Part of this thread is working out what finitism is.

We need a philosopher of math. They're rare.

I see Harry found Sam's new thread.

We have @jgill.

Cool. How did he explain finitism? Aristotle?

Take a look for yourself.

You don't remember? :joke:

But otherwise yes, "1" obviously doesn't refer to anything at all. — StreetlightX

"1" represents an idea of quantity. Does it follow that "1" refers to an idea of quantity? I would say the answer to that depends on what you mean by "refer".

StreetlightX" refers to StreetlightX, whereas "1" refers to "1". — ZzzoneiroCosm

Being a pedantic arsehole, I am driven to correct you on this: "1" refers to 1.

So the rule is that for every number, one can add one. The rule only generates one new number. One has to see the rule in a different way in order to understand infinity: imagine a number bigger than any number the rule could generate... — Banno

All you've done is offered two distinct definitions of "number". Under the first definition, we get a bigger number by adding a number. Following that rule, there is no way to get a number bigger than what is given by that procedure. Under the second definition, we must assume that there are other numbers, not derived in the first way. The bigger number in the second definition will never make it into the first set of numbers, so the two are in that sense incompatible. It's not a huge problem, just like different possible worlds, so long as we recognize the points of incompatibility. If the incompatibility goes undetected there may be a problem because each is named "number".

Here's the issue though. We can count anything we would ever need to count using the first rule, so what's the point of the second? It doesn't help us to count anything, all it says is that no matter how high we count we can still count higher. But we already knew that, because we know that we can keep adding a number. So it doesn't allow us to do anything more than we can do with the first rule, nor does it tell us anything we didn't already know from the first. It is completely useless, and on top of that it gives us a new kind of "number" which is incompatible with the rules in the system of counting. Looks like an axiom designed for equivocation to me.

Anther way to approach it that the rule "For every number, you can add one. to make a bigger number" is not generating all the numbers, but only the integers. We can find infinity by calculating 1 divided by 3, as a decimal; or by asking what number times itself makes 2. — Banno

You're looking at this in completely the wrong way. A whole number is undivided. The integers are a special formulation of whole numbers, allowing for the inclusion of zero and negatives. Now you want to divide these whole numbers into parts. These are fractions. So why not call them "fractions", because that's what they are? Instead, you want to call them "numbers". Same problem as above, we now have some sort of numbers which are incompatible with the other "numbers". Why do that? You're just creating confusion and a recipe for equivocation again. If the "numbers" are the counting numbers, and we can (in theory) divide these numbers into parts, then why call the parts "numbers" as well?

Do you recognize that "one" is a fundamental unity? If you divide one in half, this does not give you two, it gives you two halves. Why would you want to represent a half as a number, when it's clearly not a number, it's a half? Some flamboyant mathemagician artist comes along and says let's make an axiom whereby a half, along with all other fractions become numbers, wouldn't that be cool. No it wouldn't be cool because there's a big problem, some fractions cannot be represented as numbers. Instead of recognizing, well that was a mistake, let's leave a distinction between numbers and fractions, the mathemagicians just try to cover up the mistake with more and more complex axioms.

SO we learn how to count, and then we learn how to do other things with counting. — Banno

This not quite right. You should say that we learn how to count, then we learn how to do other things with numbers. The other things are not counting. We could call the other things "art", but a lot of it is more like a magic show, illusions, smoke and mirrors, deception.

This is both funny and profound. — jkg20

Yeah, it's kind of fully sick Zen.

We can count anything we would ever need to count using the first rule, so what's the point of the second? — Metaphysician Undercover

Well, with a bit of work it allows us to find an instantaneous velocity... among other things.

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. — jkg20

The reification consists in the idea that there are an infinite set of numbers. The ordinary procedure of counting has no logical limit to it; so, it can go on forever and hence comes the illusion of infinite series. Counting can go on forever, but there is never a point where infinity is attained, or even more closely approached.

Ah, the comfortable refuge of (supposed) authority!

Well, with a bit of work it allows us to find an instantaneous velocity... among other things. — Banno

Smoke and mirrors.

Smoke and mirrors. — Metaphysician Undercover

You keep saying that, as if it were an argument.

But we have:

There is no velocity at an instant. — Metaphysician Undercover

And yet, we can Calculate Instantaneous Velocity

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

I am driven to correct you on this: "1" refers to 1. — Janus

My thought was that "1" might refer to "1". A circularity. Not that "1" might refer to 1.

"1" represents an idea of quantity. Does it follow that "1" refers to an idea of quantity? I would sy the answer to that depends on what you mean by "refer". — Janus

If "1" refers to an idea, then it is an idea shared. Else your idea of 1 would not be the same as mine.

So what sort of thing is that?

You keep saying that, as if it were an argument. — Banno

I made the argument, and addressed your reference.. You rejected my argument with nothing more than "you're wrong". Sorry but it's you who has presented no argument.

Here's my argument: you and physics cannot both be right. Which to choose, which to choose....

I explained already, the uncertainty principle demonstrates that physicists are not really calculating instantaneous velocity. Physics is wrong, they are not calculating instantaneous velocity. They might call it that, but it's clearly not what it is. Otherwise there'd be no uncertainty in the question of the momentum of a particle when it is at a specific place at a specific point in time.

I can see why you would want to change the topic.

↪bongo fury So you want to argue that — Banno

One thing at a time?

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"? — bongo fury

To which,

↪bongo fury There are infinities. — Banno

Ought that have clarified for the competent reader that @Pneumenon meant "then we can get infinite extensions from it, too"?

Just hoping not to misunderstand either one of you.

I'm lost. Not sure what you are asking.

Did "then we can get extensions from it" mean "then we can get infinite extensions from it"?

So... you are asking what I think Pneumenon meant that Wittgenstein meant at https://thephilosophyforum.com/discussion/comment/404886 ?

We were trying to decide if the version of Wittgenstein in the article thought there couldn't be infinite extensions... I think...

Which is why I said

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. — Banno

So... you are asking what I think Pneumenon meant that Wittgenstein meant at — Banno

No! Only whether the word-string "then we can get extensions for it" was a misprint of "then we can get infinite extensions for it"?

A different reading of it (as not a misprint) seemed plausible, so I thought I should check.

And ...?

I can see why you would want to change the topic. — Banno

You're the one who changed the topic. Instead of wanting to discuss the issue, what it is that is represented by the formula they call "instantaneous velocity", you changed the subject to a question of who's wrong, physics or meta.

Your refusal to address the issue is getting rather boring. Instantaneous velocity is an average, we went through this yesterday. There is no such thing as a determination of velocity at a point in time. That's obvious, nothing moves when no time passes, so to determine any velocity requires a period of time. If this does not make sense to you, and you won't take it from me, do some reading as to what "instantaneous velocity" really is, it's an average.

Instantaneous Velocity
The quantity that tells us how fast an object is moving anywhere along its path is the instantaneous velocity, usually called simply velocity. It is the average velocity between two points on the path in the limit that the time (and therefore the displacement) between the two points approaches zero.

https://openstax.org/books/university-physics-volume-1/pages/3-2-instantaneous-velocity-and-speed

Notice the decisive phrase, the time "between the two points approaches zero". If it was truly an instant, there would be no time, the value for t would be zero, and the equation would be useless.

If "1" refers to an idea, then it is an idea shared. Else your idea of 1 would not be the same as mine.

So what sort of thing is that? — Banno

A sort of thing that is.

Instantaneous velocity is an average, — Metaphysician Undercover

The page you referenced quite explicitly sets out the difference between average velocity and instantaneous velocity.


Fuck, there's even a diagram.

A sort of thing that is. — ZzzoneiroCosm

Abstract object.