- What intrigues me most are the similarities between the ideas behind the Koch Snowflake, the Sierpinski forms and other 'early' fractals with the philosophical paradoxes of Zeno. These paradoxes have mostly been put on the shelf and trotted out in philosophy classes as comedic examples of how logic can deny reality. However, I have always had the feeling that Zeno was actually trying to show how reality confounds human logic. Plato and Aristotle did not understand, and are on record as being dismissive of Zeno's teachings as nonsense. Socrates, however, was left speechless, and as was his odd way, wandered off in silence, never to speak of Zeno or his paradoxes again, as far as we know. Philosophers down the ages have dallied with the paradoxes, laughed at their absurdity, and tossed them aside. Some mathematicians have seen the paradoxes as mathematical problems, and through that lens used them to create the science we know as calculus. Some few philosophers have seen the paradoxes as metaphysical problems, but to date have not cracked the mainstream with their thoughts. I see fractals in Zeno's Paradoxes.
- The paradox of Achilles and the Tortoise is exactly the same “problem” that is presented by the mathematical function of the iteration of the 19th century “mathematical monsters” – specifically and most directly applicable, the Cantor Set. In both Zeno's paradox of Achilles and the Tortoise, and in Georg Cantor's 1833 mathematical monster, the line segment that has a third of it's length removed an infinite number of times, the resulting reality defies mainstream human logic. In other words, the mainstream mathematical model of reality is in fact not a model of reality, but simply a model of human fantasy. To be sure, human fantasy is a creative force, and many astounding ideas have been built upon human fantasy, including Euclidean geometry and Newtonian physics. But in all of these human fantasies there has always been unresolved paradoxes, fudges like irrational numbers and the concept of entropy. This is, I posit, because these models do not reflect reality, but only define human thoughts. They are naught but daydreams.
- There is an exchange between Plato and Diogenes that is interesting in this context of scientific models, reality, and fractals.
- Plato was teaching his theory of ideas and Forms, and, pointing to the cups on the table in front of him, said, “while there are many cups in the world, there is only one `idea' of a cup, and this cupness precedes the existence of all particular cups.” At which point Diogenes interrupted by saying: "I can see the cup on the table, but I can't see the `cupness'".
Plato, in a condescending answer replied: "That's because you have the eyes to see the cup, but,” tapping his head with his forefinger, "you don't have the intellect with which to comprehend `cupness'."
Diogenes then walked up to the table, examined a cup and, looking inside, asked, "Is it empty?" Plato nodded in the affirmative, whence Diogenes then asked: "Where is the `emptiness' which procedes this empty cup?"
Plato collected his thoughts for a few moments, but before he gathered an answer Diogenes reached over and, tapping Plato's head with his finger, said "I think you will find here is the `emptiness'."
- Just as when Mandelbrot's theory of the fractal nature of reality finally resolved those 19th century math monsters into obvious expressions of basic reality (i.e. the way “nature” works), I think that if we continue to learn to accept our experiences over what we have been taught, we will continue to unlock the mysteries of nature and the reality of the universe.
- Yogi Swamigal once said: “You have eyes, you have ears, you have nose, you have hands? What are you doing with them? What is the use of eyes, if you don't see? What is the use of ears, if you don't hear?”
Thursday, April 07, 2011
Posted by OnlyEd at 4/07/2011 07:55:00 AM