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[increpare]

One over infinity (the multiplicative inverse of infinity?) only gives you zero when you write it on a math exam.

Actually there are some conventions where it's fine to write it (in measure theory we did it I recall).  Good joke though

Oh, GV beat me to it.  Yeah: you can formally append infinity to your basic number system, but it won't interact too well with 'proper', finite numbers (you can't cancel it with itself, multiplying it by zero gives an indeterminate answer, blah blah blah).

[Chris Whitman]

Oh, GV beat me to it.  Yeah: you can formally append infinity to your basic number system, but it won't interact too well with 'proper', finite numbers (you can't cancel it with itself, multiplying it by zero gives an indeterminate answer, blah blah blah).

Yes, but you're thinking of constructing infinite and infinitesimal elements via sequences of real numbers or something similar: something formally defined.

My point is you can't just say that infinity is not a number but include it in equations anyway, and then say that math is spooky and infinity makes no sense.

[increpare]

Yes, but you're thinking of constructing infinite and infinitesimal elements via sequences of real numbers or something similar: something formally defined.

No, I'm thinking of a formal extension of the cheap and cheerful variety.  You affix a new symbol ∞ (and -∞ as well I guess) to the real numbers, say, and you have to explicitly say how it interacts with +,x, and -.

Assuming r is a non-zero real number, one defines

r+∞=∞
r-∞=-∞
1/∞=0
∞+∞=∞
∞*r=∞
∞*∞=∞
∞/∞ undefined
∞+(-∞) undefined
et cetera.

(I think you leave 1/0 undefined actually...given the sign ambiguity)

You lose the ability to factorise and cancel terms, and there's nothing interesting in the definition in and of itself, but it's good enough for basic integration theory

My point is you can't just say that infinity is not a number but include it in equations anyway, and then say that math is spooky and infinity makes no sense.

And a very valid point it is as well

[Chris Whitman]

All right! I figured we were still talking about nonstandard reals.

Anyway, as you've pointed out, the result isn't even a field (actually, it isn't even an additive group), so... it isn't very useful. Even then, though, you are using a well-defined notion of infinity which is not 'spooky.'

[andy wolff]

Interesting topic. I myself am completely uneducated, and as such have (possibly interesting) views and ideas about what has been said, but little basis on which to make claims. Also I'm completely incoherent.

It is now time for A LARGE BLOCK OF TEXT.

You guys who are arguing that time doesn't exist (or rather that only present exists, not past or future) don't seem to understand either time in the same way that I do or the concept of Duration which has been mentioned here. Think of time in relation to this idea of Duration and you will probably realize that conscience as we know it is only capable of percieving things as reality in increments rather than streams, though some of us are also able to conceive 'streams' in theoretical concepts.
As such memory is in part what we percieve of the past and in part what we perceive of the present. When you're thinking about a memory you are still experiencing the present and an infinitesimal amount of the future (Transition: you experience your motion through time, slightly). While it can't be proven that time exists in a greater reality than our perception, it can be proven that it exists in our perception to some extent. For example, Evidence exists. Memory is a type of this evidence. Other types are manifestations of physical reality, or rather humanity's discovering of them and its transferrence of these ideas of their existence. With Evidence, we can have a concept of time other than our almost undetectable sense of the passage of time separate from memory and consciousness and other than our incremental sense of the present. I say our sense of the present is incremental because I think of it like a clock cycle on a computer; while it may seem like one is having a constant thought, it is possibly just a long string of small incremental thoughts. Though I would probably argue this because it seems off somehow to me.

I also wanted to explain my current understanding of physical, temporal, and further reality, for some reason. I myself am noticing that the progression of dimensions contains patterns and concepts far beyond the capability of limited (by point of view and shallowness of perception) intelligent understanding. Wanted to, but didn't.

Here I started to go into multi-dimensional geometry but in writing it I wasn't really going anywhere so I stopped and deleted what could have been really interesting. It wasn't though.

You'll have to forgive my nonsensical rambling, if you read it. I realise that much of it is pure blasphemy and logical fallacy, but it was good to ramble. I may ramble some more, later.

[agj]

∞+(-∞) undefined

Does this mean that not all infinites are the same? Because if they are, then shouldn't the result be 0?

While it can't be proven that time exists in a greater reality than our perception, it can be proven that it exists in our perception to some extent. For example, Evidence exists. Memory is a type of this evidence.

Then again, there is no evidence that memory is actually a bunch of recollections of the past, we only assume so. But if you meant to say that time does exist subjectively, then certainly.

[brog]

∞+(-∞) undefined

Does this mean that not all infinites are the same? Because if they are, then shouldn't the result be 0?

If we take ∞+(-∞) = 0 in the arithmetic that increpare defined, then:
(∞ + ∞) + (-∞) = ∞ + (∞ + (-∞))
∞ + (-∞) = ∞ + 0
0 = ∞

And then it completely collapses, because x+0 = x for all real numbers x, but 0=∞ and x+∞=∞, therefore all numbers are equal to ∞.
Which would be silly.
If you just leave it undefined then the system can still be useful.  If everything's equal.. not very interesting.

This isn't saying anything about whether "all infinities are the same" in a general sense (they aren't), it's just a made up system* in which you can work with infinite values.


* Or is it?  Since this is a metaphysics thread I guess we can argue about whether imagined things are found or created if you like.

[GregWS]

"Infinity" not being one number makes perfect sense to me (at least when we're talking about infinity in the mathematical sense).  An infinitely big number could be different in different equations, so all we can do is treat it like "an infinitely big number."

[muku]

Hell, I Like Cake, are you some kind of polymath? The range of topics on which you have something sensible to say is downright intimidating.

Also, thanks for the effort to take this thread from handwaving to hard math.

[increpare]

"Infinity" not being one number makes perfect sense to me (at least when we're talking about infinity in the mathematical sense).

I'm inclined to be as skeptical of your sense of infinity as I was of GV's (that is to say, interestedly so). 

You seem to be implying that something can be a 'number', while having a 'value' that can change depending on the context.  I'm wondering how you can separate these two concepts?

[GregWS]

I'm not saying that infinity is a number in math, it's more of a partially defined variable.  It's not x, and it's not 2, it's something in between.

And like I said before, I'm only speaking of infinity in the mathematical sense, not in the metaphysical sense.

[agj]

If you just leave it undefined then the system can still be useful.  If everything's equal.. not very interesting.

Thanks, yeah, that makes sense.

[increpare]

I'm not saying that infinity is a number in math,

erm.

An infinitely big number could be different in different equations

Okay, even given that your reply just there was a qualification to this, could you give me an example of something that would for you constitute an example 'infinity' in maths, and give an example of how it might change from one context to another?

FWIW, it's possible to completely pin down the notion of certain classes of 'infinite numbers' (some Connes-style ones, some Surreal numbers, some Ordinalities/Cardinalities), to tell when they're equal or not equal, to put them on a more-or-less equal footing with finite numbers in whatever context they come up with, even to express them explicitly  (for example, one can talk unambiguously of the "cardinality of the natural numbers", or express the infinite surreal number {1,2,3,... | }  ).  To speak of them all as being representations of a single 'infinity' seems...not to be the best way of phrasing it (though I'm open to discussion on this point).

That said, there are interesting complications when dealing with the class of numbers known as the Surreal numbers, where you can in principle deal with equalities and inequalities, but in practice never explicitly exhibit any particular infinite or infinitesimal numbers. 

In some sense, what I'm saying is that when I read 'infinity is a number in math', I feel there's something you're missing out on: rather, there are many different classes of objects in mathematics, which can be thought of as representing 'infinite objects' in various contexts.

[_Tommo_]

Hell, I Like Cake, are you some kind of polymath? The range of topics on which you have something sensible to say is downright intimidating.

Also, thanks for the effort to take this thread from handwaving to hard math.

In fact, he saved me from studying analisys today

[Valter]

Increpare, the problem is that when using infinity in math, it's not used as a number, it's used as a variable. As I said before, infinity is just an infinitely huge number. Infinity can grow at different rates.

how many numbers are between 0 and 1? Infinity

How many number are between 0 and 100? Also infinity.

See? Infinity can be  different values, so equations with more than one infinity are called "indeterminate forms", because there's no way of telling which infinities are higher. Like infinity - infinity, infinity divided by infinity, and things like that.

[brog]

how many numbers are between 0 and 1? Infinity

How many number are between 0 and 100? Also infinity.

See? Infinity can be  different values

These are the same value.  The map f(x)=x*100 takes the first set to the second in a one-to-one fashion, showing that they have the same number of elements.  See Part I of I Like Cake's dissertation.

Infinity can be  different values, so equations with more than one infinity are called "indeterminate forms", because there's no way of telling which infinities are higher. Like infinity - infinity, infinity divided by infinity, and things like that.

These forms show up as limits; this is a different concept to "the number of numbers between a and b".  You're not putting an 'infinite value' into an equation, you're look at how it behaves as the values you put into it get larger and larger.  See Part II of I Like Cake's dissertation.

These are different concepts, and I think you're confusing them.

[Movius]

The fact that this thread has degraded into an argument about maths confirms the unalterable truth that Metaphysics is useless.

[diwil]

I'm not really good with math, but I'm good with logic. Even if it's abstract logic. I like discussing metaphysics, such as; "if time was like water."

[brog]

The fact that this thread has upgraded into an argument about maths confirms the unalterable truth that Metaphysics is useless.

I kind of disagree.  Maths and metaphysics are quite closely related; they both study things that don't "exist" in a tangible sense, but are still somehow "real".  We can't test hypotheses with physical experimentation, so we have to use abstract reasoning to deal with them.
The main difference is that mathematics works by precisely defining abstract objects and then reasoning logically about them, and metaphysics (in my opinion) seems to usually get stuck on arguing about what the definitions should be in the first place.  Maths has been far more successful in producing results, so I think it's a good place to start from when studying metaphysics.

[increpare]

I kind of disagree.  Maths and metaphysics are quite closely related; they both study things that don't "exist" in a tangible sense, but are still somehow "real".  We can't test hypotheses with physical experimentation

Elementary combinatorial, geometrical, and arithmetical theorems and hypotheses can be quite handily tested and investigated using real, physical objects (or even abstractions of them, like numbers).  Admittedly, if the real experiment produces results contradicting a well-verified theorem, the the criticism will probably lie with the model, rather than the underlying logical structure which created the theorem.

Experimentally-based results are not accepted as proof, and for good reason, but at the same time that doesn't render them entirely meaningless.

There is also the field of experimental mathematics, about which I don't know too much, but in principle I find it quite interesting in and of itself.

(Constructivism is something I've always been meaning to learn a little about.  The little I know I got from the topos-theoretic end of intuitionism). 

The whole word of mathematical logic and set theory is achingly distant from my specialisation, alas.