We can imagine things without a concrete instantiation. — Metaphysician Undercover

I find that hard to believe. How is it possible to imagine a unicorn by not picturing a unicorn? How is it possible to imagine the number 6 without picturing six things. How is it possible to imagine beauty without picturing something beautiful?

It is true that when I imagine a unicorn I could picture the word "unicorn", but this is still a concrete instantiation.

When you imagine a unicorn, if you are neither picturing a unicorn nor the word "unicorn", what exactly are you imagining? What exactly is your "Intentionality" directed at?
===============================================================================

We can imagine things without a concrete instantiation. That's how artists create original works, they transfer what has been created by the mind, to the canvas — Metaphysician Undercover

Problematic. If an artist, no matter whether Monet or Michaelangelo, could create something that hadn't existed before, this would be the same problem as to how something can come from nothing, the same problem as to how there be an effect without a cause.

An artist may reorganise existing parts, a blue line, a tree, a sky or a yellow mark, into a new whole, such as a painting of Water-lilies. An artist may change the relationship between parts that already exist, but the artists cannot create the parts out of nothing.

In fact, your challenge would be to find an artwork which included a part that did not already exist in some previous artwork.

The artist finds new relationships between existing parts. They don't create the parts.

The artist imagines new relationships between "concrete instantiations" of existing parts. The artist pictures new relationships between existing parts, and if successful, then applies them to the canvas.

"Infinity of infinites" in natural language

In natural language, we know the meaning of the word "infinity", yet infinity is unknowable by the finite mind, meaning that "infinity" must refer to something knowable.

Something very large is knowable, such as the number of grains of sand on a beach, the number of water molecules in a glass of water or the number of people living in a city.

We are able to understand "Infinity" as like "the number of grains of sand on a beach".

Therefore, our understanding of "infinity" is not literal but rather as a figure of speech, specifically, a simile.

If "infinity" means "like the number of grains of sand on a beach", then the expression "an infinity of infinities" becomes an "like the number of grains of sand on a beach" of "like the number of grains of sand on a beach". This is ungrammatical

IE, in natural language, the term "infinty of infinites" is ungrammatical.

That's a bijection which cannot be carried out, cannot be completed. It's a nonsensical proposition. — Metaphysician Undercover

Nonsensical?

Seems you confusing ideas about set theory. Or ignorance about the subject.

Set theory is part of mathematics, but of course you can be with your idea that it's "nonsensical".

Sorry, but I did assume this thread was about mathematics.

Which math textbook says "infinite" means "not finite"? — Corvus

You said, "Problem with Set Theory is that their concept "infinite" means "finite""

What set theory textbook, or any reference in set theory or mathematics, says that 'infinite' means 'finite'?

Meanwhile, many textbooks in mathematics, including set theory, analysis, algebra, topology, computability, probability and discrete mathematics give the definition of 'infinite' as 'not finite'. What is your purpose in asking if you're not thinking of reading one of them?

Two pairs of definitions:

x is finite iff x is one-to-one with a natural number
and
x is infinite iff x is not finite

x is Dedekind finite iff x is not one-to-one with a proper subset of x
and
x is Dedekind infinite iff x is not Dedekind finite

But with the axiom of choice (as with the most common set theory in mathematics, which is ZFC) we have:

x is finite iff x is Dedekind finite
thus
x is infinite iff x is Dedekind infinite.

More specifically:

Without the axiom of choice, we have:

If x is finite then x is Dedekind finite
thus
If x is Dedekind infinite then x is infinite

With the axiom of choice, we have both:

If x is finite then x is Dedekind finite
thus
If x is Dedekind infinite then x is infinite

and

If x is Dedekind finite then x is finite
thus
If x is infinite then x is Dedekind infinite

/

Meanwhile you are posting flat our disinformation when you post ""Problem with Set Theory is that their concept "infinite" means "finite"".

And you'll not show any textbook or article or lecture notes in set theory or mathematics that say 'infinite' means 'finite'.

Indeed, even in plain language, the prefix 'in' with 'infinite' is taken in the sense of 'not'. x is infinite if and only if x is not finite.

You need to stop posting confusions and disinformation.

You said, "Problem with Set Theory is that their concept "infinite" means "finite""

What set theory textbook, or any reference in set theory or mathematics, says that 'infinite' means 'finite'? — TonesInDeepFreeze

I have already quoted from Wittgenstein from his writings "infinite" in math means "finite", and he adds that the mathematicians discussions will end. It is obvious you have not read the post.

But my point is not about "infinite" is "finite" or whatever. My point was that the concept "infinite" means something totally different, and math's infinity in set theory doesn't exist. This is not what other folks says, or may some folks did, I don't know. But that is just my idea. I don't need any supporting comments on that from anyone, when I think that is the case.

But you are quoting from the old and outdated mathematician Dedekind on the concept of "infinity", and it means "not finite". To me it just sounds vacuous word game to say infinity is not finite, but "not finite". It is a concept which doesn't exist in reality. It is an abstract concept for describing motions, actions and operations.

Anyway, Dedekind's set theory had faults and limitations. Here is what ChatGPT says about his Set Theory and concept of Infinity.

"Dedekind's set theory, while foundational and influential, does have some limitations and criticisms. Here are a few:

Axiomatic Foundation: Dedekind's set theory lacks a formal axiomatic foundation comparable to other set theories like Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC). Without a clear set of axioms, Dedekind's set theory may be seen as less rigorous or formal by contemporary standards.

Treatment of Infinity: While Dedekind made significant contributions to the understanding of infinity, his treatment of infinity in set theory may be considered less systematic compared to later developments, such as Cantor's work on transfinite numbers and ZFC set theory. Some critics argue that Dedekind's definition of infinite sets as those that can be put into one-to-one correspondence with proper subsets of themselves is not as precise or comprehensive as later formulations.

Lack of Explicit Axioms: Dedekind's set theory does not provide a set of explicit axioms like those found in ZFC set theory. This lack of a formal axiomatization can make it difficult to establish the foundational principles of Dedekind's theory and to reason rigorously about sets within this framework.

Scope and Development: Dedekind's set theory was developed in the late 19th century and may be seen as lacking some of the conceptual developments and formalizations that occurred in later set theories. While his work laid important groundwork for the development of modern set theory, it may not encompass the full range of concepts and techniques found in more contemporary approaches." -ChatGPT

I would have expected your reply to my question from the reputable and well known modern math textbooks which says "infinite" is "not finite", as you have been insisting as the case. But it doesn't matter. To me, infinity is an abstract concept which has no entity, and shouldn't be used for naming the set elements or sets. It doesn't reflect the reality accurately, and is a vacuous concept. Infinity only makes sense when it is describing motions, actions or operations. Or it can be used in the poetry or metaphor as a figure of speech. That is fine.

I am not claiming anything on the math theory. I am just pointing out the contradictions and false information in your posts, and replying to them. It would be a gross distortion of the fact and over exaggeration to state anything more than that about my replies.

For a mathematical antirealist, does any of this constitute hypocrisy?

I can't see the relevance. Your game clearly involves real objects, pebbles, or in the case of your presentation, the letters. Would the antirealist insist that these are not real objects? — Metaphysician Undercover

Earlier you said (for example):

In set theory it is stated that the elements of a set are objects, and "mathematical realism" is concerned with whether or not the things said to be "objects" in set theory are, or are not, objects.

and

However, it's hypocrisy to say "I'm a mathematical antirealist" and then go ahead and use set theory.

By a 'mathematical antirealist' I meant someone who thinks maths is invented, not discovered. Or someone who thinks that your "objects" in set theory only exist in our minds, or as pebbles or ink or pixels, etc.

The whole of number theory or set theory can be reduced to a game with pebbles like the one I described. More colours of pebbles, more rules, but just rows of pebbles and precisely defined ways of rearranging them. It is thus possible to do number theory or set theory without mentioning numbers, or sets, or any other mathematical objects, or using a natural language at all. Tricky, but possible.

You can interpret some patterns of pebbles as objects of various sorts, but treat them as mental crutches, vague hand-wavy ideas, expressed in natural language with all its confusions and ambiguities, which can guide your intuition. Or you can believe they really exist somewhere. Either way, I don't see any hypocrisy.

I get the feeling you have no experience working with formal systems, and have no real understanding of metamathematics. I can't explain your inability to see the the relevance of my game otherwise.

Regarding the "=" sign, it was invented in 1557 by Robert Recorde — Michael

Robert was the first known usage in a printed work, but he did not invent it. The symbol was used in Italy before Robert.

Looks a bit like he has 1+1=2 mixed up with somethign like "1+1" ="2"?

I have already quoted from Wittgenstein from his writings "infinite" in math means "finite", — Corvus

I can't find anything of the sort in this thread. You quoted him, in another thread, as saying

"Let us not forget: mathematician's discussions of the infinite are clearly finite discussions. By which I mean, they come to an end." - Philosophical grammar, p483. Wittgenstein. — Corvus

Which is very far from what you attribute to him here.

But you will double down, again.

You quoted him, in another thread, as saying — Banno

Too many threads on infinity. You found it OK. Anyway, it wasn't far.
Tone was in the thread, and he would have seen it.

Anyway, Cantor and Dedekind wouldn't have opposed to infinity in set theory, because they made them up. It was Frege, Russell, Quine who had reservations on it even if didn't oppose to it. Wittgenstein sounds he was against it.

What set theory textbook, or any reference in set theory or mathematics, says that 'infinite' means 'finite'? — TonesInDeepFreeze

You misunderstood. It meant that Wittgenstein said that mathematician's infinite means finite in his writings. See the quote above.

Then you said, infinite is not finite, but "not finite". I asked for the textbook definition for infinite in math. Again, my point on it is that, infinity is an abstract concept which has no referent object.

Hmm. You misattributed a position to Wittgenstein. He did not say that "infinite" means "finite".

Now go back to this:

You said, "Problem with Set Theory is that their concept "infinite" means "finite""

What set theory textbook, or any reference in set theory or mathematics, says that 'infinite' means 'finite'? — TonesInDeepFreeze

You misunderstood. — Corvus

No, Tones took up what you said, asking you to justify it. You are in error, both in claiming "Problem with Set Theory is that their concept "infinite" means "finite" and in attributing anything like that to Wittgenstein.

This is your modus operandi.

Hmm. You misattributed a position to Wittgenstein. He did not say that "infinite" means "finite". — Banno

"Let us not forget: mathematician's discussions of the infinite are clearly finite discussions. By which I mean, they come to an end." - Philosophical grammar, p483. Wittgenstein. — Corvus

Wasn't he saying clearly mathematician's infinite are finite?

No, Tones took up what you said, asking you to justify it. You are in error, both in claiming "Problem with Set Theory is that their concept "infinite" means "finite" and in attributing anything like that to Wittgenstein. — Banno

Yup, that was my interpretation of Wittgenstein. What is your ground for saying it error?

This is your modus operandi. — Banno

Describe "infinity" in clear and actual way in understandable language, and I will tell you about your modus operandi.

Wasn't he saying clearly mathematician's infinite are finite? — Corvus

What? No.

Yup, that was my interpretation of Wittgenstein. What is your ground for saying it error? — Corvus

My ground involves reading what Wittgenstein says: "mathematician's discussions of the infinite are clearly finite discussions. By which I mean, they come to an end." He is not saying that infinity is finite, but that the discussions of mathematicians are finite.

As I said above, you will double down. You will also seek to obfuscate and change topic. But here, your error is clear. The subject of the quote is not the infinite, but mathematician's discussions of the infinite.

Edit: here it is, posted while I was writing the above - the attemtp to change topic:

Describe "infinity" in clear and actual way in understandable language, and I will tell you about your modus operandi. — Corvus

My ground involve reading what Wittgenstein says: "mathematician's discussions of the infinite are clearly finite discussions. By which I mean, they come to an end." He is not saying that infinity is finite, but that the discussions of mathematicians are finite. — Banno

So which discussion is not finite in that case? Does any discussion under the sun go on forever? It doesn't make sense.
Are you possibly suggesting Wittgenstein would have meant that obvious cliche in his writings?

So which discussion is not finite in that case? Does any discussion under the sun goes on forever? it doesn't make sense. — Corvus

You are descending into incoherence. No discussion is not finite. A double negative that you deserve. Yes, no discussion goes on forever.

With the possible exception of attempting to have you admit an error.

As I said above, you will double down. You will also seek to obfuscate and change topic. But here, your error is clear. The subject of the quote is not the infinite, but mathematician's discussions of the infinite. — Banno

This part is your usual modus operandi, which is ad hominem and straw man.

This part is your usual modus operandi, which is ad hominem and straw man. — Corvus

:rofl:

I have shown that you misattributed a remark to Wittgenstein. Cheers.

You are descending into incoherence. No discussion is not finite. A double negative that you deserve. Yes, no discussion goes on forever.

With the possible exception of attempting to have you admit an error. — Banno

So it is evident your interpretation on W. was wrong.

I have showen that you misattributed a remark to Wittgenstein. Cheers. — Banno

You haven't even explained what "infinity" means. W. would have said, there is no meaning on which things that cannot be described in words.

With the possible exception of attempting to have you admit an error. — Banno

How can anyone admit error when the other party is pushing his wrong ideas with the misinterpretation of Wittgenstein, and inability to explain fully what the world "infinity" means, when asked?
How can one admit error when he is not in error but the other party is?

And so it goes.

Here is what you quoted:
"Let us not forget: mathematician's discussions of the infinite are clearly finite discussions. By which I mean, they come to an end."

This clearly does not support your contention:
'I have already quoted from Wittgenstein from his writings "infinite" in math means "finite"'

You are flailing about.

Ok, you can interpret him whatever way you want. But it doesn't make sense. That is the point. It is not just mathematician's discussions which end. All discussions end. That is too obvious.

What Wittgenstein must have meant was the concept of infinity in mathematics. It was a contentious topic at the time. He didn't agree with it. That is the way I understood him on the point. It was just reflecting my point very nicely for the definition of infinity. I am not trying to change your views or ideas. Just telling you about it because you wanted the argument.

Ok, you can interpret him whatever way you want. — Corvus

it's not a question of interpretation. It's clear that the subject of "mathematician's discussions of the infinite are clearly finite discussions" is mathematician's discussions of the infinite, and not the infinite. Bolding, to display the distinction.

That clearly does not support your contention that Wittgenstein said mathematicians take the infinite to be finite.

No one, not I, not Wittgenstein, and not, apparently, your good self, is suggesting that mathematical discussions are not finite. Now I do not know if this is an issue of comprehension on your part, or a another attempt at using rhetoric to change the topic. The first point here is that you misrepresented Wittgenstein. The second point here is that you refuse to acknowledge your error. The third point is that this is an approach you have repeated in this thread and elsewhere. And not only you, but various others, many of them having contributed to this thread, adopt a similar lack of accountability.

But now I am kicking the pup. Enough, perhaps.

The first point here is that you misrepresented Wittgenstein. The second point here is that you refuse to acknowledge your error. The third point is that this is an approach you have repeated in this thread and elsewhere. And not only you, but various others, many of them having contributed to this thread, adopt a similar lack of accountability. — Banno

Not only your reading on Wittgenstein is wrong, but also you seem to be misunderstanding many things in philosophy. It is not just this thread, but also in many other threads you seem to be claiming things from your misunderstandings and misrepresentation of the facts. Therefore you seem to be going around the circles on the points not getting clear to the point with no depth and no accuracy in many occasions.

Plus you seem to be tending to take sides of the posters regardless of right or wrong of the points, but who you think your cliques are. It is visible many times, and hard to miss it.

I would be disappointed with Wittgenstein if what he meant in the quote was truly "mathematician's discussions are finite, and they all end." to mean the discussions as per se, as you keep on insisting.

But I know your insistence comes from your misunderstanding of Wittgenstein, and what he meant was the concept of infinite in mathematics is actually "finite", henceforth his usual aphoristic claim, "their discussion will end."

Not only your reading on Wittgenstein is wrong... — Corvus

How?

Here it is again:

It's clear that the subject of "mathematician's discussions of the infinite are clearly finite discussions" is mathematician's discussions of the infinite, and not the infinite. Bolding, to display the distinction. — Banno

Set your understanding out, or retract.

I have already quoted from Wittgenstein from his writings "infinite" in math means "finite" — Corvus

(1) Please link to where you quoted Wittgenstein writing that 'infinite' in mathematics means 'finite'.

(2) Wittgenstein doesn't speak for mathematics anyway. Whatever Wittgenstein wrote, it wouldn't change that fact that mathematics does not define 'infinite' as 'finite', which would be utterly ridiculous, as mathematics defines 'infinite' as 'not finite'.

But my point is not about "infinite" is "finite" or whatever. — Corvus

Whatever your point is, what you claimed that in mathematics, 'infinite' means 'finite', which is a wildly ridiculous claim and blatant disinformation.

My point was that the concept "infinite" means something totally different, and math's infinity in set theory doesn't exist. — Corvus

That's a different claim from the claim that, in mathematics, 'infinite' means 'finite'.

But you are quoting from the old and outdated mathematician Dedekind on the concept of "infinity", and it means "not finite". — Corvus

You are hopeless as far as rational discussion.

You asked me what textbooks in mathematics define 'infinite' as 'not finite'. The answer to that question is that just about every textbook in mathematics that gives a mathematical definition of 'infinite' gives the definiens as 'not finite', or sometimes the Dedekind definition that is equivalent with 'not finite' in mathematics. The formal mathematical definition 'not finite' goes back to Tarski and the definition 'one-to-one with a proper subset of itself' goes back to Dedekind; but that in no way vitiates that still, the current definition is 'not finite' or its Dedekind equivalent. You challenged me as to what mathematics textbooks say; my answer to that is not vitiated by the fact that the definition is long standing in mathematics. What is wrong with you?

Moreover, obviously, in even just an ordinary context, 'the 'in' in 'infinite' is a negation, so it's 'not finite' and not 'finite'. Again, what is wrong with you?

And now you are pasting Chat GPT (!) quotes that you don't even understand. You can't learn set theory from Chat GPT! What is wrong with you?

"Dedekind's set theory lacks a formal axiomatic foundation comparable to other set theories like Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC). Without a clear set of axioms, Dedekind's set theory may be seen as less rigorous or formal by contemporary standards." - Chat GPT — Corvus

You are such an intellectual incompetent.

I am not talking about Dedekind's theory. I'm talking about a particular definition. And that definition is used is equivalent with 'not finite' in the ZFC that you just mentioned. Indeed, any informal theory is less rigorous than formalized ZFC. So what? That doesn't change the fact that Dedekind's formulations cannot or have not been formalized subsequent to his own writings.

Again, since you missed this:

Definition of 'infinite' in mathematics:

x is finite iff x is one-to-one with a natural number.

x is infinite iff x is not finite (by the way, that is sometimes called 'Tarski's definition)

Another definition of 'infinite' in mathematics:

x is infinite iff x is one-to-one with a proper subset of x (Dedekind's definition)

Those are provably equivalent in set theory with the axiom of choice (such as ZFC). Without the axiom of choice, we can only prove: If x is one-to-one with a proper subset of itself then x is not one-to-one with a natural number.

In any case, with both those definitions, it is blatant that 'x is infinite' is defined as 'x is finite'.

Moreover, now you are taking recourse to the notion of formalization, when just a few posts ago you were trying to dispute me when I mentioned a key advantage of formalization! What is wrong with you?

"While Dedekind made significant contributions to the understanding of infinity, his treatment of infinity in set theory may be considered less systematic compared to later developments, such as Cantor's work on transfinite numbers and ZFC set theory." - Chat GPT — Corvus

Cantor was more systematic about sets, but Cantor also was not a formal theory and had problems that needed to be rigorously resolved by formal set theory.

Anyway, this has no bearing on the fact that the set theoretical definition of 'infinite' is not ridiculously, as you claim, 'finite', nor on the fact that both Tarski's and Dedekind's definition obtain in current mathematics.

"Some critics argue that Dedekind's definition of infinite sets as those that can be put into one-to-one correspondence with proper subsets of themselves is not as precise or comprehensive as later formulations." - Chat GPT — Corvus

Ask Chat GPT who it thinks those critics are and to quote them.

Dedekind's definition incorporated into axiomatic set theory is absolutely precise. And in ZFC it is exactly as comprehensive as Tarski's definition, since in ZFC they are equivalent, as I stated that equivalence explicitly in my previous post.

You don't know jack aboutany of this. You just want to be right about disdain for set theory, so you're willing to enter any specious and counterfactual argument you can come up with, including inapposite quotes from.. Chat GPT (!).

"Dedekind's set theory does not provide a set of explicit axioms like those found in ZFC set theory. This lack of a formal axiomatization can make it difficult to establish the foundational principles of Dedekind's theory and to reason rigorously about sets within this framework." - Chat GPT — Corvus

Again, that does not vitiate that nevertheless his work has been formalized subsequent to his own writings and that include his definition of 'is infinite'.

You are foolishly quoting Chat GPT without even a basis to understand the quotes, their context or their import or lack thereof for our conversation.

"While his work laid important groundwork for the development of modern set theory, it may not encompass the full range of concepts and techniques found in more contemporary approaches." — Corvus

So what? Contemporary mathematics still uses his definition of 'is infinite' as it is equivalent with 'not one-to-one with a natural number', i.e. 'not finite', in ZFC.

The quote below deserves attention as among worst:

I would have expected your reply to my question from the reputable and well known modern math textbooks which says "infinite" is "not finite" — Corvus

The textbooks I can cite you are not just reputable, but they are among the most standard, most used, and most referenced textbooks in current use.

I asked you why you want me to name one if you are not interested in looking at it. Indeed, not interested in looking at any of the many I can cite.

On my desk right now, I have a stack of modern textbooks, some of them regarded as quintessential references, in various mathematical subjects, as they all define 'infinite' as 'not finite'. What do I get if I list their titles and authors for you? You're not going to look them up. So what's the point? Or, how about this: I'll list them all, then you can admit that that you don't know what you're talking about when you say that mathematics defines 'infinite' as 'finite' but rather that mathematics defines 'infinite' as 'not finite'.

I am not claiming anything on the math theory. — Corvus

You're lying in our face. You claimed that in math 'infinite' means 'finite'.

I am just pointing out the contradictions and false information in your posts, and replying to them. — Corvus

You have shown no contradiction or false information in my posts. Rather, you have shown that you are ignorant of the subject, confused about the subject, disinformational in things you've said about the subject, specious in your arguments about the subject, willing to resort to ridiculously intellectually incompetent posturing by quoting Chat GPT (!) that you don't understand and as the quotes don't even approach impeaching anything I've said, and flat out lying when you say that you didn't make a claim about mathematics when plainly you did.

It would be a gross distortion of the fact and over exaggeration to state anything more than that about my replies. — Corvus

Nope. If anything it's understatement to say what I've said about your intellectual incompetence and dishonesty.

Tone was in the thread, and he would have seen it. — Corvus

So what? It doesn't say that mathematics takes 'infinite' to mean 'finite'. And even if it did (which it does not), it doesn't represent mathematics or mathematicians, since they very certainly do NOT take 'infinite' to mean 'finite'.