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.