Ultrafinitism

[Telos]

Ultrafinitism (or ultraintuitionism) is a philosophy of mathematics that denies the infinite set N of real numbers, because this set is not a completed infinity.

So many self-help and new age books rely on the words "infinite potential" to market their ideas. It's sexy and powerful. However, I think those words are dubiously vacuous together. Potential is ability that has not been actualized (or "has not been completed"). Upwards infinity is also incomplete. They are really saying "incompleted incompleteness." And those words are not sexy.

New age spirituality strives for completeness, at-one-ment, and to actualize one's potential. To say one has infinite potential denies their right for completeness.

[beavis]

Ultrafinitism is easily disproven. If there is no infinite set of real numbers, there must be only a finite quantity of real numbers, so there must be 2 numbers that are the biggest real numbers (and not equal to each other). The average of those 2 numbers must also be real, but cant possibly be in the set. Contradiction, the finite set must include all real numbers, but a real number exists not in the set.

I agree the talk about "infinite potential" is ignorant, but its not just in new-age stuff. Old religions claim lots of infinite things, mostly about Zeus and Allah.

[Leannain]

i didn't know maths had  philosophy  

[Telos]

If there is no infinite set of real numbers, there must be only a finite quantity of real numbers

Between any two real numbers there are infinitely many more real numbers. This infinity has bounds.

so there must be 2 numbers that are the biggest real numbers (and not equal to each other). The average of those 2 numbers must also be real, but cant possibly be in the set.

The average of these two numbers would be exactly halfway between them and therefore within the set. You can average infinity if you know its bounds.

The infinity between real numbers is at least a completed infinity.

It's difficult to incorporate this formulaically, but optimistically I want to have infinity and also have an apex. I want to have my cake and eat it to. ;)

[Tyciol]

Math aside, the reason infinite improvement is marketed is that we like to believe our potential is limitless. That's my view, it's not founded but true or not it does result in the best improvement.

[beavis]

Telos, its a proof by contradiction. I assume the claim of ultrafinitism is true, that no "infinite set N of real numbers" exists, and find a logical contradiction, proving that the opposite is true, that there is an infinite quantity of real numbers.

You cant say the average of X and Y is in the set, because in a finite set (size 2 or more) with no duplicate numbers, there are exactly 2 biggest numbers. If the average of X and Y was in the set, X would equal that average, because it would be the "new" second-biggest number. That average cant be in the set, but is a real number. Contradiction.

If you allow the average of 2 real numbers to also be a real number, then the quantity of real numbers increases unboundedly, all instantly.

Between any two real numbers there are infinitely many more real numbers. This infinity has bounds.

What are those bounds? Obviously the bounds are wrong because any number in the bounds plus more than the size of the bounds must be outside the bounds but still a real number.

[Telos]

There is nothing more than the size of the bounds!

Unicorns don't exist! (They're just horses with rods on their foreheads!) Reductio Ad Absurdum does not apply, because it is not very constructive!

lol


Bah, it's no big deal anyway.

[beavis]

Things dont have to be constructive to apply.

[Telos]

Beavis, ultrafinitism is a constructive philosophy (aptly known as "constructivism')

[beavis]

if they dont know how to construct something, they know it cant exist?

[Telos]

No, just that it hasn't been constructed. It is difficult to imagine constructing an unbounded infinity, however.

[Tyciol]

I believe infinity to exist, it's just so obvious with examining math, the only reason it's not considered real is just these silly technicalities of not being able to make or define it very well.

[Ybom]

if they dont know how to construct something, they know it cant exist?

No, they know it doesn't exist because it hasn't been fully constructed yet. Just because it has a pseudo-immediate potential of being constructed doesn't mean it will be constructed, and until it has been constructed it doesn't exist.

A really good example of this, to me, would be the really nice Chinese restaurant that was being built next to Busch Gardens close to my house. Supposedly it was going to be the finest Chinese quisine in my area, in the $20 a plate range. Everyone was looking forward to eating there, until one day they put up barracades and it had a For Sale sign in front. Yes it had a large potential of existing, but today we were silly in thinking that it had essentially already existed in some point in the future. It hadn't been fully constructed yet!

There is nothing more than the size of the bounds!

Unicorns don't exist! (They're just horses with rods on their foreheads!) Reductio Ad Absurdum does not apply, because it is not very constructive!

lol

Bah, it's no big deal anyway.

However, is it possible to consider unicorns as existing because they flourish in people's minds? Isn't that the whole purpose of unicorns, to stir our imaginations?

[Dakmor]

No, they know it doesn't exist because it hasn't been fully constructed yet. Just because it has a pseudo-immediate potential of being constructed doesn't mean it will be constructed, and until it has been constructed it doesn't exist.

It is true that the potential of being constructed doesn't mean it will be constructed but also it doesn't mean that it won't be constructed. So they cannot know it doesn't exist.
This the reasoning people use to prove the religions.

[Dakmor]

I believe infinity to exist, it's just so obvious with examining math, the only reason it's not considered real is just these silly technicalities of not being able to make or define it very well.

Can you define it??