Indeed, sometimes one's choices don't matter to what happens eventually. On occasion the path forks and one's in a dilemma which one to take, one then does, after pondering deeply upon the options, only to find out later that both paths reunite farther down. Such things do happen. Makes me wonder if free will means anything at all!

Yes, so on this assumption some eastern mystical traditions are poor mystical practice. This is not to say that they are not excellent meditation, or mind control technique’s etc. The notion that anyone can achieve enlightenment simply by acting in exactly the correct way (as in the infinite monkey analogy) is misleading.

Also inline with the other replies I would suggest that infinity is a peculiarity of intellectual thought and cannot be applied to real world situations. For example a pendulum would decelerate & accelerate at infinite rates at the end of each swing. It might be possible to Map this mathematically, but in the real world it is clearly not what happens. Likewise I have yet to come across an example of infinity occurring, or applying to real world situations.

It is because, for the Greeks, a number is a count (arithmos). It tells us how many of whatever thing you are counting. There can be no counting without a unit of the count, some one thing that is counted, apples, oranges, or fruit. An infinite or unlimited amount is not a number, it does not tell us how many. — Fooloso4

:ok: Yet infinity seems as natural as getting 2 by adding a 1 to 1. A simple procedure (+ 1) can cause so much havoc in our minds. The only option is to deny the existence of infinity, but then a large chunk of modern math would need to be consigned to the rubbish heap. That's what a book on philosophy on math says anyway. Calculus will probably be the first casualty.

Also interesting is the whole number sequence: 0, 1, 2,...

From 0 to 1: That's something from nothing! Creatio ex nihilo.

From 1 to 2: Doubling ( × 2)

From 2 to 3 and 3 to 4 and n to n + 1 hereafter, the ratio $\frac{n + 1}{n}$ approaches unity (1).

Then there's this: x + 1 = x. The solution is x is nothing. No finite number exists that satisfies this equation.

However, with a little algebra that equation becomes $1 + \frac{1}{x} = 1$ and we all know $\lim_{x\to\infty}\frac{1}{x} = 0$. In other words x = $\infty$.


So, x = nothing (no, not zero, nothing), and x = $\infty$. That means $\infty + 1 = \infty$. Mathematics breaks down!

My simple musings.

Also interesting is the whole number sequence: 0, 1, 2,...

From 0 to 1: That's something from nothing! Creatio ex nihilo. — Agent Smith

For Greek mathematics 2 is the first number. Two tells us how many ones or units of the count. They did not have the concept of zero. There cannot be an infinite number of things because a number tells us how many. But this is not to reject what is unlimited, that is, without number.

Modern physics would not be possible without symbolic mathematics. In modern number theory a number 'n' need not be the number of any particular thing existing or not.

For Greek mathematics 2 is the first number. — Fooloso4

You're joking, right?

You're joking, right? — Agent Smith

Nope. We retain a vestige of this. When we say we have a number of things we do not mean one thing.

The notion of arithmos emerges from the experience of counting. When we count, we always have a multiplicity of things before us. When faced with a single thing, we do not countit. If we say that it is “one,” we are speaking about its unity or we are asserting that it exists. One is not many. Therefore, “one”is not an arithmos.The first arithmos is “two. — An’ a one, an’ a two …

So, x = nothing (no, not zero, nothing), and x = ∞. That means ∞+1=∞. Mathematics breaks down! — Agent Smith

And this is an example of mysticism? Obsession with lemniscates will lead to no good. Please see your psychoanalyst. :chin:

That's exactly what it means, it's basic calculus. — Pantagruel

The problem though is that a curved line is fundamentally different from a straight line, because the curved line, no matter how short the segment, will always require two dimension, and the straight line will always be on one dimension. So there will always be angles between the sides, no matter how many there are. If basic calculus denies this difference between the straight line and the curved line, it must be mistaken.

Well, I won't say you're wrong. There must've been a very good reason why the Greeks were so reluctant to incorporate infinity into their math. Even Archimedes & Eudoxus, two people who were among the first to employ the method of exhaustion simply stopped/limited their calculations at/to an arbitrarily large but finite number (Archimedes used, if memory serves, a 96-sided polygon to approximate a circle when calculating ππ). — Agent Smith

Things like this have proven to be very useful, just like the calculus that Pantagruel refers to, it's very useful. I would call these useful lies.

What, may I ask, are the specific issues you have with ∞∞? Is it the paradoxes (Cantor's mind probably couldn't parse them and ergo, his brain crashed) or something else? — Agent Smith

What if I said that the issues I have with infinity are infinite? Look at the issue in my reply to Pantagruel for example. The concept of infinity is used to blur the fundamental difference between the straight line and the curved line. The two are incompatible, but employing infinity makes it seem like they are not incompatible. There are countless examples of similar instances where infinity is used to blur the incompatibility between two fundamentally incompatible things, like discrete and continuous for another example. It's just a basic error carried out for convenience sake, like representing a circle as a 96 sided polygon.

The method of exhaustion, refined to the notion of limits in calculus, is clearly stated as an approximation by mathematicians. A 96-sided polygon isn't a circle but is merely circle-like, that's all. Something is better than nothing, oui? Infinity simply inreases the accuracy of our calculations and I guess that's why it's such a big deal.

Nevertheless, you have a point. Now that I think about it mathematics, though described as the queen of the sciences by Gauss, is like trying to understand women (curves) in very manly (straight lines) terms.

Good day!

And this is an example of mysticism? Obsession with lemniscates will lead to no good. Please see your psychoanalyst — jgill

:lol: Good call. The lemniscate, if you'll recall, was the reason Cantor lost his marbles. Of course Kroenecker was being nasty; probably that was the proverbial last straw that broke the camel's back.

Makes sense in a weird sorta way.

We say "a" or "an" when there's only one of something e.g. " an apple" or "a dog".

Obsession with lemniscates will lead to no good. Please see your psychoanalyst. — jgill

Or, if you are feeling mischievous, hand him some $2^{\aleph_0}$ and tell him the computable numbers are countable.

The lemniscate, if you'll recall, was the reason Cantor lost his marbles. — Agent Smith

Well it makes for a lovely myth (sort of like Nietzsche going mad from his denial of God). He moved on from the lemniscate to the Hebrew alphabet before going mad, btw.

Infinity simply inreases the accuracy of our calculations and I guess that's why it's such a big deal. — Agent Smith

The circle is the limit of a polygon with n-sides. Each polygon is therefore an approximation of the circle. No polygon is itself the circle, of course. Or let $f(n) = 1/n$, then we can use a compact notation for taking limits and say $f(\infty) = 0$, while $\forall n \in \Bbb N \; \; f(n) \gt 0$. Note that the function need not achieve the limit but only 'eventually' (for large enough n) get and stay arbitrarily close.

Well it makes for a lovely myth (sort of like Nietzsche going mad from his denial of God). He moved on from the lemniscate to the Hebrew alphabet before going mad, btw — jas0n

$\aleph_0$. That's the only infinity that makes sense to me; kinda feel like a time traveler (physically in the 21^st century but mentally a mathematical troglodyte)

That's the only infinity that makes sense to me; kinda feel like a time traveler (physically in the 21st century but mentally a mathematical troglodyte) — Agent Smith

Do you mean $\aleph_0$, the cardinality of the natural numbers?

arbitrarily close — jas0n

Yep, that's the phrase I was looking for. Infinity is used to get as close as possible to a target (curves/females). Have you heard of The Teakettle principle

Do you mean , the cardinality of the natural numbers? — jas0n

Aye!

Hage you heard of The Teakettle principle — Agent Smith

Looked it up, and it's a big part of math. 'We'll transform this into a quadratic equation, which we covered last week...'

Aye! — Agent Smith

As you probably know, the old timers of math tended to feel that way...that only 'potential' infinity was respectable. But beyond what is accepted formally (say you embrace the symbol game of an infinite tower of differing infinities), an ancient problem remains. What does it all mean? To what does it all refer? How does it hook up with the rest of life?

Infinity is used to get as close as possible to a target (curves/females). — Agent Smith

Indeed. Try to compound interest more and more often, and you'll naturally bump into probably the most famous number (for insiders) which is $e$.

A 96-sided polygon isn't a circle but is merely circle-like, that's all. — Agent Smith

It all depends on how you define "circle-like". Clearly it is not at all "circle-like" when we compare a curved line to straight lines at angles to each other. If the defining thing is practical purpose, then it is circle-like, because it serves the purpose in practice.

But since the representation, which serves the purpose, is really nowhere near like the thing it is meant to represent, we can see a deep problem, if truth is supposed to be correspondence. Our theories which work in practice, and are confirmed and validated by the fact that they do work, (experimentation), might still be far from the truth.

As you probably know, the old timers of math tended to feel that way...that only 'potential' infinity was respectable. But beyond what is accepted formally (say you embrace the symbol game of an infinite tower of differing infinities), an ancient problem remains. What does it all mean? To what does it all refer? How does it hook up with the rest of life? — jas0n

I don't know, but @T Clark might have something to say. The metaphysics, the ontology, of infinity, may not be as important as how useful it is to us. Figuring out if there are actual infinities or if they're just potential infinities would be the icing on the cake, yes?

circle-like — Metaphysician Undercover

The word "like", in my humble opinion, furnishes the required degree of freedom to claim that (say) a 192-sided polygon is an approximation of a circle. Your eyes, for sure, will find it really difficult to tell them apart, even your sensitive finger tips will fail in this task.

I believe it's this very issue that you raise that makes infinity so attractive/appealing to mathematicians; You wouldn't be able to tell the difference between an $\infty$-sided polygon and an actual circle. Invoking Leibniz's 2^nd law of identity (the identity of indiscernibles), I'd say it's all good; for all intents and purposes, won't you agree?

What if your die had an infinite number of sides, do you think it would be circular? — Metaphysician Undercover

N.B. Something is better than nothing is the key principle at play here. We can reject infinity for many reasons, but look at all the good it's doing!

Looked it up, and it's a big part of math. 'We'll transform this into a quadratic equation, which we covered last week...' — jas0n

I wish! :smile:

The metaphysics, the ontology, of infinity, may not be as important as how useful it is to us. Figuring out if there are actual infinities or if they're just potential infinities would be the icing on the cake, yes? — Agent Smith

Yes. And for me the issue of whether there are 'really' various infinities leads inexorably what we could mean if we say so. All roads seem to lead to the 'problem' of the meaning of 'meaning.'

You wouldn't be able to tell the difference between an ∞∞-sided polygon and an actual circle. — Agent Smith

IMO, there would be no difference at all. The phrase 'infinite-sided polygon' is typically interpreted as a circle. (Nonstandard interpretations are possible, of course.)

Have you heard of The Teakettle principle — Agent Smith

Even having been a prof of mathematics I learn something about the subject on this forum. Never came across this. :smile:

Your eyes, for sure, will find it really difficult to tell them apart, even your sensitive finger tips will fail in this task. — Agent Smith

The senses deceive us, that's why we have logic.

I believe it's this very issue that you raise that makes infinity so attractive/appealing to mathematicians; You wouldn't be able to tell the difference between an ∞∞-sided polygon and an actual circle. Invoking Leibniz's 2nd law of identity (the identity of indiscernibles), I'd say it's all good; for all intents and purposes, won't you agree? — Agent Smith

I'd disagree, this is not a correct application of the identity of indiscernibles. An infinite sided polygon is an incoherent object. It could not have any identity.

We can reject infinity for many reasons, but look at all the good it's doing! — Agent Smith

Until we reject it, we'll never figure out how much bad it is doing. How much bad does it take to negate how much good?

Even having been a prof of mathematics I learn something about the subject on this forum. Never came across this. — jgill

It's physicists taking a dig at mathematicians. The Teakettle Principle, in my humble opinion, is basically a variation of don't reinvent the wheel principle, but there's more to problem solving than just standard/formulaic solutions, oui?

It (The Teakettle Principle) makes a whole lot of sense, but then it's ridiculous to follow the principle mechanically. That's the gist of the joke as far as I can tell.

All I can say is you're not incorrect, but as I pointed out, infinity allows approximations that turn out to be useful when dealing with feminine geometric objects (curves). Mathematicians seem to have isolated infinity to certain domains in math by way of damage control (infinity is like nuclear power - useful, yes, but extremely dangerous) i.e. they've been retained in areas where the accompanying paradoxes aren't as problematic. This is just a hunch of mine - it seems a reasonable course of action taking into account the givens.

IMO, there would be no difference at all. The phrase 'infinite-sided polygon' is typically interpreted as a circle. (Nonstandard interpretations are possible, of course.) — jas0n

Yep! Thanks for letting me know. @Metaphysician Undercover will find this tid bit right up his alley.