Zero and infinity

1. Nov 4, 2009

apeiron

Zero and infinity are both symmetry states. Every change (that is arithmetical operation) leaves them essentially unchanged. 50 times zero is zero. Likewise 50 times infinity is still infinity.

Zero represents nothing. Infinity represents everything. Hence - judged on their deep mathematical structure - nothing and everything are equivalent states of affairs. They are both symmetries.

Of course we would want to say they are not quite equivalent. They seem positive and negative like charge. Nothing is the lack of all possible things, and everything is contrariwise, the presence of all possible things.

But this just points to a greater lurking symmetry. Like charge, positive and negative can "fill each other in", cancelling to become a greater symmetry. So zero plus infinity yields an even deeper level of mathematical symmetry. Or translated to ontology, everything and nothing are really the same thing at a deeper level of existence.

This deeper symmetry state ought to have its own name (I like vagueness).

So where is the flaw in this logic?

2. Nov 4, 2009

fatra2

Not true. Take the summation operation.

$$0 + 1 \ne 0$$

$$\infty + \text{anything} = \infty$$

There is a simple difference between the two.

Cheers

3. Nov 4, 2009

Pythagorean

I dont think infinity is necisarily everything either. On an axis, as we go to infinity, we bypass some number, a, but we don't consider it part of infinity.

4. Nov 4, 2009

apeiron

Of course, quite right.

I was really just thinking about self-actions in terms of zero and infinity, so 0 + 0, etc. Adding other numbers immediately breaks the symmetry.

But what about $$\infty - 1?$$ Does that still leave infinity, or does that too break the symmetry?

However I agree the argument is unlikely to work because of the way the number-line is defined - anchored at zero and open at infinity.

Infinity would have to be paired with infinitesimals to compare mathematical apples with apples I guess. The limit of largeness and the limit of smallness. An asymmetrical pair in fact.

Nothing and everything can still be defined as symmetry states I believe. But not so simply.

Stiil it is then interesting that we have zero to stand as a mathematical representation of nothing, but what symbol to handle everything?

5. Nov 4, 2009

JoeDawg

Zero is not nothing. Zero is wherever you start counting from.

6. Nov 4, 2009

vectorcube

infinite can be potential infinite or complete infinite. The latter deals with transfinite numbers. Neither notion is vague.

7. Nov 5, 2009

wofsy

There are different infinities that are non-commensurate. There is no largest infinity so the idea of infinity as everything is impossible.

8. Nov 5, 2009

arildno

Meaningless blather.
Nope.
So?
In your word of blather, perhaps.
Really?
All integers is an infinite set. Are they "everything"?
Every third square is an infinite set. Are they everything?

Correction:
NOT "deep mathematical structure", rather, within the conceited, superficial world of blather you have created.

Nope.
Is a member of "everything" also a member of "nothing"?

Symmetric with respect to what? Rotational motion perhaps? Or, symmetric with respect to a line??
Now, you are blathering again.
Hmm..EVERYTHING, perhaps?

9. Nov 5, 2009

wofsy

when you say infinity what do you exactly mean?

10. Nov 5, 2009

apeiron

Blather (bla-thur); defn.: Ideas I don't get because they are too outside my limited experience and that scares me. A word with magic properties in philosophical discussion as repeated often enough, it gives the impression that more considered responses are unneccesary in scholarly debate.

11. Nov 5, 2009

Pattonias

nice

12. Nov 5, 2009

apeiron

People used to start counting from 1. Then zero was felt to be necessary.

Mathematical representation has evolved in ways that humans have felt logical and useful (note: useful). Which shows choices were made along the way (so other thoughts were possible).

Anyway, zero was indeed introduced as a better starting point. And you can see how it smuggled in a whole ontological perspective.

First people just thought in terms of somethingness. We start with 1. Then they realised that the existence of something demands a context. So they tried to imagine removing the 1 to discover what was left. They felt they were left with a void. Nothing. An empty space that 1 used to fill.

This view was reasonable and useful in some lights. But it was not the only choice, nor the most fundamental choice IMHO.

The alternative is to think 1 exists by development, by constraint. So rather than subtracting 1 to find a nothing (well, never actually nothing as now you have a void, a naked space of some kind) you would think about how rigorously do "devolve" that 1.

To have something is an asymmetric state of affairs. It is 1 against a backdrop of nothing - and also everything, as we seem to have also an infinity of numbers, and an even more troubling continuity of the number line (the reals, the irrationals, etc). So how can we devolve 1 to find its more symmetric origins?

We have made maths a very simple game that any kid can master (but not me as I demonstrate, I just can't seem to stick with simple rules). However this was achieved by discarding an equal amount of the actual ontological complexity.

This is what makes me laugh about all the protestations about the objectivity, necessary truth and platonic uniqueness of mathematics. We have made a story so simple that it seems it would have to be the case no matter what. 1+1=2 is something that just has to be true anywhere. But making choices about paths to take is the definition of subjective. It is impossible to go crisply in some direction without equally definitely leaving all other available directions behind.

13. Nov 5, 2009

arildno

And there is not a single mathematician who says this, either.

Maths is like games, picking a set of rules and see what follows from those rules (according to chosen rules of valid inference).

Trivially, many such games exist.

But these banalities do not warrant that what you've written about "nothing" and "infinity" is deep and "true".

14. Nov 5, 2009

BobG

You're over-romanticizing the invention of zero. It was initially invented in India and by the Mayans (independently) as a placeholder in numbers. Before zero was invented, an empty space sat between the two numbers. You could tell the difference the between 11 and 101, but it started to get pretty hard to tell the difference between 101, 1001, 10001, and so on.

I don't thing people paid much attention to zero, itself, until they started to think about negative numbers, number lines, and Cartesian graphs.

15. Nov 5, 2009

apeiron

Agreed I am glossing over a compex social history of ideas big time. I just intend to emphasise what is most relevant here from a modern philosophical viewpoint.

We could go back before greeks to the babylonian "zero" that was the spacer to mark hierarchical transitions in measuring scale. The change from minutes to seconds in Babylonian astronomy.

Number started as a formal symbol system with tally sticks and scratches on clay. So with the idea of 1. Then as counting became more elaborate, the need to mark an absence, a gap, of some kind became increasingly urgent. The idea of such gaps or nothings would have been part of philosophy at least vaguely of course. But then marking it with a counting system would have had a wonderful way of making such notions seem definite, crisp, even objectively existent.

So zero has a long social history by which it became as generalised as possible. As globally symmetric as possible. Look at it from any (mathematical) direction and we seemed to be seeing the same object.

And what is relevant here is the way it migrated from being the gap between somethings to becoming the start of all things. What was once a humble spacer where counting systems needed to skip over some kind of gap became instead the anchoring origin - the non-place from which all other places could be counted. What is more symmetric, more singular, than an origin point in counting?

1-ness still retains special magic as the identity number. And zero is the nothing from which all somethings are held to originate. It defines the direction in which we measure (the directions that are infinitely open).

This is the ontology that I challenge. If we go back and accept dichotomies are more basic, then for zero to be correct, then its antithesis must also be correct. So what is that antithesis (or does zero need to be redifined)?

Arildno will have another spluttering fit at the thought conventional maths models can be challenged, that alternatives might get explored, but why not? Especially if it is possible to do it in a rigorous way based on existing bodies of philosophical thought and employing recent mathematical advances like fractals, lie algebras, networks, phase transitions, non-linear dynamics, category theory, etc?

God may have invented the integer, but then people invented god. So time to quit worshipping at the feet of the holy 1, the even holier 0, the quite ecstatically mystical 1+1=2.

Come on. Chant after me, arildno, vectorcube, qsa, and all you other guys, number is necessary truth, maths is transcendant, maths from its platonic heaven cast the shadow on the cave wall that made us and our ensemble of realitiies. Woo hoo, wave the incense, shake your funky theorems.....

16. Nov 5, 2009

arildno

a) "1-ness still retains special magic as the identity number."
Eeh??
b) "And zero is the nothing from which all somethings are held to originate."
Held by whom??

c) "It defines the direction in which we measure (the directions that are infinitely open)"

Zero defines directions??

17. Nov 5, 2009

apeiron

With your greater knowledge of math, perhaps you can answer the simple question I posed earlier.

If we subtract 1 from infinity, would we still have infinity?

We can add 1 and infinity remains, but what kind of answer do we get if we treat infinity as a "starting point" and begin to descend?

18. Nov 5, 2009

Staff: Mentor

Locked pending moderation.