10/13/2007, 10:05 AM
Dear Jaydfox,
You are right! Nevertheless, we have to pay attention to the fact that mathematical analysis (M. A.) is supplying us with very important tools, such the concepts of "function" and "continuity", which are absolutely indispensable for a correct analysis of a problem but that, nevertheless, are only (... extremely important) tools. In fact, it is clear that \( f(z)=z^2 \) is a "function", but its inverse \( f^{-1}(z) \) is not a "function", the "continuity" of which cannot be analyzed. Nevertheless, we also perfectly know that an inverse operation of the "square-of-x" exists and that it is the two-valued "square-root-of-x". The graph of the second object (it is NOT a "function") can be easily obtaind by a simple change (x<->y) of variables, without any other kind of modifications. All of us know that this is true, but we cannot express such a thing with a standard mathematical language.
For instance, I know that sqrt(4) = {-2,+2}, but I was never able to find a pocket calculator that gives this correct result (not even "Mathematica" does). It is tacitely understood that we can simply calculate the "principal branch", in order to get sqrt(4) = 2 and that, after that, we smartly duplicate the operation on the symmetrical (negative) branch, for obtaining the required second value (-2). This fact disturbed me since I was almost a child. I was always thinking that the adults were not serious people!
Your example of the circle (with r = 2) is also amazing. According to the standard M. A. a circle cannot represent a "function" and we cannot verify its "continuity". On the contrary, the little devil I have in my brain suggests that its graph is absolutely continuous and that it is a two-valued "function". Its derivative must also be continuous and "two-valued".
Réné Thom, with his "Théorie des Catastrophes" analyzed these strange (but, at the same time, quite normal) mathematical objects and was able to classify what he called elementary "catastrophes". But, as far as I know, he didn't succeed to formulate a consistent new extension of the standard M. A. for covering these important objects.
I discussed with Henryk about that and he tried to convince me that "something exists already" (e.g.: Riemann surfaces) and that mathematics is not just a philosophical opinion. I agree in principe. But, on the practical ground, ... I doubt!
GFR
jaydfox Wrote:The slog is not a function, nor is tetration. This is important to understand. The logarithm is not a function, in the sense that analytically continuing it from any point by integrating its derivative yields multiple values for any point. Yet exponentiation is in fact a function, so long as you're clear on how you define the base.
For example, \( f(z)=z^2 \) is a function, but the inverse \( f^{-1}(z) \) is not a function.
You are right! Nevertheless, we have to pay attention to the fact that mathematical analysis (M. A.) is supplying us with very important tools, such the concepts of "function" and "continuity", which are absolutely indispensable for a correct analysis of a problem but that, nevertheless, are only (... extremely important) tools. In fact, it is clear that \( f(z)=z^2 \) is a "function", but its inverse \( f^{-1}(z) \) is not a "function", the "continuity" of which cannot be analyzed. Nevertheless, we also perfectly know that an inverse operation of the "square-of-x" exists and that it is the two-valued "square-root-of-x". The graph of the second object (it is NOT a "function") can be easily obtaind by a simple change (x<->y) of variables, without any other kind of modifications. All of us know that this is true, but we cannot express such a thing with a standard mathematical language.
For instance, I know that sqrt(4) = {-2,+2}, but I was never able to find a pocket calculator that gives this correct result (not even "Mathematica" does). It is tacitely understood that we can simply calculate the "principal branch", in order to get sqrt(4) = 2 and that, after that, we smartly duplicate the operation on the symmetrical (negative) branch, for obtaining the required second value (-2). This fact disturbed me since I was almost a child. I was always thinking that the adults were not serious people!
Your example of the circle (with r = 2) is also amazing. According to the standard M. A. a circle cannot represent a "function" and we cannot verify its "continuity". On the contrary, the little devil I have in my brain suggests that its graph is absolutely continuous and that it is a two-valued "function". Its derivative must also be continuous and "two-valued".
Réné Thom, with his "Théorie des Catastrophes" analyzed these strange (but, at the same time, quite normal) mathematical objects and was able to classify what he called elementary "catastrophes". But, as far as I know, he didn't succeed to formulate a consistent new extension of the standard M. A. for covering these important objects.
I discussed with Henryk about that and he tried to convince me that "something exists already" (e.g.: Riemann surfaces) and that mathematics is not just a philosophical opinion. I agree in principe. But, on the practical ground, ... I doubt!
GFR
