Finitism made precise

[Skepdick]

I reject infinities. What is it that I am rejecting exactly and precisely?

I reject the claim that you can always add 1 to a mathematical object to get an even larger object.

"Always" implies an infinite source of 1s. Infinitism is circular and pre-supposes itself. It also smuggles time into mathematics through the English language. Most mathematicians deny this fact.

Finitism: You can add 1 until you run out of 1s to add.

Tick. Tock. Tick. Tock... Mathematicians think they have an infinite clock.

[Iwannaplato]

I reject infinities. What is it that I am rejecting exactly and precisely?

I reject the claim that you can always add 1 to a mathematical object to get an even larger object.

"Always" implies an infinite source of 1s. Infinitism is circular and pre-supposes itself. It also smuggles time into mathematics through the English language. Most mathematicians deny this fact.

Finitism: You can add 1 until you run out of 1s to add.

Tick. Tock. Tick. Tock... Mathematicians think they have an infinite clock.

I don't think mathematicians treat infinity as a number, not that this is what you said.

Can you present a situation where we have a number and we can't add 1 to it?

Or perhaps the 'always' means that one is always adding to something, which implies infinite longevity.

But if it meant 'in any given situation,' then I ask my question.

[Skepdick]

I don't think mathematicians treat infinity as a number, not that this is what you said.

Sure. They treat it at a never-ending source of numbers. Take 1 from infinity. And another one. And another one.
Infinity is still same size.

And. you are getting something out of nothing.

Can you present a situation where we have a number and we can't add 1 to it?

Any and every situation in which you are prevented from doing so. You run out of resources.

You ask for 1 more and the source says "sorry - out of stock. Come back later".

Or perhaps the 'always' means that one is always adding to something, which implies infinite longevity.

It's the same confusion about the way infinities behave.

Adding to something doesn't change its size. It's an infinite void. A black hole.
Taking from something doesn't change its size. Infinite source. Of whatever it is the source of.

But if it meant 'in any given situation,' then I ask my question.

It means that and it's just true. A tautology. By definition.

You need one more second. But you die now.

[Iwannaplato]

Sure. They treat it at a never-ending source of numbers. Take 1 from infinity. And another one. And another one.
Infinity is still same size.

And. you are getting something out of nothing.

I didn't understand this. Do you mean they use infinity to justify whipping out some enormous number, one that doesn't apply to anything counted or even estimated to be real in some way? Something like that?

Any and every situation in which you are prevented from doing so. You run out of resources.

You ask for 1 more and the source says "sorry - out of stock. Come back later".

OK.

It's the same confusion about the way infinities behave.

Adding to something doesn't change its size. It's an infinite void. A black hole.
Taking from something doesn't change its size. Infinite source. Of whatever it is the source of.

Could you give me an example?

And it seems like you are saying that mathematicians are just making stuff up. I think at least some would agree. Then very oddly some of this made up stuff ends up applying to reality or being useful. Some, not all of math. Like imaginary numbers seem to be useful in quantum mechanics. Or non-Euclidian geometry in Einstein's stuff.

I don't know what that all means. Could be luck. I wouldn't know how to analyze the statistics involved: well, make up shit long enough some of it may relate to reality.

[Skepdick]

Sure. They treat it at a never-ending source of numbers. Take 1 from infinity. And another one. And another one.
Infinity is still same size.

And. you are getting something out of nothing.

I didn't understand this. Do you mean they use infinity to justify whipping out some enormous number, one that doesn't apply to anything counted or even estimated to be real in some way? Something like that?

Don't even call it a number. It's a wrapper, exactly. It encapsulates/hides/blackboxes some magic inside.
Think of it as a blackbox with a button. Push button - get number 1.
There never comes a time when you push the button and you get nothing.

Could you give me an example?

Practically - maybe, but this is much much easier conceptually.

Imagine a bag of golf balls. You keep adding golf balls but it never overflows.
And it's the exact same bag from which you can keep taking balls from and it never overflows.

infinity - x = infinity
infinity + x = infinity

For any and all numbers (that English speakers are familiar with)

And it seems like you are saying that mathematicians are just making stuff up.

We all are. That's what conception means. Creating ideas.

I think at least some would agree. Then very oddly some of this made up stuff ends up applying to reality or being useful. Some, not all of math. Like imaginary numbers seem to be useful in quantum mechanics. Or non-Euclidian geometry in Einstein's stuff.

Yep yep. That's the way it goes, the problem is that there's all sorts of bugs/creatures waiting to fall out the woodwork. For example when you talk about something that doesn't exist. Nothing. Or use the number 0.

Try to follow what it refers to. Oh. Nothing. It's the logical error. The nothing from which the something comes from.

I don't know what that all means. Could be luck. I wouldn't know how to analyze the statistics involved: well, make up shit long enough some of it may relate to reality.

Yeeep.

[Iwannaplato]

Don't even call it a number.

I didn't mean that infinity was a number, if that's what you meant. I was just thinking you meant that once infinity is on the table, then a quadillion to the power of itself ten times is on the table in a way it wouldn't be otherwise.

And it seems like you are saying that mathematicians are just making stuff up.

We all are. That's what conception means. Creating ideas.

Emphasis on 'just'. That it has no relation to reality.

Yep yep. That's the way it goes, the problem is that there's all sorts of bugs/creatures waiting to fall out the woodwork. For example when you talk about something that doesn't exist. Nothing. Or use the number 0.

Try to follow what it refers to. Oh. Nothing. It's the logical error. The nothing from which the something comes from.

This I didn't really get. Nor what I didn't respond to.

I don't know what that all means. Could be luck. I wouldn't know how to analyze the statistics involved: well, make up shit long enough some of it may relate to reality.

Yeeep.

OK, so does it strike you as amazing? This is purely intuitive. As I said I wouldn't know how to analyze or even create statistics on this, and I'd be skeptical even experts could (on what grounds? dunno).....but
it seems like quite often things that people just make up fiddling with numbers and symbols ends up being useful in math. I have sympathy for the mathematical realists.

[Skepdick]

Don't even call it a number.

I didn't mean that infinity was a number, if that's what you meant. I was just thinking you meant that once infinity is on the table, then a quadillion to the power of itself ten times is on the table in a way it wouldn't be otherwise.
[/quote]
That's probably a bad metaphor because it assumes the standard real numbers. Yes in any number system without infinitesimals that's exactly how it would behave.

But with infinitesimals it would get just that much bigger.
Or just that much smaller. Never quite fully deflating and never quite fully inflating.

Emphasis on 'just'. That it has no relation to reality.

The ideas have some properties that are useful in practice. Space/time complexity. Predictive power. Behaviour that can be run on computers.

It's "scientific" in that it's solid in some particular notion of solid. This notion excludes deconstruction/skepticism.

Then it all deflates.

This I didn't really get. Nor what I didn't respond to.

Let me try another way. Mathematics is intentional logical errors. Choice. Axioms which save time.

We know there's an error, but we turn a blind eye. Like reifying nothing(0) making it something. If you ever asked me to show you 0 like atheists demand evidence for God I can't show you anything other than the writing of the Prophets and to assure you that I believe in 0.
But it's also true that I believe in non-existing things. because 0 is nothing and nothing doesn't exist.

It's a self-contradictory idea but it gives us the principle of induction. Useful!

OK, so does it strike you as amazing? This is purely intuitive.

Yes. I am 100% committed to intuitionism/computer science.

As I said I would know how to analyze or even create statistics on this, and I'd be skeptical even experts could.....but
it seems like quite often things that people just make up fiddling with numbers and symbols ends up being useful in math. I have sympathy for the mathematical realists.

In the pub sympathy is all they get. Soon as you get the philosophy hat on - it's mortal combat

It's just a silly game. Fun game. Amuzing game.

Pick a team. Don't take it serious, your ego's going to take a bruising

[Magnus Anderson]

I reject the claim that you can always add 1 to a mathematical object to get an even larger object.

What that means is that, for each integer that you can imagine, you can come up with a larger integer ( e.g. by adding 1 to it. )

If you think that's false, all you have to do to prove it is to show us an integer for which this isn't the case.

You haven't done that.

Instead, what you did is state that the argument in favor of the claim that there is no such thing as the largest integer that can be conceived is a circular one, and thus, a faulty one. The reason you think it's a circular one is because you think that it starts with the premise that there is an unlimited number of 1s. I don't think that's true.

How do you prove that a given quantity exists?

But before that, what does it mean to say that a quantity exists? It certainly does not mean that there is a real life example of such a quantity. It also does not mean that such a quantity can be realized at some point in the future nor that it was possible to realize at some point in the past. Instead, what it means is that such a quantity can be conceived, that it is a logically possible quantity. Not all quantities are logically possible. For example, the quantity that is greater than 1 and less than 1 is a logically impossible quantity.

As such, to prove that a quantity exists, in a mathematical sense of the word, you have to show that it is a logically possible quantity, one that is not a contradiction in terms, not an oxymoron. And in order to do that, you have to verify that its associated concept is free from contradictions. If you want to argue the opposite case, you have to point to a contradiction.

I reject infinities.

When you reject infinities, you reject the ability to answer questions such as "How many natural numbers are there?" It's a handicap, that's for sure.

[Skepdick]

I reject the claim that you can always add 1 to a mathematical object to get an even larger object.

What that means is that, for each integer that you can imagine, you can come up with a larger integer ( e.g. by adding 1 to it. )

If you think that's false, all you have to do to prove it is to show us an integer for which this isn't the case.

You haven't done that.

Burden of proof is not on me. It's on you to show where all these 1s you keep adding are coming from. How many of them do you have?

Prove that "you can always add 1" is true. From what premises does it follow?

Instead, what you did is state that the argument in favor of the claim that there is no such thing as the largest integer that can be conceived is a circular one, and thus, a faulty one. The reason you think it's a circular one is because you think that it starts with the premise that there is an unlimited number of 1s. I don't think that's true.

You are contradicting yourself. Suppose you have counted up to the number X by starting at 0 and then add 1, add 1, add 1. Suppose you run out of 1s.

If you never run out of 1s - of course there is no largest integer.
But if you do run out of 1s - of course there must be a largest integer.

How do you prove that a given quantity exists?

By constructing it with all the materials at your disposal.

If you can't construct it from the materials available to you - it's does't exist.
If you can construct it from the materials available to you - it does exist.

But before that, what does it mean to say that a quantity exists?

In Mathematics anything defined to exist - exists. If you define an "infinite number of 1s" to exist - then they exist.

If you remain silent about it and just keep adding 1s.... what do you have behind your back there? Show me!

It certainly does not mean that there is a real life example of such a quantity. It also does not mean that such a quantity can be realized at some point in the future nor that it was possible to realize at some point in the past. Instead, what it means is that such a quantity can be conceived, that it is a logically possible quantity. Not all quantities are logically possible. For example, the quantity that is greater than 1 and less than 1 is a logically impossible quantity.
As such, to prove that a quantity exists, in a mathematical sense of the word, you have to show that it is a logically possible quantity, one that is not a contradiction in terms, not an oxymoron.

That depends on what you think the exclusion criteria for possibility are. If it's only contradiction - then nothing is impossible in mathematics.
Define the impossible and it becomes possible.

If you think you can drink 2 cans of soda when you only have 1 - that's impossible.

Classical logic/Mathematics doesn't have resource constraints. Everything is unbounded by indunction/infinity limits.

Linear logic has resource constraints.

And in order to do that, you have to verify that its associated concept is free from contradictions. If you want to argue the opposite case, you have to point to a contradiction.

The thing you are confusing most here is mixing up logic and mathematics. "You can always add 1" is the principle of induction. How do you contradict yourself by counting? After which number comes a contradiction?

The problem is in the presupposition. "You can always add 1". Is that a falsifiable or an unfalsifiable statement?

What if I sometimes can't add 1?
What if I eventually can't add 1?

When you reject infinities, you reject the ability to answer questions such as "How many natural numbers are there?" It's a handicap, that's for sure.

What handicap? The answer is "as many as you can manufacture with the resources at your disposal". And if you don't want to manufacture numbers - "as many as you can imagine".

[Magnus Anderson]

Burden of proof is not on me. It's on you to show where all these 1s you keep adding are coming from.

What makes you think that? I'd rather say the burden of proof is on the one making the claim. And you're making a claim. You're claiming that we cannot always add 1 to an integer to get an even larger integer. You started this thread. Not me. You are very clearly arguing in favor of finitism. As such, it is up to you prove your case. Merely shooting down other people's arguments, even if you're successful at it, which you aren't, won't cut it.

Suppose you have counted up to the number X by starting at 0 and then add 1, add 1, add 1. Suppose you run out of 1s.

If you never run out of 1s - of course there is no largest integer.
But if you do run out of 1s - of course there must be a largest integer.

Have you ever counted to a centillion? If the answer is "No", how do you know that a centillion is a legitimate number? Your argument against infinity can also be used against any arbitrarily large number.

By constructing it with all the materials at your disposal.

If you can't construct it from the materials available to you - it's does't exist.
If you can construct it from the materials available to you - it does exist.

What does it mean to "construct it [ a quantity ] with all the materials at your disposal"?

Are you seriously suggesting that, in order to prove that a quantity such as one centillion exists, I have to gather a centillion number of objects in front of me or imagine a centillion sheep in my head?

In Mathematics anything defined to exist - exists.

Not quite. As an example, square-circles do not exist even though the term "squre-circle" has a very clear definition. Declariing that they exist can't change that fact.

That depends on what you think the exclusion criteria for possibility are. If it's only contradiction - then nothing is impossible in mathematics.

Again, not true. Square-circles are a very clear example. A number that is greater than 1 but less than 1 is another example. There are many such examples.

When you reject infinities, you reject the ability to answer questions such as "How many natural numbers are there?" It's a handicap, that's for sure.

The answer is "as many as you can manufacture with the resources at your disposal". And if you don't want to manufacture numbers - "as many as you can imagine".

You're answering the wrong kind of question. In other words, you misunderstood the question.

[Harbal]

I reject infinities.

Might you change your mind at some point, or will you reject them for ever?

[Skepdick]

I reject infinities.

Might you change your mind at some point, or will you reject them for ever?

When you finish counting to infinity I'll change my mind.

[Skepdick]

Burden of proof is not on me. It's on you to show where all these 1s you keep adding are coming from.

What makes you think that? I'd rather say the burden of proof is on the one making the claim. And you're making a claim.

No, I am not. I am rejecting a claim.

You're claiming that we cannot always add 1 to an integer to get an even larger integer.

It's not a claim. It's a rejection of a claim.

I am rejecting the claim that 1 can always be added to an integer.

You started this thread. Not me. You are very clearly arguing in favor of finitism. As such, it is up to you prove your case. Merely shooting down other people's arguments, even if you're successful at it, which you aren't, won't cut it.

I am not shooting down anything. I am rejecting a presupposition as unjustified.

The pressupposition that you can "ALWAYS add 1"

Have you ever counted to a centillion? If the answer is "No", how do you know that a centillion is a legitimate number? Your argument against infinity can also be used against any arbitrarily large number.

That has nothing to do with anything. My argument is that you can't ALWAYS add 1.

Take any N. There comes a point where you run out of 1s to add to your arbitrary N.

What does it mean to "construct it [ a quantity ] with all the materials at your disposal"?

Produce a Mathematical proof which computes the quantity you are speaking about.
If you want to construct the number 4 then 1+1+1+1 will do as a construction.

If you want to construct the number 0, then what's the minimal number of CPU cycles you need?

Are you seriously suggesting that, in order to prove that a quantity such as one centillion exists, I have to gather a centillion number of objects in front of me or imagine a centillion sheep in my head?

No. You just have to imagine a continuous path from your premises/axioms to your number.

If that path requires an infinite number of steps - sorry.

Not quite. As an example, square-circles do not exist even though the term "squre-circle" has a very clear definition. Declariing that they exist can't change that fact.

That's not true. Circles are squares in a Taxicab geometry.

https://en.wikipedia.org/wiki/Taxicab_geometry

Again, not true. Square-circles are a very clear example. A number that is greater than 1 but less than 1 is another example. There are many such examples.

Here's a square circle. The distance from the centre to any red dot is constant - a radius.

You're answering the wrong kind of question. In other words, you misunderstood the question.

Did I? Count the natural numbers and tell me how many there are. How much time do you think you need to finish counting?

[Magnus Anderson]

Let me try another way. Mathematics is intentional logical errors. Choice. Axioms which save time.

We know there's an error, but we turn a blind eye. Like reifying nothing(0) making it something. If you ever asked me to show you 0 like atheists demand evidence for God I can't show you anything other than the writing of the Prophets and to assure you that I believe in 0.
But it's also true that I believe in non-existing things. because 0 is nothing and nothing doesn't exist.

It's a self-contradictory idea but it gives us the principle of induction. Useful!

"Show me 0" is a meaningless statement, and thus, a meaningless request, one that cannot be fulfilled.

In fact, any statement of the form "Show me X" where X is a number is a meaningless statement, a request that cannot be fulfilled.

Can you show me "1"? Or "2"? You can't. It's a request that makes no sense.

What you can ask instead is "Show me X Ys" where X is a number and Y is a physical object of some sort.

It makes no sense to show you "1" but it makes sense to show you "1 horse". In fact, since "1 horse" is something that exists, I can show it to you.

It makes no sense to show you "0" but it makes sense to show you "0 horses". However, since "0 horses" is the same as nothing, it's not something I can show you.

However, the fact that I can't show you "0 horses" does not mean that "0" is a useful but self-contradictory idea.

Even though I can't show you "0 horses", I can show you places where there are exactly 0 horses. Such places can be perfectly accurately represented with the help of the concept of zero. Take your pocket, for example. It can be perfectly accurately described with the statement "A place where there are exactly 0 horses". Note that the statement isn't saying "A place where nothingness exists". You have to be a serious literalist to misinterpret it that way.

Furthermore, the universe itself has many numerical properties. The number of people living on Earth, for example, is 8 billion. The number of unicorns in the world, which is a numerical property of the universe, is exactly zero. Similarly, the number of elephants in your pocket is zero just as well.

[Skepdick]

"Show me 0" is a meaningless statement, and thus, a meaningless request, one that cannot be fulfilled.

Nonsense. What comes before the number 1 in your head?

Consider yourself shown.

In fact, any statement of the form "Show me X" where X is a number is a meaningless statement, a request that cannot be fulfilled.

There are an infinite number of linguistic representations I can show you that refer to the number. You can find it in your own head...

Can you show me "1"? Or "2"? You can't. It's a request that makes no sense.

Of course, I can. 3-2, 4-3, 5-4, 6-5. If you evaluate the expression using the standard arithmetic you were taught at school - you get 1
6-4, 5-3, 8-6. If you evaluate the expression using the standard arithmetic you were taught at school - you get 2.

What you can ask instead is "Show me X Ys" where X is a number and Y is a physical object of some sort.

Ehhh, are you denying that by saying "think of the letter after B" I am not showing you a letter of the alphabet? Don't be silly!

I only have to show it to your perception, not to your senses.

It makes no sense to show you "1" but it makes sense to show you "1 horse". In fact, since "1 horse" is something that exists, I can show it to you.

So when I say 2+2 you aren't thinking of the number 4? So weird!

It makes no sense to show you "0" but it makes sense to show you "0 horses". However, since "0 horses" is the same as nothing, it's not something I can show you.

"0 horses" makes absolutely no sense to me.

-> 3 eggs
-> 2 eggs
- 1 egg
??? -> NO egg

However, the fact that I can't show you "0 horses" does not mean that "0" is a useful but self-contradictory idea.

Yes it does. In exactly the same way I showed you why talking about "nothing" results in contradictions. Nothing isn't anything.

But in Mathematics we have made it something because it's super useful.

Even though I can't show you "0 horses", I can show you places where there are exactly 0 horses.

That's the entire universe then.

Such places can be perfectly accurately represented with the help of the concept of zero.

Why would you even represent the entire universe with "0 horses"?

Take your pocket, for example. It can be perfectly accurately described with the statement "A place where there are exactly 0 horses". Note that the statement isn't saying "A place where nothingness exists". You have to be a serious literalist to misinterpret it that way.

For any location in the entire universe: I would be confused as to why you are claiming it has "0 horses".

You have to be a very weird person who talks funny and dreams of horses.

Furthermore, the universe itself has many numerical properties. The number of people living on Earth, for example, is 8 billion. On the other hand, the number of unicorns in the world, which is a numerical property of the universe, is exactly zero. Similarly, the number of elephants in your pocket is zero just as well.

The number of unicorns is most definitely not zero.

There's at least one in your head.