The issue we've encountered is that the axiom of extensionality is simply false. — Metaphysician Undercover

The axiom of extensionality is

If A and B are sets, then A = B iff every element of A is also an element of B, and vice versa. — Open Logic

It tells us how to use the "=" sign. It is an instruction, and so is not the sort of thing that can be false. You either follow the instruction or you do not. If you do not follow the instruction you are not participating in the logic of sets.

The law of identity has various forms, but in set theory it is that

A=B iff both A⊆B and B⊆A. — Open Logic

This is a consequence of extensionality, not an axiom.

What Meta is doing is refusing to use "=" in the way the rest of us do. It's as if someone were to insist that the Rook move along a diagonal, all the while pretending that they had made a profound discovery about chess in doing so. Meta is simply not playing the game right.

To this sin Meta adds that of mischaracterising Wittgenstein. The private language argument is not that a symbol cannot be identical to an internal sensation, But that internal sensations cannot be treated in the way we treat other objects.

That sounds much like my thinking on days that do not start with the letter "T". Abstractions as a fabrication of grammar...

Increasingly I find learning novel logic systems quite difficult, as if there is too much background missing. The topic is surprisingly different to when I first studied it, much more of an emphasis on computing, far more integrated than it once was, and my short term memory is not what it was fifty years since. When in a masochistic frame of mind I'll work through bits of the Open Logic text. It's often very simple things that hold one up - it seems to be the text of choice for undergrad logic courses and for some of the more advanced stuff. There is a lot to be said for the discipline of having a tutor to help work through examples, especially where small bits of jargon catch one out.

Church is an interesting choice.

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.

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.

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.

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.

We also have at hand that classic rebut to Feyerabend: If anything goes, everything stays.

That is, if we drop the notion of truth as a valid assessment of our utterances in favour of the will to power or some such, we are endorsing the powerful, reinforcing their hegemony.

Post modernism cannot speak truth, therefore it cannot speak truth to power.

This leads me to think that social constructivism/constructionism is not necessarily postmodern in the philosophical sense, even if these distinct approaches are lumped together in the popular imagination. — Jamal

Very much so. One problem is that PoMo has spiritual objections to truth, but mathematics takes a far more pragmatic approach. So mathematicians will insist that certain statements are true. PoMo, not so much.

There are whole worlds between platonic realism and post modern relativism.

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.

The mechanical act is not identical because the act of writing B takes longer than writing A. More letters and punctuation is used. — NOS4A2

Yeah, for that reason the emphasis in speech act theory is usually the utterance, not the sentence. But the point holds.

It’s becoming more and more clear that people are searching for acts in the text and not in the actor. — NOS4A2

That's not correct.

Here it is again, set out so the actor is clear:

NOS4A2 pressed buttons on a keyboard (if that is what you indeed did)
NOS4A2 made marks on a screen
NOS4A2 made a sequence of letters
NOS4A2 wrote "Any advice?"
NOS4A2 asked a question
NOS4A2 asked for advice
NOS4A2 elicited responses from Banno and others

NOS4A2 did things with words.

Which of these is false?

As if we haven’t already heard plenty from the likes of Sokal. Reactionary anti-postmodernist chatter from mathematicians , scientists and politicians is no less common than pomo investigations of mathematics. — Joshs

Yeah, what would mathematicians know about maths?

The article I shared was about as sympathetic as you might expect, and more than I expected. It takes an example from the literature,

Absolutism is deliberately replaced by cultural relativism, as if 2 + 2 = 5 were correct as long as one’s personal situation or perspective required it to be correct — White 2009,

...and points out that

First of all, cultural relativism is out of context in this setting. When postmodernists claim that a mathematical truth is never absolute, they mean it is to be interpreted relative to a background. Certainly 2 x 5 = 1 is true in mod (3) arithmetic. No sane mathematician or educator would go around redefining addition or any other mathematical construct because his or her “personal situation” requires it to be correct. The Platonic fact that the sum of the interior angles of a triangle being exactly 1800 was challenged neither because the personal situation of Lobachevski nor because the personal perspective of Riemann warranted it, but because the resulting geometries turned out to be no more or no less correct that the Euclidean one. — Ilhan M. Izmirli

But no doubt you have a different opinion?

ok, good. And the next step is to agree that there is something fishy here. Which is what I am saying. It’s incomplete.

I suspect that postmodernists talking about mathematics woudl be a dime a dozen. Google supports this.

But a mathematician talking about post modernism... that might be interesting.

A poster who starts out in a thread by declaring "end of story" does not bode well. — TonesInDeepFreeze

That paragraph kinda set up for the gross oversimplification that was to come though.

I did suspect that you may have dissipated in a quantum puff for a while, but here you are, making it less likely that heat death has already happened.

Seems to me the difference between and others here is that he is pretty convinced by the Boltzmann discussion, while the others are more comfortable acknowledging that it is interesting but very far form conclusive.

The SEP article on Action was updated recently and addresses many of these issues. I have quite a bit of sympathy for Davidson's view (from Anscombe, mentioned above), that flicking the switch, turning on the light and alerting the burglar (a by now standard example) is one act, with several descriptions. But that is about theory of action, not so much speech acts. There is much overlap, of course, but as I've argued, it doesn't seem to mater much if the locution and illocution are seen as one act or two; but that what is of import is the distinction between content and force.

an agent provocateur. — creativesoul

Maybe. If it gets a few more folk to learn a bit of philosophy of language it might be for the greater good.

Yes. The dynamic is complex, yet remarkably the site remains fairly stable. It seems to me that amongst recent recruits there is little background in philosophy, logic, maths or even physics, but perhaps it always seems so.

The blatant misrepresentation seen here is a very different thing to the psycoceramics. The latter on occasion does force one to explain or re think.

All by way of repeating that the bad posts do elicit good replies.

But I wonder if the general reader is able to tell the one from the other.

Hilary Putnam? — TonesInDeepFreeze

To some extent the misunderstanding of various authors may be the result of our friends being autodidactic. The supposition that somehow the SEP article on Wittgenstein's philosophy of mathematics supports psycoceramic views might be a result of shallow reading of such tertiary sources. These topics are vast, needing careers, rather than degrees, to understand the topic, let alone make a significant contribution.

Anyway, respect to the mods for what they do. While it might be a little bit better, it could easily be a whole lot worse.

...you make out as if what you and your cliques say are the only truth, and the rest of the world are false. — Corvus

Sad that the "clique" with which you are in disagreement is that of the mathematicians. Hm.

Anyway, time to move on. Long ago.

Way back, I wrote of @Corvus:

But you will double down, again. — Banno

Even I was not expecting such recalcitrance. That was 24 hours and three pages ago. Those three pages are replete with Corvus' squirming and flailing.

There are interesting and controversial ideas in Wittgenstein's anti-platonism, which could make an excellent thread. 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.

That's a limitation on @Jamal's otherwise excellent forums. A more proactive moderation might improve the philosophy being done hereabouts. But so many of the better posts are, as and have shown in this thread, responses to ineptitude.

And so it goes.

You may try for, literally, years and he will not understand. — TonesInDeepFreeze

Indeed; I did; he doesn't.

I performed one visible act, did one measurable thing, but you saw two visible acts, or me doing two visible things. So did I really perform two acts, or are you describing the same act in two different ways?

If I were to record myself writing I would see one act. I can point to it, witness it again and again. I am unable to see two. — NOS4A2

Again, you wrote a sentence and you asked a question. You can call it two acts, if you like, or one act with two descriptions. What is salient is that there is a difference between writing a sentence and asking a question.

I can’t say I made any such action. — NOS4A2

So did you utter "Any advice?" without intending that other folk might respond?

That would be odd. Indeed, that might be enough to render your utterance not a question.

How is it different? — NOS4A2

Simply in that someone might make the very same marks as part of, say, a random scribble - thumping on the keyboard, perhaps - and hence, not intending to elicit a response, not have asked a question.

Yep.

If you're not asking a question when you ask "Any advice?"then what are you doing? Are you quoting a question? Maybe you're pretending to ask a question? Yet, there it is: a question. So where in space and time has this illocutionary act occured? — NOS4A2

I agree entirely, and add that the where and when of the illocutionary act is the same as the where and when of the locution: In your having written "Any advice?" you performed both a locution and an illocution; which is to claim no more than that you both wrote a sentence and you asked a question.

I didn't elicit any reply from you... — NOS4A2

Yes, you did, and continue to elicit replies by your responses. I would not have posted this, had you not posted that.

But words can't act. — NOS4A2

Again, I quite agree. But you can do things using word - that's the point.

there is one act and one act only, the locution, in this case the writing. — NOS4A2

...which involves both making marks and asking a question. Again, the issue is that asking a question is different to making a mark, and this difference is well worth marking, and hence the terms locution and illocution.

Your claim was out of point from the start, because you see the discussion in the quote as discussion in talking. It is the concept of infinity in Mathematics he was meaning, which doesn't exist, hence not speakable and is meaningless. If you are still hanging on that "discussion" and make song and dance about it, you are not in the game. — Corvus

It's difficult to make anything sensible from this. The point I am making is simple, you misrepresented Wittgenstein's view. He is saying that mathematical discussions are finite, not that infinity is finite - an absurdity that seems peculiar to you.

You are amazing! — TonesInDeepFreeze

"Incorrigible" would be more accurate.

if you like. Folk to attach too much to truth. There are true sentences.

I do wonder if his inability to understand speech acts is related to his extreme individualism.

I have.

I’ve addressed your post and comments directly.

More misrepresentation. Pathetic.

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.

Perhaps; think the context part of the use, but the distinction might be made.

However the issue here seems more central to our basic accounts of language. seems to hold that while we can make marks and sounds, we do not ask questions or make statements; or that acts are bodily movements, and hence that questions and statements are not acts; or something, that i have not been able to fathom.

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.

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.

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.

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.

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

When one definition simultaneously addresses every possible objection... — PL Olcott

...one should proceed with extreme scepticism.

You have not understood Quine. I don't think you have understood the analytic/synthetic distinction. And I don't think that on this topic you are "open to learning", as teachers sometimes say. You have produced the answer without first making sense of the question - something you already did in your previous threads.

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.

It is indistinguishable from the locutionary act. — NOS4A2

Well, no, it isn't. Making marks and asking a question are very different acts. Here are the same marks: "Any advice?". I am not using them here in order to asking a question. So there is a difference between making the marks and asking the question; which is to distinguish between the locution and the illocution. And neither is "invisible", what ever that might mean in this context.

Your response is no act of mine. — NOS4A2

No, but your eliciting a response is an act of yours. Just as your post elicited this reply. I would not have written this were it not for your post, and hence this post is an act resulting from your act.

You spoke them or wrote them. No others acts have occurred or are apparent or can be measured. — NOS4A2

You also made statements and asked questions. Are these not acts you performed? Why not?

...the theory ought to be reworked to include “listening acts”, the acts of a listener. — NOS4A2

Speech act theory is embedded in social discourse, implicitly and explicitly addressing the place of utterances in social activity. Perlocutions include the acts of the listener.

From recollection, you maintain a form of hyper-individualism, which it seems makes it difficult for you to see the social aspects inherent in speech acts. I remain unable to see what your objection is.

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.

yeah, that must be it.