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.
. It has since been shown that a knot cannot exist in any dimension 
. Two distinct knots cannot have the same