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Can anyone explain what the big deal with infinities is?

Started by Crispy Nov 6, 2004 at 8:05 AM 92 replies 13k views

Original Post

Nov 6, 2004 at 8:05 AM

NOTE: yes, I did sleep through my math lectures. However this is only a rhetoric question and doesn't relate to anything other than the desire to raise some commotion. By definition, in mathematics infinity is a number that is greater than any real number. I can understand that. However, since mathematics is, by definition, a system that has the capability of describing any (real) number regardless of its nature, doesn't this introduce a paradox? Furthemore, how would mathematics go about describing the last real number (before infinity)? To me, this would be a far more fascinating issue than trying to describe what infinity is. Convergence? In that case where's the thin line between real in infinite? Since by nature mathematics is an "analog" science, isn't it also the case that which ever way you go (positive, negative, more precise, less precise), you're bound to be stopped by infinity at one point. Doesn't this infer that mathematics, in fact, is based on the concept of infinity (more precisely, it's an outtake of one segment from a space that exists without size or nature - like a nox that has no walls?), which makes it a pseudoscience really because as far as I know, there isn't a plausible explanation as to what infinity really is or whether it actually does exist. Since such a concept only exists in our heads (as far as we know), wouldn't that suggest that we're all disillusioned? Or perhaps crazy? Just a little something to ponder.

benryves

Nov 6, 2004 at 8:10 AM

∞ is indeed weird. But isn't ∞-1=∞ (I'm no maths expert), so that could be wrong. What I find odder still is that x/x=1, 0/x=0 and x/0=∞ - so what does 0/0=? It's "undefined", which seems a bit of a cop out to me. [grin]
Mathemeticians are weird people. Let's leave it at that!

Andrew Russell

Nov 6, 2004 at 8:16 AM

x/0 is undefined, not infinity, and it's for good reason - not a cop out.

and yes, inf-1 = inf (IIRC)

look up wolfram mathworld if you really need answers.

benryves

Nov 6, 2004 at 8:17 AM

Quote:
Original post by Andrew Russell
x/0 is undefined, not infinity.

I stand corrected. Good point!

Etnu

Nov 6, 2004 at 8:20 AM

We should just treat math like people's estates; if you try to divide something by zero, the state just takes everything.

Torkel

Nov 6, 2004 at 8:28 AM

Quote:
Original post by Crispy
Furthemore, how would mathematics go about describing the last real number (before infinity)? To me, this would be a far more fascinating issue than trying to describe what infinity is. Convergence? In that case where's the thin line between real in infinite?

There is no "last number", since if you managed to find the "last number", last number+1 would be larger, and also real. Infinity is more of a concept than a number.

Andrew Russell

Nov 6, 2004 at 8:29 AM

Quote:
Original post by Crispy
Furthemore, how would mathematics go about describing the last real number (before infinity)?

[damnit - forum ate my long post about this - retyped...]

OK, there is something called the Well Ordered Principal (this is IIRC), which goes something like (or, at least, can be expressed like) this:

x < x+1

This basically shows that there is always a real number greater than another real number, and therefore you will never reach the "end" of real numbers.

Then you have infinity. This is defined as being greater than any real number:

x < x+1 < inf

This shows that you can't reach infinity, you will always be less than it.

Now, if we try to get inf-1:

x+1 < inf
take x = inf-1
(inf-1)+1 < inf
inf < inf

you get a contradiction - hence inf-1 is not possible.

jfclavette

Nov 6, 2004 at 9:00 AM

Yeah. Let X tend toward inf

x * 2 = inf
x * 3 = inf
x * 2 > x * 3
inf > inf

infinity is only a concept. There are varying degrees of infinity and often in calculus, you have to try and remove inf/inf indeterminations, for example.

MdLeadG

Nov 6, 2004 at 9:16 AM

Quote:
Doesn't this infer that mathematics, in fact, is based on the concept of infinity (more precisely, it's an outtake of one segment from a space that exists without size or nature - like a nox that has no walls?), which makes it a pseudoscience really because as far as I know, there isn't a plausible explanation as to what infinity really is or whether it actually does exist.

Wasn't it Plato who (initially) believed infinity was ... heresy(?) I can't remember exactly - but I *highly* recommend Infinity and the Mind. Very very deep book. Discusses the history and evolution of infinity mathematically and philosophically. Good read.

Anyway, my point was even though long ago cultures did have a concept of infinity, or absolute, alot of them had strong beliefs that it was ... not something that would be expressed in maths or geometries - (because they believed that God himself was the absolute, therefore inexpressible).

Torkel

Nov 6, 2004 at 9:22 AM

Fun fact: In ancient Egypt, Infinity == one million.

Extrarius

Nov 6, 2004 at 9:34 AM

Crispy: Mathematics is a 'pure science', which means it can't be proved false becaquse it has no relation to reality at all. All that relating to reality stuff is taken by chemistry, physics, etc. Math is just exploring a bunch of rules we made up to be internally consistent. They might be based on observations (things like the constant π), but they're still just things we made up.

Its hard to prove an authorative definition wrong =-)

Agent1

Nov 6, 2004 at 10:36 AM

As has been pointed out, infinity is not a number.

My Calculus professor mentioned an incident with his son that illustrates this well...

<Son> "My teacher said that infinity is the biggest number."<Professor> "Tell her that infinity plus one is bigger."

MdLeadG

Nov 6, 2004 at 11:06 AM

Quote:
Original post by Crispy
... However, since mathematics is, by definition, a system that has the capability of describing any (real) number regardless of its nature, doesn't this introduce a paradox?

Infinity introduces all kinds of paradoxes (which IMO are the most fun :P). Most are solved by proving something that was supposedly infinite, is not. Since the universe is infinite, there certainly already exists a complete desription of infinity, but since the mind is not infinite, I don't think we'd be able to grasp it. Much like trying to think like a god - just not possible (supposedly).

Agony

Nov 6, 2004 at 11:14 AM

Quote:
Original post by Agent1
As has been pointed out, infinity is not a number.

I will reiterate that again for emphasis. Understanding this will prevent all sorts of erroneous ideas about infinity.

Infinity is a mathematical concept. It has various well defined properties under various conditions. But it does not reliably have the same well defined properties that numbers have. People often seem to come to the conclusion that if it's more ambiguous than a number, then it is 100% ambiguous, and thus worthless. But that is an incorrect leap of logic.

It is a concept that is similar to, but not equivalent to, the concept of a number.

Conner McCloud

Nov 6, 2004 at 12:10 PM

Quote:
Original post by Agent1
My Calculus professor mentioned an incident with his son that illustrates this well...
<Son> "My teacher said that infinity is the biggest number."<Professor> "Tell her that infinity plus one is bigger."

Psh...every third grader knows that.

"Did not!"
"Did too!"
"Did not!"
"Did too!"
"Did not times infinity!"
"Did too times infinity plus one!"
"Did not times infinity plus two!"

And so forth

CM

jsgcdude

Nov 6, 2004 at 1:54 PM

Quote:
Math is just exploring a bunch of rules we made up to be internally consistent.

This simply isn't true as Gödel proved that no system of logic is 100% internally consistant ie it can never be used to prove itself.

Well, thats not exactly right.

Tron3k

Nov 6, 2004 at 1:59 PM

Quote:
Original post by MdLeadG
Since the universe is infinite, there certainly already exists a complete desription of infinity, but since the mind is not infinite, I don't think we'd be able to grasp it. Much like trying to think like a god - just not possible (supposedly).

Ridiculous. Your mind doesn't have to be infinite to comprehend infinity. Are people with bigger minds able to comprehend bigger numbers?

MdLeadG

Nov 6, 2004 at 1:59 PM

Quote:
Original post by jsgcdude
This simply isn't true as Gödel proved that no system of logic is 100% internally consistant ie it can never be used to prove itself.

Yeah, basically he proved that mathematics itself is incomplete. (which is a very good and inspiring thing, to say the least). There can be no ultimate truth machine, and the mind is not infinite.

MdLeadG

Nov 6, 2004 at 2:06 PM

Quote:
Original post by Tron3k
Quote:
Original post by MdLeadG
Since the universe is infinite, there certainly already exists a complete desription of infinity, but since the mind is not infinite, I don't think we'd be able to grasp it. Much like trying to think like a god - just not possible (supposedly).
Ridiculous. Your mind doesn't have to be infinite to comprehend infinity. Are people with bigger minds able to comprehend bigger numbers?

The size of your mind makes absolutely (pun intended) no difference. When you try and have infinity inside your mind, you basically end up having an infinite regress (look up Godel and set thoery). As was stated before, infinity is not an actual number, it is a concept - finite mass cannot hold infinite information, which it would *NEED* to do in order realise infinity to the full (or at all, really)

NotAnAnonymousPoster

Nov 6, 2004 at 2:11 PM

Infinity rarely appears in mathematics. It's made use of in set theory and an entire arithmetic exists for it, but this definition sees the concept as an extension of the natural numbers, not the reals.

The definition that infinity is a number greater than any real doesn't appear in the standard theories, though it does appear in some non-standard ones. The presence of the infinity symbol and the phrase "tends to infinity" when discussing limits is misleading. In standard analysis, limits and convergence have nothing to do with infinity, and the use of the word is a left over from the early calculus.

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