In honour of the life of Benoit Mandelbrot, I’d like to discuss the philosophical implications of he famous set.
The most obvious way in which it moved forward the philosophical yardsticks is the often noted fact that fractals resemble natural objects. By the 19th century scientists were well on the way to explaining much of the natural world’s observable phenomena. However, there ability to describe the shapes observed in nature was well beyond the grasp of geometry. With the advent of fractal geometry, suddenly shapes such as mountains, rivers, clouds, trees, and so on, became geometric objects. Prior to that geometry only described shapes which were relatively uncommon in nature: spheres, parabolas, dodecahedrons, and such.
The implication of this leap is significant: when a shape can be described by mathematics, then it is reasonable to infer that the process that created the shape can be described by mathematics. Generalizing this observation leads to a far more profound implication lies in a basic observation about the Mandelbrot set.
The filigree of whorls and paisley forms of the set are incredibly beautiful and intricate.
The set neither random nor repetitive, and is in fact infinitely complex. On seeing it with no explanation, one might guess that it was either the work of some ingenious but obsessive artist, or a creation of nature.
Yet all of this astounding and magnificent complexity is rendered by iterating a formula simple enough for most high school students to understand:
This is the philosophical foundation on which Stephen Hawking is building when he says "The universe can and will create itself out of nothing".
The key to the creative power of such simplicity is in the iteration. In Mandelbrot's words "Bottomless wonders spring from simple rules endlessly repeated". In exactly the same way as biological complexity and diversity have arisen spontaneously from the endless repetition of the simple rules of natural selection, the endless repetition of the simple rules of physics led to the creation of the universe. In fact, one can regard natural selection itself as an extension of the repetition of the laws of physics.
The most obvious way in which it moved forward the philosophical yardsticks is the often noted fact that fractals resemble natural objects. By the 19th century scientists were well on the way to explaining much of the natural world’s observable phenomena. However, there ability to describe the shapes observed in nature was well beyond the grasp of geometry. With the advent of fractal geometry, suddenly shapes such as mountains, rivers, clouds, trees, and so on, became geometric objects. Prior to that geometry only described shapes which were relatively uncommon in nature: spheres, parabolas, dodecahedrons, and such.
The implication of this leap is significant: when a shape can be described by mathematics, then it is reasonable to infer that the process that created the shape can be described by mathematics. Generalizing this observation leads to a far more profound implication lies in a basic observation about the Mandelbrot set.
The filigree of whorls and paisley forms of the set are incredibly beautiful and intricate.
The set neither random nor repetitive, and is in fact infinitely complex. On seeing it with no explanation, one might guess that it was either the work of some ingenious but obsessive artist, or a creation of nature. Yet all of this astounding and magnificent complexity is rendered by iterating a formula simple enough for most high school students to understand:
zn+1 = zn2 + c
The fact that such complexity can arise mechanically from such simplicity demonstrates that an intelligent creator is not necessary to explain any degree of apparently sophisticated organization found in nature. Furthermore, it demonstrates this fact with the absolute certainly that is only possible in the realm of mathematics.This is the philosophical foundation on which Stephen Hawking is building when he says "The universe can and will create itself out of nothing".
The key to the creative power of such simplicity is in the iteration. In Mandelbrot's words "Bottomless wonders spring from simple rules endlessly repeated". In exactly the same way as biological complexity and diversity have arisen spontaneously from the endless repetition of the simple rules of natural selection, the endless repetition of the simple rules of physics led to the creation of the universe. In fact, one can regard natural selection itself as an extension of the repetition of the laws of physics.
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