Space, Time and Infinity

[uwot]

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:

...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?

[Seremonia]

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).

[surreptitious57]

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

[Seremonia]

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.

[Lawrence Crocker]

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.

[Wyman]

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.

[Lawrence Crocker]

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.

[Lawrence Crocker]

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]

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.

[Philosophy Explorer]

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

[Lawrence Crocker]

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]

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?

[Lawrence Crocker]

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.

[Advocate]

>I consider space as being infinite. Looking out at the night sky I see the Universe but importantly I see a black background. The background means, to me, that space will go on forever. Yet, I can't truly believe it, even though it seems that it is necessary.

The concept of space is the correlation of our external and internal (proprioceptive) senses. Space in the external sense (the application of the idea, taken to its logical extreme) is infinite because Actuality is infinity. There are no such thing as Ends outside of minds with purposes.

>If there were no universe there still should be empty space and that space would be infinite. Even though I am saying it I still can't wrap my mind around it. It's too much space. It goes on forever. Does it make sense to say that is has an origin? I think not. For, what would be the origin of the origin? Who makes the makers? We can't get rid of the ideas of eternity and infinity.

"The Universe means all that is. Any idea which tries to render that concept smaller so other concepts can fit is automatic fail. Our brains are about the size of a football (American) and literally everything we understand is a simplified metaphor. There are also no beginnings outside the purposes of a mind. Actuality is infinite in all directions, sizes, and scales.

>The same goes for time. We can conceptually go back in time forever. And we can go forward forever, it just doesn't make any sense. with the idea of eternity and infinity we are negating our own existences it seems.

Change is the universal substrate of Actuality and time is measured (experienced) change. You can understand matter, causality, space, energy, mass, entropy, etc. in the same context. The past is remembered experience. The future is anticipated experience.

>Infinite space and eternal times are impossible to believe in the face of our own existence. Yet it seems that it is necessary.

Brain the size of a football, meet literally everything else! We cannot even understand a football completely because there's meta-information that would make the space required larger than our brains.

>Are space and time mere concepts in the mind?

Concepts in the mind represent reality. Everything in the mind (the patterns in the brain) has correlates in reality. A unicorn is a horse plus a horn.

[Advocate]

[quote="Lawrence Crocker" post_id=199953 time=1430342009 user_id=10843]
Kant concluded that space and time were forms imposed by our minds upon reality. One of his arguments, the "First Antinomy of Pure Reason" in his [i]Critique of Pure Reason[/i] anticipates some of your thoughts here.

The physicists tend to say that, although we can conceptualize infinite space and time infinite in both directions, the geometry of space (including whether it is infinite) and the career of time are physical questions.
[/quote]

Imposed is a problem there. Recognized is appropriate. The infinity of space is evidenced by the fact that every time we increase the resolution of our instruments we find more; either outward/upward or inward. Meanwhile we have never found an end to anything, only changes of state.