The Nature of Infinity
In the mathematical world, infinity is a concept that has been accepted as highly applicable and essential to mathematics. Yet it is not something at exists in the physical world, the finite world, the ‘real’ world. If we are to regard Eddington number 15 747 724 136 275 002 577 605 653 961 181 555 468 044 717 914 527 116 709 366 231 425 076 185 631 031 296 (this was copied from wikipedia.org). This is traditionally 10^80. The Eddington number is supposed to be the number of protons in the universe. In theory, there are not anymore quantities in the universe larger than this. So, do numbers stop at Eddington Number?
If we are to use the platonic world of forms, and regard the world of mathematics and the physical world as completely separate entities, then numbers are viewed as adjectives. There is nothing in the physical world that is three for example. Something may have threeness, or a group may consist of the elements of three, but the number three doesn’t exist as a physical entity. But what does this tell us about infinity (∞)?
The first thing to do is try to find a place in the physical world where infinity exists. This is not taken completely under the idea of Sir Roger Penrose that we simultaneously exist in three worlds: the physical, the mental, and the mathematical. If Penrose is correct, then infinity does not exist in the physical world, and only in the mathematical, and that infinity is something that acts in ways we cannot know in the physical world.
The only thing that I can see that is supposed to be infinite in the physical world is the idea of perpetual motion. Since perpetual motion is something that is not found in the physical world, infinity does not exist in their. If something cannot exist ad perpetuum or ad infinitum.
My point is that infinity is a concept so difficult to grasp with is it is something that doesn’t exist in the physical world. This brings my to my next point that is infinity a number or a concept. There is clearly something wrong with a number where x+1=x. If I am to understand correctly there are different sets, levels or numbers of inifinity in set theory. Well if x+1=x and y+1=y and y≠x then either y‹x or x‹y .
I have hitherto claimed that numbers exist independent of space and time. This is in accord with the Platonic view of the universe, but I claim not to be a Platonist but an Existentialist. But back to numbers. If numbers exist independent of space and time, then does ∞ exist in the same way also? I would assume so because, if Penrose is to be believed as I think I hope he ought to be in this case, then the mathematical world does not have to deal with space and time as the two other worlds do. Yet if numbers are ideas, or if at least infinity is an idea it ought to exist in the realm of the mind and the realm of mathematics. I am sure there is some ambiguity or paradox here but I care not to address it at the present time.
Finally, I would like to say that infinity is either an idea or a number or both, assuming all numbers are ideas. Regardless, infinity is something that will continue to make us stretch our brain for centuries to come, or at least as long as humans will exist for we heretofore have barely begun to understand the beauty in the idea of ∞.
If we are to use the platonic world of forms, and regard the world of mathematics and the physical world as completely separate entities, then numbers are viewed as adjectives. There is nothing in the physical world that is three for example. Something may have threeness, or a group may consist of the elements of three, but the number three doesn’t exist as a physical entity. But what does this tell us about infinity (∞)?
The first thing to do is try to find a place in the physical world where infinity exists. This is not taken completely under the idea of Sir Roger Penrose that we simultaneously exist in three worlds: the physical, the mental, and the mathematical. If Penrose is correct, then infinity does not exist in the physical world, and only in the mathematical, and that infinity is something that acts in ways we cannot know in the physical world.
The only thing that I can see that is supposed to be infinite in the physical world is the idea of perpetual motion. Since perpetual motion is something that is not found in the physical world, infinity does not exist in their. If something cannot exist ad perpetuum or ad infinitum.
My point is that infinity is a concept so difficult to grasp with is it is something that doesn’t exist in the physical world. This brings my to my next point that is infinity a number or a concept. There is clearly something wrong with a number where x+1=x. If I am to understand correctly there are different sets, levels or numbers of inifinity in set theory. Well if x+1=x and y+1=y and y≠x then either y‹x or x‹y .
I have hitherto claimed that numbers exist independent of space and time. This is in accord with the Platonic view of the universe, but I claim not to be a Platonist but an Existentialist. But back to numbers. If numbers exist independent of space and time, then does ∞ exist in the same way also? I would assume so because, if Penrose is to be believed as I think I hope he ought to be in this case, then the mathematical world does not have to deal with space and time as the two other worlds do. Yet if numbers are ideas, or if at least infinity is an idea it ought to exist in the realm of the mind and the realm of mathematics. I am sure there is some ambiguity or paradox here but I care not to address it at the present time.
Finally, I would like to say that infinity is either an idea or a number or both, assuming all numbers are ideas. Regardless, infinity is something that will continue to make us stretch our brain for centuries to come, or at least as long as humans will exist for we heretofore have barely begun to understand the beauty in the idea of ∞.
posted by Brian Hillman at 4:04 PM
1 Comments:
I believe that Aristotle would argue that the best way to understand the idea of infinity is to attempt to understand it through our experience with infinite things. As you have stated there are no individual physical objects in the realm of human experience with which we can begin to understand infinity. This leads us to the first problem of understanding. Human beings may only understand an object or concept through our prior experiences with a similar or identical object of which we are attempting to understand. Along this line of reasoning we attempt to apply our sensual and mental experiences to a concept like infinity.
For a parallel example we could ask: where are the bounds of the universe located? Also, if the universe contains within it all things, and it is also expanding what exactly is it expanding into? When we posit these questions we have applied the notion of the boundaries of the physical world to an infinite idea. Our sensual experiences tell us that there must be a line or border to the universe that can possibly be traveled to and is the end of the universe because that is all that we know and all that we can understand. But applying our sensual experiences to the quandary does not provide us with sufficient answers to the nature of the universe nor will they provide answers for the nature of infinity.
Finally, to make my point: infinity is a finite concept that we apply to an infinite idea. If somehow we could break down the Eddington number into whatever made up all of the protons in the universe, and whatever made up those objects, and somehow understood the most basic building block of all matter--For Aristotle it is the Unmoved Mover--and took each and every one of those and compiled them, that number would equal to infinity. But then our ever inquisitive minds would ask: what exactly makes up the Unmoved Mover i.e. those basic building blocks. And again we would attempt to apply a notion like infinite divisibility--a mental or mathematical experience--to these objects, and again we would derive insufficient answers to the original question.
Therefore infinity is not a number; it is solely a concept, a boundary for humans to wrap their minds around so they no longer have to contemplate the infinite divisibility of all things. To quell our inquisitive mind we create concepts like infinity in order to move on to other areas of understanding. We do this in all aspects of life. Another example is "Why are we here and how did we get here?" Most people are satisfied with a story of a supreme being who created them and has a heaven for which they can reside once they perish. When human beings subscribe to those types of notions they quell the inquisitiveness and the occasional fear associated with these concepts in order to advance their thoughts into other aspects of existence. Unfortunately most humans are satisfied with these simple explanations of existence, and infinity among other things. Most do not even question them. But as our dear friend Socrates pointed out: “The unexamined life is not worth living.”
Unfortunately I do not feel as though I have adequately explained myself. For lack of a better excuse it is extremely late. But I imagine you will easily infer the explanation for which I have insufficiently posited. Another unfortunate factor is that you have no idea who I am. But I stumbled across your blog last year and have read it rather faithfully since. This isn’t the first time I have felt compelled to write, but it is the first I’ve had time for.
Jeff Schussler jms122@pitt.edu
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