khuram

Some Points About Nature of Infinity:

Posted by khuram on May 6, 2007

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Some Points About Nature of Infinity:

Issue of the nature of infinity is quite confusing. There are theories in Mathematics, which suggest that a finite line and an infinite line, both will be having one to one, and onto correspondence between their respective points. It means that as per these theories (e.g. Georg Cantor; 1845-1918), number of points of a finite line has to be exactly equal to the number of points of an infinite line. Apparently, this theory made no proper sense to me. Actually I believe in the “discrete” nature of reality rather than considering it to be “continuous”. Soon I shall try to explain this point in my next posts. For the purpose of this post, we can just assume that reality is “discrete” and is not “continuous” in any way. Keeping in view this assumption, below I try to discuss some of the supposed characteristics of infinity:


1. The set of all positive integers (i.e. infinite) is smaller (in cardinality) than the set of real numbers between 0 and 1.

My Comments:

May be true. But I doubt in the “real” existence of real numbers. As I assume (which I shall prove later on) that reality is “discrete” and not “continuous”, so what I think is that continuous numbers can exist only in certain abstract mathematical relations but any such kind of continuity cannot exist in our Physical world. There cannot be infinite discrete numbers between 0 and 1.


2. The set of real numbers between 0 and 1 is equal (in the sense that there exists a one to one and onto mapping) to the set of real numbers between 0 and 2.

Actually this one to one and onto mapping of points is considered to be existing even between a finite line and an infinite line. It means that according to mathematics (theory of Georg Cantor; 1845-1918), the numbers of points on a finite line have its number of points exactly equal to the number of points of an infinite line.

It can be seen that this theory is older than the emergence of Quantum Physics. I no more consider this theory to be valid. A finite line must be having finite number of discrete points and an infinite line must be having infinite number of discrete points. There cannot be one to one correspondence between finite points and infinite points. Quantum Physics has even calculated the minimum possible (or absolute minimum) distance. There is no anything like perfect continuity in our physical world.

Secondly, Geometry still uses old Greek concept of point. It is defined something like an abstract point, which occupies no space. The same Geometry defines ‘line’ something like as a “combination of points”. What I think is that this pure abstract mathematics cannot be applied to the physical world. If a ‘point’ has no space at all then how any ‘line’ (i.e. combination of points) can have any space…??? I think that a combination of ‘space less’ points cannot have any length. Abstract Mathematics says that a ‘line’ has length but it does not have any width. Anyways, there is need to have a Quantum or Discrete Geometry as well.

Basic forms, and a definition of Infinity:

A line can be started from a definite point and can be considered to be extending to infinity on one direction. Such a line can be considered to be ‘infinite’ line. But remember that this ‘infinite’ line has a ‘definite’ origin. This line is infinite only on onward side but this line is not infinite on backward side.

On the other hand, there also can be a line, which can be considered to be extending infinitely towards both sides. This line is also infinite. But this line is infinite towards onward and backward sides both. So there can be one directional infinity as well as there can be two directional infinity. Similarly, there can be multiple directional infinity as well and in the same way, there can be an all-directional infinity also.

One type of infinity can be smaller or larger than other one. Meaning of infinity is Never Ending on one or more sides. A thing which can end on all sides (like a finite line), cannot have never ending points in it.

What is meaning of 1/0?

If you have to divide $100 among zero people, it only means that you are not going to disburse any sum to anyone at all. You can distribute $100 to as many (i.e. never ending) zero persons you like. 1/0 is only Abstract Mathematics. It is good only in abstractions. It cannot be as it is applied to real physical world. More precisely, 1/0 is not the case of “never ending”. Actually it is the case of “never happening”.

The following objection was raised on my above-mentioned points:


“There cannot be infinite discrete numbers between 0 and 1”.
You may well be right. But if I take rationals to be discrete, as I can count them through a one to one and onto mapping with the set of integers, then there are indeed infinite discrete numbers between 0 and 1. So your frame of reference and mine are very different. We cannot discuss much and can only agree to disagree on our frames of references.

My response was:

I already have dealt with the issue that a smaller and a larger “line” have exact one to one correspondence between their respective number of “points”. I made diagrams to see if really there was such one to one and onto mapping of points or not.

The actual mistake in official theory lies in the definition of “point” in Geometry. Geometry considers “point” as a “space less” particular location. Since this “point” is space less, so it is having no “length” at all. The same Geometry considers “line” as a linier combination of “points”. The same Geometry also considers that a line possesses a non-zero length but it doesn’t possess any width. But if the constituents of line i.e. “points” had no length at all then how just the combination of those “points” could result in any non-zero length…???

Anyways, this definition of “point” may be right definition in pure Abstract Mathematics. But this is not right for our physical world because there can be no space less physical entity in our spatial world. A “physical point” would be having some “space”. In fact, all the “points” would be having same or uniform non-zero space. Let’s say the length of one point is 1 and the length of a finite line is 100. It means that this line has only and only 100 (discrete) points. Points of this line CANNOT have one to one and onto mapping with the points of that line whose length is 200 or infinite.

What is the mistake of official theory…???

The official theory actually draws one to one and onto mapping not between the individual points of a shorter and a longer (or infinite) line. It actually draws one to one and onto mapping between fractional parts of individual points of shorter line with the complete individual points of longer or infinite line.

Now I give you a task. Consider the “length” of a “point” to be 1. Now take two lines. First line being shorter and second line longer.

Length of first line = 100

Length of second line = 200

In this way, there are 100 “points” in first line and there are 200 “points” in second line. Now try to draw one to one and onto mapping between all the points of first line with all the points of second line. Be careful that do not take fractional or overlapping parts of individual points of shorter line!

Believe me, you will not be able to do it, because it is an impossible task. So come out of the fantasy of old Abstract Mathematics where length of “point” is zero and sum of many zeros (i.e. a “line”) is non-zero positive length. In a physical world, any real “point” will be having non-zero positive length.

This is not the case of just difference of frames of references. It is a matter of clear-cut mistake of 0 + 0 + 0 = 3

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13 Responses to “Some Points About Nature of Infinity:”

  1. Patrick said

    I have considered this view recently. In my research on random number generator functions used in many computer applications, the appearance of randomness in such functions points to the possibility that we indeed might live in a truly discrete reality. Consider that random number generators create a number space that is entirely predictable though appears random. If our own reality consists of an almost innumerable amount of factors or particles then a complex mixing of those factors could create a sense of randomness. Though if we do live in a discrete reality, the complexity of mixed things would effectually be random as the computing power required to predict or prove this fixed nature with the number of elements involved would be so immense that doing so would be impractical. At the scale at which we are finding the universe to exist at, whether things are truly discrete or continuous is of no consequence as the the immensely complex mixing of discrete values approximate our concept of randomness. If things are random, then perhaps the inverse is true in that random mixing/interaction eventually approximates the abstract as patterns emerge from the chaos.

  2. khuram said

    Thanks Patrick, for sharing your valuable findings. I think if reality is discrete, then it has still far reaching consequences. I have worked on those issues in draft form. Hopefully soon i’ll present those things also in this blog.

    Thanks again for your insightful points.

    Regards!

  3. mindloop said

    Hey khuram, I took a large span of time to respond to all of your material presented. It wouldn’t really fit in a comment, so I thought I’d display it on my blog. Here is my in-depth response: http://mindloop.wordpress.com/2007/07/07/re-some-points-about-nature-of-infinity/

    Also, Patrick, I talked a little about how reality on a macroscopic level might look continuous and approximation continuous universes for spans of time, but I do not think that a discrete reality could approximate a continous analogy for an indefinite interval of time, because not all numbers are “computable” or constructible.

    In fact, because of the linear symbolic represent-ability of mathematical constructions, the number of constructible numbers is “only” \aleph_0 . So “almost all” (a mathematically rigorous term you can look up yourself) numbers are not constructible.

    One can infer that in a potentially oracle-inductive-like universe, or something similar, is the case with probability “almost” 1. (In a naive sense, I admit.) Because of how late it is, I will not continue my discussion tonight. Errr… I mean this morning.

  4. […] July 7th, 2007 in Uncategorized Here is someone who, in light of “paradoxes” about infinity and a bit of Cantor’s […]

  5. Gyvor said

    I enjoyed your writings. I found your blog because I share a similar interest and was searching for people who have similar ideas to mine. I too beleive that there exists only two basic thing. That is something (matter) and nothing (space). I beleive in the big bang. I beleive at the beginning, of this “current” universe in which we live, all matter was compressed together into a sphere. It was compressed together because matter is attracted to matter. The atoms within kept compressing together until they could not move in that direction anymore and reversed direction outward resulting in the explosion of matter in all directions. The attraction of matter caused matter to group together to form galaxies and solar systems etc but these creations are still moving outward into the nothingness of space. They will slowly all acumulate together until they once again form the one singular sphere of matter that existed before the big bang. This cycle will repeat itself infinitly. Okay, so the fundemental question is why is matter attracted to matter. This force is the force that allows this infinate cycling of the universe to happen.
    Well here is the though experiment I came up with. Consider there is a single bowling ball of matter and nothing else in existence. How could we tell if it was moving? How could it have mass? How could it have size?—-IF there is nothing else to relate itself too. Now place a tiny marble into this picture. All of a sudden this bowling ball is relative to something. It can have movement , it can have mass, it can have size. It can exibit its reality because it is now relative to something. To exibit its reality it must reveal its energy relative to the second existing object. If the bowling ball was moving then it would exibit this energy as movement away from the marble. If it was still it would exibit this energy by attracting the marble (which has less mass=less energy). This is the primal force from which all other forces are created. Gravity, weak, strong, electromagnetic are all variations of this force. The only difference is in the size of matter they refer too. This primal force as I decribed above has to occur in order for reality to exist, it occurs by default because you have two choices, something or nothing. And nothing cannot exist because by its very nature nothing is actually something.

  6. khuram said

    @ Gyvor

    It’s nice to know that you are having similar ideas to mine. Thanks for your comments on my blog. I do consider space as a “form of nothingness”. Having this view, I go against the established view that “spacetime” is a physical fabric like structure which can have even curvatures etc.

    So far I have not written on exact this issue on my blog. But you can check my recent debate on Philosophy Forum on this issue on following link:

    http://forums.philosophyforums.com/threads/criticism-on-the-idea-of-space-time-four-dimensions-27921.html

    Anyways, thanks again for your valuable comments. Your suggestion/ guidance will always be honoured and appreciated.

    Regards!

  7. Julia Set said

    It is always interesting to find people who share the same questions.

  8. Julia Set said

    Have a look at our discussion on similar topics.

  9. Julia Set said

    http://jeanreve.spaces.live.com/

  10. Shameela said

    Assalam u Alaekum

    Great stuff.. Alhamdulillah

    You are right that Infinity means “Never ending” and Anything/0 is never happening…

  11. asha said

    very nice.i like it

  12. eshwari said

    nice.good keep it up!

  13. eshwari said

    good.keep it up!

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