The Nature of Infinity: The Continuum
The classic introduction into the nature of infinity is given by the function y = 1 / x as shown in the figure (where x and y can be real, or rational numbers). One can see that when x = 1 then y = 1 and that this point, by symmetry, forms the middle of the function curve. The curve then extends forever along both the x and y axis, approaching each axis but never becoming equal to x = 0 or y = 0.
Infinity which one could argue would come about with 1 / 0 is not a real number. Such a result would cause a discontinuity in the function causing a break resulting in two functions instead of one.
The key point here is that if the curve is thought to be composed of individual points then the finite interval between 0 and 1 holds as many points as the infinite interval between 1 and infinity! We have a contradiction here. How can the same number of points exist within two differently sized intervals?
The solution is not to consider the function line as composed of a bunch of points but instead to realize that it represents something much deeper and significant which mathematicians call a continuum. A continuum can stretch or shrink to accomodate whatever a function or systems demands of it. Continuums also make themselves obvious in the mathematical process of diferentiation in which a function interval is made as small as needed to reach a state of changelessness for the system. Once this state of changelessness is reached the system has no need to make it smaller.
Einstein called our space and time a Space-Time Continuum in recognition of the above observations. All physical quantities at the fundamental level are either integers or continuums (the spaces). Because of the continuum the mathematician Georg Cantor, who did so much work with infinity, emphasised that there was no such thing as the infintesimal (the infinately small) in either mathematics or metaphysics. Thus when I read that certain physicists claim that physical theories fail when something is shrunk to an infinately small point I wonder if they really understand the foundation of mathematics. An example of this is when an electron is shrunk down to be infinately small (a point particle). In this case the charge density becomes infinately large producing an infinately large potential energy.
Now consider how this idea of a continuum comes about. It is not deduced and niether is it derived from empirical experimentation. It was derived from an inductive process after the recognition of an internal contradiction. That required the development of a deeper unifying concept, not the surrender to blind acceptance that contradictory ideas can both be valid. Only by going deeper do we gain understanding.
Infinity which one could argue would come about with 1 / 0 is not a real number. Such a result would cause a discontinuity in the function causing a break resulting in two functions instead of one.
The key point here is that if the curve is thought to be composed of individual points then the finite interval between 0 and 1 holds as many points as the infinite interval between 1 and infinity! We have a contradiction here. How can the same number of points exist within two differently sized intervals?
The solution is not to consider the function line as composed of a bunch of points but instead to realize that it represents something much deeper and significant which mathematicians call a continuum. A continuum can stretch or shrink to accomodate whatever a function or systems demands of it. Continuums also make themselves obvious in the mathematical process of diferentiation in which a function interval is made as small as needed to reach a state of changelessness for the system. Once this state of changelessness is reached the system has no need to make it smaller.
Einstein called our space and time a Space-Time Continuum in recognition of the above observations. All physical quantities at the fundamental level are either integers or continuums (the spaces). Because of the continuum the mathematician Georg Cantor, who did so much work with infinity, emphasised that there was no such thing as the infintesimal (the infinately small) in either mathematics or metaphysics. Thus when I read that certain physicists claim that physical theories fail when something is shrunk to an infinately small point I wonder if they really understand the foundation of mathematics. An example of this is when an electron is shrunk down to be infinately small (a point particle). In this case the charge density becomes infinately large producing an infinately large potential energy.
Now consider how this idea of a continuum comes about. It is not deduced and niether is it derived from empirical experimentation. It was derived from an inductive process after the recognition of an internal contradiction. That required the development of a deeper unifying concept, not the surrender to blind acceptance that contradictory ideas can both be valid. Only by going deeper do we gain understanding.
Comments
Lally - I think the key idea is that if you look for points in a continuum you will find them, even an infinite amount. At least for real numbers some proof exists stating that between any two numbers you can always find another. I do not remember if that holds up for rational numbers. So really a continuum consists of the set of real numbers (or more general). The number PI (the ratio of the circumference to the diameter of a circle) is an irrational number meaning it belongs in the set of real numbers.
Cantor was able to show that if one could count in an orderly way (enumerate) numbers from variously defined number sets that one could get ratios of infinities. So not all infinities are the same!
Another point (no pun intended!) of all this is that any information processing system must have some lower scale in which changes are meaningless to it, which other parts of its system do not detect. So in this way space and time become discrete but this discretness is not synchronised in the universe (or the brain for that matter). That is to say no global clock exists to indicate when states should change everywhere.
Oh, my brain is doing push-ups and crunches!
It's been a while, but I think I followed the concept of that, if not the detail. And it felt delicious. I used to be an earth and space science nerdlette, but left it all (including, gulp, my beloved geometry) to go into music. How tough is it to get into this group? I bought my copy of Einstein (Isaacson) and one of my fav books is FLATLAND. Let me know?
In regards to groups, just join what ever group you found this post in since they all have open enrollment.