Please quote any passage in that article that you think claims that Wittgenstein said that in mathematics 'infinite' means 'finite'.
— TonesInDeepFreeze
It is his metaphor
— Corvus

Asking a second time, what quote in the article do you claim supports your claim that Wittgenstein said that mathematics takes 'infinite' to mean 'finite'? — TonesInDeepFreeze

It is a metaphor from my point of view. It is obvious, and I have kindly explained it to you above.
It a way of expression saying, when something is so bad, one could say "Well it's f***ing great." Anyone can see and use it to describe the situation with cynicism. If you have problem understanding it, I cannot help you.

If you are asking in which article he said it, I recall it was from a book I don't own. But I saw it in the internet somewhere. I will try to find it, and update on the book title and page. I don't have the information off hand. I couldn't have made the quote from my own imagination. To me, it sounded a genius in the expression at the time of reading it.

However, it explains the historical background of the concept of infinity how controversial the concept was in detail. — Corvus

It was controversial when they didn't know better. It's not controversial now because they know better. Those opposed to set theory now are, for the most part, non-mathematicians who don't know better but think they do.

Let mathematicians argue about set theory. Anyone else just isn't equipped to understand the matter.

It was controversial when they didn't know better. It's not controversial now because they know better. Those opposed to set theory now are, for the most part, non-mathematicians who don't know better but think they do. — Michael

Many still believe it is controversial, and I do too. No one is saying it is illegal to use it, but just pointing out the existence of the controversy and also reservation on the theory. No one can deny that.

In real life infinite set doesn't exist. I start putting something in a box forming a set, soon the box gets full or the object runs out. It doesn't go on filling the box forever.

Many still believe it is controversial, and I do too. — Corvus

Many mathematicians?

Many still believe it is controversial, and I do too. No one is saying it is illegal to use it, but just pointing out the existence of the controversy and also reservation on the theory. No one can deny that. — Corvus

This goes back to what I said here:

I wonder if mathematical realists and mathematical antirealists have different views about mathematical infinity. I'm a mathematical antirealist. I have no problem with mathematical infinity. The "existence" of infinite sets does not entail the existence of infinities in nature (whether material or Platonic).

Infinite sets have a use in mathematics. That's all that matters. Reading more into them is a mistake.

Many mathematicians? — Michael

Not sure on Mathematicians, but if they are logical, I would presume they would.

nfinite sets have a use in mathematics. That's all that matters. Reading more into them is a mistake. — Michael

Maybe. I don't see much practical point apart from filling in and adding more pages of the textbooks making them heavier.

I gave the Mark Twain / Samuel Clemens example as an illustration, not an argument, of the distinction between sense and denotation. — TonesInDeepFreeze

The problem being, that contrary to your claim, there are no things denoted in mathematics therefore mathematics is not "extensional" in the way of your analogy. @Michael agrees that mathematics deals with values rather than things. And since values are inherently intensional the mistake you made ought to be easily avoided by Michael.

If one rejects the view that abstract objects exist (and obviously, as abstractions, they don't exist physically), then, of course, the left term and the right term in an identity statement cannot refer to abstract objects. But that is a different objection than objecting to taking '=' as standing for the identity relation.

And if one objects to calling whatever mathematics refers to as 'objects', then we note that the word 'object' is a convenience but not necessary, as we could say 'thing' instead, or 'value of the term', or 'denotation of the term', or even none of that, and just say 'members of the domain of discourse' so that 'T = S' is interpreted as, for any model M for the language, M(T) is M(S). — TonesInDeepFreeze

It is not matter of whether abstractions exist as physical objects, it is a matter of whether abstractions exist as "objects", or "things" in any rational, coherent sense of the word. The law of identity states that a thing is the same as itself, and we can satisfactorily replace "thing" with "object", or vise versa, making them interchangeable for the sake of discussion. Now the issue is whether there is an identity relation (consistent with the law of identity) expressed by "=" in mathematics.

So, the demonstration and reason why, there is not a "thing" or "object" which is referred to by a numeral such as "1" or "2", and why that supposed "thing" would be incoherent and irrational if it was a thing which is referred to, is explained by my example of "1+1=2". If the two 1's both refer to the very same thing, then there is only one thing represent by those two 1's. Therefore no matter how many times we represent that same thing, we cannot have an equivalence with 2. So it ought to be very clear to you that "1" cannot refer to an object or thing because this would render mathematics as incoherent. Even the simple minded ChatGPT understood this example, and in the other thread where Banno presented this to it, it was very clear to say that in mathematics "=" commonly represents equality, "not identity".

Moreover, there is a difference between what is meant in mathematics by '=' and what one thinks mathematics should mean by '='. Whatever one thinks mathematics should mean by '=' doesn't change the fact that in mathematics '=' stands for identity. — TonesInDeepFreeze

This is exactly the problem which I've been repeating over and over. In common usage of mathematics, "=" signifies equality. GPT corroborated, even though you dispute its authority on common usage of mathematics. However, some mathematical theory, such as set theory defines "=" as signifying identity, regardless of how it is actually used in mathematics. This produces the problem you mention. Some people such as yourself, think that "=" should signify identity, because this would make it consistent with the theory they support, even though the fact remains that in mathematical usage "=" continues to represent equality rather than identity.

Do you agree, that when it is the mathematicians themselves, who are insisting on what "=" should mean, with complete disregard for how it is actually used in mathematics, there is a problem? This is a common epistemological problem demonstrated by Plato in the Theaetetus. Epistemologists have an idea of what "knowledge" should mean, 'JTB', and this supports their epistemological theory. However, as Plato demonstrated we cannot actually exclude the possibility of falsity pervading knowledge, so the T of JTB doesn't actually represent a true definition of "knowledge" according to what the word is actually used for. It simply represent what some epistemologists think "knowledge" should mean. Likewise, "=" does not mean identity in mathematics, it represents equality, despite the fact that some mathematicians think it should represent identity because that's what their theory states.


My criticism remains unaddressed. Let me put it more clearly. Since we are discussing values, not physical objects as in the case of your example, there is no such thing as an extensional reading of "1+1 = 3-1". That constitutes a misinterpretation.

I suggest you to read an elementary school book on set theory. There indeed are infinite sets and there can be a bijection between these sets. It's not just "mistake" like you think. — ssu

Bijection is a specific procedure. If you think that an infinite bijection can be carried out, such that you can produce a conclusion about the cardinality of a supposed infinite set, then you ought to be able to demonstrate this bijection. This would demonstrate that you have made a valid conclusion concerning the set's cardinality. And by "demonstrate" I mean to actually perform this bijection, not to simply represent it with a symbol or symbols, as if it has been performed. The latter does not qualify as a demonstration because one can make a symbol to represent any impossible conception, like a square circle, or whatever. Are you prepared to make that demonstration?

But an attempt at any such conversation in these fora would quickly be derailed by those who cannot grasp equality and those who misattribute and fabricate willy-nilly. — Banno

This I agree with. There is a serious problem with those who conflate equality and identity to "fabricate willy-nilly". We seem to be in much agreement in this thread, which is unusual. You have already pointed out the problem with people like Tones and Michael who claim to be doing mathematics when they are not. These two have displayed a need to refer to non-mathematical examples like Twain=Clemens, and the president of the United States, to demonstrate their supposedly "mathematical" principles.

there is no such thing as an extensional reading of "1+1 = 3-1" — Metaphysician Undercover

There is. The extensional reading of "1 + 1" is the number 2. The extensional reading of "3 - 1" is also the number 2. And the number 2 is identical to the number 2.

Also – and correct me if I'm wrong @TonesInDeepFreeze – but "1 + 1" doesn't actually mean "add 1 to 1". Rather, it means "the number that comes after the number 1". And "3 - 1" means "the number that comes before the number 3".

The number that comes after the number 1 is identical to the number that comes before the number 3.

Axiom
Jane is standing between John and Jack, with John on our left and Jack on our right

Inference
The person to the right of John is identical to the person to the left of Jack

The inference is valid even though Jane, John, and Jack are not physical people and are not abstract entities that exist in some Platonic realm.

It seems very straightforward to me.

How do your views square with indispensability? — TonesInDeepFreeze

Wouldn't his views triangle with indispensability even? Corvus seems to be arguing for some kind of anti-realism about at least some mathematical entities, it seems.

Since we are discussing values, not physical objects as in the case of your example, there is no such thing as an extensional reading of "1+1 = 3-1" — Metaphysician Undercover

When you say "values" it seems you refer exactly to what is supposed to be the extensional reading of 1+1 or 3-1. So, if we are discussing values, saying that 1+1 is the same as 3-1 is correct, as both represent the same value, even if not the same operation.

We seem to be in much agreement in this thread, — Metaphysician Undercover

I have some sympathy for anti-realist views in maths, I've expressed this elsewhere over several years. These stem from reading Wittgenstein. The problem is that both you and @Corvus badly misrepresent Wittgenstein in an attempt to subjugate his name to your psycoceramics.

So far neither of you have been able to cite anything like an endorsement of either your eccentric and unsound view of equity nor Corvus' confusing finite and infinite. Nor will you.

But the result is that we are unable to have a significant discussion of constructivist views of maths.

he extensional reading of "1 + 1" is the number 2. — Michael

That's nonsense, you cannot read "1+1" as "2" because that's obviously a misreading. There is an operation signified by "1+1" and this implies that the reading of it must be intentional. It would absolutely be a misreading of "1+1" to read it as "2". And to get 2 out of 1+1 is intensional as well.

Also – and correct me if I'm wrong TonesInDeepFreeze – but "1 + 1" doesn't actually mean "add 1 to 1". Rather, it means "the number that comes after the number 1". And "3 - 1" means "the number that comes before the number 3". — Michael

See, this is proof that your reading of "1+1" is intensional. "The number that comes after the number 1" is clearly intensional, and that's how you read "1+1". You cannot read "1+1" as two because that would be a misreading. Only "2" gets read as two.

When you say "values" it seems you refer exactly to what is supposed to be the extensional reading of 1+1 or 3-1. So, if we are discussing values, saying that 1+1 is the same as 3-1 is correct, as both represent the same value, even if not the same operation. — Lionino

That's right, but Michael and I already went through this discussion. The values which are produced by "1+1"and "3-1" are only created by carrying out the operations referred to by "-", and "+". The expressions "1+1" and "3-1" refer to those procedures, not the values produced as a conclusion to the procedures. To conclude that "1+1" and "3-1" both produce the same value requires that the operations referred to be carried out correctly. Therefore, that "1+1", and "3-1" each produce the same value is dependent on correctly carrying out the operations which are represented by the expressions. What is represented by the expressions is the operations, not the values which result as a conclusion.

The problem is that both you and Corvus badly misrepresent Wittgenstein in an attempt to subjugate his name to your psycoceramics. — Banno

I like that description "psychoceramics". It makes me feel like I belong to a group, the psychoceramicists, rather than just a lone wolf.

But the result is that we are unable to have a significant discussion of constructivist views of maths. — Banno

Oh you poor little boys, can't keep yourselves from being distracted by the antics of a couple of psychocermacists.

I like that description "psychoceramics". It makes me feel like I belong to a group, the psychoceramicists, rather than just a lone wolf. — Metaphysician Undercover

Fair enough. I doubt, were you to get together, that you would find much agreement apart from the "cliques" being wrong, and your martyrdom.

What about our interest in crackpots like Tones?

You only picked out the usage of the infinity in the book for insisting your point in this thread. — Corvus

I read the chapter about the history of set theory and philosophy about it. I haven't posted anything to dispute of it nor, in certain parts, anything to affirm it.

Included in that chapter, the author explains the importance of formalization, very much along the lines I did earlier in this thread, on which point you disputed.

I read much of the rest of the book, as it interests me in the particular way that it develops a class theory.

Anyway, the bulk of the book is an intro to set theory, covering material I have studied in similar textbooks, though, as mentioned, I'm tempted to go back over that material with this book, as I am interested in the authors particular way it develops a class theory.

I read it from the start to the end. — Corvus

But you missed the definition of 'infinite' that completely agrees with the one I mentioned but that you challenged me to cite a textbook that uses that definition. So, I am still baffled why you challenged me to cite a textbook when your own favorite book on set theory, which you claim to have read, is one of many many textbooks that give the definition you challenged me to show that it is in a textbook.

And I highly recommend that you reread that chapter on the history of set theory and philosophy about it, so you will see how the author and I are aligned on the subject of formalization, as you instead displayed that you don't understand it and as you objected to my remarks about it in your usual style of confusion, strawman and non sequitur.

What about our interest in crackpots like Tones? — Metaphysician Undercover

That's a beam calling the mote a beam.

You obviously have problem understanding metaphors and ordinary use of English language. You seem to bite into a little words in the expressions, and as if one has to stick to the every word and comma in the sentence in the legal contract. I tend to write with metaphorical and simile expressions and idioms a lot just like other ordinary English users. You can't seem to understand that. — Corvus

I understand metaphor.

I didn't demand perfection in what you said.

You said that mathematics regards 'infinite' to mean 'finite'. That's not a metaphor. If you meant that it was a metaphor or that you didn't actually mean to say that mathematics regards 'infinite' to mean 'finite', then you could have conveyed that the first time I told you that mathematics does not regard 'infinite' to mean 'finite'.

Then you deflected to say that it's something that Wittgenstein said. So, that deflects from you making the claim to you claiming that Wittgenstein made the claim. But the Wittgenstein quote, whiles perhaps ironic and acerbic, does not as presented without more context, say that mathematics regards 'infinite' to mean 'finite.

So then you deflected again to say that the Stanford article supports your claim about Wittgenstein. But you fail to give any quote from the Stanford article.

It might be that Wittgenstein meant that mathematics regards 'infinite' to mean 'finite', but you have not shown that he did, and even if he did, merely that he did would not show that mathematics regards 'infinite' to mean 'finite'.

So now you deflect to a claim that is false (that I don't understand metaphor) and one that is both false and a strawman (that the reason I don't agree with you is that I demand perfection of expression). And, this is while you've been complaining about ad hominem, as your quote above is itself ad hominem. And while you've completely skipped my detailed points about ad hominem in posting.

What about our interest in crackpots like Tones? — Metaphysician Undercover

On the contrary, when I check Tone's arguments, they are very mainstream. Almost painfully so. I find that admirable; Tones has corrected my excesses.

This post: by way of example, setting out the issues clearly and historically.

degraded the discussion into a comedy
— Corvus

The ridiculousness is courtesy of you. Maybe not comedy, but still risible is the claim that set theory takes 'infinite' to mean 'finite'.
— TonesInDeepFreeze
You start your post with throwing insults to others before even going into the points under discussion. What courtesy are you talking about? — Corvus

You lied about me when you said I started with insults. I gave you the links that prove that you're lying about that. And even showed that you first made an insult against another poster.

The record of posts shows that I posted without personal remarks, and for a while, until it became clear that you are posting in bad faith - from ignorance, confusion, strawman, evasion of refutations.

For the second time, you are lying when you claim that I started posts in this thread (for that matter, any thread) with insults. Meanwhile look in the mirror for a change - there's a huge steel beam across your eye.

And the courtesy I'm talking about, Mr. Metaphor who can't discern irony, is just what I said it is: that you provide comedic relief when you go through all the ridiculous contortions you do just to avoid simply recognizing that set theory does not define 'infinite' as 'finite'.

If you are asking in which article he said it, I recall it was from a book I don't own. But I saw it in the internet somewhere. — Corvus

I searched 'Wittgenstein mathematics infinite means finite'. Of course, there are hits with all those terms, but the one I saw come up with a preview close to your claim is this thread itself.

And, again, the levels:

You claimed that mathematics regards 'infinite' to mean 'finite'. You claimed falsely.

Then you claimed that Wittgenstein said it. But the quote you adduced did not say that mathematics regards 'infinite' to mean 'finite'.

Then you said that Wittgenstein meant it as metaphor. But, at least without context, it is not clear what the metaphor would be there. And even if the claim that mathematics regards 'infinite' to mean 'finite' is
metaphor, it doesn't relieve that you did not present as metaphor yourself.

Then you claimed that the Stanford article supports that Wittgenstein said that mathematics regards 'infinite' to mean 'finite'. But there you cannot give a quote in which the article says that or even implies it.

Then you claim that your original claim that mathematics regards 'infinite' to mean 'finite' was merely metaphorical. But since it doesn't read as metaphor, when you were first told that mathematics does not regard 'infinite' to mean 'finite', you could at that time just say that indeed you do not claim that mathematics regards 'infinite' to mean 'finite'.

Meanwhile, you challenged me to cite a book that defines 'infinite' as 'not finite', while your own favorite book itself gives that definition.

Meanwhile, you lie about the very record of posts in this thread.

And you hypocritically decry ad hominem, while you use ad hominem. And you ignore the detailed remarks I said about ad hominem.

And all of that is in your pursuit to take down set theory, while you know virtually nothing about it, are confused about it, misrepresent it, and ignore explanations given you about it.

/

So someone might say, "Oh, but Tones, why are you going on, prosecuting this one little item?"

Because every time I catch this crank in his intellectual dishonesty (even to the extent of lying about the record of posts) he comes back with even more intellectual dishonesty. It is worth making that clear as yet another object lesson about the perniciousness of Internet crankery.

Values are not "inherently intensional".

One may reject ideation and communication premised in abstract objects. But the notion of identity is not even limited to abstract objects. Whatever things one does countenance as existing, named by, say, T and S, we have T = S if and only if T is S. That is what '=' means when it is used in contexts of ordinary identity theory, logic, mathematics and other contexts to. If one wishes to use it with another meaning in another context, then, of course, fine. But that doesn't justify saying that in logic and mathematics it is not used just as logic and mathematics says it is used.

'=' doesn't even require a mathematical theory as its context or even the acceptance of abstractions, but rather that in identity theory, for whatever things, abstract, concrete, physical or are being looked at on your desk right now, the statement of identity is that of being the same thing.

Again, more exactly:

If 'T' and 'S' are terms, then

'T = S' is true if and only if T is S.

And whether 'T' and 'S' stand for abstract things, abstract objects, values that are abstract things, values that are abstract objects, concrete things, physical things, or whatever things you are looking at right now on your desk.

And how can anyone, even a crank, not understand:

1+1 = 2

'1' refers to the number one. And since '1' and '1' are the same numeral, it would be redundant, though obviously true, to say that both '1' and '1' refer to the number one.

Then, '1+1' refers the SUM of the number one with the number one. And that SUM is the number two. Or in more formulated mathematics, '1+1' refers to the successor of the number one; and the successor of the number one is the number two.

The denotation of '1+1' is not two of the number one, but rather it is the SUM of the number one with the number one.

'2' refers to the number two, and '1+1' refers to the sum of the number one and the number one, which is the number two. So '2' and '1+1' both refer to the number two, so '2' and '1+1' refer to the same number. That is, 1+1 is 2, which is expressed as:

1+1 = 2.

It is difficult to reason with someone about mathematics who doesn't understand that 1+1 is 2.

(Yes, I can hear sane and rational people saying, "Really, Tones? You spent your precious time tonight explaining to a grown person that 1+1 is 2?")

/

Taking Chat GPT as an authority, or even remotely reliable, as does the crank is pathetic.

Anyway, I couldn't resist. Chat GPT told me that:


1+1 equals 2.

1+1 is 2.

/

In mathematics and logic 'equality' and 'identity' mean the same.

/

The reason I say that '=' stands for the identity relation is not that "it would make it consistent with the theory I support". Rather, in mathematics, not just in set theory, ordinarily '=' stands for equality, which is identity.

But the crank is now arguing that in mathematics it's equality, which is not identity, which is untrue.

/

In ordinary mathematics, bijection is not a "procedure". Rather a bijection is a certain kind of function.

And we do prove the existence of certain bijections.

And one does not "perform" a bijection.

Here's a bijection:

{<1 1>}

One does not "perform" it.

Here's a bijection:

{<k j> | k in N & j = 2*k}

One does not "perform" it.

(But maybe there's no point in explaining this to someone who does not understand 1+1 = 2.)

/

The crank misrepresents again by claiming I mentioned intensionality regarding a person's name to support a mathematical claim. I explained exactly the role of mentioning the example of intensionality and that it was not an argument that such an example applies mathematically. But the crank skipped that so that he could misrepresent my point.

when I check Tone's arguments, they are very mainstream — Banno

Just to be clear:

I enjoy reading classical mathematics; I find great wisdom in mathematical logic; I admire the rigor of logic and mathematics; I admire the astounding creativity in logic and mathematics; I recognize that classical mathematics is the basic mathematics used for the sciences; I recognize the objectivity in mechanical checking of proofs (and generally that at least in principle, if time were taken to fully formalize then proofs are mechanically checkable); I admire the intellectual honesty of logicians, mathematicians and many philosophers of logic, mathematics and language; I admire the simplicity of the axiomatization of mathematics from the set theory axioms, especially the relative simplicity, as axiomatizing alternative mathematics is often much more complicated; and I enjoy, though I am haunted by, the philosophical problems that arise from classical mathematics.

But I do not claim that classical mathematics is the only "true" mathematics; or that there can't be a better mathematics; or that it is wrong to have philosophical objections to classical mathematics. Indeed, with my limited time and limited talent for mathematics, I do very much enjoy learning about alternative logics and alternative mathematics, and I very much admire and relish the wisdom, creativity, and productivity of the alternatives, and also the great philosophical debates around classical and non-classical mathematics.

Also – and correct me if I'm wrong TonesInDeepFreeze – but "1 + 1" doesn't actually mean "add 1 to 1". Rather, it means "the number that comes after the number 1". And "3 - 1" means "the number that comes before the number 3".

The number that comes after the number 1 is identical to the number that comes before the number 3. — Michael

Each of these is true if and only if each of the others is true:

S = T

S equals T

S is identical with T

S is T

the denotation of 'S' = the denotation of 'T'

the denotation of 'S' equals the denotation of 'T'

the denotation of 'S' is identical with the denotation of 'T'

the denotation of 'S' is the denotation of 'T'

/

All of the below are identical with one another. All of the below are equal to one another. All of the below are the same as one another.

1+1

the sum of 1 and 1

1 added to 1

1 plus 1

the successor of 1

3-1

the difference of 3 and 1

1 subtracted from 3

3 minus 1

the predecessor of 3

2

two

the denotation of '1+1' [but not the Godard movie '1+1']

the denotation of 'the sum of 1 and 1'

the denotation of '1 added to 1'

the denotation of '1 plus 1'

the denotation of 'the successor of 1'

the denotation of '3-1'

the denotation of 'the difference of 3 and 1'

the denotation of '1 subtracted from 3'

the denotation of 'the predecessor of 3'

the denotation of '2'

the denotation of 'two'

Axiom
Jane is standing between John and Jack, with John on our left and Jack on our right

Inference
The person to the right of John is identical to the person to the left of Jack

The inference is valid even though Jane, John, and Jack are not physical people and are not abstract entities that exist in some Platonic realm. — Michael

Yep.

How do your views square with indispensability? — TonesInDeepFreeze

That reminds me, I still am interested in how he thinks Putnam's indispensability view jibes with his own views.

"The number that comes after the number 1" is clearly intensional, — Metaphysician Undercover

That quote could be written only by someone who does not understand what is meant by 'extensional' and 'intensional'. A name is not just one of extensional or intentional. Rather, a sentence has both an extensional aspect (the denotation) and an intensional aspect (the connotation).

/

'1+1' does not stand for an operation. It stands for the result of an operation applied to an argument.

'+' stands for a function ('operation' if you insist).

'1+1' stands for the value of the function applied to the argument <1 1>.

Spot on. Crucial in understanding Wittgenstein's views on mathematics, in which the extension of mathematical terms becomes problematic.

My way of making sense of it is that in the modern sense we understand the extension of "2+3" as 5; and the intension of "2+3" as the algorithm, or perhaps the program, it prescribes for us to follow. Hence "2+3" and "4+1" give different algorithms for us to follow, but each will give the same answer - different intension, same extension.

Wittgenstein worked with a more directly platonic notion of extension - the thing that "2+3" points two - and it was at least partially his rejection of this Platonism - what could such a "thing" be? - that led him to his somewhat more contentious views. Roughly, his view prevented him from accepting that there are infinite mathematical extensions.

My own suspicion, which is without a strong formal argument, is that all mathematical entities might be best understood as sets of instructions - that in effect there are no extensions in mathematics. I hold to this view on Tuesdays and Thursdays, the remainder of the time thinking that it makes no nevermind if we do treat the results of these processes as if they are real; in a fashion not unlike how we treat money as real despite it being only a series of transactions.

I had a go at articualting this in the thread "'1' does not refer to anything" four years ago: that mathematical entities are things we do, not things we find.

But here we are getting into the sort of discussion that I think will prove impossible with present company.

It is fine to say that mathematics should be done intensionally. But cranks go wrong when they claim of the classical mathematics they're criticizing that it does do it intensionally, or that it must be done intensionally, or that it is inconsistent for not doing it intensionally.

Indeed it is a fine idea that we may talk about a program that outputs successively longer finite sequences rather than talking about an infinite sequence. But easier handwaved than axiomatized. There have been proposals for intensional mathematics (especially, for example, Church), but it's not an easy thing, the devil is in the details; it's not realized by crank handwaving, confusions and illogic.

Yep. Further the extent to which the formalisation of intensional logic capture "sense" as used in natural languages remains unclear. But it is an interesting approach.

There's some hint here of Wittgenstein's idea of following a rule as implementing a practice, of "continuing in the same way", but this is very speculative. An area well worth keeping one eye on, I think.

I'm not suggesting mathematics should be done intensionally, so much as puzzling over what the distinction between intension and extension amounts to. It hints at something pivotal, but well beyond my ken.

It's also fine to have a philosophical stance that there are no abstract objects. But being true to that stance then requires eschewing even everyday locutions about mathematics and everyday thinking about many things. For that matter, at least for me, the use of language in thought and communications is to provide frameworks for dealing with so-called concrete experience, not merely to remark that one observes the so-called concretes.

Mathematics does not pretend to be isomorphic with all the concrete objects and particles of science. Nor that mathematics is a factual report about concretes. Rather, mathematics provides an idealized framework that we can choose to use in different ways, including providing an axiomatization for the formulas we do use for the sciences. Mathematics is an armature for knowledge about concretes; it is not supposed to be itself a report of those concretes. The armature is not itself the things you put in it.

And the way I understand - other mileages may vary - frameworks, whether mathematical, philosophical or conceptual in any field of study, is that they should facilitate fluid thinking and communicating, and to avoid, if possible, having to stretch oneself in contortions such as having to grasp for convoluted expressions to avoid saying the word 'object' in an utterly natural way when talking about things such as numbers, or to have to eschew the economy of conceptualizing numbers as things rather than to commit to imagining that a number is born and dies, off and on and off and on, every time someone thinks of it and then stops thinking of it or that there even is no 'it' they are thinking of but only physical events in a brain, or that, wait, what is the notion of 'event' anyway without abstraction?

On the other hand, if one wants to try to think of mathematics and formulate it and communicate it but without reference to abstractions or abstract objects, I say have at it. But that doesn't make everybody else wrong for thinking of numbers as things and saying such ordinary things as "the sum of two and two is four". And especially classical mathematics is not crippled by the mere wish of a crank, without a concrete proposed alternative, that there is an unannounced, unarticulated physicalist replacement.

I find it a crude notion that each mathematical mention must correspond to represent each, every and any of the concretes and particles that are themselves present to us mentally as constructs in a conceptual framework. A framework is not an assertion, and its value is being able to conceptually and/or practically cope with or predict experience. Such frameworks may be preferred or not in how well they conceptually and/or practically cope with or predict experience, but also in the satisfaction derived from the conceptual order and beauty they provide. When I am confused by too many facts all at once, or about, for example, how things work, I am relieved of that confusion by a framework that allows me to put that experience in order, to process it. I may be confused by the behaviors of other people, for example. But then I may posit such things as traits, goals, etc. I don't posit that those are concrete things. They are abstractions, they are posited as a conceptual armature so that a person's actions don't appear to me as a random jumble but rather my framework allows me to think of those actions in a narrative and to make predictions about them on that basis. When I there are of numbers mentioned, I don't have to think of them as popping in and out of existence each time they are mentioned or not, but rather I have an armature in which numbers don't do that. And when I there are a lot of numbers involving some problem, either conceptual or practical, that I want to solve, I have a system of principles about numbers that allows me to find the answers I want. That system is an abstract armature, not a concrete thing. It's not required that each concept, each abstraction itself corresponds to a particular concrete.

Meanwhile, maybe there is a way, but I don't know of it, to avoid that thought and language themselves presuppose that 'object', 'thing', 'entity', 'is', 'exists', etc. are basic and that explication of them cannot be done without invoking them anyway. When I say "What is that thing in the sink?" I presuppose even the concept, which itself is an abstraction, that there are things, that concretes are things, and even the notion of 'concrete' is an abstraction. And I don't see anyone who can talk about experience ('experience' also an abstraction) without eventually invoking utter abstractions such as 'object' and 'is', whether referring to abstractions or concretes.

The crank will mangle what I wrote, misrepresent it, presume to knock down strawmen of it. Likely, I won't have to time to compose a response, especially to the sheer volume of his confusions.