I don't want to watch a video right now.

Cantor's diagonal argument says that any list of reals is incomplete. We can prove it directly by showing that any list of reals (not an assumed complete list, just any arbitary list) is necessarily missing the antidiagonal. Therefore there is no list of all the reals. — fishfry

Exactly.

I'll ask again:

But if you start from that there is no bijection, and then prove it by:
If there is a bijection then there is a surjection
There is no surjection.
Therefore, there is no bijection.

Isn't that a proof by contradiction?
— ssu — ssu

I gave you a very detailed answer. I can't do better than what I already wrote. Or, if you like, let me know what you don't understand in my post.

Only countably many interpretations of each sentence. — fishfry

I'm talking about interpretations for languages as discussed in mathematical logic.

There are uncountably many sets, so there are uncountably many universes for interpretations.

Or, another way: Consider just one uncountable universe. Let the language have at least one individual constant. Then there are uncountably interpretations as each one maps the constant to a different member of the universe.

I don't propound the notion that that approach could be adapted for natural languages too, but it doesn't seem unreasonable to me.

a trip to the moon on gossamer wings — fishfry

Seeing just that one phrase from the great song made my night. Such a soul satisfyingly beautiful song by a gigantically great composer.

We're talking about different things. I'm talking about formal theories and interpretations of their languages as discussed in mathematical logic, and such that theories are not interpretations. — TonesInDeepFreeze

enderton page ref please or st*u. second time i'm calling your bluff on references to your magic identity theory.

Exactly. — TonesInDeepFreeze

Stop agreeing with me, that's no fun!

(edit) So you see I do know some logic after all!

I don't propound the notion that that approach could be adapted for natural languages too, but it doesn't seem unreasonable to me. — TonesInDeepFreeze

ok

Seeing just that one phrase from the great song made my night. Such a soul satisfyingly beautiful song by a gigantically great composer. — TonesInDeepFreeze

You're alternately insulting and praising me. Make up your mind!

enderton page ref please or st*u. second time i'm calling your bluff on references to your magic identity theory. — fishfry

Did you mean for that to be in the 'Infinity' thread?

In that thread, you've now seen that I already had given you the Enderton pages yesterday and I gave them to you even though you had not asked for them. There's no bluff and never has been. I've been giving you post after post of correct corrections, information and explanations. It's not my fault that you regard that as inimical.

I know you're kidding. But underneath there lies an actual point for me, which is that I don't think you know how insulting you are in certain threads when you read (if it can be called 'reading') roughshod over my posts, receiving them merely as impressions as to what I've said, so that you so often end up completely confusing what I've said and then projecting your own confusions onto me.

But I do appreciate that you quoted Cole Porter's so charming and magical lyric. And there was another special musical moment for me today, so my evening was graced.

Thank you, @fishfry

It seems that from you I get extremely good answers. Yes, Lawvere's fixed point theorem was exactly the kind of result that I was looking for. It's just typical that when the collories are discussed themselves, no mention of this. I'll then have to read what Lawvere has written about this.

And that not necessary is important for me. This is what @TonesInDeepFreeze was pointing out to me also. I'll correct my wording on this.

It seems that from you I get extremely good answers. — ssu

Thank you.

Yes, Lawvere's fixed point theorem was exactly the kind of result that I was looking for. It's just typical that when the collories are discussed themselves, no mention of this. I'll then have to read what Lawvere has written about this. — ssu

If you're interested in this stuff, do you know the nLab Cafe? It's a category theory wiki. Here's their page on the theorem

It's all very categorical. Like a new paradigm for thinking about math.

And that not necessary is important for me. This is what TonesInDeepFreeze was pointing out to me also. I'll correct my wording on this. — ssu

I'm not sure how the subject came up. It's interesting to know that all these diagonal type proofs can be abstracted to a common structure. They are all saying the same thing.

I know you're kidding. But underneath there lies an actual point for me, which is that I don't think you know how insulting you are in certain threads when you read (if it can be called 'reading') roughshod over my posts, receiving them merely as impressions as to what I've said, so that you so often end up completely confusing what I've said and then projecting your own confusions onto me. — TonesInDeepFreeze

If I crossed any lines, I apologize. But I think you are equivocating the word "insult." If I tell you, Tones, you are a low down rotten varmint who cheats at cribbage!" that's an insult.

But if I don't happen to dwell on every word you write; and if I often find your expository prose convoluted and unclear, especially when you lay out long strings of symbols without any context; my eyes do glaze over, and I do skip things.

That is not an insult. It's just me being me, reacting to whatever you wrote that made my eyes glaze. The fault is all mine, But that's who I am and how I am. I am not insulting you.

Can you see the difference between:

(a) Me actively and directly insulting you; and

(b) Me just being my highly imperfect self, doing something that annoys you.

Surely you can see the difference.

But I do appreciate that you quoted Cole Porter's so charming and magical lyric. And there was another special musical moment for me today, so my evening was graced. — TonesInDeepFreeze

Well that's good, so let's go with the grace.

I'm not sure how the subject came up. — fishfry

From the OP at least I made the connection.

It's interesting to know that all these diagonal type proofs can be abstracted to a common structure. They are all saying the same thing. — fishfry

That's what really intrigues me. Especially when you look at how famous and still puzzling these proofs are...or the paradoxes. Just look at what is given as corollaries to Lawvere's fixed point theorem:

Cantor's theorem
Cantor's diagonal argument
Diagonal lemma
Russell's paradox
Gödel's first incompleteness theorem
Tarski's undefinability theorem
Turing's proof
Löb's paradox
Roger's fixed-point theorem
Rice's theorem

Of course in mathematics a lot theorems have corollaries, but I would just point out to what these theorems are about: limitations in proving, limitations in computation and a paradox, that basically ruined naive set theory and spurred the creation of ZF-logic. All coming from a rather simple thing.

Going back to the OP and the article given there, perhaps in the future it will be totally natural (or perhaps it is already) to start a foundation of mathematics or a introduction to mathematics -course with a Venn diagram that Yanofsky has page 4 has. Then give that 5 to 15 minutes of philosophical attention to it and then move to obvious section of mathematics, the computable and provable part.

Going back to the OP and the article given there, perhaps in the future it will be totally natural (or perhaps it is already) to start a foundation of mathematics or a introduction to mathematics -course with a Venn diagram that Yanofsky has page 4 has. Then give that 5 to 15 minutes of philosophical attention to it and then move to obvious section of mathematics, the computable and provable part. — ssu

IMO those concepts are far too subtle to be introduced the first day of foundations class. Depending on the level of the class, I suppose. Let alone "Introduction to mathematics," which sounds like a class for liberal arts students to satisfy a science requirement without subjecting them to the traditional math or engineering curricula. Truth versus provability is not a suitable topic near the beginning of anyone's math journey. IMO of course.

Hello

IMO those concepts are far too subtle to be introduced the first day of foundations class. — fishfry

Agreed.

What children would really benefit from, is someone to teach them hope, preferably of the most irrational kind, i.e. the stronger, the better.

The mathematics class is clearly not suitable for that, but the mathematics teacher could actually be. But then again, in that case, he is not teaching math but trying to keep the students teachable. That is another job altogether.

Adults cannot teach hope to the children anymore.

Even the children's own (usually hopelessly divorced) families are no longer able to do that. You cannot teach what you don't have. That is why the children grow up believing that there is no hope.

The culture most excelling at "scientifically" inspired hopelessness, is communist China, but the West is clearly not far behind.

Nowadays the young Chinese want to "tang ping" (Chinese: 躺平; lit. 'lying flat') and believe that you should "bai lan" (Chinese: 摆烂; pinyin: bǎi làn; lit. 'let it rot').

The Chinese youth also increasingly believe in the "10 no's" (or the 10 don't") and insist that they are "the last generation". That is obviously a completely true, self-fulfilling prophecy.

The Chinese communist party react by trying to censor and ban public expressions of nihilism or absurdism, even though these things are the natural end point of believing that only pure reason can be a legitimate source of meaning.

There is much more to the struggle with the absurd than just sleeve tattoos, piercings and blue hair. The people who are the most in need of hope, are the least likely to find any.

If someone else does not keep them teachable, then all teaching will be in vain. There no longer exists anybody who can do that.

Truth versus provability is not a suitable topic near the beginning of anyone's math journey. IMO of course. — fishfry

Me too.

IMO those concepts are far too subtle to be introduced the first day of foundations class. Depending on the level of the class, I suppose. Let alone "Introduction to mathematics," which sounds like a class for liberal arts students to satisfy a science requirement without subjecting them to the traditional math or engineering curricula. — fishfry

There's a lot that in mathematics is simply mentioned, perhaps a proof is given, and then the course moves forward. And yes, perhaps the more better course would be the "philosophy of mathematics" or the "introduction to the philosophy of mathematics". So I think this forum is actually a perfect spot for discussion about this.

Of course it would be a natural start when starting to talk about mathematics, just as when I was on the First Grade in Finland the educational system then had this wonderful idea of starting to teach first grade math starting with ...set theory and sets. Ok, I then understood the pictures of sets, but imagine first graders trying to grasp injections, surjections and bijections as the first thing to learn about math. I remember showing my first math book to my grand father who was a math teacher and his response was "Oh, that's way too hard for children like you." Few years later they dropped this courageous attempt to modernize math teaching for kids and went backt to the "old school" way of starting with addition of small natural numbers with perhaps some drawings and references about a numbers being sets. (Yeah, simply learning by heart to add, subtract, multiply and divide by the natural numbers up to 10 is something that actually everybody needs to know.)

Truth versus provability is not a suitable topic near the beginning of anyone's math journey. IMO of course. — fishfry

It sure is interesting. And fitting to a forum like this. If you know good books that ponder the similarity or difference of the two, please tell.

First Grade in Finland the educational system then had this wonderful idea of starting to teach first grade math starting with ...set theory and sets — ssu

Called the New Math in the USA. I can't even imagine this in grade one. I taught elements of it in college algebra courses in the 1970s - but not for long.

These are all mathematical truths, but they're not very interesting mathematical truths. — fishfry

Here is what ChatGpt has to say about mathematical truth:

In mathematics, truth is typically understood within the framework of logical consistency and proof. Here are a few key aspects of truth in mathematics:

Logical Consistency: Mathematical statements and propositions must be internally consistent. This means that there should be no contradictions within a mathematical system. For example, in Euclidean geometry, the parallel postulate is consistent with other axioms, but in non-Euclidean geometries, different parallel postulates lead to different but internally consistent geometries.

Verification through Proof: In mathematics, a statement is considered true if it has been proven using rigorous logical arguments based on accepted axioms and definitions. The process of proving involves demonstrating that the statement follows logically from these axioms and previously proven statements (lemmas).

Objective Reality: Mathematical truth is independent of human beliefs or opinions. Once a mathematical statement has been proven, it is universally accepted as true within the mathematical community. This aspect of objectivity distinguishes mathematical truth from truths in other domains, which may depend on subjective interpretation or observation.

Unambiguity: Mathematical statements are precise and unambiguous. Each term used in mathematics is defined rigorously, and the rules of inference and logical operations are well-defined. This clarity ensures that the truth of mathematical statements can be objectively assessed.

Scope of Truth: In mathematics, truths are often considered to be eternal and immutable once proven. For example, the Pythagorean theorem, once proven, remains true indefinitely and universally applicable within the domain of Euclidean geometry.

In essence, truth in mathematics is grounded in rigorous logical reasoning, proof, and adherence to accepted axioms and definitions. It is a fundamental concept that underpins the entire discipline, allowing mathematicians to build upon previously established truths to explore new areas and make further discoveries.

That needs work.

It leaves out that for the most used overall system for mathematics, it is not the case that every truth is provable.

It leaves out that the concept of mathematical truth is actually not formulated in terms of proof. Rather, proof and truth are formulated separately, but then mathematics shows that, for first order logic: A statement is provable from a set of premises if and only if the truth of the premises entails the truth of the statement.

It leaves out that the greatest objectivity is in the fact that it is machine checkable whether, at least in principle, a given formal sequence that is purported to be a proof is actually a formal proof.

That needs work — TonesInDeepFreeze

It is a tad simplistic. But it is as far as I went in that direction in my career; as for infinity, I never quite reached it for it lay beyond bounds. It's good you and fishfry are more up to date. Thanks for your service.

There's a lot that in mathematics is simply mentioned, perhaps a proof is given, and then the course moves forward. And yes, perhaps the more better course would be the "philosophy of mathematics" or the "introduction to the philosophy of mathematics". So I think this forum is actually a perfect spot for discussion about this. — ssu

I agree that's suitable for this forum. Just not for "Intro to Math," which I interpreted as "Last math class the liberal arts majors will take," or something like the Discrete Math class they teach these days to math and computer science majors.

Of course it would be a natural start when starting to talk about mathematics, just as when I was on the First Grade in Finland the educational system then had this wonderful idea of starting to teach first grade math starting with ...set theory and sets. Ok, I then understood the pictures of sets, but imagine first graders trying to grasp injections, surjections and bijections as the first thing to learn about math. I remember showing my first math book to my grand father who was a math teacher and his response was "Oh, that's way too hard for children like you." Few years later they dropped this courageous attempt to modernize math teaching for kids and went backt to the "old school" way of starting with addition of small natural numbers with perhaps some drawings and references about a numbers being sets. (Yeah, simply learning by heart to add, subtract, multiply and divide by the natural numbers up to 10 is something that actually everybody needs to know.) — ssu

That sounds like the "New Math" they had when I was in school. I loved it but it was a failure in general.

I don't think they teach basic arithmetic anymore. It's a problem in fact.

It sure is interesting. And fitting to a forum like this. If you know good books that ponder the similarity or difference of the two, please tell. — ssu

There's always Gödel's Proof by Nagel and Newman. And Gödel, Escher, and Bach: An Eternal Golden Braid by Hofstadter. Actually I only leafed through it once but everyone raves about it. I'm not up on the literature of pop-mathematical logic. Or real mathematical logic, for that matter.

Here is what ChatGpt has to say — jgill

Et tu? ChatGPT doesn't know anything about mathematical philosophy. It just statistically autocompletes strings it's been fed.

That sounds like the "New Math" they had when I was in school. I loved it but it was a failure in general.

I don't think they teach basic arithmetic anymore. It's a problem in fact. — fishfry

There's many things they don't teach in school when looking at what my children have to study. Usually the worst thing is when the writers of school books are too "ambitious" and want to bring in far more to the study than the necessities that ought to be understood.

Here's the general theorem in the setting of category theory. It's called Lawvere's fixed point theorem. Not necessary to understand it, just handy to know that all these diagonal-type arguments have a common abstract form. — fishfry

I looked at this. Too bad that William Lawvere passed away last year. Actually, there's a more understandable paper of this for those who aren't well informed about category theory. And it's a paper of the same author mentioned in the OP, Noson S. Yanofsky, from 2003 called A Universal Approach to Self-Referential Paradoxes, Incompleteness and Fixed Points. Yanofsky has tried to make the paper to be as easy to read as possible and admits that when abstaining from category theory, there might be something missing. However it's a very interesting paper.

In it he makes very interesting remarks:

On a philosophical level, this generalized Cantor’s theorem says that as long
as the truth-values or properties of T are non-trivial, there is no way that a
set T of things can “talk about” or “describe” their own truthfulness or their
own properties. In other words, there must be a limitation in the way that T
deals with its own properties. The Liar paradox is the three thousand year-old
primary example that shows that natural languages should not talk about their
own truthfulness. Russell’s paradox shows that naive set theory is inherently
flawed because sets can talk about their own properties (membership.) Gödel’s
incompleteness results shows that arithmetic can not talk completely about
its own provability. Turing’s Halting problem shows that computers can not
completely deal with the property of whether a computer will halt or go into
an infinite loop. All these different examples are really saying the same thing:
there will be trouble when things deal with their own properties. It is with this
in mind that we try to make a single formalism that describes all these diverse
– yet similar – ideas.

The best part of this unified scheme is that it shows that there are really no
paradoxes. There are limitations. Paradoxes are ways of showing that if you
permit one to violate a limitation, then you will get an inconsistent systems.

And I would really underline the last chapter above. The issue is about limitations and if you end up in a paradox, you simply have had an inconsistent system to start with. Usually in the way that your premises or the "axioms" you have held to be obviously true, aren't actually true, not at least in every case. Hence an outcome similar to Russell's paradox is simply a logical consequence of this. Also understanding that these are limitations doesn't mean that the consistency of mathematics is brought to question. I think on the contrary: you simply have to have these kind of limitations for mathematics to be logical and consistent.

(If anybody is interested, there are some classes by Yanofsky in Youtube, for example Outer limits of reason. I haven't watched them yet, so I cannot rate them.)

Et tu? ChatGPT doesn't know anything about mathematical philosophy. It just statistically autocompletes strings it's been fed. — fishfry

Here is a quote from Reddit that brings some clarity to the subject of "truth" in mathematics these days:

Godel's completeness theorem, applied to group theory, says that any statement that's true for every group can be proved from the axioms of group theory. Similarly, there is more than one model of ZFC. The existence of various models of ZFC is analogous to the existence of different groups. Some statements are true in one model of ZFC and false in another. Such a statement is independent of ZFC.

I'm an antique. Truth for me is associated with proof.

proof implies truth, but truth does not imply proof.

Suppose we have a consistent set of axioms for mathematics (the set theory axioms will do nicely). Then if the axioms are true then all theorems derived from those axioms are true. But there are truths not derivable from the axioms.

In other words, whatever is provable is true. But it's not the case that whatever is true is provable.

But there are truths not derivable from the axioms. — TonesInDeepFreeze

People who believe that pure reason is the only source of meaning will never accept this, no matter how often you hammer it into their heads.

Even if we had the axioms of the physical universe, most of its facts would still be inexplicable. Stephen Hawking already pointed that out, but apparently nobody cared:

https://www.hawking.org.uk/in-words/lectures/godel-and-the-end-of-physics

What is the relation between Godel’s theorem and whether we can formulate the theory of the universe in terms of a finite number of principles? One connection is obvious. According to the positivist philosophy of science, a physical theory is a mathematical model. So if there are mathematical results that can not be proved, there are physical problems that can not be predicted.

With the overwhelming majority of facts inaccessible even from the perfect axiomatic theory of the universe, it is clear that the "God of gaps" conjecture is simply nonsensical.

Hence, all of this is clearly very unpopular.

I'm an antique. Truth for me is associated with proof. — jgill

If you cannot accept the true nature of the truth, you may need its false nature for your worldview. I don't know how much you are invested in positivism, if at all. A positivist will never accept the truth about the truth.

In other words, whatever is provable is true. But it's not the case that whatever is true is provable. — TonesInDeepFreeze

'Not everything that counts can be counted, and not everything that can be counted counts'

People who believe that pure reason is the only source of meaning will never accept this, no matter how often you hammer it into their heads. — Tarskian

Any examples of those people come to mind?

Any examples of those people come to mind? — Wayfarer

Positivists are like that:

https://g.co/kgs/3yTBNh7

a philosophical system that holds that every rationally justifiable assertion can be scientifically verified or is capable of logical or mathematical proof

(and that therefore rejects metaphysics and theism)

.

In fact, it may actually be ok to reject metaphysics and/or theism but not for positivist reasons.

These people really exist. David Hilbert was one and he even wanted proof for positivism:

https://en.m.wikipedia.org/wiki/Hilbert%27s_program

Statement of Hilbert's program
...
Completeness: a proof that all true mathematical statements can be proved (in the formalism).
...
Gödel's incompleteness theorems, published in 1931, showed that Hilbert's program was unattainable for key areas of mathematics.

A lot of people simply ignore Godel's work and continue to behave as if positivism makes sense. I cannot readily pinpoint anybody in particular but I know that the false belief is widespread. The problem is certainly not imaginary.

Positivists are like that. — Tarskian

Sure. I’ve always rejected positivism, although for different reasons. I see positivism as being a kind of undercurrent in modern thought. But I don't know if Hilbert fits the bill. Hilbert's work in mathematics and his foundational program, known as Hilbert's program, aimed to provide a solid foundation for all of mathematics by formalizing it and proving its consistency using finitary methods. This goal aligns more with a foundationalist approach rather than with positivism per se.

Positivism, particularly as developed by the Vienna Circle in the early 20th century, emphasizes empirical science and the idea that meaningful statements are either empirically verifiable or logically necessary.

Hilbert was more concerned with the internal consistency and formalization of mathematics rather than the empirical verification of mathematical statements. His program sought to ground mathematics on a set of axioms and prove its consistency through purely syntactic means, without reference to empirical content.

I did a unit on A J Ayer's Language Truth and Logic, which is a canonical text of positivism, and found it immensely annoying. I was pleased to learn that it had became evident not long after its publication, that Ayer's style of positivism was self-contradictory, because the kind of verificationism that he insisted on, could neither be validated nor falsified by empirical methods. So it failed its own criteria! My tutor said it was like the mythical Uroboros, the snake that eats itself. 'The hardest part', he would say with a wink 'is the last bite.'

But while there are some overlaps in the emphasis on formalism and logic, Hilbert's aims were distinct from the broader philosophical tenets of positivism.

So I agree with your rejection of positivism, but not for your reasons.