeration of them but there is obviously an enumeration: by length and then by lexicographic order. So they are "countable but not computably countable." That's the very best you can do along these lines.

But again, I am not primarily making a cardinality argument. My two objections remain: One, the constructive line is not Cauchy-complete; and two, that constructivists must necessarily deny the possibility of random bitstrings.

Bit about the univalence axiom was in one of Mephist's posts, no idea what happened there. — fdrake

Yes in fact at the time I couldn't understand how your name got in there. Might have messed up the editing at my end.

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. — simeonz

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.

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). — Mephist

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. — Mephist

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. — Mephist

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). — Mephist

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. — Mephist

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. — Mephist

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. — Mephist

Ok.

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). — Mephist

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 — fdrake

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 ). — fdrake

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.

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.
— fdrake

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. — fdrake

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. — fdrake

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. — fdrake

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. — fdrake

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. — fdrake

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.

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.

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. — Metaphysician Undercover

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. — Metaphysician Undercover

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". — Metaphysician Undercover

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? — Metaphysician Undercover

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". — Metaphysician Undercover

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. — Metaphysician Undercover

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". — Metaphysician Undercover

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?

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? — Mephist

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. — Mephist

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. — Mephist

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.

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. — Mephist

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.

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?

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

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?

Especially considering the algebraic approach that you presented. I find it intuitively satisfying without considering set theoretic foundations. And the Dedekind cut is satisfying if one can admit sets of rational numbers (intuitively self-supporting, IMO). — mask

Please explain this to @Metaphysician Undercover! I've had no luck.

Now those are strange entities, unlike the essentially finite square root of 2 (as you've already noted.) A dark ocean of infinitely informative numbers that can't be named is far more poetic and disturbing than little old 2‾√. — mask

Yes indeed. I think of them as the "dark matter" of the real number line. We can't name them, we can't compute them, we can't use them for anything. But without them, there aren't enough reals to be Cauchy-complete. The reals lose their essential property. And without dark matter, the galaxies would fly apart. There are more things in heaven and Earth, Horatio, / Than are dreamt of in your philosophy -- Shakespeare.

Hillary said she'd obliterate Iran.
— fishfry
...in response to an Iranian nuclear attack on Israel. — Wayfarer

True, but she received a lot of criticism at the time, including from Obama. Her warmongering is what cost her the Dem nomination. So yes, she was making a hypothetical point. But her language revealed her character. If you criticize me for taking her remark out of context, would you level the same charge at Obama?

Obama: Clinton's 'obliterate' Iran statement too much like Bush

https://www.cnn.com/2008/POLITICS/05/04/dems.election/

The problem with this perspective is that once you've carried out an action, and determined that it was a mistaken action, you cannot simply undo the mistake. So "we never should have gone in" does not justify "we should leave". — Metaphysician Undercover

In practical terms I agree. If we left there would be a bloodbath. This is how we stayed so long in Vietnam. We didn't want to "waste" all the lives and money already lost. So we wasted plenty more before Congress finally had enough and cut off funds for the war.

So as I said, we're screwed if we stay in Iraq and screwed if we leave. The disaster was inevitable the day we invaded. No way out.

ow Trump has lobbed a stink bomb in there — Punshhh

For what it's worth I supported Obama's Iran deal, imperfect as it was. For someone who wants to get us out of the wars, Trump's Iran policy has been counterproductive IMO.

I'm not partisan, I'm over the pond. My interest is how the Middle East is going to kick off. My only concern is that there isn't a nuclear conflagration in the region. Its looking more likely now. Especially if this is the beginning of the US moving out of the region. — Punshhh

I'm an old, as the kids say. In grade school -- full disclosure, this was during the Eisenhower administration -- we used to talk about current events. The Middle East was about to blow up. I remember something we all agreed in class: That when WWIII starts, it will start in the Middle East.

Now it's all these many years later, and the Middle East is still about to blow up and we all agree that when WWIII starts, it will start in the Middle East.

You have a good insight about the bloody mess that's going to follow when the US leaves. That's the problem. I'm a peacenik and we never should have gone in. I think we should leave tomorrow morning. But when we do, things are going to be really really bad. And the US will be blamed. Colin Powell said about invading Iraq: "You break it, you bought it." And we broke it. We're screwed if we stay and screwed if we go. That is the legacy of Bush's war, aided and abetted by the NYT, Hillary, Joe Biden, and the rest of the Democratic establishment. No wonder there's a populist uprising, imperfect as its leader may be.

Just dropping in to trigger the checkbox liberals. Sorry you didn't get your war. I'm old enough to remember when Dems didn't pray for war just to make the president look bad. And the House wants to restrict the president's war powers? Where the hell were they in 2002? All of a sudden the House gives a shit about the endless wars? Pelosi signed off on the torture. Never forget that. It helps to keep you focussed on reality and not the partisan bullshit you get fed by the media. Hillary said she'd obliterate Iran. She was the warmonger in chief from Waco to Benghazi by way of Serbia. Don't forget that either. Hillary lost because she was the war candidate and Trump was for peace. The Dems won't admit that.

According to physics, the most fundamental stuff of science is fields, not particles. The Standard Model lays out 13 fields which exist throughout the universe, oscillate and interact with one another to generate everything else. Particles are packets of energy in the fields described by quantum mechanics. There are three other unknown ones: Dark Energy, Dark Matter and Inflation. — Marchesk

I was just talking about this in another thread, and realized that I'm confused on a point of physics. I know two things:

* The 13 fields. I definitely can't name them all. There's gravitation, and electroweak, which is electromagnetism plus the weak nuclear force; then the strong nuclear force, and the Higgs field. I don't know about any others. But basically a field in physics is a thingie -- usually a vector or a tensor -- attached to each point of space. So you have a vector field or a tensor field evolving over time. Basically multivariable calculus on a LOT of steroids, into the realm of differential geometry and general relativity.

* In quantum theory, everything is a probability wave. An electron, for example, is not to be thought of as a point-like thing at all. It's not a tiny little charge of electricity located at some coordinates in space. What it is, is a probability distribution that determines the chance that if someone happened to look at the electron, they'd find it in that location. If nobody looks, it doesn't have a location. Or it has all locations. Same thing. Once an observation is done, the electron is found to be in some position or state. How this all works is an open problem at the intersection of physics and metaphysics.

I think in my mind I've conflated the fields with the probability waves. Can anyone fix my physics? How do the probability distributions interact with the various force fields, if it's ok to call them that. Gravity and electromagnetism and so forth.

"The square root of two" has no valid meaning in the rational number system. This means that taking a square root is not a valid operation. — Metaphysician Undercover

Let me restate your quote as a formal argument.

P: The square root of two" has no valid meaning in the rational number system.

C: This means that taking a square root is not a valid operation.

The conclusion doesn't follow from the premise. A valid conclusion would be, "This means that taking a square root is not a valid operation in the rationals. And of course that is correct.

One can, however, conceive of and build, with logical correctness. systems of numbers in which there IS a square root of 2.

I have an example on my mind, I'll toss it out there.

Do you know modular arithmetic, or the "integers mod 5" and so forth? Telling time is just the integers mod 12. If it's 11 now, what time will it be in 2 hours? The answer is 1. We add "mod 12," which means first do normal addition, then subtract out the largest multiple of 12 we can. In fact you've alluded to this phenomenon. When we divide two integers we get a quotient and a remainder. In modular arithmetic, we only care about the remainder.

We can do the same trick with any modulus, as it's called. Consider the integers mod 7. They consist of the symbols 0, 1, 2, 3, 4, 5, and 6, with addition and multiplication mod 7.

In the integers mod 7 we can add, subtract, and multiply. In general we can't divide. So the integers mod 7 are a perfectly valid system of numbers, not unlike the integers, but not like them either. [They're a quotient of the integers if you took abstract algebra].

Now, what is 3 x 3 in the integers mod 7? Well, 3 x 3 = 9 normally; and in the integers mod 7, the number 9 corresponds to the number 2.

So 3 x 3 = 2. That is, 3 is a number that, when squared, results in 2. So in the integers mod 7, 3 is the square root of 2. Just to startle people I'd go as far as to write

$3 = \sqrt 2$

Like every statement in math, its truth value depends on the context. In the context of the integers, the statement is false. In the context of the integers mod 7, the statement is true.

[By the way what about -3? Well in the integers mod 7, we have -3 = 4. That's because 3 + 4 = 0. Now 4 x 4 = 16 = 2, after we've subtracted off 14. So even in the integers mod 7, it's true that if x is a square root of something then so is -x. That's a general rule that you can deduce just from the laws of basic arithmetic. If you took abstract algebra, we're talking about the ring axioms].

Now that's interesting, but it doesn't solve the problem of having a square root of 2 that also knows about the rationals. But there's a perfectly good number system called $\mathbb Q[\sqrt 2]$ that is:

* A number system where we can add, subtract, multiply, and divide (except by 0); and

* It contains an exact copy of the rational numbers; and

* It contains a square root of 2.

There's no question that such an object exists in mathematics. It has mathematical existence by virtue of the fact that we can (1) characterize it axiomatically; and (2) construct it out of bits and pieces of set-theoretic operations. And even though you don't like set theory we can do the same thing in category theory or homotopy type theory or without any foundation at all simply by writing down the ring axioms and modding out the ring of polynomials having integer coefficients, by the ideal generated by the polynomial $x^2 - 2$. I know you don't like technical stuff by I'm pointing out that I don't need set theory to build a square root of 2.

Now if we are talking about mathematics; and an object has mathematical existence, by what right do you require some other standard of existence?

By the way in the integers mod 5, we have

2 x 2 = 4 = -1.

So the integers mod 5 have a square root of -1.

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

I see your point.

You are saying that there are 2 books and two fish and 2 schools of thought; but there is no 2 in the abstract.

Well, imagining or mentally conjuring up a "thing" that is 2, by itself, is one of the greatest intellectual leaps of humanity. As I've noted before, you appear to reject the very concept of abstraction.

The invention of the concept of number was a great leap forward for mathematics and also for civilization. That let us study 2 + 2 without having to say 2 fish plus 2 fish and then having to re-calculate 2 elephants plus 2 elephants, and then still not being sure about 2 birds plus 2 birds.

It's the power of abstraction that allows us to handle all these cases at once.

You reject abstraction. You're not wrong. It's just a nihilistic philosophy of math and of civilization. Everything about our lives is abstraction. We can't live without abstraction. How do you live without abstraction? How do you function, not believing in numbers?

Now if you want to say, "Yes but you admit numbers aren't real, they're only an abstraction!" I respond: Yes exactly. And traffic lights aren't real either, they're only an abstraction. Law is an abstraction. Government is an abstraction. Science is an abstraction. The Internet is an abstraction. Humans have the power of abstraction. It's how we crawled out of caves and built all this.

Well, in the current theory of the physical world (standard model, or whatever variant of it you prefer) all atoms of the same element are supposed to be EXACTLY the same (indistinguishable, even in principle, with absolute precision), right? — Mephist

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

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

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

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

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

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

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

There's a famous theory of Wheeler and Feynman, "not to be taken seriously" but evidently mathematically possible, that the reason all elecrons are the same is that there is only one electron in the universe. It moves rapidly backward and forward in time; and that's why whenever we see it, it appears to be in a different place. Like a point moving up and down from below a flat plane to above it. Every time it crosses the plane, you'd see an instance of the point. You'd think there are lots of points, when in fact it's only one point traveling perpendicular to your reality.

https://en.wikipedia.org/wiki/One-electron_universe

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

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

The most accurate physical theory in the world, Feynman's quantum electrodynamics, has predicted some quantity or other to 12 decimal places. I read that somewhere. 12 decimal places is pretty good. But mathematically, it's not exact at all. If you had 12 decimal places of pi it wouldn't be pi.

Let me add that modern physics no longer thinks about "particles" like electrons and atoms. Rather, everything is interacting probability waves. An electron isn't a pointlike thingie. An electron is a probability wave, smeared all over the universe. When we observe it, we find that it appears to be in a particular location defined by a probability distribution.

There's even a current thread on this site on that very subject. "Fieldism versus materialism." We don't have particles or things or objects anymore. Just probability waves. Very strange, what the wise physicists are up to lately.

https://thephilosophyforum.com/discussion/7414/modern-realism-fieldism-not-materialism/p1

So the short answer to your question is, no. We can never know or measure an exact distance in the physical universe.

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

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

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

So, in a model of the physical world where all distances have to be multiple of a given fixed length (I don't know if such a model exists, but let's assume this as an hypothesis), there cannot be squares
made of unit lengths. — Mephist

Yes ok. I happen to have visited a world like that once. Manhattan. It's composed of a grid of mutually perpendicular streets and avenues. (Not entirely, but for purposes of discussion).

How far is it from 1st street and 1st Avenue to 2nd street and 2nd avenue? Well it's not $\sqrt 2$, because you can't drive or walk diagonally through the buildings. Rather, the distance is 2. You have to walk one block north and one block west.

There's a name for this: The taxicab metric. In the taxicab metric, the unit circle is a square. Next time some philosopher tells you there are no square circles, you can go, "Oh yeah? There are in the taxicab metric!" and thereby confound him.

But you know I still don't agree with you about squares. Of course there are square blocks in New York City. Actually they're rectangles because the streets are closer together than the avenues, but let's ignore that for sake of discussion. There are square blocks. You just can't walk along the diagonal! Your distance is the sum of your vertical and horizontal travel.

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

I don't know what these unit lengths are made of: they are simply the building blocks of my model, the same as the "strings" of string theory or the "material points" of Newtonian mechanics! — Mephist

Does the taxicab metric help your thinking?

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

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

The ultimate nature of our physical world is wide open to speculation, informed and otherwise. Even our wisest don't know.

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

Try to find M and N such that the sticks arrive at the same point. Since M/N is irrational, you can't do it, — Mephist

All physical measurement is approximate. You can't have a physical stick of length 1. It's not only impossible, it's meaningless. There is no physical apparatus in the world, even in theory, that could do any better than to say that "The length of the stick is 1 +/- .00005 with 99.343% certainty. I'm making up the numbers but that is what the nature of physical measurement is: a number, an error tolerance, and a probability that the true value is within the tolerance.

In the real world you can't measure the diagonal and you can't measure the sides. You can't measure anything with absolute precision. In classical physics, you can't but God can. In quantum physics, even God can't. To clarify that: in classical physics, we can't possibly measure the exact length of a stick, but at least in theory the stick does have a specific length. In quantum physics, a stick has no length at all until we measure it; at which point, the classical problem of the inexactness of physical measurement kicks in.

Well, the "issue" of the irrationality of the diagonal of the square is the one that ancient greeks recognized: you cannot find any unit length that enters both in the side and in the diagonal of the square an integer number of times (no matter how little you take your unit length).

So there cannot exist any fundamental minimal length of physical space — Mephist

No, that does not follow. The irrationality of $\sqrt 2$ is a purely mathematical fact. It tells us nothing about the physical world.

I'm not trying to change your mind about whether or not the mind is algorithmic, I'm just commenting on the progression of technologies-people-think-the-mind-is-like that you mentioned. — Pfhorrest

Ok. I just don't think that when Newton thought of a point moving through continuous space he was thinking, "I could simulate this with a computer." We know he wasn't thinking that.

the main thrust of my point ("everything is computation" is just an evolution of "everything is a machine", — Pfhorrest

I stand by what I wrote and you didn't change my mind. I agree that you can SIMULATE continuous systems with discrete ones, and SIMULATE discrete systems with continuous ones.

This doesn't bear on the contemporariness of the idea that we're computers or the universe is a computer. A computer means something very specific: an algorithm, or "effective procedure," as defined by Turing in 1936. We are not algorithms. But whether we are or aren't, your point -- valid though it may be -- doesn't bear on the question.

Pneumatic digital computers are an actual thing. Voltage is continuous too, that doesn't stop us from using "high voltage" and "low voltage" as discrete states, switching between them, and making digital computers out of that. The same can be done with water, air, basically anything that flows. — Pfhorrest

Right. You can SIMULATE a continuous system with a discrete one. But their fundamental nature is different.

A flow, as conceived by the ancients and also by Newton, is a continuous path. If you imagine a continuous path from 0 to 1, it passes through all the uncountably many real numbers, including the noncomputable ones.

A computable path from 0 to 1, by contrast, can only pass through the computable real numbers, of which there are only countably many. No algorithm can represent or express a noncomputable number.

But we're a little off topic. I expressed an opinion as to why I don't believe in the miraculous contemporariness of the simulation hypothesis. You didn't actually comment on that.

Your solution involves a violation of the fundamental laws of logic, the law of identity (as explained on the other thread), therefore I reject it. — Metaphysician Undercover

At the time, I responded thoughtfully to your ideas. You never once engaged with the points I made. Now weeks later you're still repeating your claims without ever having responded to the points I made. It's not productive to engage with you.

Computers are a kind of machine. A kind of machine that can actually be implemented as flows of water, — Pfhorrest

Conflating the discrete with the continuous; the computable with the noncomputable.

I haven't read the thread but saw the title and just wanted to put in my two cents.

Postmodernists observe that logic has often been used to suppress truth and oppress people. Who hasn't had the experience of some authority figure or bureaucracy using logic and rationality to do you some moral wrong? "We're only following the rules."

We live in the midst of populist revolutions going on in many countries. The wise elites who supposedly "know what's best" have been mucking things up badly, and people are starting to notice.

Logic is no guide to truth. Logic only tells you which conclusions follow from the given premises. The question is: Who controls the premises?

One argument that I find compelling is a meta-argument, It's the currency, or the contemporariness, or the topicality of the simulation hypothesis that I find suspect.

In ancient Rome they had great waterworks. They thought about the mind in terms of flow. I've read that the word nous, for soul, and pneumatic, come from the same root, from the idea that mind is a flow.

After Newton we thought the universe was a machine.

And now that we live in the age of computation, we think the mind is an algorithm.

What are the odds that we, of all the generations that have ever lived and that ever will live, are privileged to be the ones to discover the ultimate nature of the mind and of the universe?

Slim and none, I'd say.

Why should the world or the mind be an algorithm? The first thing Turing discovered after he defined computation, is that there are naturally stated problems whose solution cannot be computed, even in principle, even with arbitrary amounts of computational power.

I am one who believes that the mathematical phenomenon of noncomputability is the key to the next revolution in physics. I don't think you can explain the world with algorithms. And again ... what a coincidence that would be if it turned out to be true. One day the Web explodes into mainstream culture, and the next day everyone decides, "Oh yeah we must be algorithms too."

I am not a believer.

So, combining "metaphysical" with "actual" means someone is thinking a metaphysical thought? Or does the expression imply an interaction with physical reality? I am going on a classical definition of the expression. What do you really mean? Please clarify with examples. Thanks. — jgill

I didn't understand a word @aletheist wrote but I was embarrassed to admit it.

I do not understand what metaphysical actuality is. Are the natural numbers metaphysically actual? The completed set of natural numbers? The square root of 2? Chaitin's constant, which is known to be noncomputable? Is a brick metaphysically actual? How about an electron? A quark? A string?

Since I have found you to be a normally clear-headed and insightful participant here, I fear that your persistence in dealing with Metaphysician Undercover lately may be producing some unfortunate side effects. — aletheist

LOLOL.

perhaps you misread my previous post. — aletheist

I think I still don't know what actuality means.

But you said that potential infinity has metaphysical actuality. Don't the natural numbers (as modeled by the Peano axioms) contradict that?

In that vein, do you recognize that there's a conceptual distinction between an "actual infinity" and a "potential infinity"?
— Relativist
Yes, it corresponds to the difference between metaphysical actuality and logical possibility. Again, mathematical existence refers to the latter, not the former. — aletheist

I wonder if I understand that. Potential infinity is often taken to be the collection (but not set) of the natural numbers 0, 1, 2, 3, ... It's potential in the sense that given n you automatically have n+1; but you never have all of them taken together at once in a set.

But how is that metaphysical actuality? There's nothing in the physical world that corresponds to the endless sequence of natural numbers. It seems to me that 0, 1, 2, 3, ... and {0, 1, 2, 3, ...} are equally abstract. One can accept or reject the axiom of infinity; but either way you end up with a structure whose "actually" is certainly in question. Yes? No? I have no idea.

You would not remember anything so it will feel like living a whole new life. — Devans99

Ah, that's a good feature. Solves the problem.

How though? It's not like your memories will transfer between lives. You won't notice you're living the same life for the 20 billionth time. If your memories trasferred then you're by definition living different lives — khaled

Ok! No memory between lives. I get it.