I've pretty much given up on Harry. Too hard to make sense of his posts.

Is that settled science? — ZzzoneiroCosm

What?

abstract entities come form our talking about what we do

I presume you do not mean that abstract entities are generated from our talking about what we do, since that would make them things anyway. So what do you mean by "come from"?

Hm. We can choose to include or exclude whichever laws of logic we like, so long as we live with the consequences. Hence,

Equality and identity are distinct ideas... — Metaphysician Undercover

Well, that is up to you. If you can make a coherent system along these lines, then go for it. That's what the Tractatus says about logic.

I presume you do not mean that abstract entities are generated from our talking about what we do, — jkg20

But one of the things we do, is to talk. And we can talk about our talk.

Every few years I set up a forum game in which players take it in turns to add a new rule. Offten the consequences are mediocre. Sometimes they are extraordinary.

But you can start anywhere... and you get the same number.

Perhaps - but it is so difficult to articulate.

But you can start anywhere... and you get the same number.

I don’t get it. Maybe I’m counting wrong, because I’ve never used “1” to refer to the third count in a counting sequence.

I don’t get it. — NOS4A2

Yeah, I noticed that.

Re Wittgenstein's finitism, for me it always just fell out from his view that mathematics is nothing over and above a human activity, and since we are finite, nothing we can construct is going to be infinite. Not sure whether that's a good argument or not, but it seems to be the bare bones of it. The idea that sets have only finite extensions is in any case not generally accepted, ZF set theory even has an axiom that includes infinte sets from the outset. Of course, W's reading of this would presumably be that the axioms just give recursive rules we can use to continually come up with new, disinct members to add to a set, but we always have to stop doing that at some point and just say "and so on" or, the more mathematically acceptable, "...". As you said, sure extensions are finite if you define extensions as finite, and Wittgenstein defined them as finite, but others did not. He thought they were making a mistake. They did not.

mathematics is nothing over and above a human activity, and since we are finite, nothing we can construct is going to be infinite. — jkg20

I'm flummoxed that Wittgenstein's argument might be so artless. And so, I'm asking for something more.

So we have him baulking at the the diagonal and rejecting incompleteness as a result. Yet I am in agreement with his constructivist views, as set out above. I was bothered that the one might necessitate the other; however it seems now that they are unrelated. At least, no one here seems to have shown such a relation - but then so many of the replies appear not to have been on topic.

So I might be wrong.

@jgill? What sort of thing are numbers?

If numbers are limited to aspects of the act of counting, then the fact that it's impossible to count to infinity, which is finitism's point in a nutshell, becomes more significant.

So we have him baulking at the the diagonal and rejecting incompleteness as a result. Yet I am in agreement with his constructivist views, as set out above. I was bothered that the one might necessitate the other; however it seems now that they are unrelated. — Banno

I get the sense from the SEP article that Wittgenstein will allow what modern philosophers of mathematics call potential infinites. It looks like Witty would be okay with, e.g. saying that the successor function can be applies an infinite number of times.

I'm not sure how problematic this really is. "There is no set of all the real numbers" is only true from this perspective if we regard the set as an extension. What's stopping us from just paraphrasing it as shorthand for what can be done with an intention and a finite extension(s)?

EDIT: nevermind, I derped. Obviously you can't do that with the reals 'cause they're uncountable. So you have to junk the reals.

...we should take some pains to distinguish constructivism from intuitionism.

As I've suggested, I don't see how constructivism is committed to finitism

Nor do I see that constructivism is committed to rejecting the Law of Excluded Middle. Rather, a constructivist approach would say that including the law leads us this way, excluding it leads us that way; and which way you choose depends on what you are planning to do.

Another way to put it is, contrary to Brouwer, mathematics just is the language of mathematics.

Nor is mathematics just a creation of the mind; that's too solipsistic. Mathematics is a collaborative enterprise, not something in individual minds.

Maybe I'm just dense, but: the reals are uncountably infinite, which means, if we're Wittgensteinian constructivists, that the set of real numbers doesn't exist, since it can't be paraphrased as shorthand for something that can be done with an intension and a finite set of extensions.

But you said that those two things are unrelated. How come? How do you save the set of reals if it can't be constructed recursively?

Side note on ∞:

There's an anecdote attributed to Wittgenstein showing how an infinite past seems uniquely counterintuitive.

Wittgenstein overhears someone saying "5, 1, 4, 1, 3. Done."
He asks what that was about, and they respond that they just finished reciting π backward.
"But, how old are you?"
"Infinitely old. I never started, but have been at it forever and finally finished."

Not logically impossible or inconsistent, notes both James Harrington and Craig Skinner, but a strong intuitive argument nonetheless.
The moment they were done reciting seems random, there seem to be no sufficient reason their recitation was done at one time and not another, any other. And likewise for any of the other digits.
So, with our expectation violated, we tend to reject the thought experiment, and out goes an infinite past.

Anyway, looks like Wittgenstein doesn't accept ∞ as such.

Not sure I buy the rationale here, but intuitionist physics is a worthwhile pursuit I think:

Does Time Really Flow? New Clues Come From a Century-Old Approach to Math.
Natalie Wolchover
Quanta Magazine
Apr 2020

If you can make a coherent system along these lines, then go for it. — Banno

I take that as an invitation. Equality always requires a qualification, the same quantity, the same quality, the same size, shape, degree, etc.. Without that qualification "equal" is meaningless. Identity indicates "the same" absolutely, without qualification. The former, equality, Is an identification by means of reference to properties. The latter, identity, is an identification of the object itself, regardless of properties. Can you apprehend the difference between pointing to an object, thereby identifying that object without reference to any properties (identity), and stating such and such properties, and finding whatever object, or objects, which are indicated by (equal to) that description.

Notice Tractatus 4.12721, a multiplicity of objects fall under a formal concept, so a formal concept itself cannot be an object. A number "1", "2", as a formal concept cannot be an object. The law of identity indicates an object, it cannot indicate a multiplicity of objects which are equal by the terms of the concept, which is what is signified by a formal concept.

"1" has the superficial grammar of a noun, but this is misleading.

Rather "1" is to be understood through its role in the process of counting. It is understood in learning how to count, not in pointing to individuals. — Banno

"1" refers to an abstraction , as do all natural numbers.

Consider the abstraction, "3" . Three-ness is a property that is held by some states of affairs: those consisting of 3 objects.

"+1" is a successor relation that holds between two consecutive natural numbers. In the real world, this maps to a relation between states of affairs. For example it's a relation between a collection of 3 apples and a collection of 4 apples.

'Hilbert said that in a proper axiomatization of geometry, “one must always be able to say, instead of ‘points, straight lines, and planes’, ‘tables, chairs, and beer mugs’”' — Baden

This may work in some weird way, but this would also bastardize the language. The concepts "point, straight lines, planes" have at least some semblance to human envisioning what these things mean in geometric concepts. While "tables, chair and beer mugs" would also work if used consistently, there are already assigned meanings for these words that are vividly different from the assigned approximate meanings of point lines and planes in our language.

In other words, a person could rewrite entire books of science, philosophy and literature, by assigning to each word's meaning a totally different existing word, which would lose its original meaning. This is good exercise is logic and in theoretical thinking of the use of language, but would amount to nothing more. Therefore it is not done. Notice, that no textbook of geometry uses "tables, chair an beer mugs" for "points, straight lines and planes." There is no other reason for the lack of wicked bastardization, but the fact that some words are more conducive to conjure up a meaning for a newly introduced concept.

jgill? What sort of thing are numbers? — Banno

One is the sound of a single finger snapping. :cool:

Like staring at the sun, looking too hard into the foundations of mathematics can damage the mind's eye.

Rather "1" is to be understood through its role in the process of counting. It is understood in learning how to count, not in pointing to individuals.

And of course this goes for other mathematical entities, too. They are things we do, not things we find. — Banno

In a running race (like Marathon races) people are assigned each a different number. For identification purposes.

Would you call those numbers (one included, and other mathematical entities included, such as "2", "4", etc.) things we do, not hings we find?

The runners could be called "A" "B" "C" ... "AAZAET", etc. or they could be identified with a scale of colours.

This opening post has pretended to define the true nature of "1", but alas with impoverished thinking. Language uses its components in many ways, and to try to restrict a multiply-used component to fewer uses than the language already employs for that component, is a proposition that is obviously wrong.

Let me explain. The Opening Post appeals to the masses to use the word only in the meaning that the writer of the OP allows. But the word has long ago grown beyond that meaning only. The OP ignores other valid meanings to prove its wrong point, and declares the other valid meanings wrong. This amounts to nothing less than trying to redefine the language.

The moment they were done reciting seems random, there seem to be no sufficient reason their recitation was done at one time and not another, any other. — jorndoe

That's why they finished when they did. That point in time when they actually finished was just as valid as any other point in time to finish, since any other point in time would have been equally as valid a finishing point as the actual one.

The reason that is sufficient to explain why they ended when they did, is that 1. They could have ended any time, reasonably, and 2. the time they ended at was in the set of "any time", and 3. unsaid, but assumed, and fulfilled the requisite, that there is only one time that the recitation ends. It can't end, for instance, two different times. Or 345 different times. It can only end one time.

I'm flummoxed that Wittgenstein's argument might be so artless. And so, I'm asking for something more. — Banno

I do not think you are likely to get it. As far as I recall Wittgenstein himself did not in the end think very highly of his lectures on the foundations of mathematics.

I think 1 is a pointer, when purely thought, void of 1X.

What is a pointer?

It's something that points a mind to an object/subject.

Like a lazer pen - not to say we have cycloptic vision.

I never actually agreed with the number's significant role.

1 points to 2-9 in base 4 and 0 can be understood relative to 1, base 4 is a category of 1, and so is base 8. 1, has it's use.

In this case it's not a pointer but a medium of communicate object/subject relativity. That is 1X, interpreted impurely.

It's just not so significant...

we pretend that integers are real things, and this leads us on to more complex ways of talking about integers, and so a sort of recursion allows us to build mathematics up from... nothing. — Banno

Sure, maths as fiction with a super-coherent plot.

And with illustrations, too. Kind of, Alice in Wonderland.

I've pretty much given up on Harry. Too hard to make sense of his posts. — Banno

Yet Sam26 was able to make perfect sense of what I said, and everyone else I have had a conversation with on these forums, was able to make perfect sense of what I said. You are the only one that has a problem making sense of what I say.

I don’t get it.
— NOS4A2

Yeah, I noticed that. — Banno

But when others can't make sense of what you said then that's their problem. :roll: I said the same thing as NOS4A2, yet you understood them. These are the symptoms of a delusional disorder.

But you can start anywhere... and you get the same number. — Banno

...the same number of what? In starting anywhere, you'd change the context of your counting, and would be counting something different, so how would you get the same number?

One could argue, probably successfully, that Wittgenstein was not a finitist, i.e., he never held to the idea that the finite character of language meant that there weren't infinite processes or methods. He was mainly interested (at least it can be argued) in the problem of the grammar of the infinite method or procedure. 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.

These are the symptoms of a delusional disorder. — Harry Hindu

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.)

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

True. Wouldn't that be appropriate for a behaviorist?

Well. I agree with Banno.

People esteem 1 too much when it's clear that we have done wrong by it.

It would be better if 1 didn't exist so referring to it using 'it' would be false.

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