Space, Time and Infinity

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uwot
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Re: Space, Time and Infinity

Post by uwot » Thu May 07, 2015 6:10 pm

van Keister wrote:Do you understand anything about modern geometry especially fractal geometry? If you did then I believe you wouldn't be worshipping "infinity" as something devine as you do.
I know next to nothing about fractal geometry, it's not my field, nor interest; but I do know you cannot cite an instance of me "worshipping "infinity"".
This is more my sort of thing:
van Keister wrote:...our ideas of space/time and infinity are local and not as mystifying as phenomenologists have us believe.
What is mystifying about the descriptions of phenomenologists?

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Seremonia
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Re: Space, Time and Infinity

Post by Seremonia » Mon May 11, 2015 9:36 am

Wyman wrote:
Seremonia wrote:
Philosophy Explorer wrote:Seremonia said:

"Secondly, infinity is 'not limited by another thing'."

Are you aware there are different levels of infinity? There's Aleph null, Aleph one, Aleph two, etc. So, in a sense, infinity limits itself where Aleph null, e.g., is limited by Aleph one. So let me ask you what you mean by limited?

PhilX
As previously stated, limited means, it's dependent upon (in any possible ways) something else.
Seremonia, have you ever studied calculus? Between any two real numbers (or any two points on the real number line), there are infinitely many numbers. And yet, one can determine exactly what the area bounded by a curve is, or the volume of any bounded solid (or on the differential side, instantaneous velocity, or the slope of a curve at any point, or a thousand other applications) by calculations involving quantities that approach infinity or approach infinitely small quantities. So mathematics, even at the level of calculus, deals extensively with infinity - infinite sets, open and bounded, infinite series with finite values, etc., etc., etc..
Hahaha, yes, it's like zeno paradox.

Let me put this way, suppose there are infinitely many numbers in between two pointers. Well, those (infinitely) many numbers must be considered as real. In the sense, there are real things as much as those represented by infinite numbers in between two pointers (things). Now, can you calculate total length (content or something) of all of them? Consider we can do it, then we found the result that it has length (content or something) as far (much) as "X". It has limitation indeed. But looking inside them, "the infinite" says "there are unlimited numbers of (content or something).
An axiom: something (without additional assertions) can’t transcend beyond something itself. From one liter water (without additional assertions) can’t be poured as much as 1 gallon water.
So we have an understanding here. There is such unlimited inside a limitation. It's like there is a glass of tea and we can pour a tea from it as much as 1 gallon, which is impossible (unless there is something -tea - from outside the glass adding continuously to the glass).

The point is, while we are doing philosophical analysis, it must be relied on reality. Although we can think about it but we can't bring it.

But, although it's a tricky thinking, for me, we can still associate it with reality, as long as we could understand it properly. For me, this kind of understanding, actually it must be associated to another kind of understanding. I accept that kind of zeno paradox as an understanding that we are all one, and there is no separation in between all (but it takes another explanation which is irrelevant in this discussion).
Last edited by Seremonia on Mon May 11, 2015 11:16 am, edited 1 time in total.

surreptitious57
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Re: Space, Time and Infinity

Post by surreptitious57 » Mon May 11, 2015 10:50 am

Numbers are not real as they are mathematical symbols and mathematics is abstract by definition
Now there are real numbers but the word real in that context means some thing entirely different
And so talking of numbers in a real sense is completely meaningless for they do not exist in reality

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Seremonia
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Re: Space, Time and Infinity (numbers & real things)

Post by Seremonia » Mon May 11, 2015 11:03 am

Seremonia wrote:Well, those (infinitely) many numbers must be considered as real. In the sense, there are real things as much as those represented by infinite numbers in between two pointers (things).
What i mean numbers are real is, that it must be considered having relation with real things to be involved in discussion.

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Lawrence Crocker
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Re: Space, Time and Infinity

Post by Lawrence Crocker » Mon May 11, 2015 12:55 pm

surreptitious57 wrote:Numbers are not real as they are mathematical symbols and mathematics is abstract by definition
If there are infinitely many numbers, numbers cannot be physical symbols as there are only finitely many symbols.

There cannot even be a symbol (which must be of finite length) for the typical transcendental number.

That no abstract entities exist needs to be argued. What it is for pi to exist is something different than for pies to exist, but that is not enough to show that pi lacks existence.

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Re: Space, Time and Infinity

Post by Wyman » Mon May 11, 2015 2:07 pm

If there are infinitely many numbers, numbers cannot be physical symbols as there are only finitely many symbols.
How do you know that? The axiom of infinity assumes that there are infinite sets. What limit is there to the symbols by which we represent the natural numbers? For any n, if you give me the symbol for n, I'll give you n+1 - by induction, there are infinitely many separate symbols.

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Lawrence Crocker
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Re: Space, Time and Infinity

Post by Lawrence Crocker » Wed May 13, 2015 8:03 pm

I took "symbols" in the post to which I was responding to be physical entities. I have no problem with an infinity of abstract symbols, such as the numerals in your example. Some abstract numerals, of course, will never exist physically -- a point of the strict finitists.

More important, with respect to the claims of the post to which I was responding is that the real numbers cannot be symbols as there are not even enough abstract symbols for that. At least with the usual constraints, there can only be aleph null such symbols.

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Lawrence Crocker
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Re: Space, Time and Infinity

Post by Lawrence Crocker » Wed May 13, 2015 8:26 pm

I should have added that, although it cannot be that numbers are symbols (unless there are at most aleph null numbers), the idea that numbers are the creatures of symbolic activity is respectable. One of the chief tasks of this view is to flesh out what the creatures are and how mathematical activity creates them.

Wyman
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Re: Space, Time and Infinity

Post by Wyman » Wed May 13, 2015 8:38 pm

So you're saying that there may be infinitely many numbers, but there are not infinitely many 'physical objects?' I don't know whether that is true or not; I'm doubtful that it is provable. But that there are more real numbers than possible physical objects (which I suppose have to be countable) doesn't imply that there are not infinitely many physical objects.

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Re: Space, Time and Infinity

Post by Philosophy Explorer » Wed May 13, 2015 10:05 pm

Lawrence Crocker wrote:I took "symbols" in the post to which I was responding to be physical entities. I have no problem with an infinity of abstract symbols, such as the numerals in your example. Some abstract numerals, of course, will never exist physically -- a point of the strict finitists.

More important, with respect to the claims of the post to which I was responding is that the real numbers cannot be symbols as there are not even enough abstract symbols for that. At least with the usual constraints, there can only be aleph null such symbols.
Question for LC: which "abstract" numerals exist physically?

I would refer you to Wiki's article about Georg Cantor's diagonal proof for the greater infinity of the real numbers (specifically due to the transcendental numbers). It's a proof based on nine numerals.

PhilX

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Lawrence Crocker
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Re: Space, Time and Infinity

Post by Lawrence Crocker » Thu May 14, 2015 12:59 pm

I understand "abstract" to imply "not physical." So understood there are no physical abstract symbols. Of course, people have made a certain large but finite number of physical symbols, instantiating (a smaller number of) abstract symbols.

There is no mathematical proof of the finitude of physical symbols. That this universe has only finitely much stuff in does, however, seem to be the current consensus among cosmologists. If some form of multiverse hypothesis is true, then there might be infinitely many physical symbols in some other universe or summing across all universes. For our purposes as symbol users, the existence of only finitely many physical symbols of any relevance seems pretty safe.

Cantor's proof demonstrates the existence of sets whose elements cannot be put into one to one correspondence with unique symbols (even abstract ones). That Cantor accomplished this proof using only finitely many symbols is the sort of thing that makes it prudent to add the qualification that, although the real numbers cannot be symbols, they might be "creatures of symbolic activity." There are a lot of positions in philosophy of mathematics lying between strong Platonism and strong conventionalism. The "creatures of" formula would apply to some of these. (The strongest forms of conventionalism are mathematically untenable; the strongest forms of Platonism are metaphysically extravagant. Regrettably, many of the intermediate positions, including "creatures of symbolic activity," suffer from vagueness at crucial points, relying heavily upon metaphors that resist explication.)

Wyman
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Re: Space, Time and Infinity

Post by Wyman » Fri May 15, 2015 2:33 pm

That this universe has only finitely much stuff in does, however, seem to be the current consensus among cosmologists. If some form of multiverse hypothesis is true, then there might be infinitely many physical symbols in some other universe or summing across all universes. For our purposes as symbol users, the existence of only finitely many physical symbols of any relevance seems pretty safe.
Here's a question: does the continuity of the physical world necessitate an infinity of objects, since the continuity of the number line (real numbers and geometrical space) needs the axiom of infinity? Can we describe the universe by mathematical models which depend on the idea of infinity and continuity while maintaining that what is described is finite?

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Lawrence Crocker
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Re: Space, Time and Infinity

Post by Lawrence Crocker » Sat May 16, 2015 12:43 pm

Wyman wrote:Here's a question: does the continuity of the physical world necessitate an infinity of objects, since the continuity of the number line (real numbers and geometrical space) needs the axiom of infinity? Can we describe the universe by mathematical models which depend on the idea of infinity and continuity while maintaining that what is described is finite?
Whether the physical world is truly continuous is an open question given the apparent importance of the Planck constant, but we can assume that away for now. If we do absolutely need real numbers to describe physical reality, then there is a respect in which there is a physical infinity of the small. Whether this counts as an infinity of physical objects will be more a matter of how inclusive we want to be with the concept of "object." Were there "particles" of smaller and smaller sizes infinitely, then we might have a physics of an infinite numbers of objects in a finite space. If the only things that can divided infinitely are more like distance or charge strength, then, although we could talk about infinite objects, that might be a little eccentric, at least for the physicist -- perhaps not for the mathematician. The physicist might want to say that the mathematician is projecting abstract objects onto physical reality -- objects that are not "really there" in that their status as objects plays no role in the physics.

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