I don't know what hyperbole you have in mind. Maybe 'nobody'. Because you seemed adamant with all-caps, and, as I recall, three variations of 'no', I didn't know it was hyperbole. So I merely replied to it at face value. Of course I would not have begrudged you then declaring it was only hyperbole. But still, I don't think what was hyperbolized was correct, even if given non-hyperbolized restatement. — TonesInDeepFreeze

You would not have begrudged me then, but you will begrudge the living bejeebus out of me now? LOL. Does the phrase, "Give it a rest," have any resonance with you?

Anyway, your response again misses my point. My point that you just quoted is not to take issue with your hyperbole, but rather to point out how your more recent argument goes wrong. — TonesInDeepFreeze

I don't remember making any recent argument with you other than that it's pointless to argue about how many mathematicians believe this or that, and that in any event I have happily conceded that "plenty" of them disagree with what I wrote.

Okay, but my point quoted above was not about that. — TonesInDeepFreeze

I may be lost by now. I have no idea what we're discussing.

Do you mean the hyperboles "blow up the moon" and "AIDS denier"? — TonesInDeepFreeze

No, those are literal facts of record. I supplied the relevant Wiki links.

If so, that's fine that you say now it was hyperbole. But I did take your comments at least to be a claim that a view that mathematical truth is not confined to model-theoretical is on its face preposterous even outlandish. I said that a lot of mathematicians don't view truth as merely model-theoretic, and you replied to the effect that there are intellectually talented people who believe a number of crazy things. It is reasonable for me to say that believing that truth is not merely model theoretic is not that kind of crazy, if it is even crazy at all. — TonesInDeepFreeze

I will stipulate that "plenty" of people don't think mathematical truth is merely model-theoretic. I have been stipulating this for some time, but evidently it's just not enough for you. I hereby stipulate that my use of "nobody" in that context was hyperbole, that I didn't mean for it to be taken literally; and that in retrospect I would have been better off saying that the "complement of plenty" of mathematicians have never given the matter five minutes of thought in their entire lives.

Do you find this satisfactory, or are further mea culpas necessary?

Some set theorists have pointed how we can reduce some axiomatic assumptions and still get the mathematics for the sciences. And even if ZFC is too productive, that doesn't refute that a good part of the interest in the axiom of infinity is to axiomatize (even if too productively) the mathematics for the sciences. — TonesInDeepFreeze

In other words its adoption is pragmatic. But ok your point was to include axiomatizing science in the discussion and I'm happy to agree. But of course that is the interesting point. The axiom of infinity is in contradiction with known physics; yet physics is based on the real numbers, and the theory of the real numbers requires the axiom of infinity. Truly it's a mystery. There is, by the way, a field of study in its very early stages called constructive physics, which attempts to build physics within the framework of constructive mathematics. Exactly to try to get around some of these issues. Not around infinity, necessarily, but around nonconstructive mathematics.

The field is new enough that there are only papers, and no Wiki entry. But at least a handful of people are thinking about the problem. Mathematics is way too much and way too false about the real world for it to serve as a suitable foundation for physics. That's my interpretation, not necessarily anyone else's.. But again this is Wigner's point, the UNREASONABLE effectiveness of math in the physical sciences. (There I go with the all caps again, a bad typographical habit for sure). Math is so "out there" that it's unreasonable that it finds such indispensable (Putnam & Quine's word) application in the world. It's a mystery.

No, not that I don't at all know. Rather, I don't know that it hasn't been axiomatized at all (i.e. hasn't been axiomatized to any extent whatsoever or with no progress toward axiomatizing it). That was in response to what you wrote, "Physics has not been axiomatized at all." — TonesInDeepFreeze

LOL. You said, "I don't know that it hasn't been axiomatized AT ALL," your caps. Which could mean:

a) You don't know AT ALL if it hasn't been axiomatized; or

b) You don't know if it's been axiomatized AT ALL, as opposed to in its entirety.

I pointed out that parts of science have been very nicely axiomatized, such as Newtonian gravity. That would be an agreement with (b). Whereas (a) refers to the state of your knowledge.

But surely we can agree that ALL of science is very far from being axiomatized. At least I hope we can agree on that.

Mathematics has been created by human beings, with physical bodies, physical brains, living in the world. It has no means to escape the restrictions imposed upon it by the physical conditions of the physical body. Therefore it very truly is bound by the world. Your idea that mathematics can somehow escape the limitations imposed upon it by the world, to retreat into some imaginary world of infinite infinities, is not a case of actually escaping the bounds of the world at all, it's just imaginary. We all know that imagination cannot give us any real escape from the bounds of the world. Imagining that mathematics is not bound by the world does not make it so. Such a freedom from the bounds of the world is just an illusion. Mathematics is truly bound by the world. And when the imagination strays beyond these boundaries, it produces imaginary fictions, not mathematics. — Metaphysician Undercover

The concept of infinite infinities is already part of mathematics today. Therefore, in your dubious distinction between mathematics and “imaginary fictions”, your placement of infinite infinities on the side of "imaginary fictions" makes no sense; infinite infinities is already on the side of mathematics. Your attempted stipulations to the contrary are pointless.

You would not have begrudged me then, but you will begrudge the living bejeebus out of me now? — fishfry

No. I did not begrudge you hyperbole. Rather, (1) I explained why previously it was not unreasonable for me not to infer that you were writing hyperbolically, and (2) That even factoring for hyperbole, I disagreed with the non-hyperbolic claim behind the hyperbole.

Does the phrase, "Give it a rest," have any resonance with you? — fishfry

I don't see that when you continue to reply, with both repeated points and arguments and new points and arguments that I should then not also reply.

I don't remember making any recent argument with you other than that it's pointless to argue about how many mathematicians believe this or that — fishfry

The arguments are in your posts recently made.

I may be lost by now. I have no idea what we're discussing. — fishfry

You quoted me:

It may have been tedious, but my entries have been responses to your continued posting on the subject. It seems odd to me that one would continue to reply, and repeat some points that are essentially the same, but then complain that responses on point to your own replies are thereby tediousness. — fishfry

And to that you replied that you were being hyperbolic (presumably when you said that nobody takes as axioms as true other than model-theoretically). So I replied that your hyperbole is not connected to what you quoted by me.

Do you mean the hyperboles "blow up the moon" and "AIDS denier"?
— TonesInDeepFreeze

No, those are literal facts of record. I supplied the relevant Wiki links. — fishfry

I addressed the additional matter of outlandishness; I didn't thereby declare that I am forever changing any subject or not changing it.
— TonesInDeepFreeze

I used a figure of speech called hyperbole. — fishfry

Since the context was outlandishness, when you replied that you were being hyperbolic, my reply was wondering whether you meant that you were being hyperbolic when you compared "blow up the moon" and "no AIDS exists" to a view that mathematical truth is not limited to the model-theoretic. So, since that is not what you meant, now again, I don't know what connection you intended between my point about outlandishness and your having been hyperbolic in saying that nobody regards axioms as true except in the model-theoretic sense.

No, not that I don't at all know. Rather, I don't know that it hasn't been axiomatized at all (i.e. hasn't been axiomatized to any extent whatsoever or with no progress toward axiomatizing it). That was in response to what you wrote, "Physics has not been axiomatized at all."
— TonesInDeepFreeze

LOL. You said, "I don't know that it hasn't been axiomatized AT ALL," your caps. Which could mean:

a) You don't know AT ALL if it hasn't been axiomatized; or

b) You don't know if it's been axiomatized AT ALL, as opposed to in its entirety. — fishfry

I just explained very clearly that I meant that I don't know that physics has not been axiomatized to any extent. I'll say it yet another way: I don't know that there is no extent to which physics has been axiomatized.

(1) YOU said, "Physics has not been axiomatized at all". So I replied "I don't know that physics has not been axiomatized at all".

Obviously I'm replying to YOUR OWN sense of YOUR OWN statement "Physics has not been axiomatized at all". Especially in context, I didn't take YOUR OWN statement as ambiguous.

I pointed out that parts of science have been very nicely axiomatized, such as Newtonian gravity. That would be an agreement with (b). Whereas (a) refers to the state of your knowledge. — fishfry

You didn't mention Newtonian gravity in the post to which I first replied.

Here is the exchange:

If 'to make interesting' includes 'to provide an axiomatization of the mathematics for sciences'.
— TonesInDeepFreeze

This could never be true. Physics has not been axiomatized at all. They can't even reconcile quantum mechanics and relativity. And the idea that set theory could ever be a foundation for physics seems to me to be an unlikely stretch. But at least that is an interesting and substantive topic in the philosophy of math and science. — fishfry

Especially in context, I took you to mean that there has been no progress in axiomatizing physics.

If someone says, "The house has not been built at all", then one would understand that to mean that there has been no progress in building the house, that no one has completed even the first phase in construction of the house.

So when you said, "Physics has not been axiomatized at all", I took you to mean the obvious sense that there has been no progress in axiomatizing physics.

If you don't mean that sense, then what sense did you mean when you said "Physics has not been axiomatized at all"?

The axiom of infinity is in contradiction with known physics — fishfry

I don't know that. The axiom of infinity says there is an inductive set and, with other axioms, entails that there is an infinite set. Set theory doesn't say that there is an infinite set of particles or that physical space extends infinitely outward or whatever. Also, is it definitively established that there are not infinitely many particles or that space does not extend infinitely outward?

I supplied the relevant Wiki links. — fishfry

I'm glad to know there's still hope for Wikipedia!

I want to bounce something off of you.

Scenario A (This universe):

1. Speed limit: 186000 miles per second (light speed)

2. If infinite energy is applied on an object, that object will attain light speed.

Scenario B (Another universe)

3. Speed limit: None! Go as fast as you can.

4. If infinite energy is poured into an object, that object will attain infinite speed

Basically, in terms of energy (infinite energy in both cases), there's no difference between light speed (this universe/scenario A) and infinite speed (another universe/scenario B).

Light speed then is an actual infinity. It's completed (1860000 miles per second) and it's equivalent to infinity (another universe/scenario B)

As a side note a speed limit (light speed, 186000 miles per second in this universe) violates the law of conservation of energy. What's happening to the energy I'm exerting on an object if its speed doesn't change proportionately?

Scenario A (This universe): — TheMadFool

I don't know anything about physics. Nothing I've said here pertains to the physical universe.


2. If infinite energy is applied on an object, that object will attain light speed.[/quote]

There's no such thing as "infinite energy" as far as contemporary physics is concerned, nor is it sensible that an object could attain light speed by any means at all.

Sorry can't be of any assistance here, this is speculative physics and seems to contradict known physics.

ArXiv.Org : 161 math papers on Wednesday. 4 in Logic (foundations).

No one I knew in my corner of the mathematical community had anything much to say about set theory or foundations, other than, perhaps, something along the lines of "Oh yeah, neat how Peano Axioms do that."

In my Intro to Grad Math course in 1962 when we got into the second half of Halmos Naive Set Theory most of us lost interest. I remember thinking, "chains, towers, etc.- sounds medieval".

I don't know that. The axiom of infinity says there is an inductive set and, with other axioms, entails that there is an infinite set. Set theory doesn't say that there is an infi — TonesInDeepFreeze

nite set of particles or that physical space extends infinitely outward or whatever.

If an inductive set that's not physical "exists," what does that mean to you? I've stated repeatedly to @Meta that it exists "mathematically," and he has correctly challenged me on this. I can play formalist and say it's just a formal game, like chess. I concede that this is not entirely satisfactory. If you play Platonist, I can ask you where your inductive set lives, and what else lives there with it? The Baby Jesus? The Flying Spaghetti Monster? Captain Ahab? Platonism is full of conceptual problems. As is formalism. But you're the one claiming that an inductive set "exists," so I would ask you what you mean by that.

Also, is it definitively established that there are not infinitely many particles or that space does not extend infinitely outward? — TonesInDeepFreeze

It's unknown, but disallowed by all physical theories except the highly speculative ones like Eternal Inflation, positing infinite time and an actual infinity of universes. Even that theory's inventor no longer believes in it, but the papers keep getting published anyway. Physics is in a heck of a state these days. Einstein and the other early 20th century giants cared about ontology. Modern physicists lose their careers for talking about ontology. Sad state of affairs.

No. I did not begrudge you hyperbole. Rather, (1) I explained why previously it was not unreasonable for me not to infer that you were writing hyperbolically, and (2) That even factoring for hyperbole, I disagreed with the non-hyperbolic claim behind the hyperbole. — TonesInDeepFreeze

"I explained why previously it was not unreasonable for me not to infer that you were writing hyperbolically" broke my parser. And that hurts!

I don't see that when you continue to reply, with both repeated points and arguments and new points and arguments that I should then not also reply. — TonesInDeepFreeze

You're right. If I say I am not replying, that would constitute a reply. I shall henceforth simply not reply to this inane self-referential conversation. When you something substantive, as opposed to looping back on the syntax of whatever I may have said, I'll reply.

(several paragraphs later)

Well that didn't leave much. I think there's a a potentially interesting conversation about the axiom of infinity.

If an inductive set that's not physical "exists," what does that mean to you? — fishfry

I'm not claiming any particular sense of existence. Nor am I disputing any particular sense of existence. In context of the question whether set theory is inconsistent with physics, I am interested in the context of formal axiomatization (I'll just say 'axiomatization'). In Z we have the theorem: Ex x is infinite. I would think that that would provide an inconsistent axiomatization T of mathematics/physics only if T has a theorem: ~Ex x is infinite.

If you play Platonist — fishfry

I don't.

But you're the one claiming that an inductive set "exists," — fishfry

I claim that set theory has a theorem: Ex x is an inductive set. I don't opine as to what particular sense we should say that provides. I do tend to think that whatever that sense is, it is at least some abstract mathematical sense. And I appreciate that there are variations held by different people. I can "picture" in my mind certain notions such as "the least inductive set is an abstract mathematical object that I can hold in my mind as "picked out" by the predicate of being a least inductive set". I find it to be a coherent thought for myself. But I don't have any need to convince anyone else that such a view of mathematical existence should be be generally adopted or even considered coherent by others.

"I explained why previously it was not unreasonable for me not to infer that you were writing hyperbolically" broke my parser. And that hurts! — fishfry

Then your parser is weak handling double negation. I chose double negation because it best suits the flow of how I think about the proposition. With less negation: I explained why previously that it was reasonable for me not to infer that you were writing hyperbolically.

If I say I am not replying, that would constitute a reply. — fishfry

Just to be clear, my replies were not merely to you saying that you are not replying.

inane self-referential conversation — fishfry

My part is not inane. And whether or not you think that conversation about conversation should be eschewed, I don't think that way.

When you something substantive, as opposed to looping back on the syntax of whatever I may have said — fishfry

I did not merely "loop back on the syntax of whatever you may have said". It's interesting that you want an end to posting about the conversational roles themselves, but you want to do that while still getting in your own digs such as "inane" and dismissive mischaracterization such as "looping back on the syntax".

when we got into the second half of Halmos Naive Set Theory most of us lost interest. — jgill

And I have not the least interest in the subject of corporate financing. Go figure. But I do make it a point to go into discussion threads about business and economics and make my boredom with the subject well known.

Sure the pieces are made of atoms, but there is no fundamental physical reason why the knight moves that way. — fishfry

Yes there is a physical reason for this. The pieces cannot be floating in air, nor can they randomly disappear and reappear in other places, nor be in two places at once. There are real physical restrictions which had to be respected when the game was created. So a board and pieces, with specific moves which are physically possible, was a convenient format considering those restrictions.

This is the problem with your claim that mathematics is not bound by real world restrictions. You can assert that it is not, and you can create completely imaginary axioms, such as a thing with no inherent order, but when it comes to real world play (use of such mathematics) if these axioms contain physical impossibilities, it's likely to create problems in application. The creator of chess could have made a rule which allowed that the knight be on two squares at once. or that it might hover around the board. But then how could the game be played when the designated moves of the pieces is inconsistent with what is physically possible for those pieces?

I have full respect for the notion that mathematical axioms might be completely imaginary, like works of art, even fictional, but my argument is that such axioms would be inherently problematic when applied in real world play. You seem to think that it doesn't matter if mathematical axioms go beyond what is physically possible, and it's even okay to assume what is physically impossible like "no inherent order". And you support this claim with evidence that mathematics provides great effectiveness in real world applications. But you refuse to consider the real problems in real world applications (though you accept that modern physics has real problems), and you refuse to separate the problems from the effectiveness, to see that effectiveness is provided for by principles which are consistent with physical reality, and problems are provided for by principles which are inconsistent.

That math is inspired by the world and not bound by it? To me this is a banality, not a falsehood. It's true, but so trivial as to be beneath mention to anyone who's studied mathematics or mathematical philosophy. — fishfry

I've explained to you very clearly why it is false to say that mathematics is not bound by the real world. Perhaps if you would drop the idea that it is a banality, you would look seriously at what I have said, and come to the realization that what you have taken for a banal truth, and therefore have never given it any thought, is actually a falsity.

But it's still a formal game. — fishfry

I've gone through this subject of formalism already. No formalism, or "formal game", of human creation can escape from content to be pure Form. You seem to be having a very hard time to grasp this, and this is why you keep on insisting that there's such a thing as "pure abstraction". Content, or in the Aristotelian term "matter" is what is forcing real world restrictions onto any formal system. So when a formal system is created with the intent of giving us as much certainty as possible, we cannot escape the uncertainty produced by the presence of content, which is the real world restriction on certainty, that inheres within any formal system.

That's an interesting point. Yet you can see the difference between representational art, which strives to be "true," and abstract art, which is inspired by but not bound by the real world. Or as they told us when I took a film class once, "Film frees us from the limitations of time and space." A movie is inspired by but not bound by reality. Star Wars isn't real, but the celluloid film stock (or whatever they use these days) is made of atoms. Right? Right. — fishfry

Let's take this analogy then. Will you oblige me please to see it through to the conclusion? Let's say that abstract art is analogous to pure, abstract mathematics, and representational art is analogous to practical math. Would you agree that if someone went to a piece of abstract art, and started talking about what was represented by that art, the person would necessarily be mistaken? Likewise, if someone took a piece of pure abstract mathematics, and tried to put it to practice, this would be a mistake.

Bear in mind, that I am not arguing that what we commonly call pure math, ought not be put to practice, I am arguing that pure math as you characterize it, as pure abstraction, is a false description. In other words, your analogy fails, just like the game analogy failed, the distinction between pure math and practical math is not like the distinction between abstract art and representational art.

I recognize the difference between pure and applied mathematics. And you seem to reject fiction, science fiction, surrealist poetry, modern art, and unicorns. Me I like unicorns. They are inspired by the world but not bound by it. I like infinitary mathematics, for exactly the same reason. Perhaps you should read my recent essay here on the transfinite ordinals. It will give you much fuel for righteous rage. But I didn't invent any of it, Cantor did, and mathematicians have been pursuing the theory ever since then right up to the present moment. Perhaps you could take it up with them. — fishfry

I do not reject fiction, I accept it for what it is, fiction. I do reject your claim that pure mathematics is analogous to fiction. Here is the difference. In fiction, the mind is free to cross all boundaries of all disciplines and fields of education. In pure mathematics, the mathematician is bound by fundamental principles, which are the criteria for "mathematics", and if these boundaries are broken it is not mathematics which the person is doing. And, these boundaries are not dreamt up and imposed by the imagination of the mathematician who is doing the pure mathematics, they are imposed by the real world, (what other people say about what the person is doing), which is external to the pure mathematician's mind. This is why it is false to say that pure mathematics is not bound by the real world. If the person engaged in such abstraction, allows one's mind to wander too far, the creation will not be judged by others (the real world) as "mathematics". Therefore if the person wanders outside the boundaries which the real world places on pure mathematics, the person is no longer doing mathematics.

edit: This again is the issue of content. If the content is not consistent with what is judged as the content of mathematics, then the person working with so-called "pure" abstractions cannot be judged as doing mathematics. Therefore the abstractions cannot be "pure" as there are restrictions of content, as to what qualifies as mathematics.

The concept of infinite infinities is already part of mathematics today. Therefore, in your dubious distinction between mathematics and “imaginary fictions”, your placement of infinite infinities on the side of "imaginary fictions" makes no sense; infinite infinities is already on the side of mathematics. Your attempted stipulations to the contrary are pointless. — Luke

My argument is that such things are wrongly called mathematics, due to faulty conventions which allow imaginary fictions, cleverly disguised to appear as mathematical principles, to seep into mathematics, taking the place of mathematical principles. And obviously, it's not a stipulation but an argument, as I've spent months arguing through examples.

I don't know anything about physics. Nothing I've said here pertains to the physical universe.

— fishfry

2. If infinite energy is applied on an object, that object will attain light speed.

There's no such thing as "infinite energy" as far as contemporary physics is concerned, nor is it sensible that an object could attain light speed by any means at all.

Sorry can't be of any assistance here, this is speculative physics and seems to contradict known physics

Too bad. Thanks for replying though.

But I do make it a point to go into discussion threads about business and economics and make my boredom with the subject well known. — TonesInDeepFreeze

Among the two or three participants? I would think it might provide viewer relief. But to each his own. :roll:

jgill posted a while back about the tiny percentage of overall math papers that are devoted to set theory. Few working mathematicians give any of these matters the slightest thought. — fishfry

Just thought I'd supply recent information regarding.

Not sure. Einstein did joke that the moon doesn't need to exist when we aren't looking towards where it should be. Would the idea of infinity be like a reality simulation in which things are only added to the information data base when it is needed for observation?

It took me a moment to understand your comment. Then it dawned on me you are returning to the OP! Thanks. :cool:

I'm not claiming any particular sense of existence. Nor am I disputing any particular sense of existence. In context of the question whether set theory is inconsistent with physics, I am interested in the context of formal axiomatization (I'll just say 'axiomatization'). In Z we have the theorem: Ex x is infinite. I would think that that would provide an inconsistent axiomatization T of mathematics/physics only if T has a theorem: ~Ex x is infinite. — TonesInDeepFreeze

Of course that is vacuously true, since there is no axiomatic formulation of physics. The axiom of infinity is inconsistent with known physics since there is no principle of modern physics that stipulates the existence of any infinite set, let alone an inductive set of natural numbers. Since it's known that there are only $10^{78}$ hydrogen atoms in the observable universe, I'll take that as evidence that contemporary physics can not accept the axiom of infinity as a physical principle. You can't equivocate your way out of this. Perhaps you could simply acknowledge that the axiom of infinity is plainly at odds with known physics, yet a cornerstone of modern mathematics.

I claim that set theory has a theorem: Ex x is an inductive set. I don't opine as to what particular sense we should say that provides. — TonesInDeepFreeze

Why not? What prevents you from dipping a toe in the water and taking a stand? It's clear that there aren't infinitely many physical objects, except in the most speculative physical theories having no experimental support. So why not say something like, "The axiom of infinity is a formal statement that, as far as we know, is false about the world, yet taken as a fundamental truth in mathematics. And I account for that philosophically as follows: _______." Ducking the question doesn't help.

I do tend to think that whatever that sense is, it is at least some abstract mathematical sense. — TonesInDeepFreeze

Yes. Very good. The axiom of infinity is taken as true in "some abstract mathematical sense." My point exactly, on which we are now in agreement. There are models of set theory in which it's true; at least if there are any models at all.

And I appreciate that there are variations held by different people. I can "picture" in my mind certain notions such as "the least inductive set is an abstract mathematical object that I can hold in my mind as "picked out" by the predicate of being a least inductive set". I find it to be a coherent thought for myself. But I don't have any need to convince anyone else that such a view of mathematical existence should be be generally adopted or even considered coherent by others. — TonesInDeepFreeze

You've come to be in agreement with me. The only way the axiom of infinity can possibly be accepted as true or meaningful is in the context of purely abstract math, and NOT physics. Hence the axiom of infinity is a statement that is clearly false about the world, yet taken as a basic truth in math. My point exactly.

Then your parser is weak handling double negation. I chose double negation because it best suits the flow of how I think about the proposition. With less negation: I explained why previously that it was reasonable for me not to infer that you were writing hyperbolically. — TonesInDeepFreeze

No longer responding to this line of discourse (Nlrttlod).

Just to be clear, my replies were not merely to you saying that you are not replying. — TonesInDeepFreeze

Nlrttlod.

My part is not inane. And whether or not you think that conversation about conversation should be eschewed, I don't think that way. — TonesInDeepFreeze

Eschewed and espit out. Masticated to death. Munched and crunched.

I did not merely "loop back on the syntax of whatever you may have said". It's interesting that you want an end to posting about the conversational roles themselves, but you want to do that while still getting in your own digs such as "inane" and dismissive mischaracterization such as "looping back on the syntax". — TonesInDeepFreeze

Nlrttlod.

No. I did not begrudge you hyperbole. Rather, (1) I explained why previously it was not unreasonable for me not to infer that you were writing hyperbolically, and (2) That even factoring for hyperbole, I disagreed with the non-hyperbolic claim behind the hyperbole. — TonesInDeepFreeze

Nlrttlod.

I don't see that when you continue to reply, with both repeated points and arguments and new points and arguments that I should then not also reply. — TonesInDeepFreeze

Nlrttlod to this and to the rest of it.

My argument is that such things are wrongly called mathematics, due to faulty conventions which allow imaginary fictions, cleverly disguised to appear as mathematical principles, to seep into mathematics, taking the place of mathematical principles. And obviously, it's not a stipulation but an argument, as I've spent months arguing through examples. — Metaphysician Undercover

What "disguise"?

Of course it's an attempted stipulation, since you are attempting to stipulate what the mathematical conventions should be (because you consider them "faulty"). Regardless, does your claim that mathematics should conform to "real world boundaries" (whatever that could mean) apply not only to infinite infinities, but also to infinity, zero, negative numbers, imaginary numbers and the like?

Yes there is a physical reason for this. The pieces cannot be floating in air, nor can they randomly disappear and reappear in other places, nor be in two places at once. There are real physical restrictions which had to be respected when the game was created. So a board and pieces, with specific moves which are physically possible, was a convenient format considering those restrictions. — Metaphysician Undercover

Right. Just like mathematicians can't fly by flapping their arms. The rules of chess are arbitrary and constrained by physical law, as are the axioms of set theory.

This is the problem with your claim that mathematics is not bound by real world restrictions. You can assert that it is not, and you can create completely imaginary axioms, such as a thing with no inherent order, but when it comes to real world play (use of such mathematics) if these axioms contain physical impossibilities, it's likely to create problems in application. — Metaphysician Undercover

Well this is Wigner's point. Some aspects of mathematics is so obviously fictional that it is UNREASONABLE that math should be so effective in the physical sciences. I don't expect to be able to personally explain how this works, but I hope you would agree that this is a mystery that more than one clever person has tried to sort out. It is UNREASONABLE that math is so effective in the physical sciences. You can't hang that problem around my neck as if it's mine personally. Everyone knows about the problem.

The creator of chess could have made a rule which allowed that the knight be on two squares at once. or that it might hover around the board. But then how could the game be played when the designated moves of the pieces is inconsistent with what is physically possible for those pieces? — Metaphysician Undercover

In that respect math is even more free to be fictional, since sets and other mathematical entities are not bound by the laws of physics. If you drop a set near the earth, it doesn't fall down, unlike a bowling ball or a chess piece. You are making my point for me. Math is MORE free to be utterly fictional even than chess.

But actually your point is wrong. Physical chess pieces are not required. If I say "e4" to a chess player they know exactly what that means, and it is not necessary to physically push the white King's pawn forward two squares. Indeed, people play chess blindfolded, keeping the position entirely in their mind. Physical pieces are not necessary to play chess.

I have full respect for the notion that mathematical axioms might be completely imaginary, like works of art, even fictional, — Metaphysician Undercover

All right! Took a long time but I appreciate that you have acknowledge this.

but my argument is that such axioms would be inherently problematic when applied in real world play. — Metaphysician Undercover

Take it up with Wigner. I can not personally take responsibility for this mystery, which is indeed problematic. Some aspects of math are patently false, yet so amazingly useful in the physical sciences. Your argument? I think Wigner beat you to it. I hope you take this point. You are only repeating a well-known problem in the philosophy of math.

You seem to think that it doesn't matter if mathematical axioms go beyond what is physically possible, and it's even okay to assume what is physically impossible like "no inherent order". — Metaphysician Undercover

It's not only ok, it's one of the cornerstones of modern math. And again, this isn't me advocating for such a position. This is me describing the facts of modern math. Pick up any book on set theory. Look at the axiom of extensionality. I am not advocating for its rightness or wrongness; only reporting to you that extensionality is the first rule of set theory. Sets aren't affected by gravity or electric charge either. You have a problem with that?

And you support this claim with evidence that mathematics provides great effectiveness in real world applications. — Metaphysician Undercover

I hardly need to supply evidence. I hope you don't deny it. Do you deny it?

But you refuse to consider the real problems in real world applications (though you accept that modern physics has real problems), and you refuse to separate the problems from the effectiveness, to see that effectiveness is provided for by principles which are consistent with physical reality, and problems are provided for by principles which are inconsistent. — Metaphysician Undercover

On the contrary. I agree with Wigner than parts of math are so obviously fictional and divorced from reality that it is UNREASONABLE, Wigner's famous word, that math should be so effective in the physical sciences. You act like nobody ever thought of this before but it's a famous essay and a famous mystery. There are no sets, at least not as conceived by set theory. Yet set theory founds the real numbers which underly most of physics. I "refuse to consider" this mystery? Of course not. I point you to Wigner's essay time after time after time, and you come back and act like this is some great mystery that you yourself have personally uncovered and that I deny.

I've explained to you very clearly why it is false to say that mathematics is not bound by the real world. — Metaphysician Undercover

If you drop a set near the earth, it doesn't fall down. Sets have no gravitational or inertial mass. They have no electric charge. They have no temperature, velocity, momentum, or orientation. In what sense are sets bound by the real world?

Perhaps if you would drop the idea that it is a banality, you would look seriously at what I have said, and come to the realization that what you have taken for a banal truth, and therefore have never given it any thought, is actually a falsity. — Metaphysician Undercover

I've given the matter quite a bit of thought and expressed my thoughts in this conversation. It's a banality that mathematical objects are not bound by the laws of nature, except that -- stretching a point -- mathematical objects are products of the human mind and the human mind is bound by the laws of nature. So perhaps ultimately there's a physical reason why we think the thoughts we do. I'd agree with that possibility, if that's the point you're making.

I've gone through this subject of formalism already. No formalism, or "formal game", of human creation can escape from content to be pure Form. — Metaphysician Undercover

I gather that "pure Form" is a term of art in philosophy with which I'm not familiar.

You seem to be having a very hard time to grasp this, and this is why you keep on insisting that there's such a thing as "pure abstraction". — Metaphysician Undercover

As opposed to what? Impure abstraction? There are abstract things in the world. SEP has an article on the subject.

Content, or in the Aristotelian term "matter" is what is forcing real world restrictions onto any formal system. — Metaphysician Undercover

Matter has a very specific technical meaning in physics. Matter has gravitational mass. Mathematical objects don't. You can't weigh a topological space. I can't speak to what Aristotle might have thought, but he also believed that bowling balls fall down because they're like the earth and thing go to their natural place. We no longer take him seriously on that. I don't know why you think I should take him seriously on whether mathematical sets, which Aristotle didn't know about, have mass. You don't think sets have mass, do you? In this post you are pursuing a line of argument I find nonsensical. Of course mathematical objects are not bound by the laws of nature.

So when a formal system is created with the intent of giving us as much certainty as possible, we cannot escape the uncertainty produced by the presence of content, which is the real world restriction on certainty, that inheres within any formal system. — Metaphysician Undercover

That was a little word salad-y. If you mean that mathematical sets are sort of like bags of groceries but definitely not completely like bags of groceries, of course I agree. For one thing, mathematical sets can contain infinitely many elements.

Let's take this analogy then. Will you oblige me please to see it through to the conclusion? — Metaphysician Undercover

Ok.

Let's say that abstract art is analogous to pure, abstract mathematics, and representational art is analogous to practical math. — Metaphysician Undercover

The analogies are only good to a point, but ok I'll play along. For example abstract art is not indispensable to the formulation of modern physics, whereas abstract math is.

Would you agree that if someone went to a piece of abstract art, and started talking about what was represented by that art, the person would necessarily be mistaken? — Metaphysician Undercover

Of course not. Art critics do it all the time. I may not agree or even understand, but people who care about such things see meaning in abstract art, even real world meaning. After all Moby Dick is a work of fiction yet cautions us to not follow our obsessions to our doom.

Likewise, if someone took a piece of pure abstract mathematics, and tried to put it to practice, this would be a mistake. — Metaphysician Undercover

What???? Are you joking? Physicists do it every day of the week and twice at grant proposal time. Group theory, part of the subject matter of an undergrad math major course called Abstract Algebra, is the heart and soul of particle physics these days. Functional analysis and differential geometry, two highly abstract areas of math, are the foundation of quantum mechanics and general relativity, respectively.

Your questions are wildly off the mark. Physical scientist apply the most abstruse and abstract areas of mathematics constantly, as part of their daily work.

Bear in mind, that I am not arguing that what we commonly call pure math, ought not be put to practice, — Metaphysician Undercover

The physicists will be relieved.

I am arguing that pure math as you characterize it, as pure abstraction, is a false description. — Metaphysician Undercover

I quite agree. Let me say that again I quite agree. Therefore it is a MYSTERY that it is so UNREASONABLY effective. Wigner Wigner Wigner. These thoughts have already been thought, this problem is well known. Much of abstract math is false as false can be, as regards the physical world; and yet, those same parts of math are indispensable -- Quine and Putnam's word -- in the physical sciences. It's a puzzler alright. But a very well-known puzzler.

In other words, your analogy fails, just like the game analogy failed, the distinction between pure math and practical math is not like the distinction between abstract art and representational art. — Metaphysician Undercover

I don't think you made your case. You didn't make your case at all.

I do not reject fiction, I accept it for what it is, fiction. I do reject your claim that pure mathematics is analogous to fiction. Here is the difference. In fiction, the mind is free to cross all boundaries of all disciplines and fields of education. In pure mathematics, the mathematician is bound by fundamental principles, which are the criteria for "mathematics", and if these boundaries are broken it is not mathematics which the person is doing. — Metaphysician Undercover

Mathematical principles are historically contingent, and the greatest advances have been made when someone transcends and violates the established principles of their time. Negative numbers, irrational numbers, non-Euclidean geometry, transfinite numbers and set theory. Cantor caused a revolution. His radical ideas on transfinite quantities was received with great skepticism bordering on horror. Today his ideas are taught to high school students. Cantor was told that what he was doing was not mathematics. His great opponent, Kronecker, who had actually been Cantor's teacher, said, "I don't know what predominates in Cantor's theory – philosophy or theology, but I am sure that there is no mathematics there." https://en.wikipedia.org/wiki/Controversy_over_Cantor%27s_theory

In math, violating the "fundamental principles" is how progress is made.

And, these boundaries are not dreamt up and imposed by the imagination of the mathematician who is doing the pure mathematics, they are imposed by the real world, (what other people say about what the person is doing), which is external to the pure mathematician's mind. — Metaphysician Undercover

I've just shown that some of the greatest advances in math have been made by blowing up the opinions of the world. What happens is that the opinions of the mathematical world change. Or as Planck said, scientific progress proceeds one funeral at a time. Meaning that the old guard die off and the young Turks readily adopt the radical new ideas.

This is why it is false to say that pure mathematics is not bound by the real world. If the person engaged in such abstraction, allows one's mind to wander too far, the creation will not be judged by others (the real world) as "mathematics". Therefore if the person wanders outside the boundaries which the real world places on pure mathematics, the person is no longer doing mathematics. — Metaphysician Undercover

They're creating radical new mathematics. As has happened innumerable times throughout history.

Your insufficient knowledge of mathematics and its history causes you to have such mistaken ideas, that there are eternal principles that never change. On the contrary, each generation blows up the ideas of the past. Standards of rigor, what counts as a proof, what counts as a number, what counts as mathematics, is constantly changing and is a matter of historical contingency.

Just thought I'd supply recent information regarding. — jgill

Much appreciated.

I'm not claiming any particular sense of existence. Nor am I disputing any particular sense of existence. In context of the question whether set theory is inconsistent with physics, I am interested in the context of formal axiomatization (I'll just say 'axiomatization'). In Z we have the theorem: Ex x is infinite. I would think that that would provide an inconsistent axiomatization T of mathematics/physics only if T has a theorem: ~Ex x is infinite.
— TonesInDeepFreeze

Of course that is vacuously true, since there is no axiomatic formulation of physics. — fishfry

I thought it was obvious that since we don't have in front of us an axiomatization, then my question is hypothetical regarding whatever proposed axiomatization might be presented. Such an axiomatization would have mathematical axioms and also extra primitives and axioms for physics. I don't see that such a theory would have to be inconsistent. For example, "Ex x is infinite" is not inconsistent with "~Ex x has infinitely many particles" (I'm just using 'particles' as a placeholder for whatever technical rubric would actually be used).

The axiom of infinity is inconsistent with known physics since there is no principle of modern physics that stipulates the existence of any infinite set, — fishfry

That doesn't entail inconsistency. Just because a theory doesn't have a certain principle doesn't entail that adding that principle causes inconsistency. But if physics had a principle that it is not the case that there exists an infinite set, then yes, there would be inconsistency. But even if physics had a principle that there are not infinitely many particles, that is not itself inconsistent with the existence of infinite sets, such as infinite sets of numbers if numbers are not axiomatized to be particles.

contemporary physics can not accept the axiom of infinity as a physical principle. — fishfry

I never said that it would be a physical principle. It would be a mathematical theorem to which are added primitives and axioms for theorems of physics.

I claim that set theory has a theorem: Ex x is an inductive set. I don't opine as to what particular sense we should say that provides.
— TonesInDeepFreeze

Why not? — fishfry

If I wanted to take a normative stance on the subject then I would.

why not say something like, "The axiom of infinity is a formal statement that, as far as we know, is false about the world, yet taken as a fundamental truth in mathematics. And I account for that philosophically as follows: _______." — fishfry

Because I am not motivated to do that. Moreover, I am not inclined to accept that my thoughts (including lack of conclusions) about a subject needs to conform to a Procrustean format "this and that, and I account for that philosophically ______."

Ducking the question doesn't help. — fishfry

I haven't made claims that I've ducked supporting.

The axiom of infinity is taken as true in "some abstract mathematical sense." My point exactly, on which we are now in agreement. There are models of set theory in which it's true; at least if there are any models at all. — fishfry

Of course, a consistent theory has models. But nothing I've said then commits me to adopting a position for or against the proposition that mathematical statements are true only relative to models.

You've come to be in agreement with me. The only way the axiom of infinity can possibly be accepted as true or meaningful is in the context of purely abstract math — fishfry

Nothing I've said commits me to such a claim. I said that I have a sense of the abstract meaning of mathematical statements (also, I can add, that certain axioms fit my intuitive concept of what sets are as abstractions). That doesn't entail that I also must go on to say that my sense previously described must be the ONLY correct, meaningful or useful sense, and especially it does not require that I take a stance that the notion of model-theoretic truth is the ONLY correct, meaningful or useful sense.

and NOT physics — fishfry

Again, I haven't claimed that the axiom of infinity "says" anything about physical objects or has anything to do with physics other than it provides the mathematics used for the sciences such as physics.

The axiom of inifinity is non-controversial, as it merely amounts to the inductive convention of calling a finite tree a "tree", a finite list a "list, a finite set a "set" etc. Nobody who talks about "lists", "trees" or "sets" in ordinary language implies a completed totality of such objects, and neither does the use of the axiom of infinity in a proof, because as we recall proofs by definition have finite derivations and use every axiom finitely.

The real numbers however, are nonsensical with respect to experimental physics and engineering, where their literal definition is at odds with respect to how the formalism is actually used. There, real numbers aren't used literally in the sense of referring to infinitely precise quanitities, but are used non-rigorously or "non-standardly" to refer to indefinite and imprecise quantities and taken together with noise and error terms. For this reason, in conjunction with the rapid ascent of automated theorem proving and functional programming that are based on type theory, the awkward, misleading and practically false language of real analysis can only die fast.

There are different formulations that may have equivalences, and there are complications throughout, but I know of no proof nor mention in the article you cited that shows the equivalence of AC with LEM in intuitionistic set theory. The SEP article does say "each of a number of intuitionistically invalid logical principles, including the law of excluded middle, is equivalent (in intuitionistic set theory) to a suitably weakened [italics in Bell's earlier article] version of the axiom of choice. Accordingly these logical principles may be viewed as choice principles." But the question was not that of various choice principles but of AC itself, and we have not been shown a proof that AC and LEM are equivalent in intuitionistic set theory. — TonesInDeepFreeze

yes, originally I was speaking roughly in relation to that article while making what i considered to be a tangential point in relation to the thread topic. As an axiom, LEM when interpreted in the Set category by the usual Tarskian approach, is an axiom of "finite choice" in the sense of asserting 'by divine fiat' the existence of a choice function for every relation into a finite set, i.e. that every finite set is 'choice '. Stronger choice principles additionally assert the existence of choice-sets that are the non-constructive infinite unions of the finite choice sets.

I don't know your personal use of the terminology. Of course even without AC every finite set of nonempty sets has a choice function. I don't know whether LEM is needed for that (at least on first glance I don't see that it is).

For this reason, in conjunction with the rapid ascent of automated theorem proving and functional programming that are based on type theory, the awkward, misleading and practically false language of real analysis can only die fast. — sime

Well, I'm glad I won't be around to see that.

Has an actual, real live physicist posted on this thread? There have been a lot of assumptions about physics, interesting opinions, but I wonder what people in the profession have to say about the number systems they employ. fishfry provided a link to a novel paper on constructivism in physics that shows there is some degree of interest in the subject in the physics community. Kenosha Kid is a quantum researcher. :chin:

http://tph.tuwien.ac.at/~svozil/publ/1995-set.pdf

https://www.sciencedirect.com/science/article/abs/pii/S0960077996000550

and from a set theorist:

http://logic.harvard.edu/EFI_Magidor.pdf

Has an actual, real live physicist posted on this thread? There have been a lot of assumptions about physics, interesting opinions, but I wonder what people in the profession have to say about the number systems they employ. fishfry provided a link to a novel paper on constructivism in physics that shows there is some degree of interest in the subject in the physics community. Kenosha Kid is a quantum researcher. :chin: — jgill

Hello. Was, not is. Sold my soul for a bag of gold and sick guitar skills.

I haven't followed the thread, sorry, and responding to the OP 19 pages in seems weird. But I'll chuck a thought in and you can just pretend I wasn't here if I'm off-track.

I feel loathe to claim that physicists "believe" much. Physics is grounded on curiosity; if you're chock full of beliefs, why go and ruin it by learning things? I'm probably particularly minimalist when it comes to beliefs though. I don't "believe" in quarks and Higgs bosons and even quantum theory in the way I "believe" in electrons, protons, evolution and the special theory of relativity.

When we endorse a theory, what we're saying is "whatever is actually going on, it's got to be something like _this_". Sometimes the theory is compelling enough to base an actual belief upon, but really it's all a work in progress.

In terms of the boundary of the universe, I'm intrigued but uncertain. My intuition is that the universe is temporally bounded and from relativistic considerations, that being so, we should expect space to be bounded also. But that's unjustified and falls short of a belief.

I'd be a bit more confident about saying that the universe is either infinite or has periodic boundary conditions, i.e. some surface of a finite hyper-object. I don't think when you get to the end there's a wall, like in The Truman Show. :)

Infinity crops up in mathematical physics all the time but it's usually a mathematical artefact not a physical one. It could be a trick, like in integration, where you integrate over infinity knowing that the thing you're integrating attenuates to zero. Or it could be a consequence of representing something difficult as a power series or some other kind of expansion, like Feynman diagrams. Speaking of, quantum field theories a rife with infinities, requiring a renormalization procedure to get rid of them. Feynman invented renormalization and refered to it as a trick. As the fourth greatest physicist of all time, we should take him seriously, not renormalization.

The most overt infinity is probably the singularity: a point of finite energy but infinite energy density. It's perfectly feasible there's no such thing, that black holes are tiny but finite, but there'd be an intriguing question of what's holding it up since, by that point, even the strongest known pseudo-force in existence -- statistics (Pauli exclusion) has given up the ghost! But there's no shortage of theories in which black holes do all manner of crazy things (like birthing new universes).

Thing is, if there's infinity out there, we're not likely to encounter it. Infinity is either a limit we can't reach or a sign that something's broken. The visible universe is large but finite. The effects of black holes that we can observe are large but finite. If it is out there, either it or old age will kill you before you find it.

I am not informed in physics. So anything I say about axiomatizing physics might very well need to be corrected or qualified. However, I can at least address the question as to basic logic.

Just for a minimal start, it is clearly evident that it is not in principle inconsistent merely to add non-mathematical axioms to ZFC, even if the non-mathematical axioms include a declaration that the non-mathematical domain is finite.

For the most simple example:

To ZFC, add a 1-place predicate symbol: P (intuitively, Px means x is a particle). Add an axiom E!x Px. Define: p = the unique x such Px. So the set of particles is finite.

Or, instead of having the axiom E!xPx, we could have: ExPx & ~Ey(y is infinite & Axey Px). Then there are an indeterminately many particles but not infinitely many of them.

[EDIT: The above two paragraphs are not what I meant:

Add a 1-place predicate symbol: R (intuitively, Rx means x is a particle). Add an axiom E!xAy(yex <-> Ry). Define P = {y | Ry}. Add axiom: P is finite.

And delete the second paragraph.]

Those are not inconsistent theories. [EDIT: That is not an inconsistent theory.]

In Z set theories, we may define:

x is a class <-> (x = the-empty-set or Ey yex)

x is a set <-> (x is a class & Ey xey)

x is a proper class <-> (x is a class & ~ x is a set)

x is an urelement <-> ~ x is a class

Then we have theorems:

Ax x is a class

Ax x is a set

~Ex x is a proper class

~Ex x is an urelement

That entails that every particle is a set. And we might not like that. But we could say "Oh well, that is an artifact of abstraction that won't hurt the physical theorems we'll prove, similar to the fact that the set theoretic definition of numbers burdens numbers with abstract set theoretical artifacts that however don't interfere with the mathematics we will do with those numbers."

Or we could reformulate as follows:

Delete the axiom of extensionality.

Add primitive: S (Sx intuitively meaning "x is a set").

Add axiom: E!x(Sx & Ay ~yex) (there is a unique empty set).

Add axiom Axy((Sx & Sy) -> (Az(zex <-> zey) -> x=y)) (revised axiom of extensionality).

Add axiom: Px -> x is an urelement. [EDIT: Rx -> x is an urelement]

Adopt all the other axioms of ZFC.

It seems a safe bet that that is a consistent theory. Granted, it is not ZFC and not strictly speaking set or even class theory (such as NBG) that both are characterized by perhaps the most crucial property of classes and sets - extensionality. But set theory with urelements may be recognized as a reasonable variant.

And one may add additional primitives such as a function for mass, primitives about spacetime, primitives and definitions about subparticles, physical objects made from particles, and axioms about particles, subparticles, and physical objects, their masses, interactions among them in space and time, etc... And with axioms with mathematics about the physical objects, etc, hopefully deriving theorems of physics.

I don't see that there would be a correct argument that merely in principle such a theory must be inconsistent.

/

See Suppes's 'Introduction To Logic' pgs 291-305 for another example: axioms for Particle Mechanics.

Some aspects of mathematics is so obviously fictional that it is UNREASONABLE that math should be so effective in the physical sciences. — fishfry

The vast majority of mathematical principles, like pi, and the Pythagorean theorem which we discussed, are not fictional, and that is why mathematics is so effective. So there is nothing UNREASONABLE in the effectiveness of mathematics. You can argue from ridiculous premises, as to what constitutes "true", and assume that the Pythagorean theorem is not true, as you did, but then it's just the person making that argument, who is being unreasonable.

If you drop a set near the earth, it doesn't fall down. Sets have no gravitational or inertial mass. They have no electric charge. They have no temperature, velocity, momentum, or orientation. In what sense are sets bound by the real world? — fishfry

The word "set" is a physical thing, which signifies something. And it only has meaning to a human being in the world, with the senses to perceive it. Therefore sets, as what is signified by that word, are bound by the real world.

except that -- stretching a point -- mathematical objects are products of the human mind and the human mind is bound by the laws of nature. So perhaps ultimately there's a physical reason why we think the thoughts we do. I'd agree with that possibility, if that's the point you're making. — fishfry

Now you're catching on. Consider though, that the physical forces of the real world are not the "reason why" we think what we do, as we have freedom of choice, to think whatever we want, within the boundaries of our physical capacities. The physical forces are the boundaries. So we really are bounded by the real world, in our thinking. We do not apprehend the boundaries as boundaries though, because we cannot get beyond them to the other side, to see them as boundaries, they are just where thinking can't go. Therefore it appears to us, like we are free to think whatever we want, because our thinking doesn't go where it can't go.

You will not understand the boundaries unless you accept that they are there, and are real, and inquire as to the nature of them. The boundaries appear to me, as the activity where thinking slips away andis replaced by other mental activities such as dreaming, and can no longer be called "thinking". We have a similar, but artificial boundary we can call the boundary between rational thinking and irrational thinking, reasonable and unreasonable. This is not the boundary of thinking, but it serves as a model of how this type of boundary is vague and not well defined.

Further, we have a boundary between conscious and subconscious, and this is closer to being a real representation of the boundary of thinking. The subconscious activity of the mind is not called "thinking", So dreaming is not thinking. You can see that the descriptions of these two types of boundaries are somewhat similar, being modeled on some form of coherency. A form of coherency marks the difference between reasonable and unreasonable, and a different type of coherency marks the difference between waking mental activity (thinking), and dreaming (mental activity outside the bounds of thinking). In the latter division, you ought to see clearly that it is the physical condition of the body (being asleep), which provides the boundary that we model with a description as to what qualifies as thinking, and what does not. However, there is a coherency or lack of it, within the mental activity, which corresponds with the physical boundary which we model as the difference between awake and asleep. In the case of reasonable/unreasonable, we have cases of physical illness, and intoxication, which demonstrate that the boundary, which is a boundary of coherency, has a corresponding physical condition.

So here's the point. We have mental activity which is thinking, and mental activity which cannot be classified as thinking. Therefore there must be a boundary to thinking which separates it from that other activity. The difference is described as a difference in coherency But, since there are real world physical differences which correspond with the described boundaries of coherency, I propose that it is the real world physical boundaries which impose upon our mental activity, the inclination to create corresponding mental boundaries of coherency. So for instance, the law of non-contradiction is a boundary of coherency. To violate the law is to put oneself outside a boundary of coherency. But the law of non-contraction is a statement of what we believe to be impossible, in the real physical world. So it is the acknowledgement of this physical impossibility, as being impossible, which substantiates the assumed mental boundary of coherent/incoherent.

Now, you are proposing a type of thinking "pure mathematics", which is not at all bounded by the real physical world. How could there possibly be such a thing? As I explained, the mental activity which is called thinking, is already bounded in order that it be separate from the mental activity which is not thinking, and there is a corresponding physical condition, being conscious, or awake, which provides the capacity for thinking. Since this physical condition is required for, as providing the capacity for, thinking, then thinking is necessarily bounded by the real physical world. A person cannot go in thought, beyond the capacity given to that person by one's physical body.

Suppose we allow that thinking might move past the boundaries of coherency, (which I admit we have created), to be not at all bounded by coherency. The problem is that there is a corresponding real world boundary which is responsible for the creation of the boundary of coherence. Do you allow that subconscious mental activity, and dreaming, are thinking? See, it makes no sense to say that thinking can go beyond the bounds imposed upon it by the real physical body which capacitates it.

Now suppose we say that there is a special type of thinking, "pure mathematics", which we give that privilege to. How can we even call this thinking? We create boundaries of coherency to define what "thinking" is, keeping "thinking" within the range of conscious mental activity, but now you want to allow a special type of thinking which is not bound by this rule. In reality, "pure mathematics" is a special type of thinking, so it has stricter binding of coherency than just thinking in general does. And, corresponding with those binds of coherency are features of the real physical world.

In math, violating the "fundamental principles" is how progress is made. — fishfry

I agree with this fully. But the need to violate fundamental principles just means that what was once considered to be a boundary of coherency can no longer be consider such. It does not negate the real physical boundaries, which the boundary of coherency was meant to represent. The boundary of coherency did not properly correspond, with the real world boundary, and therefore it needed to be replaced. The need to replace fundamental principles is evidence of faulty correspondence.

I've just shown that some of the greatest advances in math have been made by blowing up the opinions of the world. What happens is that the opinions of the mathematical world change. Or as Planck said, scientific progress proceeds one funeral at a time. Meaning that the old guard die off and the young Turks readily adopt the radical new ideas. — fishfry

Yes, yes, I think you truly are catching on. The need to change mathematical principles is a feature of poor correspondence. It is not the case that pure mathematics is thinking which is not bounded, because it truly is bounded, as described. But it is thinking which does not adequately understand the real world conditions which are its boundaries. It does not understand its boundaries, that's why it might even think as you do, that it is not bounded. Therefore it often poorly represents these boundaries, and when the boundaries become better understood, the representations need to be replaced. Understanding the true real world boundaries is what produces certainty.

Thanks for the links. Here's a short excerpt from the first that is germane:

In terms of recursion theory, Bridgman’s claim can be re-interpreted such that no diverging algorithm should be allowed as legal input of any other (terminating) algorithm.One may go even further than Bridgman and assume that, since infinite entities are not operational, infinities have to be abandoned altogether.

Thanks for chiming in. Thought-provoking comments! :cool:

Here's a short excerpt from the first that is germane: " In terms of recursion theory, Bridgman’s claim can be re-interpreted such that no diverging algorithm should be allowed as legal input of any other (terminating) algorithm.One may go even further than Bridgman and assume that, since infinite entities are not operational, infinities have to be abandoned altogether." — jgill

And contrariwise in other articles. I'm sure you wouldn't want to cherry pick just gainsaying quotes.

And the article describes Bridgman's notions as stemming from Bridgman's philosophical framework. That does not preclude other frameworks, and especially doesn't entail that any form of ZFC+physics must be inconsistent. (Though the author of the article does give other vigorous arguments against set theory as a foundation for physics.)

And contrariwise in other articles. I'm sure you wouldn't want to cherry pick just gainsaying quotes — TonesInDeepFreeze

I skimmed the first article and this caught my eye. I only know renormalization from afar.