Forum members who've spent any amount of time in dialogue with Banno know he's all about force and politics. (For all his emphasis on a foundational charity.) — ZzzoneiroCosm

That, and playing language games.

If using numbers and words doesn't entail using finite objects to refer to other finite objects, then Banno isn't talking about or counting a number of anything. He would just be making ink marks on paper or making sounds with his mouth when "counting".

In other words, how is it that finite signs, as expressed by finite beings, have a sense of infinity. — Sam26

More language games.

How can you even say that one follows from the other - that one gets a sense of infinity from finite signs expressed by finite beings?

Interesting that Wittgenstein considered Russell naive.

How can you even say that one follows from the other - that one gets a sense of infinity from finite signs expressed by finite beings? — Harry Hindu

Where do you think our sense of infinity comes from? It comes from us, i.e., finite beings, we create the concepts using finite signs. We extrapolate based on the continuation of 1,2,3.. that it goes on ad infinitum. There's no mystery here.

In other words, how is it that finite signs, as expressed by finite beings, have a sense of infinity. This has more to do with Wittgenstein's later philosophy, i.e., what it means to master a technique or practice — Sam26

There is a means of grasping a rule that is not an interpretation, but is exhibited in following and going against the rule in actual cases, says (paraphrased) Witty.

Did he have anything to say about the Halting Problem? I have a sudden, strong hunch that it's related to this. Maybe I'm just seeing things, but grasping that a Turing machine goes on forever without doing any calculations seems to be a case of grasping a rule in Wittgenstein's sense.

Did he have anything to say about the Halting Problem? I have a sudden, strong hunch that it's related to this. Maybe I'm just seeing things, but grasping that a Turing machine goes on forever without doing any calculations seems to be a case of grasping a rule in Wittgenstein's sense. — Pneumenon

I don't know Pneumenon.

If using numbers and words doesn't entail using finite objects to refer to other finite objects, then Banno isn't talking about or counting a number of anything. He would just be making ink marks on paper or making sounds with his mouth when "counting". — Harry Hindu

I believe I've heard Banno say language is non-referential. Words don't refer to anything. So why would "1" be the exception?

This has more to do with Wittgenstein's later philosophy, i.e., what it means to master a technique or practice. — Sam26

Yes. However see 6.211, from the Tractatus, in which he talks of mathematical propositions being nothing unless used. That's gotta be a harbinger of things to come.

So he certainly would not have gone along with the finitism of @Metaphysician Undercover who rejects instantaneous velocity.

But,

Though commentators and critics do not agree as to whether the later Wittgenstein is still a finitist and whether, if he is, his finitism is as radical as his intermediate rejection of unbounded mathematical quantification (Maddy 1986: 300–301, 310), the overwhelming evidence indicates that the later Wittgenstein still rejects the actual infinite (RFM V, §21; Zettel §274, 1947) and infinite mathematical extensions.

and

The first, and perhaps most definitive, indication that the later Wittgenstein maintains his finitism is his continued and consistent insistence that irrational numbers are rules for constructing finite expansions, not infinite mathematical extensions.

So it's being argued that "1" has an extension, while "root 2" does not - that "1" pints to 1, while "root 2" points to a recursive rule for generating an infinite decimal. However I'm thinking, as posited in the OP, that neither has an extension.

Well, there would indeed be an error in thinking that charity was nice.

I may have been a bit rough on Harry; in my defence, on the occasions in which I have engaged with him, not much happened.

This is why I said 1 is a pointer, to imply axiom is use. — ztaziz

If it is a pointer, it can be used to point to anything.

Which seems odd.

on the occasions in which I have engaged with him, not much happened. — Banno

If using numbers and words doesn't entail using finite objects to refer to other finite objects, then Banno isn't talking about or counting a number of anything. He would just be making ink marks on paper or making sounds with his mouth when "counting". — Harry Hindu

Sometimes we do talk about infinity. When we do this, we are using finite objects - ink marks and sounds.

So...?

In other words, how is it that finite signs, as expressed by finite beings, have a sense of infinity.
— Sam26
More language games.

How can you even say that one follows from the other - that one gets a sense of infinity from finite signs expressed by finite beings? — Harry Hindu

I'm not following this at all. Are you claiming that we do not talk about infinity? OR that such talk is no more than sounds?

, ? What's this about?

Did he have anything to say about the Halting Problem? — Pneumenon

See 3.3 The Later Wittgenstein on Decidability and Algorithmic Decidability

SO a Turing Machine could be set up to calculate 1+1, and would halt - hence 1+1 has an extension; but if set up to find root 2, it would not, and hence root 2 has no extension... or something like that.

I believe I've heard Banno say language is non-referential. Words don't refer to anything. So why would "1" be the exception? — ZzzoneiroCosm

Banno thinks that "Banno" can be used to talk about Banno.

Banno thinks that "Banno" can be used to talk about Banno. — Banno

Nothing controversial about that.

It was by way of contrast with

Banno say language is non-referential. — ZzzoneiroCosm

.

Language can be about stuff. It's just that it can do other things as well. This in contrast with what might be @Harry Hindu's view - it's hard to tell - that language is only about...

Language can be about stuff. It's just that it can do other things as well. — Banno

I'm sure Harry Hindu agrees language can both be about stuff and do other things as well.

But I remember a thing about the non-referentiality of the T-sentence. I possibly assumed if the T-sentence is non-referential all language is.

Then there was the thing about the uselessness of a non-referential T-sentence.

Does a word refer?

Does a word refer? — ZzzoneiroCosm

Think on that question. I've already said that "Banno" (a word) can be used to refer to Banno.

SO what is it you are asking?

So "Banno" refers and "1" doesn't.

Maybe so. Maybe not.

"Was it for this my life I sought?
Maybe so and maybe not (Maybe so and maybe not)"

Stash, Phish

https://www.youtube.com/watch?v=1BJlSU308Wc

So he certainly would not have gone along with the finitism of Metaphysician Undercover who rejects instantaneous velocity. — Banno

The reason for rejecting "instantaneous velocity" has nothing to do with mathematics, the notion is self-contradictory. Velocity is distance covered in a period of time. There is no period of time at an instant. There is no distance covered at an instant. There is no velocity at an instant. There is no "instantaneous velocity". No matter what sophistry the mathemagician might apply, the smoke and mirrors cannot hide the contradiction from a trained philosopher.

SO a Turing Machine could be set up to calculate 1+1, and would halt - hence 1+1 has an extension; but if set up to find root 2, it would not, and hence root 2 has no extension... or something like that. — Banno

So the existence of potential infinites is secured by our ability to grasp a rule, and that rule becomes an intension in the sense used in the SEP article. If the rule allows to construct a finite extension, then we can get extensions from it, too.

So the extension of the set of integers is always finite, although it can be continued arbitrarily. And now I'm being assaulted by that giddiness of logical legerdemain that Witty talks about...

Maybe this is swinging too hard, but: the motivation for this eludes me. Abstracta are spooky, but so are ineffable rules grasped without interpretation. Why does Wittgenstein like this spook more than the Platonic spook?

If the rule allows to construct a finite extension, then we can get extensions from it, too. — Pneumenon

This is the bit that I've been unable to find clearly articulated. But it seems to be what is being argued.

Why does Wittgenstein like this spook more than the Platonic spook? — Pneumenon

It seesm to be...

There is a means of grasping a rule that is not an interpretation, but is exhibited in following and going against the rule in actual cases, says (paraphrased) Witty. — Pneumenon

There is no velocity at an instant. — Metaphysician Undercover

Yep, so you have said.

And yet, we can Calculate Instantaneous Velocity

So we conclude that either physics is wrong, or Meta is wrong.

Okay, let's try an example: the successor axiom in Peano arithmetic says that if a is a number, then so is its successor. And the induction axiom says that if s contains 0, and also every successor of every one of its elements, then s contains all the numbers.

So does Witty's constructivism make the induction axiom nonsense, or does it mean we have to construct the induction axiom from an intension and the number 0? The successor axiom, presumably, is or contains an intension.

Yeah, that's the sort of thing I've been trying to work out. I'm gathering that the point is moot.

For my own part, I'm thinking that the extension/intension juxtaposition in this context is ill-defined and confusing... or it might be just me. Anyway, hence the OP; that "1" does not have an extension; or rather that talk of extension/intension is misplaced in mathematics.

There is no velocity at an instant. — Metaphysician Undercover

That's because you confuse stopping a particle at a specific time and observing a particle at that time. Don't forget momentum. :roll:

If the rule allows to construct a finite extension, then we can get extensions from it, too.
— Pneumenon

This is the bit that I've been unable to find clearly articulated. — Banno

Just to be clear, are you both dropping (or taking as read) an "infinite"?

There are infinities.

... But no typos?