then discussion with you is pointless. — TonesInDeepFreeze

We're quite in agreement. We tend to talk past each other and I'm content to leave it at that.

The claim that "the set of particles is finite" contradicts the axiom of infinity is shockingly wrong. — TonesInDeepFreeze

I'll concede that point. But if you adopt as an axiom claims that are subject to experiment and investigation, your science won't get you very far.

Does a fetus deserve moral consideration? And when do we give the fetus moral consideration? Better question when do we give anything moral consideration? — Oppyfan

Abortion should be legal to age 21 or so. Would solve a lot of parenting problems.

Now for a serious answer, or at least a response if not an answer, I present to you the striking case of Scott Peterson, convicted in 2004 of "... the first-degree murder of his pregnant wife, Laci Peterson, and the second-degree murder of their unborn son, Conner ..."

During the trial, which took place in liberal Redwood City, California, the papers were full of talk of "unborn baby Conner." Now I ask you. Why is "unborn baby Conner" a clump of undifferentiated cells for purposes of the abortion debate, yet deserving of a name and thereby his humanity in a murder trial?

How do you convict a man of the murder of an undifferentiated clump of cells? This case always stands out for me as exemplifying the massive hypocrisy of the pro-choice position. If Laci had aborted the fetus, nobody in the San Francisco bay area would have batted an eye. Yet Scott Peterson sits in jail at this very moment, as I type these lines, for murdering that very same clump of undifferentiated cells. Well he killed his wife too, so he'd be in prison regardless. But I hope you see my point, and I wonder if some of you philosophers can help me understand why "unborn baby Conner" was even deserving of a name, let alone the status of a murder victim, in a strongly pro-choice state like California. You might say it's the mother's choice, but how can that be? If "unborn baby Conner" has human rights and can be murdered, then surely it's cold comfort to the fetus that it was his mom and not his dad who decided to kill him.

RIP unborn baby Conner. Or undifferentiated clump of cells, as the case may be.

You have ignored and outrageously misconstrued what I wrote, yet again. I didn't want to comment on the discussion itself again, but your reading confusions, as seen in this thread and other threads, is quite remarkable. — TonesInDeepFreeze

You haven't convinced me of your point in the least.

discussion with you is hopeless. — TonesInDeepFreeze

Perhaps we can mutually agree on at least this. I'm always for achieving agreement.

The set of particles is finite. — TonesInDeepFreeze

Contradicts axiom of infinity.

The mass of a particle is a positive real number. — TonesInDeepFreeze

What's a particle? What's mass? These are axioms, remember? Everything's defined in terms of a single primitive, $\in$. You've claimed you can add physics axioms to the axioms for set theory, so the burden on you is to challenge yourself to see if your idea holds water.

Looking at the complexity of their posts on other threads, I'm lucky I didn't embarrass myself. — Down The Rabbit Hole

I embarrass myself all the time around here. Thanks for the kind words.

I am saying that I haven't seen an argument that it would be inconsistent if we added to set theory, primitives and axioms for physics, and that it seems plausible that we might be able to do so. — TonesInDeepFreeze

So that I can understand what you mean, can you give an example? What kind of axiom would we add to set theory that would be an axiom for physics?

I surmise that I am not mistaken that, ordinarily, physics uses classical mathematics, which has infinite sets and is ordinarily axiomatized by set theory. That seems salient. So, from my admittedly non-expert point of view, it would seem plausible that we might combine formal mathematics with whatever parts of physics have been, or might be, formalized. At least one example of a basic portion is, as I mentioned, in Suppes's logic book, though it is not so ambitious to undertake relativity, quantum, etc. — TonesInDeepFreeze

It doesn't work that way.

It's true that physics "uses" the real numbers. And it's true that the real numbers are formalized using infinite sets. It does NOT follow that physics uses or is formalized by infinite sets.

I don't have the philosophy vocabulary to name this phenomenon, where A is a part of B and B is a part of C but A is in no way even remotely a part of C. Perhaps others can suggest the right concept.

Regardless, it's just not true and it's not valid thinking to say that "physics uses math and math uses infinite sets therefore ..." It's wrong, I just can't verbalize why.

ps -- To add a little. If you want to use the real numbers, you can just take their properties as given, namely that they are a complete totally ordered field. That uniquely characterizes them, there's only one model. (The completeness is second-order, that's why this works. I'm a bit fuzzy on that aspect of the logic but I think that's what's going on).

Now if you are doing math and you want to show that you can cook up a complete totally ordered field within set theory, you use one of the standard constructions involving the use of infinite sets. But the ontological commitments, if that's the right phrase, are of a different type. Physics uses real numbers and mathematicians formalize real numbers using infinite sets, but there's no ontological commitment in physics to infinite sets.

I believe that the idea is to try to encourage better posts, but as far as I can see it will probably just bolster certain egos — Jack Cummins

I'm with you on this. It's fine to have a button to like a post, but when you gamify discussion sites, they go downhill.

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

Much appreciated.

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.

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.

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.

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.

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.

I don't see why this is a problem for you. You hand me a proposition, and I refuse to accept it, claiming that it is false. You say, 'but I am not claiming that it is true'. So I move to demonstrate to you why I believe it to be false. You still insist that you are not claiming it to be true, and further, that the truth or falsity of it is irrelevant to you. Well, the truth or falsity of it is not irrelevant to me, and that's why I argue it's falsity hoping that you would reply with a demonstration of its truth to back up your support of it.. — Metaphysician Undercover

I am on record as holding that the axioms of set theory are neither true nor false, as they are syntactic entities whose truth or falsity can only be determined after an interpretation, or model, is provided. This is perfectly in keeping with standard practice in mathematical logic.

After all the "Abelian axiom" that xy = yx is true in the real numbers, and false in the set of 2x2 matrices whose entries are real numbers. "It is snowing" is true in Alaska in the winter, but never in San Diego. It's not possible for the axiom of replacement to be true OR false in isolation from an interpretation.

If the truth or falsity of it is really irrelevant to you, then why does it bother you that I argue its falsity? — Metaphysician Undercover

Because it shows that you misunderstand the distinction between syntax and semantics, between a formal axiomatic system and its models. So if you say an axiom is true you're wrong, and if you say it's false you're wrong! An axiom isn't true or false. Now if you would supply a model, I can tell you whether it's true in that model.

And why do you claim that I cannot argue the falsity of something which has not been claimed to be true? Whether or not you claim something to be true, in no way dictates whether or not I can argue its falsity. — Metaphysician Undercover

I couldn't talk you out of arguing that 2 + 2 and 4 represent distinct mathematical objects. I suppose I shouldn't bother with the axiom of replacement, which actually is a bit of a subtle and powerful axiom schema.

Either you are not getting the point, or you are simply in denial. — Metaphysician Undercover

Or, logically, you are in one of those two states.

Playing chess, is a real world activity, as is any activity. Your effort to describe an activity, like the game of chess, or pure math, as independent from the real world, as if it exists in it's own separate bubble which is not part of the world, is simply a misrepresentation. — Metaphysician Undercover

Doing set theory is a real world activity too, done by set theorists and undergrads the world over. Even high school students get a watered-down version of it. So my analogy holds. You're trying to say chess is "real world" because you can sit at a board and move the pieces. But it's still a formal game. You're being very disingenuous here. Sure the pieces are made of atoms, but there is no fundamental physical reason why the knight moves that way. And sure, set theorists are made of atoms too, but there is no fundamental physical reason to adopt or reject the laws of set theory.

Now, you admit that there actually is a pragmatic reason why the knight moves as it does, and this is to make an interesting game. — Metaphysician Undercover

Yes yes yes yes. That is correct. We are in agreement. People play chess for years and find that some rules give more interesting versions of the game than others. Who invented en passant pawn captures, or castling? These are obviously historically contingent developments, introduced for purely pragmatic reasons.

As are, thinking ahead here, each and every one of the axioms of set theory.

And of course playing a game is a real world activity. so there is a real world reason for that rule.[/quote[

That's an equivocation of "real world." The planets move in elliptical orbits for fundamental reasons having to do with the laws of nature (stipulating for sake of argument that there are laws of nature and that Newton and Einstein are on to something real). Chess pieces are made of wood or plastic, but their movements are not subject to the laws of nature. That is, if you drop them near the earth, they fall. But their moves within the game are arbitrary conventions of humans. Surely you can do better than to argue by equivocating these two notions.
— Metaphysician Undercover

Now if you could hold true to your analogy, and admit the same thing about pure mathematics, then we'd have a starting point, of common agreement. — Metaphysician Undercover

But I do. If you drop a mathematician from a height, he or she will fall in accordance with gravitational acceleration. But the axioms of set theory are historically contingent, pragmatically derived, matters of agreement. Like traffic lights. Red and green wavelengths are laws of nature. Which means go and which means stop is a social agreement. One which, if you violated it, can be fatal; but a social agreement nonetheless.

However, the reason for mathematical principles beings as they are, such as our example of the Pythagorean theorem, is not to make an interesting game. It is for the sake of some other real world activity. Do you agree? — Metaphysician Undercover

I agree that math is different from chess in that math is inspired by the real world (ancient bookkeeping and surveying), and has vast applications in the real world. I certainly agree that math is subtly different and that generations of philosophers have tried hard to put their finger on exactly what that means. Ideas like indispensability and so forth. Of course I agree with this point.

You keep on insisting on such falsities, — Metaphysician Undercover

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.

and I have to repeatedly point out to you that they are falsities. — Metaphysician Undercover

Since you're wrong on this point, repetition doesn't help. If you were right, you'd only have to say it once. That echoes Einstein's remark on being told that a hundred physicists disagreed with him. "If I'm wrong, one would be enough."

You actually disagree with the statement that "Math is inspired by but not bound by the world?" I propose to drill down on this because it's a clear point that we could discuss and perhaps shed some light. You disagree that math is inspired by the world? Or that it's not bound by it? I suspect you disagree with the latter. In which case I whip out non-Euclidean geometry as the classic example in support of my point.

But you seem to have no respect for truth or falsity, — Metaphysician Undercover

Not in axiomatic systems, no. Absent a model there is no truth or falsity.

as if truth and falsity doesn't matter to you. — Metaphysician Undercover

If you give me an axiomatic system plus a model, or interpretation, then truth or falsity can be determined, and matters to me.

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. — Metaphysician Undercover

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.

Therefore it very truly is bound by the world. — Metaphysician Undercover

Science fiction, abstract art, novels. Moby Dick is based on a true story of the Essex, a whaling ship sunk by a whale. But it's not bound by the story of the Essex. The characters and events are wholly made up. The point holds even more strongly for abstract art, science fiction, surrealism, and all other creative works of people.

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. — Metaphysician Undercover

Because my pencil is made of atoms? Are you now taking the cranky anti-Cantorian position? Everything since 1870 is bullshit? Is this your stance? Are you like this at the art museum too?

We all know that imagination cannot give us any real escape from the bounds of the world. — Metaphysician Undercover

Maybe you just don't have enough imagination. You seem to be taking a radical realist position of some kind whereby science fiction and abstract art and Star Wars either don't exist or aren't real or are lies that should be banned. What exactly is your position here? How far will you take this stance that imagination has no place in 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. But you do not even recognize a difference between imaginary fictions, and mathematics. — Metaphysician Undercover

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.

The reason why I can truthfully say that our discussion has never been about how math works, is that you have never given me any indication as to how it works. You keep insisting that mathematical principles are the product of some sort of imaginary pure abstraction, completely separated from the real world, like eternal Platonic Forms, then you give no indication as to how such products of pure fiction become useful in the world, i.e., how math works. — Metaphysician Undercover

Well that's Wigner's point in the Unreasonable Effectiveness paper. I don't claim to have the words or the philosophical background to give a good account of how math, which is perfectly obviously a massive fiction, can be so darn useful in the world. A lot of people have taken a shot at the question. Surely you know this.

Yet you write:
You continue to miss the point. That a vast number of mathematicians don't care about foundations doesn't imply that the vast number of mathematicians don't think axioms are true except model-theoretically, as indeed the fewer who care about foundations then reasonably we would expect the fewer who think truth is merely model-theoretic. — TonesInDeepFreeze

I used a figure of speech called hyperbole. You pointed out repeatedly that what I said was not literally correct. I conceded that "plenty" of mathematicians disagree with what I said. Yet this is still not enough for you. What more do you want?

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. — TonesInDeepFreeze

I used a figure of speech called hyperbole. You pointed out repeatedly that what I said was not literally correct. I conceded that "plenty" of mathematicians disagree with what I said. Yet this is still not enough for you. What more do you want?

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. You pointed out repeatedly that what I said was not literally correct. I conceded that "plenty" of mathematicians disagree with what I said. Yet this is still not enough for you. What more do you want?

I didn't say "axiomatization of physics". I said "axiomatization of the MATHEMATICS for the sciences" [all-caps added]. — TonesInDeepFreeze

I'm sure the standard axiomatization of math is an overkill for that.

Also, I don't know that physics has not been axiomatized "AT ALL" [all caps added]. — TonesInDeepFreeze

You don't know at all if it's been axiomatized ? That's something that can be looked up. Or you don't know if even small parts of it have been axiomatized? Your sentence was a little ambiguous. I'm sure there are axiomatizations of parts of science. Newtonian gravity has a nice axiomatization in Newton's three laws.

That's your view. My point is not nor has been to convince you otherwise. Rather, my point is that no matter that it may be your view, it is not true that nobody (or only a few) people disagree with it. — TonesInDeepFreeze

I have conceded the word "plenty." I can't continue to argue with you about what (a few, some, many, a strong plurality, a majority, an overwhelming flood) of people think. I won't respond any more to that subject. If you want to talk about whether the axiom of infinity may be meaningfully said to be true or false, that's a good conversation. If you want to argue about what people think, I can't engage on that anymore. Having conceded the word "plenty" already, I would think you would be happy, and that's as far as I'll go.

Fine. And so there's not basis to claim that nobody (or merely a few) views axioms as true in a sense other than relative to models. — TonesInDeepFreeze

99% of professional mathematicians are not involved in foundations (more or less objective number, I didn't look it up but recall @jgill's post regarding the percentage of recently published papers) and therefore have no professional opinion on the subject at all.

I have evidence from writings, conversations, and posts. From those, it is manifestly clear that it is false that "Nobody claims that the axiom of replacement or the axiom of powersets is true. NOBODY [all caps in original] says that. [,,,] [no one] is claiming the axioms of math are "true."" Then, as to what the majority of mathematicians believe, I've stated my impression based on what I have read and heard from mathematicians, while I've said that of course that impression is not scientific. — TonesInDeepFreeze

Enough. No more of this for me.

Again, that is the wrong road of argument for your position. I don't doubt that the vast majority of mathematicians don't care about foundations, in particular the model-theoretic notion of truth. But that only adds to my argument, not yours. Clearly, commonly mathematicians speak of the truth of mathematical statements, and even many mathematicians not occupied with foundations understand axioms in their field of study and often enough even the set theory axioms. So when such mathematicians say things like "the fundamental theorem of arithmetic" is true, then they don't mean it as "the fundamental theorem of arithmetic is true only in the sense that it is derivable in a consistent formal theory so that it is true in some models". — TonesInDeepFreeze

No más, por favor

It's become a point of contention only because I responded to your claim about it, and not just in popular opinion, but your claim of totality of opinion. — TonesInDeepFreeze

You made your point then got tedious and are now beyond that.

I don't care what you go to. I am making my own point that it is not the case that NOBODY (or even only a few) people regard axioms as true other than model-theoretically. — TonesInDeepFreeze

I think you've expressed yourself with sufficient conviction on the matter.

If it was meaningful for you to make the claim, then it is meaningful for me to reply to it, and to reply to your replies. — TonesInDeepFreeze

Perhaps you are taking things a bit too literally.

Views of mathematical truth don't have to be limited to what is typical otherwise. Whether or not departures from "typical" are justified, my main point was that it is not the case that all (or nearly all) mathematicians regard truth as merely model-theoretic. — TonesInDeepFreeze

As you've said.

That's your view. But it doesn't refute my point that it is not the case that all (or nearly all) mathematicians and philosophers regard axioms as true only as pertains to models. — TonesInDeepFreeze

Enough. Please.

It doesn't matter toward my point. I have not claimed nor disagreed with any notion of truth. I don't have to just to point out that it is not the case that nobody regards axioms as true except relative to models. This reminds me of an article I read today. The writer claimed that nobody finds Colbert funny. I don't have to opine whether Colbert is funny to point out that it is false that nobody laughs at his jokes. — TonesInDeepFreeze

You have pointed it out.

That opens another question. — TonesInDeepFreeze

Ok!! I'm glad to change the subject.

Whether one agrees with notions of mathematical truth other than model-theoretic, I'd be inclined not to claim that thinking philosophically or heuristically of mathematical truth as rather than model theoretic is among the ilk of proposing detonation of the moon or claiming that AIDS doesn't exist. — TonesInDeepFreeze

The question of foundations is as far from the practice of most mainstream mathematicians as blowing up the moon or AIDs denialism. If someone is classifying the finite simple groups, they are not thinking about the axiom of replacement.

— TonesInDeepFreeze

if you deny that the axiom of infinity is "manifestly false about the real world,"
— fishfry

I neither denied it nor affirmed it. — TonesInDeepFreeze

It would be fun if you did, then we could have a conversation.

Two different things: (1) "P is the case" and (2) "Nobody claims that ~P is the case". — TonesInDeepFreeze

The former being interesting, the latter tedious beyond belief.

Today when I read "Nobody thinks Colbert is funny", my first thought was not "But Colbert is funny" nor "I agree that Colbert is not funny", but rather how ludicrous it is to start an opinion article about American society with such a manifestly false claim as "Nobody thinks Colbert is funny." — TonesInDeepFreeze

So you didn't change the subject after all.

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.

I would say that if someone asks, "Is it meaningful to ask if the axiom of infinity is true or false; and if so, which?" I would be willing to argue any side of that. That the axiom of infinity both is and isn't meaningfully true or false; or that if it is, it's true; or that if it is, it's false. I could whip up a good argument for each of those three propositions meaningless, meaningful/true, and meaningful false.

(not necessarily your own): — TonesInDeepFreeze

Well then I don't feel bound to justify them. I'll let you have the last word on almost all of this. The one thing I'd like you to explain to me is that if you deny that the axiom of infinity is "manifestly false about the real world," which is a statement I actually DID make, in what sense to you regard it as physically true? Or if not physically true, how is it meaningful to say it's either true or false in some other sense, Platonic, formal, or otherwise? How is the axiom of infinity different than the way the knight moves?

Here is Maddy (linked above) quoting Hallet about infinity:

Dealing with natural numbers without having the set of all natural
numbers does not cause more inconvenience than, say, dealing with sets
without having the set of all sets. Also the arithmetic of the rational
numbers can be developed in this framework. However, if one is already
interested in analysis then infinite sets are indispensable since even the
notion of a real number cannot be developed by means of finite sets only.
Hence we have to add an existence axiom that guarantees the existence of
an infinite set.

This is a pragmatic argument. Ontologically we could do without the axiom of infinity. We adopt it on purely pragmatic grounds, in order to get a decent theory of the real numbers. That makes it neither true nor false, words that are not meaningful in this context; but rather useful, which is my position on the matter.

Yes, which makes it even more curious what one would mean by saying the axioms of ZFC are false, while proposing a theory that is equivalent to ZFC PLUS another axiom. — TonesInDeepFreeze

The axioms aren't false, either, any more than the way the knight moves in chess may be said to be true or false. It's just a rule that's been found by experience to make the game interesting.

That doesn't entail that a lot mathematicians aren't aware of axioms, including those not of set theory and those of set theory. And, again, probably most mathematicians don't get hung up on mathematical logic and its model theoretic sense of truth, yet mathematicians speak of the truth of mathematical statements.

And it's not even a given that only a few mathematicians who do understand models in mathematical logic hold that there are other senses of truth, including realism, instrumental, true-to-concept, et. al. Indeed, we know that there are mathematicians who well understand mathematical logic but still regard a sense of truth no restricted to that of "true in a model". — TonesInDeepFreeze

I truly can't argue about what the majority or substantial plurality or "some" or "a few" or whatever mathematicians believe. I have no data or evidence, neither do you. But the subject matter that most mathematicians work on, as evidenced by the number of papers published, is so far removed from foundations that I can't imagine that many mathematicians spend five minutes thinking about the subject in a year or in a career.

You made a clam about it. We don't have a scientific polling, but we can see that there are many people who don't think that mathematical truth is confined only to the model-theoretic sense. — TonesInDeepFreeze

We can talk math, or we can talk philosophy of math, but arguing popular opinion is not fruitful. What outcome are you looking for? Would you like me to go from "a few" to "a whole bunch?" I'm not sure what outcome would satisfy you. This is not a meaningful conversation.

So 'real world' is now added to the question. — TonesInDeepFreeze

That's what true and false typically mean. The axiom of infinity is manifestly false about the real world. At the very least it's inconsistent with contemporary physics. But it's an essential axiom of standard mathematics. Perhaps you can put your concept of truth into context for me such that the axiom of infinity could be regarded as even having a meaningful truth value other than it being generally accepted as an axiom of modern set theory.

Again, that you view certain notions about mathematics to be untenable doesn't entail that there are not plenty of people who don't share your view — TonesInDeepFreeze

"Plenty." Ok I can live with that. If, given that there must be 100,000 or so math professors in the world, I concede that "plenty" of them believe whatever you say they believe, would that satisfy you? Respected mathematician Alexander Abian wanted to blow up the moon; and prolific author of high-level math texts Serge Lange was an AIDs denier. The Unabomber had a doctorate in math, as did the guy who swindled the CIA during the Iraq war, Ahmed Chalabi. Mathematicians are human, they believe all sorts of things.

I'll go with "plenty" if this will mollify your sense of right and wrong here.

I don't know a lot about category theory, but it can be axiomatized by ZFC+Grothendieck-universe. — TonesInDeepFreeze

Which requires the existence of an inaccessible cardinal, the existence of which is not even provable in ZFC.

We're arguing over what other people think, we can't ever get to the bottom of that. @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. In what sense could the Peano axioms be true in the real world? There are only $10^{78}$ hydrogen atoms in the observable universe.

Many mathematicians and philosophers of mathematics regard certain axioms and theorems to be true not just relative to models. It might even be the dominant view. — TonesInDeepFreeze

My sense of the matter is as follows. The overwhelming majority of working mathematicians are not set theorists or involved in foundations. They pay no attention to set theory and would be hard-pressed to even name the axioms. It's not like your average anabelian geometer ever gives explicit though to the truth or falsity of the axiom of replacement. The question doesn't come up.

Among those who study foundations, it must be abundantly clear that the axioms are arbitrary and not literally true, since it's consistent to accept or deny Foundation, Powerset, and other axioms that are never questioned in standard math. Powerset negation is its own cottage industry these days, even though it's an extremely niche interest from a mainstream point of view.

It's hard for me to believe that anyone thinks the axioms of set theory are literally true about the world; or even about the abstract world of mathematics. There may be a few.

So at best I would say that "a few" mathematicians claim the axioms are literally true in some sense.

The foremost philosopher of set theory, Penelope Maddy, argues persuasively that the axioms are chosen pragmatically, on a variety of practical grounds. See https://www.cs.umd.edu/~gasarch/BLOGPAPERS/belaxioms1.pdf for example.

Among philosophers, who could seriously argue that the axioms of set theory are "true" in any meaningful sense; or even meaningful in any meaningful sense! I am less familiar philosophy than math, but my sense is that just as with mathematicians, most philosophers aren't concerned with the axioms of set theory at all, let alone their truth.

That's my sense of the matter. I'd go further and say that it's perfectly clear that some of the axioms, such as the axiom of infinity, are literally false. That strengthens my point to @Meta, which is that the axioms are chosen pragmatically for their utility in developing math, and not for any real-world reasons.

I would add to all that the growing importance of neo-intuitionist, constructivist, and category-theoretic approaches, in which set theory is not only false, but irrelevant. (Category theorists pay lip service to set theory by defining "small" or "locally small" categories where certain collections are required to be sets, and some of the category theorists do worry about such things, but in the mainstream of category theory, they'd don't worry too much about set theory).

But of course you said, "certain axioms," and I suppose if you want to say that high school notions like unions and intersections are true or instantiable in the real world, you'd have a point. At least for finite sets. But finite sets are not of much interest to set theorists. Finite sets belong to combinatorics.

This is not true. — Metaphysician Undercover

Ok, so the conversation is shifting now to arguing about what we've been arguing about. Slow day at Chez fishfry, so I'll play. But full disclosure, my heart's not in it.

You've been making arguments about "pure math", and "pure abstractions". — Metaphysician Undercover

Arguing in the sense of describing to you how modern math sees certain things, such as sets and order relations.

So it is you who is making a division between the application of mathematics, — Metaphysician Undercover

Me personally? You give me too much credit. Those divisions were there long before I was born. I am just talking about them. But actually we've never been talking about pure versus applied math at all, I don't know where you're getting this. Applied math is the use of math is fields like physics, economics, biology, and so forth. We haven't been talking about that at all. We've only ever been talking about pure math. The meaning of 2 + 2, the nature of sets, the nature of order relationships, how mathematicians formalize things.

"how modern math works", and pure mathematics, and you've been arguing that pure mathematics deals with pure abstractions. — Metaphysician Undercover

LOL. I'd be glad to argue that any day. Pure math deals with pure abstractions? What's your counter proposition? That's like saying barbering deals with cutting hair. There is no sensible negation to the proposition.

You've argued philosophical speculation concerning the derivation of mathematical axioms through some claimed process of pure abstraction, totally removed from any real world concerns, rather than the need for mathematics to work. — Metaphysician Undercover

On the contrary, I have argued that the choice of mathematical axioms is pragmatic. Possibly not that much in our convos but in general. The axioms are chosen because they let you build up good theories above them. I've always argued that. But "real world" concerns are not involved, that I agree with.

So your chess game analogy is way off the mark, because what we've been discussing here, is the creation of the rules for the game, not the play of the game. And, in creating the rules we must rely on some criteria. — Metaphysician Undercover

My chess analogy is perfectly apt. By what criterion is the rule for how the knight moves chosen? Why is the lower right-hand square always white ("white on the right") and never black as is often erroneously portrayed by careless prop artists in movies?

Nowhere do I dispute the obvious, — Metaphysician Undercover

Now THAT is funny. You do nothing but, starting from "2 + 2 does not denote the same mathematical object as 4," several years ago, right up to the present moment. You constantly dispute the obvious.

that this is "how modern math works". That is not our discussion at all. — Metaphysician Undercover

For the past several weeks I've been explaining to you that mathematical sets have no inherent order and you've been arguing that this is somehow "wrong." So we have definitely been discussing "how modern math works."

What I dispute is the truth or validity of some fundamental principles (axioms) which mathematicians work with. — Metaphysician Undercover

How can you argue with the truth of things that are not claimed to be true? Nobody claims that the axiom of replacement or the axiom of powersets is true. NOBODY says that. This is your own personal strawman. And it's a tremendous misunderstanding on your part that anyone is claiming the axioms of math are "true." The axioms are strings of formal symbols, true in some models and false in others. Your failure to comprehend this is a great failing of yours.

This is why the game analogy fails, because applying mathematics in the real world, is by that very description, a real world enterprise, it is not playing a game which is totally unrelated to the world. — Metaphysician Undercover

I have never had any interest in applying math to the real world. I wonder why you think I do, or should? I'm with the great British mathematician G. H. Hardy, who argued in his great essay, A Mathematician's Apology, that the beauty of an area of math is measured by how utterly useless it is; and that by this criterion his own field, that of number theory, is the most supremely beautiful area of math. How ironic, then, that number theory, which was supremely useless for 2000 years, has in only the past few decades become the core technology behind Internet security and cryptocurrencies. Hardy would spin in his grave. Hardy was played by Jeremy Irons in The Man Who Knew Infinity, highly recommended. A very rare math film that gets the math right and tells a great human story too, the tragic story of Ramanujan. A must-see for all readers of this site.

Pure math is not about the real world. Now you may not like that, and you math think it "should" be otherwise, but I am only telling you how it is. You can't argue with me about that. I don't know why you persist in trying.

So the same principle which makes playing the game something separate from a real world adventure, also makes it different from mathematics, therefore not analogous in that way — Metaphysician Undercover

It's entirely analogous. Chess is a formal game, there's no "reason" why the knight moves as it does other than the pragmatics of what's been proven by experience to make for an interesting game. And there are equally valid variations of the game in common use as well.

I have a proposal, a way to make your analogy more relevant. Let's assume that playing a game is a real world thing, *as it truly is something we do in the world, just like scientists, engineers and architects do real world things with mathematics. Then let's say that there are people who work on the rules of the game, creating the game and adjusting the rules whenever problems become evident, like too many stalemates or something like that. Do you agree that "pure mathematicians" are analogous to these people, fixing the rules? — Metaphysician Undercover

Perhaps you're thinking of engineers.

Clearly, the people fixing the rules are not in a bubble, completely isolated from the people involved in the real world play. Of course not, they are working on problems involved with the real world play, just like the "pure mathematicians" are working on problems involved with the application of math in science and engineering, etc.. — Metaphysician Undercover

My God man, pure mathematicians are not concerned with the problems of the world. And when they are, they are doing applied math, not pure.

Plato described this well, speaking about how tools are designed. A tool is actually a much better analogy for math than a game. The crafts people who use the tool must have input into the design of the tool because they know what is needed from the tool. — Metaphysician Undercover

Math is justified only by itself. That the physicists find it useful is good for them. It's not what drives math. You truly don't understand this. Now you are perfectly entitled to argue that things SHOULD be different. But you can't credibly argue that they ARE different, because they are not. Just ask a physicist about math, they'll tell you the mathematicians are off in the clouds totally untethered from the real world. As if that's a bad thing!

In conclusion, your claim that "pure mathematicians" are completely removed from the real world use of mathematics is not consistent with the game analogy nor the tool analogy. Those who create the rules of a game obviously have the real world play of the game in mind when creating the rules, so they have a purpose and those who design tools obviously have the real world use of the tool in mind when designing it, and the tool has a purpose. So if mathematics is analogous, then the pure mathematicians have the real world use of mathematics in mind when creating axioms, such that the axioms have a purpose. — Metaphysician Undercover

You didn't move me with such a weak and fallacious argument. Your argument that math is concerned with the real world was true a thousand years ago, but has not been true for a long time.

The problem is that you have been "reporting" falsely. — Metaphysician Undercover

By your ignorant measure. As measured by reality, I've been reporting accurately.

You consistently claimed, over and over again, that "pure mathematicians" work in a realm of pure abstraction, completely separated, and removed from the real world application of mathematics, and real world problems. — Metaphysician Undercover

Yes.

That is the substance of our disagreement in this thread. My observations of things like the Hilbert-Frege discussion show me very clearly that this is a real world problem, a problem of application, not abstraction, which Hilbert was working on. And, the fact that Hilbert's principles were accepted and are now applied, demonstrates further evidence that Hilbert delivered a resolution to a problem of application, not a principle of pure abstraction. — Metaphysician Undercover

Hilbert's famous quote is that ""One must be able to say at all times--instead of points, straight lines, and planes--tables, chairs, and beer mugs"" I suggest you meditate on this point.

Of course Hilbert did some applied math too. He famously offered to help Einstein finish general relativity, only for Einstein to discover, almost too late, that Hilbert was trying to finish first and gain credit. Fortunately Hilbert had an error in his calculations and in the end, graciously conceded priority to Einstein. Otherwise we'd say to this day, "He's no Hilbert!" instead of "He's no Einstein."

I can't say that I see what I'm supposed to comment on. The geometry used is the one developed to suit the application, it's produced for a purpose. With the conflation of time and space, into the concept of an active changing space-time, Euclidean geometry which give principles for a static unchanging space, is inadequate. Hence the need for non-Euclidean geometry in modern physics. — Metaphysician Undercover

Yet both theories are internally consistent. So math alone can't determine truth. I believe you've conceded my point.

Sorry fishfry, but this is evidence for my side of the argument. "The unreasonable effectiveness of math" is clear evidence that the mathematicians who dream up the axioms really do take notice, and have respect for real world problems. That's obviously why math is so effective. If the mathematicians were working in some realm of pure abstraction, with total disregard for any real world issues, then it would be unreasonable to think that they would produce principles which are extremely effective in the real world. Which do you think is the case, that mathematics just happens to be extremely effective in the real world, or that the mathematicians who have created the axioms have been trying to make it extremely effective? — Metaphysician Undercover

Of course math is inspired by the world. It's just not bound by it. A point I've made to you a dozen times by now.

Our discussion, throughout this thread has never been about "how math works". — Metaphysician Undercover

News to me.

We have been discussing fundamental axioms, and not the application of mathematics at all. — Metaphysician Undercover

I agree with that. I've never had the slightest interest in applied math. I'm with Hardy. Math is worthwhile to the extent that it's useless. Of course he was being a bit facetious, I suppose. And in the end even his belovedly useless number theory came to be indispensable to the world.

You are now changing the subject, and trying to claim that all you've been talking about is "how math works", but clearly what you've been talking about has been pure math, and pure abstraction, not application. — Metaphysician Undercover

LOL. Now that's funny. As if I've ever been talking about anything else.

@Meta surely this convo has run its course, don't you agree?

Weierstrass (my math genealogy ancestor — jgill

That's pretty cool!

Thanks for the reading material fishfry, I've read through the SEP article a couple times, and the other partially, and finally have time to get back to you. — Metaphysician Undercover

Glad you found some of this helpful. You talked about a lot of things here I'm not qualified to comment on, but I wanted to go to the end and respond to this first.

Actually, I think that often when one stops replying to the other it is because they get an inkling that the other is right. — Metaphysician Undercover

Another reason is that the conversation has gone past the point where there's any light being generated relative to the word count. And the fact that you think I'm arguing right or wrong is telling. I'm not arguing my point of view is right, I'm not even arguing a point of view. I'm telling you how modern math works. It's like this, if you don't mind a Galilean dialog.

Me: Here is how the knight moves in chess. Two squares in the horizontal or vertical direction, and one in the vertical or horizontal, respectively.

You: But that's wrong! Knights in the real world carry lances and save damsels. You can't just make up rules that don't match the real world.

Me: This is not about the real world, I'm just explaining the rules of this game.

You: You insist that you're right, but you're wrong.

Me: The rules of a game can't be right or wrong, they're just the way they are. I'm explaining, not advocating. I'm not claiming that this is the "right" way to define a knight move. I'm just telling you what the rules of chess are.

You: You're wrong, that's not how knights move.

Me. Ok, well let's agree to disagree. I prefer not to continue like this. Nice chatting with you.

You: That just shows that you know you're wrong.

Me: Oh brother.

I hope you see the parallel. I am NOT saying I'm "right" that math should be the way it is. I'm reporting to you from the front lines of math, about how things are. It's pointless for you to tell me that I'm wrong about how things are, because my report is objective. And it's pointless for you to tell me math "should" be other than it is, because I'm not the Lord High Commissioner of mathematics. I'm just telling you how the twentieth century went. It's the tediosity of holding up my end of this theater of the absurd that leads me to withdraw from the field of play.

So there's a matter of pride, where the person stops replying, and sticks to one's principles rather than going down the road of dismantling what one has already put a lot of work into, being too proud to face that prospect. — Metaphysician Undercover

That doesn't even make sense. I can take no pride in how math is, I was simply trained in universities in the modern style of the subject. Hilbert could take pride, he was one of the major instigators of the movement to abstraction. I'm only a very humble student. I'm putting my knowledge at your disposal, I'm not claiming modern math is right or wrong. I'm describing, not advocating.

I am not claiming the modern approach is right. I am only telling you how they do it. I'm telling you how the knight moves, I'm not claiming the rules of chess are "right" or "wrong." The fact that you don't see this emphasizes the futility of any time I spend typing here.

You, it appears, do not suffer from this issue of pride so much, because you keep coming back, and looking further and further into the issues. — Metaphysician Undercover

I keep thinking I might get through to you. I'm not trying to convince you the modern math approach is right. Why do you think I am?

I think Frege brings up similar issues to me. The main problem, relevant to what I'm arguing, mentioned in the referred SEP article, is the matter of content. — Metaphysician Undercover

Right. You are taking the Fregean point of view, and modern math the Hilbertian. But I don't believe I'm arguing the rightness of the modern abstract way; only trying to describe it to you.

The difference of opinion over the success of Hilbert’s consistency and independence proofs is, as detailed below, the result of significant differences of opinion over such fundamental issues as: how to understand the content of a mathematical theory, what a successful axiomatization consists in, what the “truths” of a mathematical theory really are, and finally, what one is really asking when one asks about the consistency of a set of axioms or the independence of a given mathematical statement from others.
— SEP — Metaphysician Undercover

All this is stipulated. I can't continue the convo since I haven't said anything new in weeks if not months.

In critical analysis, we have the classical distinction of form and content. You can find very good examples of this usage in early Marx. Content is the various ideas themselves, which make up the piece, and form is the way the author relates the ideas to create an overall structured unity. — Metaphysician Undercover

Karl or Groucho? You seem to still want to argue that I'm "wrong" somehow, or explain your point of view. I now understand your point of view. There's nothing I can do about it, I can't imagine doing math the way you suggest, not because one couldn't, but because Frege lost the 20th century and I know no other way.

Hilbert appears to be claiming to remove content from logic, — Metaphysician Undercover

From math, not from logic. Surely you don't claim there's content in logic "P implies Q" and "Q" imply P. There's no content there but that's one of the most ancient logical forms there is.

to create a formal structure without content. In my opinion, this is a misguided adventure, because it is actually not possible to pull it off, in reality. — Metaphysician Undercover

You say modern math is misguided as if you want me to defend the opposite proposition. I'm not defending anything. I'm just describing to you how modern math is. I'm not defending it. I'm reporting to you on what I know about it. And for what it's worth, if it's misguided, the abstract point of view not only won the 20th century, math got even more abstract in the second half of the 20th century and into the present.

This is because of the nature of human thought, logic, and reality. Traditionally, content was the individual ideas, signified by words, which are brought together related to each other, through a formal structure. Under Hilbert's proposal, the only remaining idea is an ideal, the goal of a unified formal structure. So the "idea' has been moved from the bottom, as content, to the top, as goal, or end. This does not rid us of content though, as the content is now the relations between the words, and the form is now a final cause, as the ideal, the goal of a unified formal structure. The structure still has content, the described relations. — Metaphysician Undercover

I'm not arguing the point.

Following the Aristotelian principles of matter/form, content is a sort of matter, subject-matter, hence for Marx, ideas, as content, are the material aspect of any logical work. This underlies Marxist materialism — Metaphysician Undercover

Over my head, way out of my bailiwick. Can't respond.

However, in the Aristotelian system, matter is fundamentally indeterminate, making it in some sense unintelligible, producing uncertainty. Matter is given the position of violating the LEM, by Aristotle, as potential is is what may or may not be. Some modern materialists, dialectical materialists, following Marx's interpretation of Hegel prefer a violation of the law of non-contradiction. — Metaphysician Undercover

This is all very impressive-sounding but is an alien language to me. I can't respond, I have no stake in the matter and no understanding of what you're talking about.

So the move toward formalism by Frege and Hilbert can be seen as an effort to deal with the uncertainty of content, Uncertainty is how the human being approaches content, as a sort of matter, there is a fundamental unintelligibility to it. Hilbert appears to be claiming to remove content from logic, to create a formal structure without content, thus improving certainty. In my opinion, what he has actually done is made content an inherent part of the formalized structure, thus bringing the indeterminacy and unintelligibility, which is fundamental to content, into the formal structure. The result is a formalism with inherent uncertainty. — Metaphysician Undercover

Ok. I'm sure you have the core of a nice essay there, but why are you telling it to me? And again, what do you mean "remove content from logic?" When was logic EVER about content? P and Q, remember?

I believe that this is the inevitable result of such an attempt. The reality is that there is a degree of uncertainty in any human expression. Traditionally, the effort was made to maintain a high degree of certainty within the formal aspects of logic, and relegate the uncertain aspects to a special category, as content. — Metaphysician Undercover

You're just going on without me. I don't relate any of this to anything we've been discussing.

Think of the classical distinction between the truth of premises, and the validity of the logic. We can know the validity of the logic with a high degree of certainty, that is the formal aspect. But the premises (or definitions, as argued by Frege) contain the content, the material element where indeterminateness, unintelligibility and incoherency may lurk underneath. We haven't got the same type of criteria to judge truth or falsity of premises, that we have to judge the validity of the logic. There is a much higher degree of uncertainty in our judgement of truth of premises, than there is of the validity of logic. So we separate the premises to be judged in a different way, a different system of criteria, knowing that uncertainty and unsoundness creeps into the logical procedures from this source. — Metaphysician Undercover

My point was that, having learned about the Frege-Hilbert dispute, I see that you have been arguing Frege's view and I Hilbert's. And in math, Hilbert won the 20th century. This is a matter of fact. There is nothing to argue and no right or wrong. But your discourse in this present post is alien to me, I have no idea what it's about. I am sure you are making interesting points, but they're lost on me.

Now, imagine that we remove this separation, between the truth of the premises, and the validity of the logic, because we want every part of the logical procedure to have the higher degree of certainty as valid logic has. However, the reality of the world is such that we cannot remove the uncertainty which lurks within human ideas, and thought. All we can do is create a formalism which lowers itself, to allow within it, the uncertainties which were formerly excluded, and relegated to content. Therefore we do not get rid of the uncertainty, we just incapacitate our ability to know where it lies, by allowing it to be scattered throughout the formal structure, hiding in various places, rather than being restricted to a particular aspect, the content. — Metaphysician Undercover

Modern math is what it is, and nothing you say changes that, nor am I defending it, only reporting on it. You think it's bad and wrong, ok, I'm no longer arguing the point with you if I ever was.

I will not address directly, Hilbert's technique, described in the SEP article as his conceptualization of independence and consistency, unless I read primary sources from both Hilbert and Frege. — Metaphysician Undercover

Sprichst du Deutsch?

Geometers, and mathematicians have taken a turn away from accepted philosophical principles. This I tried to describe to you in relation to the law of identity. So there is no doubt, that there is a division between the two. Take a look at the Wikipedia entry on "axiom" for example. Unlike mathematics, in philosophy an axiom is a self-evident truth. — Metaphysician Undercover

In math, axioms used to be self-evident truths, and now they're more or less arbitrary assumptions to get a given theory off the ground. No truth is claimed. I had no idea philosophers were still clinging to the old concept of axioms. No wonder they're so far behind in understanding modern math.

Principles in philosophy are grounded in ontology, but mathematics has turned away from this. — Metaphysician Undercover

It was forced on math by the discovery of non-Euclidean geometry. Once mathematicians discovered the existence of multiple internally consistent but mutually inconsistent geometries, what else could they do but give up on truth and focus on consistency?

I'm curious to hear your response to this point. What were they supposed to do with non-Euclidean geometry? Especially when 70 years later it turned out to be of vital importance in physics?

One might try to argue that it's just a different ontology, but this is not true. There is simply a lack of ontology in mathematics, as evidenced by a lack of coherent and consistent ontological principles. — Metaphysician Undercover

You say that like it's a bad thing! It was forced on math by non-Euclidean geometry. Physics is about ontology now. But of course even contemporary physics has abandoned ontology, and if you say that's a bad thing, I'd be inclined to agree with you.

You might think that this is all good, that mathematics goes off in all sorts of different directions none of which is grounded in a solid ontology, but I don't see how that could be the case. — Metaphysician Undercover

It's not good or bad, it is simple inevitable. What should math do? Abolish Eucidean or non-Euclidean geometry? On what basis?

I know you keep saying this, but you've provided no evidence, or proof. Suppose we want to say something insightful about the world. So we start with what you call a "formal abstraction", something produced from imagination, which has absolutely nothing to do with the world. Imagine the nature of such a statement, something which has nothing to do with the world. How do you propose that we can use this to say something about the world. It doesn't make any sense. Logic cannot proceed that way, there must be something which relates the abstraction to the world. But then we cannot say that the abstraction says nothing about the world. If the abstraction is in some way related to the world, it says something about the world. If it doesn't say anything about the world, then it's completely independent from any descriptions of the world, so how would we bring it into a system which is saying something about the world? — Metaphysician Undercover

As evidence I give you "The unreasonable effectiveness of math etc."

it is not the case that Frege and I do not get the method of abstraction. Being philosophers, we get abstraction very well, it is the subject matter of our discipline. You do not seem to have respect for this. — Metaphysician Undercover

You've given me not the slightest evidence that you have any idea how math works. And a lot of evidence to the contrary.

This is the division of the upper realm of knowledge Plato described in The Republic. Mathematicians work with abstractions that is the lower part of the upper division, philosophers study and seek to understand the nature of abstractions, that is the upper half of the upper division. — Metaphysician Undercover

To the extent that philosophers can't deal with mathematical practice as it is, they have no claim on such exaltation.

Really, it is people like you, who want to predicate to abstraction, some sort of idealized perfection, where it is free from the deprivations of the world in which us human beings, and our abstractions exist, who don't get abstraction. — Metaphysician Undercover

I wish. I predicated nothing. I only struggled to learn what I was taught, and I'm reporting back to you how the subject works. "People like me." Jeez man what are you going on about?

The knight moves the way it does. Or as Galileo would have said: "Yet it moves."

That's fine, and I didn't fault you for it. I merely added a point of clarification. — TonesInDeepFreeze

I appreciate your corrections, some of which were on point and some, in my opinion, perhaps not. If I was too sensitive when I read your criticisms, I'll try to grow a thicker skin. Honestly I don't know how people write books, then sit back while critics throw rocks. I could never take it. Actors too.

I prefer not to engage on each point you raised, because the back-and-forth would get in the way of the overall intent of the thread, which is to introduce the ordinals to a casual audience.

I posted to clarify certain points and to keep my mind focused a little bit on math occasionally. — TonesInDeepFreeze

That's certainly fair, and it was uncharitable of me to ascribe ignoble motives to you.

Thanks for your comments.

I think of tradeoffs virtually every time I post, since in such a cursory context of posting, I too have take some shortcuts. The fact that I added clarifications and information doesn't entail that I don't understand that an overview can't cover every technicality. — TonesInDeepFreeze

You're right, I was defensive today. I was going to write "uncharacteristically" defensive but of course that would be a lie :-)

I looked this up, evidently connected relationship is another word for a total relationship. "Today I learned."

Yes, 'x is an ordinal iff x is the order-type of a well ordered set' is a theorem. — TonesInDeepFreeze

I responded to all of your points but then thought better of it.

My challenge was to write something that would give a high-level overview of a difficult subject to casual readers without much math background. Necessarily not everything was perfectly pedantic. So you missed that point entirely.

A number of your statements were flat out wrong, such as claiming that a bijection of a well-ordered set to itself is necessarily another well-order. I had already given the counterexample of the naturals and the integers.

It's just not worth getting into it with you. Your several posts to me seemed not just pedantic, but petty, petulant, and often materially wrong. You either misunderstood the pedagogy or the math itself, often both at the same time. It's just like what you wrote a few days ago, giving a long list of topics to be studied before one can read my article, including a year's worth of abstract algebra. The challenge is to write something that can be read by casual readers WITHOUT any mathematical prerequisites. You don't seem to understand that. You might try it yourself, it's harder than it looks.

I have to tell you your post left me with a bad taste. You seem to just want to throw rocks, but you couldn't even find pebbles, so you threw grains of sand. I hope you got something out of it.

But to give a specific example just so you understand what I'm talking about, let me take your opening remark here:

"Yes, 'x is an ordinal iff x is the order-type of a well ordered set' is a theorem." followed by some picky complaint.

I led with "x is an ordinal iff x is the order-type of a well ordered set" because that's something that I can explain to a casual audience in a couple of paragraphs.

Whereas if I go with the standard textbook definition, "An ordinal is a transitive set well-ordered by $\in$, that would be immune from your petty criticism, but it would lose the entire audience immediately and never get it back. That's why I deliberately, and with thought and consideration, chose the formulation I did. You might give this point some thought. That the idea isn't to be technically pristine; but rather to choose formulations that have a hope of getting through to a casual audience. This entire concept of being understandable to a casual audience went completely over your head.

Likewise your persistent complaint that I omitted the fact that I am talking about total orders (which you called "connected" for reasons I didn't understand). Of course I did. That was a deliberate choice. One of my earlier drafts had a long, boring exposition of order theory, including partial orders, and then I just deleted it and decided to implicitly assume total orders to make the exposition more readable.This never occurred to you, you just kept harping on the point. If you would take a moment to ask yourself, "How would I explain ordinal numbers in a fair amount of depth to a casual audience," you might come to understand some of the tradeoffs involved.

Your OP a Disneyland of rides, and I not tall enough for most of them. — tim wood

To put it into context, it takes a long time to get one's mind around even the basics of the ordinals. If you think of my article as kind of a high-level tourist flight, perhaps that's more helpful. Maybe appreciate a little here and a little there.

So the main thing is that you originally had some questions about the subject based on video you saw. If you have specific questions, perhaps I can answer them.

Also I wonder if you can point me to the video. I'm curious if it's a good vid explained badly or a bad vid explained well, and maybe I'll learn something.

And yet it's countable. That seems strange. — tim wood

It is very strange. The countably ordinals are massively strange, the strangest mathematical structure I know. I looked up proofs that $\epsilon_0$ is countable and I'm not sure I understand them. That's the next thing for me to learn. $\epsilon_0$ is my own personal favorite number. I still have a very limited understanding of it.

With zero and 1, I take it a person can get to any number in {0, 1, 2,.., n}, though perhaps not efficiently. The limit of that being ω. Hmm. The only way I can understand ω or ω+1, is simply as the numbers ω and ω+1, which are just larger than any of the {0, 1,.., n}. — tim wood

$\omega$ is $\mathbb N$ in its usual order.

$\omega + 1$ is $\mathbb N$ in the funny order:

1, 2, 3, 4, 5, 5, 6, ..., 0. It's the same set of numbers, but 0 comes after everything else.

Perhaps that's the most important concept to get. The idea that we can rearrange, or permute, the elements of $\mathbb N$ to get a set with the same cardinality -- it's the same set of elements, after all -- that's also well-ordered, but that has a different order type. Because $\omega + 1$ has a largest element, and $\omega$ doesn't.

I think if we could stay here for a while and understand this one point it would be helpful. It's the heart of the whole matter, your original question of "what is a transfinite ordinal?"

The key to this whole business is to understand $\omega + 1$ as an alternate ordering of the natural numbers, one that has both a smallest and a largest element yet is the exact same set of numbers. Let's drill down there. You said you don't quite understand $\omega + 1$ so let's work on that if you're open to it. That's the key to the whole enterprise.

Maybe I need a bit more care in thinking about what a number is. Transfinite cardinals and ordinals thus not numbers in any naive sense, but in an extended sense, perfectly useful and thus perfectly good. — tim wood

They're transfinite numbers. Definitely different than plain old finite numbers or even real or complex numbers. Sort of analogous to how negative numbers were once new, and irrational numbers and then complex numbers were once new and revolutionary, and are now accepted by everyone. Transfinite numbers came out of nowhere from the mind of one person in the 1870's, and that's not very long ago in the scheme of things.

The idea was that they were simply defined into existence, — tim wood

Large cardinals require additional set-theoretic axioms, correct. That's the definition of large cardinals.

Question: is the change from ω-street to ε-street a "can't get theah from heah" transition? — tim wood

No. Everything we've discussed is provable in ZF. You don't even need the axiom of choice. There are "large ordinals" in the sense that new set-theoretic axioms are needed to prove their existence, but we haven't talked about them.

The "large countable ordinals" are ordinals for which there aren't notations, which aren't computable, and so forth. They still exist within ZF and in fact they're all countable sets. So perhaps "large countable ordinal" and "large ordinal" is misleading terminology, because they're two very different concepts.

I see the language that says you just add a successor, but what successor would that be? — tim wood

You keep taking successors and sups, successors and sups. $\epsilon_0$ is a limit ordinal, it's not the successor of any ordinal.

It's not easy to get your mind around $\epsilon_0$, I've been slowly developing intuition for it and still have a ways to go. It's the limit of the ordinals $\omega$, $\omega^\omega$, $\omega^{\omega^{\omega}}$, $\omega^{\omega^{{\omega^\omega}}}$, etc. Not easy to comprehend. These are all countable, no new axioms are required to construct them.

Even $\omega_1$, the first uncountable ordinal, exists in ZF with no choice needed.

I'm wondering if you feel like you have gotten any of your original questions answered yet, and if this thread has helped put the video you saw into context.

And one can get the "general idea" about ordinals — TonesInDeepFreeze

That's the point.

a clear understanding — TonesInDeepFreeze

I aspire to clear understanding myself. It's an ongoing process.

Here is some of the terminology (not necessarily in logical order) that one must have a very clear understanding of in order to have a clear understanding the matters in this thread. — TonesInDeepFreeze

I emphatically disagree. I wrote my post so as to not need much background at all; pretty much the basics of infinite cardinalities as vaguely understood by the average reader of this site. Whether I was successful or not, I can't yet say. But one need not be a specialist to get the general idea of the transfinite ordinals.

Seems like every popular leftist finite ordinal is coming out as trans these days. Pretty soon they’ll make it illegal to be a cis finite ordinal at all! — Pfhorrest

Set-theoretic comedy is harder than it looks :-)

Every permutation of {1 2 3} "induces" a distinct well ordering of {1 2 3}. — TonesInDeepFreeze

Yes but the same order-type, which is how we define an ordinal. Every possible permutation of a three-element set has the order type of the ordinal 3. Important to note in the present discussion I think.

An idea regarding infinity I’d like your feedback on, since you’re far more knowledgeable regarding mathematics: — javra

“Infinity” is fully synonymous to “unlimitedness”. [/quote]

One hears this a lot since it's a dictionary definition. But it's not the mathematical definition. In math, a quantity is infinite if it can't be put into one-to-one correspondence with some natural number like 1, 2, 3, ...; or else it's infinite if it can be placed into one-to-one correspondence with a proper subset of itself.

Now that's mathematical infinity. There's also physical infinity, the question of whether anything in the world is actually infinite. And there's metaphysical infinity, God, the absolute, the everything of the everything. Those aren't mathematical infinity.

All mathematical infinities (e.g., 0, 1, 2, 3, … infinity) are bounded by being other than what they are not (e.g., 0, -1, -2, -3, … infinity) and so are bounded infinities, or infinities limited to that subject of contemplation addressed. — javra

Yes, I agree. Mathematical infinity is the most limited or constrained or trivial kind of infinity, because it's the one we can reason symbolically about. And you are right, the set of natural numbers 1, 2, 3, 4, ... is infinite, but it doesn't contain any ocelots or pomegranates. So it's infinite but it's not everything. That's why I agree with you that the mathematical infinity is a very trivial and limited kind of infinity.

There is always something other relative to such infinities which demarcates them as such. Here, infinities are quantitative. — javra

Yes, agreed. In fact just the even numbers alone, 2, 4, 6, 8, ... are infinite in number, yet missing all the odd numbers. A poor excuse for infinity. I agree. But we still have a fascinating mathematical theory of infinity, which doesn't tell us anything at all about the universe or God.

Boundless, or complete, or absolute infinity, however, (though I’m not certain if this is in line with Cantor’s works) is not limited nor bound by anything; it is the same as absolute unlimitedness. It is therefore nondual in every conceivable sense of the word: there is nothing other relative to it. Hence it is not, nor can it be, numerical, for it is not quantitative. Nor can it be a quantitative understanding of “greatest” for this always stands in contrast to that which is lesser as other, as can be exemplified by X > Y, which limits the greater to X by excluding Y as the lesser other. — javra

I don't know anything about God or whether the physical universe is infinite, so I can't say. I agree with you though that mathematical infinity is subject to the constraint of being limited to mathematical objects only, and doesn't tell us anything about God or the physical universe.

And even Cantor's absolute infinity was a mathematical infinity, it was just the class of all ordinal numbers. So I don't even know how he got God from that. Probably why his theological ideas are long forgotten.

(In terms of the overall thread: Other than boundless infinity’s possible correlation to the notion of omnipresence, I don’t see how this can make the case for God as typically conceived: e.g., the greatest being among all other beings.) — javra

I don't know anything about boundless or metaphysical infinity, only mathematical infinity. As a sign I once saw on a math prof's door said: Good sense about trivialities is better than nonsense about things that matter!

Hmm. New for me: well-ordered does not mean in-order, yes? — tim wood

I'm not sure what you mean by in-order. Do you mean in its usual order? No. A well-order is an order in which every nonempty subset has a smallest element. Like the natural numbers in their usual order.

! Is it correct to think of all the well-orderings to be the same thing as all the permutations? — tim wood

No, not at all. For example we know from cardinality theory that there is a bijection from the natural numbers to the integers. So we can in fact permute the natural numbers, which are well ordered, to the integers, which aren't.

It's easier to see this in the other direction. In their usual order, the integers are:

..., -4, -3, -2, -1, 0, 1, 2, 3, 4, ...

which is NOT a well-order, because there is no first element.

But what if I reorder them like this:

0, -1, 1, -2, 2, -3, 3, -4, 4, ...

That IS a well-order, because it has the same order-type as the natural numbers. So clearly we can take a set that's not well-ordered, permute its elements, and make it well-ordered. And we could do the same in reverse.

For the natural numbers, that would just be ω! (or maybe better aleph-0!), yes? The image I have - that maybe I have to work through - is of something like a deck of card. Fifty-two of them. With 52! possible arrangements. With the cards, at least, you can't get past 52! arrangements without duplication. — tim wood

Every permutation of a finite set is a well-order. All finite sets are well-ordered and every permutation of a finite set has exactly the same order type. If I have a, b, c and then c, b, a, there's clearly an order-isomorphism between those two orders.

I suppose similarly there is an upper limit on the arrangements of NN. And I get it that each of those is countable. Why not, they're just arrangements. — tim wood

We can rearrange 0,1,2,3,4,... into 1, 2, 3, 4, ..., 0. The former has order type $\omega$, and the latter has order type $\omega + 1$. So for infinite sets, permuting does not preserve order type.

Again how to be brief? — tim wood

I might not be the best person to give advice about that :-)

Following with your construction above, it seems that ω is the ordinal associated with NN. — tim wood

With $\mathbb N$ in its usual order. But we can rearrange $\mathbb N$ to get very many different well-orders, and we can rearrange it so it's not well-ordered at all, as in the integer example I gave.

$\mathbb N$
But it also seems that ω is also associated with every infinite subset of NN.

In its usual order, but not necessarily if we reorder it.

$\mathbb N$
On this basis I (think) I can see where ω+1, ω+ω, ωω,..., come from.

$\omega + 1$ is the order 1, 2, 3, ..., 0.

$\omega + \omega$ is the even-odd order 0, 2, 4, 6, 8, ..., 1, 3, 5, 7, ...

$\omega \times \omega = \omega^2$ is a tricky one. We need $\omega$ copies of $\omega$. There's no "obvious" rearrangement of the natural numbers that does that, even though there is one. It's just not obvious. But we can use the rationals.

Consider, in the rationals, the set 1/2, 3/4, 7/8, 15/16, ... That's an instance of $\omega$.

So 1/2, 3/4, 7/8, ..., 1+1/2, 1+3/4, 1+7/8, ..., 2+1/2, 2+3/4, 2+7/8, ..., 3+1/2, 3+3/4, 3+7/8, ...,

gives us a subset of the rationals in their usual order that's an instance of $\omega$ copies of $\omega$, or $\omega^2$.

All of the countable ordinals are buried in the rationals in their usual order, so the rationals give a nice class of visualizations.

That leaves the question, how many infinite subsets of NN are there? — tim wood

$2^{\aleph_0}$, right? That's basic cardinality. Well there are $2^{\aleph_0}$ subsets all together, but only countably many infinite subsets, leaving $2^{\aleph_0}$ infinite subsets. You know the old song. $\aleph_0$ bottles of beer on the wall, $\aleph_0$ bottles of beer, you take one down, pass it around, $\aleph_0$ bottles of beer on the wall!

(And with each ω as a subset, if well-ordering does not require being in-order, then each of those yielding ω! permutations - yes?) — tim wood

What do you mean "each" $\omega$? There's only one such ordinal. It would be like saying "each 3." There's only one 3.

By in-order do you mean in its usual order? Well-orders don't require a set being lined up in its usual order. We've seen that some permutations result in well-orders and others don't, like the naturals-to-integers example. And the notation $\omega !$ is not defined at all, for the reason that you can't subtract 1 from a limit ordinal. So we couldn't sensibly define that even if we tried.

If all of this ends at ε - does it end at ε? - — tim wood

Do you mean $\epsilon_0$? No, there's $\epsilon_0 + 1,$, $\epsilon_0 + 2$, etc. We can always take successors. Remember there are two rules:

* Given an ordinal, we can always take its successor.

* Given a set of ordinals, we can always take their sup or least upper bound, in the same way we can aggregate 0, 1, 2, 3, ... into the single ordinal $\omega$ and then keep on going with successors.

The procession of ordinals never ends.

then how do you get beyond ε and still be countable? — tim wood

Keep taking successors, and when you accumulate a bunch of those, accumulate them into their sup, or least upper bound.

If it doesn't end, how do you get larger ordinals. — tim wood

Successors and sups. Successors and sups. Never ends.

I'm still reading. I suppose the answer is up ahead. I'll look for it, then, and wait for it. — tim wood

Keep asking questions. These are not easy concepts, the ordinals are a deep idea.

Can you tell me what video you saw? I'm curious to know.

If you don't mind I'll take a shot at answering your questions to @Tones.

I was thinking naively that well-ordering means the set can be and is ordered lexicographically. — tim wood

Well-ordering means that every nonempty subset has a least, or first, or smallest, element. Means the same thing. NOT like the integers, which have no first element. Does that make sense?

{a,b,c} is ordered lexicographically, and {b,c,a} isn't, but they have the same order type and it's a well-order.

But with the OP I'm thinking the "and is" isn't part of it. E.g., I thought 1,2,3 is well-ordered when presented as {1,2,3}. But now I'm thinking that 1,2,3 is well ordered in each, and all, of six variations. If the latter is true, then my "uniquely orderable" is just a mistake. — tim wood

Every permutation of a FINITE set is well-ordered and is the same order type.

That's a clever approach. Cantor would have presumably claimed that God was the sum of all those infinite infinities... Your point is that there is no such summation - there is always more to be added? — Banno

Are you referring to Cantor's absolute infinite? The point is that the class of all ordinals is not a set. Well that's not my point, it's just a fact that Cantor and others discovered, Burali-Forti in particular. But I'm not sure exactly how to relate this to your post to me here. Apologies, I'm a bit lost.

I have to pause here because you have said explicitly what I thought was an error on a video.[/url]

What vid, I'll take a look. Youtube giveth and Youtube taketh away.
— tim wood

An ordering has a first element (yes?).[/url]

A well-ordering, yes. Orderings in general need not have first elements, for example the usual < on the integers. But well-orderings always have a first, second, third, ..., element.

— tim wood

At some point candidates for the second element are exhausted, how then going forward does it remain uniquely orderable? — tim wood

Not sure I understand the question. The ordering is given, we don't have to find it. Like the usual ordering on the natural numbers: 0, 1, 2, 3, ... We are given the ordering, then we observe that it's a well-ordering in which each nonempty subset has a smallest element.

And if it does, then how do you get to ε? — tim wood

The key point is that the natural numbers start with 0 and are then produced by taking successors. The ordinals start with 0 and are produced by taking successors, and then accumulating successors in the limit, or sup operation. The ordinals are like the naturals, but with two generators -- successor and sup -- rather than just the one successor operation of the naturals.

So we go 0, 1, 2, 3, ...; and then we slurp up all of those into $\omega$ and then keep on going: 0, 1, 2, 3, ..., $\omega, \omega + 1, \omega + 2, \dots, \omega 2, \dots, \omega 3, \dots$, and all the way up to $\epsilon_0$ and beyond.

It's a little dizzying, it honestly takes a long time to get used to. But the basic idea is successors and sups, successors and sups.

Btw, for clarity on a difficult subject, you're hitting it out of the park! — tim wood

Thanks.

OK, I'm not reading all that... but thank you. — Banno

I found that shortness and clarity were inversely related. I tried best I could to make it short and clear, and in the end strove for clarity. Readers will have to say whether I succeeded. For what it's worth, ordinals are a little complicated. I've been coming back to this material periodically for years.

I recommend a book called Infinity and the Mind by Rudy Rucker. It has a fantastic visualization of the countable ordinals.

↪fishfry Well done! Handy reference for an oldtimer, Too. :cool: — jgill

Thanks!

Attempt at a coherent question here. Maybe best to leave it simple. What is an infinite ordinal? — tim wood

Such a great question. Thank you for the inspiration. I didn't want to hijack this thread with my lengthy exposition, so I posted it in the Math section over here ....

Might I approach it from a slightly physics direction. Assuming that maybe this “god” is the sum of all energy in the universe, god must be finite as the law of conservation of energy would dictate: cannot be created nor destroyed. Although finite in quantity, energy being the ability to do work one could say they must be infinite in quality - that is to say can transform from one form to the next. Cannot be destroyed cannot be created but ALWAYS changing — Benj96

Just passing by late at night, not following the convo lately. The theory of eternal inflation is a speculative physical theory that posits a a multiverse in which time is infinite in the forward direction and there are an actual infinity of universes. This is as I say speculative, and the author of the theory no longer believes in it, or no longer considers it a useful theory. Nevertheless, speculative cosmologists do consider infinite theories these days. They're probably bullpucky, but they are out there. So clearly physicists aren't troubled by conservation of energy when scribbling out these kinds of theories.

Did we really though? I think we conceived the conditions for an infinity. I can conceive 10s or maybe hundreds and infer about millions and billions, but saying I'm thinking about the impossible whole of infinity seems reaching. — Cheshire

You just need to stretch your imagination. Think of each of 0, 1, 2, 3, 4, ... where you can always "add 1." Then think of all of these together in one place, what we call the set of natural numbers {0, 1, 2, 3, 4, 5, ...}.

Now I admit that this is not a very natural thing to conceive of. But if you spend enough time studying it, it does become second nature and you can conceive of it perfectly well. I tell you honestly that I have no problem conceiving of the set of natural numbers, because I spent enough time in school studying it.

"Conceive of" is a very weak standard of existence because it's subjective. What one person can't conceive of, another person finds commonplace.

It may be true that you personally can't conceive of the set of natural numbers, but I assure you that people who study math in college end up with a very clear picture of it in their minds. And far larger sets too.

Think about the real number line that they taught you in high school. It has points at all the integers like ---, -4, -3, -2, -1, 0, 1, 2, 3, 4, ..., going to infinity in both directions. Between each whole number are the rationals like 1/2 and 2/3 and 47/99, etc., which are three rational numbers between 0 and 1. And there are all the irrationals like sqrt(2) and pi and e and so forth. Each tiny little dimensionless point on the line is the location of some real number; and there are a whole lot of those, provably more of them than there are integers.

And you learned this in high school. So you learned it, but never really spent much time conceiving it.

My thesis is that if you had simply spent a few years studying the real line and its properties, you'd end up conceiving of it perfectly well.

Now I agree that this is far short of conceiving of the infinity of God or the infinity of the cosmos, or whatever. Mathematical infinity is a very limited kind of infinity, it's the one that we can talk about using symbolic logic. But still, it's infinity and lots of people do conceive of it.

Attempt at a coherent question here. Maybe best to leave it simple. What is an infinite ordinal? — tim wood

Terrific question. I'm working on a response. I can make it simple but I can't make it short. Anything I write is going to be grossly off-topic to this thread. How do the assembled multitudes feel about that? I'll just go ahead and write up my response and post it here later or tomorrow, whenever it gets done. As far as being on topic, as I mentioned, Cantor thought that the ultimate ordinal was God. And maybe it is.

How do you go about conceiving infinity? — Cheshire

Take a class in set theory. Or just contemplate the set of natural numbers {0, 1, 2, 3, ...} that you learned about in grade school. Everybody believes in the natural numbers. So why would it be difficult to conceive infinity? We did it in grade school. "Name the biggest number." "A zillionty-zillion." "A zillionty-zillion plus one!" That's infinity. The sequence keeps going forever.

And the transfinite ordinals are what you get when you count through all the natural numbers and keep on going. I'll explain that in detail soon. Perhaps it should be in its own thread. I didn't realize we were in the ontological argument thread, this is definitely going to be a little off topic.

To what extent does a Wikipedia article substitute for reading the full text of the original works? — Pantagruel

I describe this as "knowing" versus "knowing about." To know brain surgery takes a lifetime. To know "about" brain surgery takes about thirty seconds of skimming the Wiki article on the subject.

The problem comes when so many people think they know, when they only know about.