I've been working with the Abel Transformation:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]A\{f(y)\}=\int_0^x \frac{f(y)}{\sqrt{x-y}} dy=F(x)[/tex]

[tex]A^{-1}\{F(y)\}=\frac{F(0)}{\pi \sqrt{x}}+\frac{1}{\pi}\int_0^x\frac{F^{'}(y)}{\sqrt{x-y}} dy=f(x) [/tex]

And please tolerate my appreciation of mathematical beauty by allowing me to report them here.

It occurred to me that transformations like this and others in math have profound philosophical consequences. Think about what is being achieved here: We start with a set (of funtions), and transform them into another set through the action of a transformation. But we seek to find the "inverse" transform. That is, we have the "end results" and wish to find the "starting" point (the function which was originally transformed). It seems to me that much in our endeavours to understand nature work that way: We observe consequences, and seek to discover their cause! Perhaps this is why mathematical transformations are so useful in describing nature. I am reminded of Rene' Thom:

"All creation or destruction of form, or morphogenesis, can be described by the disapperance of the attractors representing the initial forms, and their replacement by capture by the attractors representing the final form."

In the language of transformation theory, one would interpret this as our efforts to define the trajectory (transfomation) between attractors, and to discover, from the final attractor (the consequence), the initial attractor (the cause) through the creation of the inverse transform.

What do you guys think?

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# Morphogenesis and Abel Transformation

Can you offer guidance or do you also need help?

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