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

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Ontology is the study of existence. Isn't it?
No, it is the study of being, which is not necessarily synonymous with existence. For example, one view is that ontological existence (i.e., actuality) is a subset of reality (which also encompasses some possibilities and some necessities), which is a subset of being (which also encompasses fictions).

How could there be a form of existence which isn't ontological existence?
By defining "existence" in another context-specific way, obviously. There are plenty of other terms that mean something different in mathematics than in metaphysics or in other sciences.

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.
That is not just pragmatism, it is the scientific method. How else would you propose that we evaluate our hypotheses to ascertain whether they accurately represent reality?

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.
As I have explained to you several times now, no one except a platonist would claim that mathematical existence conforms to "the rigorous philosophical definition" of (ontological) existence. Everyone else understands this, so please stop belaboring your terminological objection.

I interpret this as your "epistemic stance" requires Platonic realism as a support, a foundation.
That is a misinterpretation, and you know it by now.
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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".

Yea, but I was speaking about a way to approximate space-time with discrete pieces to make computer simulations, not of the real equations. The real equations are partial differential equations defined on a continuous 4-dimensional space.

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

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

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

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

But I am afraid that's all what physics (at least contemporary physics) does: prediction. Nothing else!
Nobody knows how to make sense of the equations of quantum mechanics: physicists learned how to use them to predict the results of experiments. Maybe that is a problem, but it is a problem of physics since the beginning: Newton didn't know how to make sense of a "force" that acts from thousands of kilometers of distance.
I heard somebody say that now it's clear: everything is filled with a "field", and it's the exchange of particles of that field that transports the force. So, can you try to imagine how to generate an attractive force by exchanging an object?
The reality is that there are equations that work, and you can apply a mathematical theory made of imaginary things with imaginary rules that happen to give the right results. The real "ontological" reason why this system is able to "emulate" the experiments of the real world, nobody is able to explain. And it's not only about the use the square root of 2.
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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.

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

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

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.
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[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.

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.

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.

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?

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.

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?

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.
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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.

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

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,

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

but there are exact predictions in QM.

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).

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 )

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.

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.

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.

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)

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.
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, it is the study of being, which is not necessarily synonymous with existence. For example, one view is that ontological existence (i.e., actuality) is a subset of reality (which also encompasses some possibilities and some necessities), which is a subset of being (which also encompasses fictions).

OK, I'll assume for the sake of argument that there is a type of existence, "mathematical existence", which is a different type of existence from "ontological existence". I'll assume two different types of existing substance, like substance dualism.

By defining "existence" in another context-specific way, obviously. There are plenty of other terms that mean something different in mathematics than in metaphysics or in other sciences.

So can you tell me what fishfry didn't seem to be able to tell me. How would I define "mathematical existence"? Do all fictional things (like fishfry's example) have mathematical existence, or is it only mathematical fictions which have mathematical existence?

That is not just pragmatism, it is the scientific method. How else would you propose that we evaluate our hypotheses to ascertain whether they accurately represent reality?

That should be obvious to you, we ought to evaluate through the criteria of truth and falsity. Do you not see a difference between "accurately represent reality", and "facilitate prediction"? A significant aspect of the "scientific method" involves "observation", and observation is meant to be objective. The goal of "prediction" introduces a bias into observation.

As I have explained to you several times now, no one except a platonist would claim that mathematical existence conforms to "the rigorous philosophical definition" of (ontological) existence. Everyone else understands this, so please stop belaboring your terminological objection.

I don't understand what you are saying. How do you propose that "mathematical objects" could have existence, except through some form of Platonic realism? It's very clear, that imaginary, or fictitious objects, do not have existence as objects. If you want to assign some sort of existence to imaginary, or fictitious scenarios, it would be rather strange to say that these exist as objects.

That is a misinterpretation, and you know it by now.

I've never seen the existence of "mathematical objects" justified by any ontology other than Platonism. So there is no "misinterpretation". If you think that you can justify the existence of these so-called objects in some way other than Platonism, then I'd really appreciate the demonstration. I've actually been looking for this for years, to no avail. And, since I do not agree with the principles of Platonic realism, I've come to the conclusion that abstractions are not existent objects.

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!

But the issue is that they are not categorically distinct, as if one refers to a physical object and the other a mathematical object. It's different forms of the same thing.

Special relativity allows us to represent time as discontinuous?? Why?

Special relativity is based in a discontinuous representation of time. Each frame of reference has a unique, and discrete "time" of its own. Therefore "time" in general consists of all these different unique and particular, "times", making it discontinuous.

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

But that "rotation" is the faulty geometry we've been discussing. The circle is not real, pi is not real. Nor is the so called "continuity" assumed by this geometry real, because the idea that we can put a point anywhere in space is faulty. So general relativity has a double mistake. It starts from a discontinuous time (which ought to be represented as continuous), then it applies to this discontinuity the principles of a continuous space (which ought to be represented as discontinuous), to create the illusion of a continuous "space-time". In reality the concept of space-time consists of a faulty representation of time, with an attempt to fix it using a faulty representation of space.

Do you agree that the volume is an attribute of an object?

The "volume" is what we assign to the thing, a judgement we make based on our principles of measurement. So we cannot really say that it is something intrinsic to the object, it's a judgement, and that's why we have different standards to measure volume.

I know, this explanation is not very "philosophical"... and, to say the truth, I don't really understand why is this such a philosophical problem :yikes: But what's the problem with this interpretation?

Saying that we can assign the same volume to two distinct things, or that we can assign "5" to two distinct groups of objects, is really no different from saying that we can call two distinct things "a chair". This does not mean that there is an immaterial object which "chair" refers to, nor is there an object that "5" refers to, or an object which "volume" refers to.

But I think you are on the right track here, towards understanding what mathematics is incapable of helping us with, and that is description. Take your example of the two different pictures, where we use "5" to describe what's in the picture. From looking at one picture, and not knowing what five means, no one could ever know which aspect of the picture 5 is describing. So we look at two, and compare similarities, and perhaps deduce what "five" means. If "5" only gets meaning from referring to two or more different pictures, how is it describing anything in any of the pictures?

But I am afraid that's all what physics (at least contemporary physics) does: prediction. Nothing else!

Do you not see this as misguided, and not proper science? If the goal of science is to predict, and not knowledge, understanding, then observations will be made under that bias, therefore not truly objective. The observations will be made with the goal of being able to predict, rather than the goal of knowledge and understanding. As I explained in the other post, knowledge consists of a lot more than just being able to predict. Thales predicted the solar eclipse with very little knowledge of the solar system, and that capacity to predict did not give him a comprehensive understanding of the solar system.

Maybe that is a problem, but it is a problem of physics since the beginning: Newton didn't know how to make sense of a "force" that acts from thousands of kilometers of distance.

Perhaps this problem of pragmatism has always pervaded science to some exist, but I would argue that it has gotten much worse. There is a difference between establishing a principle, with the goal of applying that principle toward further understanding, and establishing a principle for the purpose of predicting.

The reality is that there are equations that work, and you can apply a mathematical theory made of imaginary things with imaginary rules that happen to give the right results. The real "ontological" reason why this system is able to "emulate" the experiments of the real world, nobody is able to explain. And it's not only about the use the square root of 2.

This is the same illusion propagated by the sophists of ancient Greece. The equations work, Thales predicted the eclipse. Nobody was able to explain why they work, the "real ontological reason". But that doesn't mean they all should have buried their heads in the sand, and not proceeded to investigate the real ontological reason. There's a matter of completion. Equations that work for prediction is only a part of a complete understanding.
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I believe you do not have a very thorough education in philosophy, or you would not characterize "abstraction" in this way.

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.

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).

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.

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.

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.

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,

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.

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.

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.

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.

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.

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.

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.
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@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.
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OK, I'll assume for the sake of argument that there is a type of existence, "mathematical existence", which is a different type of existence from "ontological existence". I'll assume two different types of existing substance, like substance dualism.
Wow, this keeps getting more and more ridiculous. No one is claiming that mathematical existence has anything to do with "existing substance." In mathematics--again, except for platonism--the term "existence" does not imply anything ontological whatsoever.

How would I define "mathematical existence"?
Something exists mathematically if it is logically possible in accordance with an established set of definitions and axioms. The natural numbers, integers, rational numbers, real numbers (including the square root of two), and complex numbers all exist mathematically, in this context-specific sense.

A significant aspect of the "scientific method" involves "observation", and observation is meant to be objective. The goal of "prediction" introduces a bias into observation.
Nonsense, prediction is just as much a significant aspect of the scientific method as observation. Why do we have theories? How do we come up with them? Our observations prompt us to formulate hypotheses that would explain them; this is retroduction (sometimes called abduction). We make predictions of what else we would observe, if those hypotheses were correct; this is deduction. We then conduct experiments to determine whether our predictions are corroborated or falsified; this is induction.

... I've come to the conclusion that abstractions are not existent objects.
One more time: No one is claiming otherwise.
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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?

Using abstractions (concepts) is not the same things as the act of abstraction. To conflate these two is equivocation. To define abstraction as "giving a name to something you can't kick with your foot" is woefully inadequate, for someone accusing me of having no knowledge of how "abstraction works".

Abstraction is the act which creates that supposed thing which you cannot kick with your foot. And, you cannot create something by giving a name to nothing. Therefore the "something" which is created by abstraction, must exist prior to the act of giving it a name. The name is what you kick around (with logic), but "abstraction" (verb) refers to the creation of that thing which the name "abstraction" (noun) signifies.

But let's try it the other way around. Let's assume that we make a name which refers to nothing, and manipulating that name, with logical processes (kicking it around), creates a thing which that name represents. Let's say that this is the act of "abstraction", we take a name which refers to nothing, we manipulate that name using logical processes until the name refers to something, and this creates a concept, an abstraction. If this is the case, then all we have done is given meaning to the name. There never was anything which the name referred to, and there still isn't anything which the name refers to, but kicking the name around with logical proceedings has given the name meaning.

Either way you look at it, it's false to assume that we can create something which the name refers to, by starting with a name which refers to nothing. Either there is something immaterial there (Platonic object), which the name refers to from the start, or there is nothing which the name refers to, but the name is given meaning through use. These are two distinct ways of looking at this issue. To confuse these two, and say that we start with a name which refers to nothing, and then by using the name we create "something", and this something is an immaterial object, is to say that we create something from nothing. The point being that if the name starts out as referring to nothing, and logical processes are applied, it must end up as referring to nothing, or the logic would be invalid. therefore the symbol must have meaning from the time it is presented.

In college you look at derivatives and integrals, more abstractions.

See, you're using "abstraction" as a noun here. Please do not equivocate in your demonstrations. If a name refers to something (has a definition) that definition must be upheld or else the logic is invalid due to equivocation.

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

Sure, your "casual definition" allows you to equivocate. Without equivocation you'd have no argument. Therefore you see no reason to abandon your casual definition.

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.

Are you familiar with Socrates? You have precisely described Socrates' MO. The person who knows how to do something, does not necessarily know what is being done. Such is my example of a 3,4,5 triangle. I know numerous people who can construct a right angle using the 3,4,5, formula, who have never even heard of the Pythagorean theorem. We can do the thing which the theory describes without knowing the theory. The pizza chef can very easily be making very good pizzas using marinara sauce, without knowing how to make marinara sauce, or even knowing the ingredients of the sauce. The mathematician, engineer, or physicist is most often using abstractions (concepts) without knowing what "an abstraction" is, because this is only studied in the field of philosophy. You should allow the truth of this matter. The philosopher who is trained in this, is much more likely to know what "an abstraction" is than the mathematician who uses abstractions, just like the person who is trained in making marinara sauce is much more likely to know what marinara sauce is than the person who makes pizza.

When a person (such as a mathematician) who doesn't know what abstraction is, not being trained in philosophy, starts to produce logical arguments based in unsound premises concerning the nature of "abstractions", this is called "sophistry". I call them mathemagicians, because their most famous trick is to make (mathematical) objects appear from nothing.

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.

Well, it's very clear, that it's a deficiency in spatial representation, just like the irrational nature of pi indicates a deficiency in that spatial representation. You are simply in denial. And as I said, covering up these deficiencies with complicated mathematics doesn't make them go away. That is where your denial leads you astray. Since you deny that these irrational numbers are the manifestation of a faulty spatial representation, you produce extremely complex numerical structures in a sophistic effort to cover up the truth of this fundamental flaw.

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.

You don't seem to understand the criticism. Algebra makes the same mistake as set theory, assuming that a symbol represents an object. Using algebra instead of set theory doesn't get you past my objection. From this premise, the square root of two is not a problem at all. We have a symbol, √2 it represents an object, and the problem is solved. The real question though is whether the "object" supposedly represented by √2 is a valid object. If you assume as a premise, that every symbol represents an object, then of course it is. But then that premise must be demonstrated as sound.

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

Go ahead, but no algebra or other faulty premises.

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.

My OED: symbol, "a thing conventionally regarded as typifying, representing, or recalling something..."

If your symbol does not represent something, it simply "behaves" in a particular way, then the symbol simply has meaning, due to its behaviour. We can say that it is used as "recalling something". In this way, the symbol is the thing, as you say, and as I described above. But if we define "symbol" in this way, then we cannot use algebra or set theory, which require that a symbol represents something. We'd have equivocation.

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 can say that the symbol stands for whatever you want. It is "the symbol stands for nothing" which is problematic.

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.

Do you understand that predication is made of a subject, not an object. Whether or not there is a named object is irrelevant to the act of predication. There is no problem making predications of fictional characters, these characters are known as subjects. The problem is in claiming that the fictional character is an existent object.

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.

I will grant you the existence of symbols, and you define what the symbols mean.

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.

I have no idea what the "snaggle problem" is, but if you ask me to use symbols which represent nothing, then we must dismiss any mathematical premise which assumes that a symbol represents an object. Even if we allow that a symbol may or may not represent an object, we must dismiss such premises. If we allow that a symbol has meaning, rather than that it is representative of an object, then that meaning must be defined, or else the symbol will be dismissed as having no meaning and irrelevant.

Here's the issue I see. We can get past "the symbol must have meaning attached", by assuming that the symbol represents an object, and the object represented is unknown. However, if the object represented is unknown, then we also cannot know whether the symbol actually represents an object or not. Then we might say that a symbol may or may not represent an object, but that premise is useless as an epistemological principle.

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.

Are you confessing to the use of equivocation in mathematics?

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.

Symbols are passive entities, objects. The logician manipulates, moves the symbol. The "behaviour" of the symbol is a direct result of, a representation of, the logician's actions, so what is described here is the nature of the rules, and whether the logician follows the rules. Therefore "a symbol system that behaves exactly as numbers should", expresses nothing more than a judgement of the rules of the system. What is left unknown, is "as numbers should", and this might be an arbitrary criterion.

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.

Right, but there is something here, which the name has been given to. We do not assign the name to nothing, unless it is stated that this name signifies nothing. If it is stated that the name signifies nothing, it cannot change and evolve towards signifying something, that would break the stated rule. If, instead, we allow that the name has meaning, instead of representing something, then we can allow that logic enables the meaning to "grow". It cannot contradict the earlier meaning, just expand on that. But if this is the case, then the name must have some meaning from the beginning, and a symbol with no meaning is invalid, as providing nothing to grow.

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.

Clearly, you premise that symbols have meaning, and that the meaning may expand and grow. If this is truly what you believe, how can you make this consistent with the premises of algebra and set theory which dictate that a symbol represents a thing?

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.

What I reject is inconsistency and contradictory premises, not "all of this".

Wow, this keeps getting more and more ridiculous. No one is claiming that mathematical existence has anything to do with "existing substance." In mathematics--again, except for platonism--the term "existence" does not imply anything ontological whatsoever.

Any claim of "existence" is validated (substantiated) with substance. If you think that there is a type of existence which is not substantial then please explain.

Something exists mathematically if it is logically possible in accordance with an established set of definitions and axioms. The natural numbers, integers, rational numbers, real numbers (including the square root of two), and complex numbers all exist mathematically, in this context-specific sense.

Sorry, but "possible" does not necessitate existence. Do you not recognize that "possible" refers to what may or may not be, so it is contradictory to say that possible things are existing things.

Nonsense, prediction is just as much a significant aspect of the scientific method as observation.

I didn't deny that. Both are essential parts of the scientific method. What I deny is that prediction is the goal of the scientific method. It is simply a part of it. Pragmatists allow that prediction is the goal of science.

One more time: No one is claiming otherwise.

Then how do you explain set theory, which proceeds from the assumption that a mathematical symbol represents a mathematical object? It's easy to assert "no one is claiming that an abstraction is an existent object", yet everyone backs up set theory which clearly assumes that the abstraction which a symbol represents, is an object. If these objects are not supposed to be "existent", then why attempt to back up their existence with the idea of "mathematical existence"? Due to this behaviour of yours, I can find nothing else to say other than you are boldly lying.
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Any claim of "existence" is validated (substantiated) with substance.
Perhaps in metaphysics/ontology, but definitely not in mathematics.

If you think that there is a type of existence which is not substantial then please explain.
I have already done so, repeatedly.

Do you not recognize that "possible" refers to what may or may not be, so it is contradictory to say that possible things are existing things.
Perhaps in metaphysics/ontology, but definitely not in mathematics.

What I deny is that prediction is the goal of the scientific method.
I never claimed that it is. Prediction enables us to evaluate whether our hypotheses hold up to further experimental and observational scrutiny. The goal is knowledge, which consists of beliefs (i.e., habits) that would never be confounded by subsequent experience.

It's easy to assert "no one is claiming that an abstraction is an existent object", yet everyone backs up set theory which clearly assumes that the abstraction which a symbol represents, is an object.
You keep imposing your peculiar metaphysical terminology, as if everyone else is obliged to conform to it regardless of the context. In this case, you seem to be insisting that only an ontological existent can be the object of a symbol. In mathematics, and even in ordinary language, an abstraction can also be the object of a symbol, as long as the universe of discourse is established. The objects of the names "Pequod" and "Ahab" are a boat and its captain in the fictional world of Melville's novel. The object of the word "unicorn" is a horse-like animal with one horn; the fact that no such animal exists in the ontological sense does not preclude the word from having an object at all.

Due to this behaviour of yours, I can find nothing else to say other than you are boldly lying.
Seriously? Due to this behavior of yours, I can find nothing else to say other than you are boldly ignorant (of mathematics and its terminology) and stubborn (about your rigid definitions).
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Algebra makes the same mistake as set theory, assuming that a symbol represents an object.

Alright man. It's not set theory you object to, it's 10th grade algebra. It's not abstraction you object to, it's the very concept of using the symbol '2'.

I simply can't argue with such a nihilistic position. You claim that one must know and understand the referent of a symbol before being allowed to use that symbol. That flies in the face of the entire history of science. And nobody likes flies in their face. I bid you adieu.
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Go ahead, but no algebra or other faulty premises.
How is algebra faulty? That is, algebra is a tool. Far as I know, and in my limited experience, it is a tool that works and does and accomplishes its proper tasks. But you say no. Make your case.

I'm not looking for arcane nonsense. The sense I am interested in is analogous to your saying that knives don't cut. I have knives and used as knives, they cut. Algebra, used as algebra, "cuts." So in implying that algebra is faulty, in what sense of its proper use, when used properly, does it not "cut"?
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But if we define "symbol" in this way, then we cannot use algebra or set theory, which require that a symbol represents something.
I am trying to honestly understand, but why do you propose that sets should only include apriori existing entities, and not ones defined by the processes of inference and computation themselves. That is - logic is an algorithm and our application of that algorithm manifests the imperatives in the axiomatic system. The algorithm is inaccurate in almost all practical cases, and therefore is not exactly representative of apriori existing objects.

The real question though is whether the "object" supposedly represented by √2 is a valid object. If you assume as a premise, that every symbol represents an object, then of course it is. But then that premise must be demonstrated as sound.
Let me try. Suppose that people have to compute the ratio between the lengths of the sides and the diagonal of an object that approximates a square, but lives in some unknown tessellation of space. You are not informed of the structure of the tessellation apriori and you know that the effort for its complete description before computation is prohibitive. You do however understand that the tessellation is vaguely uniform in size and has no preferred "orientation" or repeating patterns. It is random in some sense, except for the grain uniformity. The effective lengths for the purpose of the computation are the number of cells/regions that the respective segment divides. You also know that the length will be required within precision coarser then the grain of the tessellation/partitioning itself. With this information in mind, you want practical algorithm for the computation of the ratio between the sides and diagonal of a square, to an unknown precision, which is greater then the grain of the tessellation.

Now, honestly speaking, I cannot prove that the most effective generalized answer should be square root 2. But this is the kind of problem that an engineer would face. And if not theoretically, they could confirm empirically if square root 2 is a good general approximation. What would you advise? And when analyzing your algorithm, what object would you introduce to compare its convergence to other algorithms?
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Perhaps in metaphysics/ontology, but definitely not in mathematics.

Yes, I agree, in mathematics some people make the unsubstantiated claim that the symbols represent existent objects. This is called Platonic realism

You keep imposing your peculiar metaphysical terminology, as if everyone else is obliged to conform to it regardless of the context. In this case, you seem to be insisting that only an ontological existent can be the object of a symbol. In mathematics, and even in ordinary language, an abstraction can also be the object of a symbol, as long as the universe of discourse is established.

That's right, in Platonic realism the abstraction is an object, and it is believed to exist as an object. In the "universe of discourse" called "Platonism", an abstraction is an object. You seem to believe that there is some other form of ontology, some other universe of discourse, which allows that abstractions have "mathematical existence", as objects, which is not Platonism. So I am waiting for you to produce the principles which distinguish this universe of discourse from Platonism. All you have done is stated Platonist principles and lied in asserting that no one is assuming Platonism.

Alright man. It's not set theory you object to, it's 10th grade algebra. It's not abstraction you object to, it's the very concept of using the symbol '2'.

I don't object to using the symbol "2". But like any other language I might use, I want to know what the symbols are being used for. If you assert that the symbol "2" represents an object, I want a clear description of that object, so that I can recognize it when I apprehend it, and use the symbol correctly. If you are simply claiming that the symbol represents an object when you know full well that it doesn't, then you are engaged in deception.

How is algebra faulty?

I went through this already, it assumes that the symbol represents an object. This is Platonic realism which is a faulty ontology.

Far as I know, and in my limited experience, it is a tool that works and does and accomplishes its proper tasks. But you say no. Make your case.

It's very clear that it works, I never disputed this fact. However, "works", and "it is designed to help us determine theh truth" are two distinct things. That is the problem with pragmatism, if the purpose is anything other than to bring us truth, then the premises employed will reflect that other purpose instead of the goal of truth. "It works" has no necessary relationship with truth, as deception clearly demonstrates.

I'm not looking for arcane nonsense. The sense I am interested in is analogous to your saying that knives don't cut. I have knives and used as knives, they cut. Algebra, used as algebra, "cuts." So in implying that algebra is faulty, in what sense of its proper use, when used properly, does it not "cut"?

If it is being used as a system of logic employed toward determining the truth, it is faulty because it has a false premise. Platonic realism is false. Mathematical symbols do not represent objects. Aletheist seems to believe that the existence of mathematical objects can be supported by something other than Platonism, something called "mathematical existence". Perhaps you can assist and demonstrate how mathematical symbols represent some sort of objects which are other than the objects assumed by Platonic realism. As I told aletheist, I would be highly interested in this new ontology.

I am trying to honestly understand, but why do you propose that sets should only include apriori existing entities, and not ones defined by the processes of inference and computation themselves. That is - logic is an algorithm and our application of that algorithm manifests the imperatives in the axiomatic system. The algorithm is inaccurate in almost all practical cases, and therefore is not exactly representative of apriori existing objects.

Either the symbols represent objects or they do not. We might say that they may or may not represent objects, but then we would need to know whether or not they do, in order for the symbols to be useful. If we assume that there is an object represented by the symbol then the symbol is useful. But if there really is not an object represented by the symbol and we use it under the assumption that there is, then the use is deceptive.

Sorry simeonz, but your example seems to be lost on me. The question was whether whatever it is which is represented by √2 can be properly called "an object". You seem to have turned this around to show how there can be an object which represents √2, but that's not the question. The question is whether √2 represents an object.
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Yes, I agree, in mathematics some people make the unsubstantiated claim that the symbols represent existent objects. This is called Platonic realism
Indeed, that would be mathematical platonism, as I have acknowledged. However, I am not a mathematical platonist--I have quite explicitly denied that the symbols represent existent objects in the ontological sense.

You seem to believe that there is some other form of ontology, some other universe of discourse, which allows that abstractions have "mathematical existence", as objects, which is not Platonism.
Platonism is by no means the only philosophy of mathematics that employs the well-established term "existence" when referring to abstract objects. As I have clearly and repeatedly stated, for those of us who are not mathematical platonists, ontology has nothing whatsoever to do with the "existence" of such objects.

All you have done is stated Platonist principles and lied in asserting that no one is assuming Platonism.
All you have done is obtusely stuck to your rigid terminology, refusing to pay any heed to the multiple explanations that I and others have offered to correct your evidently willful misunderstanding. I see no point in wasting my time any further.
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I want to know what the symbols are being used for. If you assert that the symbol "2" represents an object, I want a clear description of that object, so that I can recognize it when I apprehend it, and use the symbol correctly.

You reject science. In science we DON'T know what something is. So we give it a symbolic name, write down the symbol's properties, and reason about it in order to learn about nature.

When Newton wrote $F = ma$ those were made up terms. Nobody knew (or knows!) exactly what force or mass is. Acceleration's not hard to define. But even then Newton had to invent calculus to define acceleration as the second derivative of the position function.

You reject all that.

Nihilism.
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Platonism is by no means the only philosophy of mathematics that employs the well-established term "existence" when referring to abstract objects. As I have clearly and repeatedly stated, for those of us who are not mathematical platonists, ontology has nothing whatsoever to do with the "existence" of such objects.

Yes, you can state that all you want, assert and insist until the cows come home, and then continue to assert some more. The challenge is yours, describe how abstractions, concepts exist as "objects", without invoking Platonism. To call the abstraction an "object" is already using a Platonic term. How are you going to show that an abstraction is an object, in any sense other than a Platonic object.

If we say that abstractions, and conceptions "exist", I have no problem with this. I very much agree that they exist. But "object" refers to a very specific type of thing, a unique individual, a particular, having an identity as described by the law of identity. Because an abstraction is not this type of thing, an object, mathematical axioms which assume that abstractions are objects, are simply wrong.

You reject science. In science we DON'T know what something is, so we give it a symbolic name, write down the symbol's properties, and reason about it in order to learn about nature.

I'm not rejecting science. When we name something which we don't know what it is, the name still represents a thing, it's just the case that we didn't know "what" that thing was at that time so we name it.. Once it is named, we know what it is, the thing called by that name. This is not the same as having a symbol which we do not know whether it refers to a thing or not.

So if you have a symbol "2", and you have apprehended a thing and assigned that symbol to this thing, then tell me something about this thing, so that I may use the symbol correctly, to refer to the thing named by that symbol.. It doesn't make sense that you would have assigned the symbol "2" to something and you know absolutely nothing about this thing which you have assigned the symbol to.

When Newton wrote F=maF=ma those were made up terms. Nobody knew (or knows!) exactly what force or mass is. Acceleration's not hard to define. But even then Newton had to invent calculus to define acceleration as the second derivative of the position function.

You reject all that.

Nihilism.

Force, mass, and acceleration are not things. They are not objects. They are concepts which describe properties. Properties are not things, and that's why you cannot point to the things which are referred to by these terms. You might provide a definition of the term, but having a definition does not make the term refer to an object. We might call it a logical subject then, the definition being what is predicated of the subject.

That I reject the notion that properties which are described by concepts like "force" "mass" and "acceleration" are themselves objects, doesn't make me nihilist. It just means that I understand the difference between an object and a logical subject.
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But "object" refers to a very specific type of thing, a unique individual, a particular, having an identity as described by the law of identity.
Again, your peculiar metaphysical terminology is not binding on the rest of us.

That I reject the notion that properties which are described by concepts like "force" "mass" and "acceleration" are themselves objects, doesn't make me nihilist. It just means that I understand the difference between an object and a logical subject.
Apparently not--an object is whatever a logical subject denotes, which can be an abstraction or a concrete existent.
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Again, your peculiar metaphysical terminology is not binding on the rest of us.

The challenge is open. All you do is assert without any justification. Where is your demonstration of an abstraction existing as an object, which is not a demonstration of Platonism?.

Apparently not--an object is whatever a logical subject denotes, which can be an abstraction or a concrete existent.

Sorry if you misunderstood, but I was talking about what is represented by the symbol in logic, and that is a subject, not an object. Whether the subject denotes an object, a number of objects, or no object at all, is irrelevant to the point.
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Cantor's theorem. |X|<|P(X)||X|<|P(X)|. This is a theorem of ZF, so it applies even in a countable model of the reals. You mentioned Skolem the other day so maybe that's what you mean. Such a model is countable from the outside but uncountable from the inside.

OK I'll stop arguing about intuitionism. But I think you didn't get my point here, so let me try one last time:
Cantor's theorem is valid in intuitionistic logic, but we know that intuitionistic real numbers are countable. In fact the theorem says: forall countable lists, there is an element that is not in the list, and we know that the set of elements missing from the list is countable because the list of all strings is countable.
Now you read the same theorem in ZFC and you interpret it as "there is an uncountable set of elements missing from the list". How do you know that the set of missing elements is uncountable? I mean: the symbolic expression of the theorem is the same, and the interpretation of the symbols is the same. How can you express the term "an uncountable set" in a language containing only the quantifiers "forall" and "there exists one" ?
And if there is no uncountable set of missing real numbers, there are no holes to fill..

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

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

For the first part of the question, I guess your question is how do you say "a finite random sequence" in intuitionistic logic. You can't! (at the same way as you can't do it in ZFC: the axiom of choice does not say "random" function). If the sequence is finite it is always computable, so you can say "there exist a finite sequence of numbers" ( the same as in ZFC ).
There is a definition of randomness as "a sequence that is not generated by a program shorter than the sequence itself" (lots of details missing, but you can find it on the web), but this is about the information content and not about the process used to choose the elements of the sequence.
About the bitstrings that aren't computable: all finite bitstrings are computable of course. So probably you mean the bitstreams that contain an infinite amount of information (not obtainable as the output of a finite program). There is no way to prove that such strings exist using a formal logic system (even using ZFC): we can interpret the meaning of Cantor's theorem in that way, and maybe there is such a thing in nature, but you cannot prove it with a finite deterministic formal logic system.
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If it is being used as a system of logic employed toward determining the truth, it is faulty because it has a false premise.
I have so many pennies in this hand, that many in that hand. How many do I have in all. If X is my left hand and Y is my right hand, then I have X+Y pennies. That's the truth of it, in the sense that X+Y = (X+Y), and the fact of it as expressed by the pennies themselves. So it appears that in appropriate use, algebra yields both truth and fact. Refute or yield. Keep in mind your statements are categorical and not conditional. Your refutation therefore should be on the same terms.
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Sorry simeonz, but your example seems to be lost on me. The question was whether whatever it is which is represented by √2 can be properly called "an object". You seem to have turned this around to show how there can be an object which represents √2, but that's not the question. The question is whether √2 represents an object.
Probably I don't understand the point of the conversation. But just to be clear. The space tessellation/partitioning was not to show how one mathematical construction can be derived from another. The tessellation corresponded to some unspecified physical roughness with uniformly spaced, but irregularly situated constituents. It aimed to illustrate that solving world-space problems imperfectly (due to efficiency constraints) results in the adoption of a modus operandi solution, whose own structure exists only in concept-space.

For analogy, similar utilitarian interpretations exist for probability and statistics. Most classical (pre-QM) applications of probability are not related to genuine physical indeterminacy, but to making decisions with imperfect knowledge of the conditions. The choices are made according to some sense of overall utility, independent of the true and objectively predictable, but unknown individual outcomes (Pignistic probability). In other words, we conceptualized indeterminacy, not because of its objective existence (aleatoric uncertainty), but due to our lack of specific knowledge in many circumstances and because introducing indeterminacy as a model was the most fitting solution to our problems.

I believe that all mathematics have utilitarian sense to them. Square root 2 is object of thought and human decision making, not of some independent world-state. Its existence may have physical grounds, but doesn't have to be perfectly accurate. It is put forward to deal with solving problems through computation. At least this is my way of thinking.

As I said, I may misunderstand the topic of the discussion altogether, which is fine. But just wanted to be sure that the intention of my example was clear. (That is - that the space tessellation was not intended as a mathematical structure, but as representation of some unknown coarseness of the physical structure, being ignored for efficiency reasons.)
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Where is your demonstration of an abstraction existing as an object, which is not a demonstration of Platonism?
Again, I do not hold than there is such a thing as "an abstraction existing as an object." I reject your peculiar terminological stipulation that an "object" can only be something that ontologically exists.

... I was talking about what is represented by the symbol in logic, and that is a subject, not an object.
No, a symbol in logic is itself either a subject or the predicate within a proposition. If it is a subject, then it denotes an object, which can be an abstraction or an existent. If it is the predicate, then it signifies the interpretant, which is a relation among the objects denoted by the subjects.
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I have so many pennies in this hand, that many in that hand. How many do I have in all. If X is my left hand and Y is my right hand, then I have X+Y pennies.

This is not a demonstration of algebra. So your example is irrelevant. Refutation complete.

What I dispute is that a symbol represents a number which is an object. In your example, X represents how many pennies in your left hand, and Y represents how many pennies in your right hand. Through abstraction we might reduce this to simple numbers, "6" and "8" for example. But those abstracted numbers is not what X and Y represent, as stated in your example. The numbers could only be produced through an abstraction from what is stated.as represented. You did not even state any numbers.

It aimed to illustrate that solving world-space problems imperfectly (due to efficiency constraints) results in the adoption of a modus operandi solution, whose own structure exists only in concept-space.

The problem I've been discussing is that whatever it is which is expressed as "a square" does not actually exist in "concept-space" because the perpendicular sides are incommensurable. There is a deficiency in the concept which makes it impossible that there is a diagonal line between the two opposing corners, when there is supposed to be according to theory. The figure is impossible, just like the irrational nature of pi tells us that a circle is impossible.

This impossibility is not a case of us not being able to do in practise what can be done in theory, due to a lack of precision. It is the opposite of this, it is a defect inherent within the theory. The figures defined by the theory are impossible, according to the theory, just like a square circle is impossible.

In other words, we conceptualized indeterminacy, not because of its objective existence (aleatoric uncertainty), but due to our lack of specific knowledge in many circumstances and because introducing indeterminacy as a model was the most fitting solution to our problems.

The problem in this situation is that the indeterminacy is created by the deficient theory. It is not some sort of indeterminacy which is inherent in the natural world, it is an indeterminacy created by the theory. Because this indeterminacy exists within the theory, it may appear in application of theory, creating the illusion of indeterminacy in the natural thing which the theory is being applied to, in modeling that natural thing, when in reality the indeterminacy is artificial, created by the deficient theory.

We can allow the indeterminacy to remain, if this form of "concept-space" is the only possible form. But if our goal truly is knowledge, then it cannot be "the most fitting solution" to our problems. When we proceed under an MO, which is essentially a habit, and we recognize that it is not the best, there is a certain laziness associated with "it works", which leaves us uninspired to seek a better operation. The usual way "works", in some instances because it is layered with a multitude of complexities, piled one on top of the other, exceptions to the rules etc.. And, regardless of these massive complexities we continue with the usual, extremely complex, way, because it works. In reality though, taking the time to analyze the fundamental problems at the base of the usual way might produce a much simpler, more efficient, and better way for revealing truth, by removing the indeterminacy from the theory.

As I said, I may misunderstand the topic of the discussion altogether, which is fine. But just wanted to be sure that the intention of my example was clear. (That is - that the space tessellation was not intended as a mathematical structure, but as representation of some unknown coarseness of the physical structure, being ignored for efficiency reasons.)

I'm not sure I actually understood your example. Maybe we can say that Euclidian geometry came into existence because it worked for the practises employed at the time. People were creating right angles, surveying plots of land with parallel lines derived from the right angles, and laying foundations for buildings, etc.. The right angle was created from practise, it was practical, just like the circle. Then theorists like Pythagoras demonstrated the problems of indeterminacy involved with that practise.

Since the figures maintained their practicality despite their theoretical instability, use of them continued. However, as the practise of applying the theory expanded, first toward the furthest reaches of the solar system, galaxy, and universe, and now toward the tiniest "grains" of space, the indeterminacy became a factor, and so methods for dealing with the indeterminacy also had to be expanded.

Now, to revisit your example, why do you assume "grain uniformity"? Spatial existence, as evident to us through our sense experience consists of objects of many different shapes and sizes. Wouldn't "grain uniformity" seriously limit the possibility for differing forms of objects, in a way inconsistent with what we observe?

Again, I do not hold than there is such a thing as "an abstraction existing as an object."

Then your beliefs are irrelevant to my concerns with algebra and set theory, which hold that the symbols represent objects.

No, a symbol in logic is itself either a subject or the predicate within a proposition. If it is a subject, then it denotes an object, which can be an abstraction or an existent. If it is the predicate, then it signifies the interpretant, which is a relation among the objects denoted by the subjects.

I'm afraid you have things backward. The symbol is itself an object. The symbol may signify a subject, and it may signify a predicate. It is impossible that the symbol "is" the subject, or "is" the predicate because then there would be no way to determine whether any given symbol is a subject or a predicate. It is only by the means of representing something (subject or predicate) that the distinction is made.
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Then your beliefs are irrelevant to my concerns with algebra and set theory, which hold that the symbols represent objects.
They do represent objects--abstractions, not existents.

I'm afraid you have things backward.
On the contrary, this is Semeiotic 101--in a proposition, the subjects denote objects, and the predicate signifies the interpretant.
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The problem I've been discussing is that whatever it is which is expressed as "a square" does not actually exist in "concept-space" because the perpendicular sides are incommensurable ... The figure is impossible, just like the irrational nature of pi tells us that a circle is impossible.
Incommensurability does not preclude (mathematical) existence. Our inability to measure two different objects (abstractions) relative to the same arbitrary unit with infinite precision does not entail that one of them is (logically) impossible.

The figures defined by the theory are impossible, according to the theory, just like a square circle is impossible.
Only according to your peculiar theory, not the well-known and well-established theory in question.
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The problem I've been discussing is that whatever it is which is expressed as "a square" does not actually exist in "concept-space" because the perpendicular sides are incommensurable. There is a deficiency in the concept which makes it impossible that there is a diagonal line between the two opposing corners, when there is supposed to be according to theory. The figure is impossible, just like the irrational nature of pi tells us that a circle is impossible.
If you view mathematics as explanative device for natural phenomena, I can certainly understand your concern. However, I see mathematics first and foremost as an approximate number crunching and inference theory. I do not see it as a first-principles theory of the space-time continuum or the world in general. I see physics and natural sciences as taking on that burden and having to decide when and what part of mathematics to promote to that role. If necessary, physics can motivate new axiomatic systems. But whether Euclidean geometry remains in daily use will not depend on how accurately it integrates with a physical first-principles theory. Unless the accuracy of the improved model of space is necessary for our daily operations or has remarkable computational or measurement complexity tradeoff, it will impact only scientific computing and pedagogy. Which, as I said, isn't the primary function of mathematics in my opinion. Mathematics to me is the study of data processing applications, not the study of nature's internal dialogue. The latter is reserved for physics, through the use of appropriate parts of mathematics.

The problem in this situation is that the indeterminacy is created by the deficient theory. It is not some sort of indeterminacy which is inherent in the natural world, it is an indeterminacy created by the theory. Because this indeterminacy exists within the theory, it may appear in application of theory, creating the illusion of indeterminacy in the natural thing which the theory is being applied to, in modeling that natural thing, when in reality the indeterminacy is artificial, created by the deficient theory.

We can allow the indeterminacy to remain, if this form of "concept-space" is the only possible form. But if our goal truly is knowledge, then it cannot be "the most fitting solution" to our problems.
As I said above, I don't think that mathematics should engage directly to enhance our knowledge of the physical world, but rather to improve our efficiency in dealing with computational tasks. It certainly is a very important cornerstone of natural philosophy and natural sciences, but it is ruled by applications, not natural fundamentalism. At least in my view.

I'm not sure I actually understood your example. Maybe we can say that Euclidian geometry came into existence because it worked for the practises employed at the time. People were creating right angles, surveying plots of land with parallel lines derived from the right angles, and laying foundations for buildings, etc.. The right angle was created from practise, it was practical, just like the circle.
This is what I mean.

Then theorists like Pythagoras demonstrated the problems of indeterminacy involved with that practise.

Since the figures maintained their practicality despite their theoretical instability, use of them continued. However, as the practise of applying the theory expanded, first toward the furthest reaches of the solar system, galaxy, and universe, and now toward the tiniest "grains" of space, the indeterminacy became a factor, and so methods for dealing with the indeterminacy also had to be expanded.
I do not see how our newfound knowledge about the universe will impact all of the old applications. How does it apply to the geometries employed in a toy factory, for example. The same computations can be applied in the same way. Unless there is benefit to switching to a new model, in which case both models will remain in active use. This is the same situation as using Newtonian physics instead of special or general relativity for daily applications. It is simpler, it works for relative velocities in most cases, and has been tested in many conventional applications. I am breaking my own rule and trespassing into natural sciences, but the point is that a computational construct can remain operational long after it has been proved fundamentally inaccurate. And therefore, its concepts remain viable object of mathematical study.

Now, to revisit your example, why do you assume "grain uniformity"? Spatial existence, as evident to us through our sense experience consists of objects of many different shapes and sizes. Wouldn't "grain uniformity" seriously limit the possibility for differing forms of objects, in a way inconsistent with what we observe?
Of course it would. I meant applications where the grain is indeed uniform, such as the atomic structure of certain materials. And even then, only certain materials would apply. The point being is - every construct which can be usefully applied as computational device in practice deserves to be studies by mathematics. As long as it offers the desired complexity-accuracy tradeoff.

I do agree that the use of mathematics in real applications is frequently naive. And that further analysis of its approximation power for specific use cases is necessary. In particular, we need more rigorous treatment that explains how accuracy of approximation is affected by discrepancies between the idealized assumptions of the theory and the underlying real world conditions. I have been interested in the existence of such theories myself, but it appears that this kind of analysis is mostly relegated to engineering instincts. Even if so - if mathematics already works in practice for some applications, and the mathematical ideals currently in use can be computed efficiently, this is sufficient argument to continue their investigation. Such is the case of square root of 2. Whether this is a physical phenomena or not, anything more accurate will probably require more accurate/more exhaustive measurements, or more processing. Thus its use will remain justified. And whether incommensurability can exist for physical objects at any scale, I consider topic for natural sciences.
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