Sunday, August 31, 2014

Pickover on Platonism…

Some extended discourse via Cliff Pickover today from his volume, "A Passion For Mathematics" (one of my favorite Pickover offerings):

"I think that mathematics is a process of discovery. Mathematicians are like archaeologists. The physicist Roger Penrose felt the same way about fractal geometry. In his book The Emperor's New Mind, he says that fractals (for example, intricate patterns such as the Julia set or the Mandelbrot set) are out there waiting to be found:

'It would seem that the Mandelbrot set is not just part of our minds, but it has a reality of its own… The computer is being used essentially the same way that an experimental physicist uses a piece of experimental apparatus to explore the structure of the physical world. The Mandelbrot set is not an invention of the human mind: it was a discovery. Like Mount Everest, the Mandelbrot set is just there.'

I think we are uncovering truths and ideas independently of the computer or mathematical tools we've invented. Penrose went a step further about fractals in The Emperor's New Mind: 'When one sees a mathematical truth, one's consciousness breaks through into this world of ideas… One may take the view that in such cases the mathematicians have stumbled upon works of God.'

Anthony Tromba, the coauthor of Vector Calculus, said in a July 2003 University of California press release, 'When you discover mathematical structures that you believe correspond to the world around you, you feel you are seeing something mystical, something profound. You are communicating with the universe, seeing beautiful and deep structures and patterns that no one without your training can see. The mathematics  is there, it's leading you, and you are discovering it.'

"Other mathematicians disagree with my philosophy and believe that mathematics is a marvelous invention of the human mind. One reviewer of my book The Zen of Magic Squares used poetry as an analogy when 'objecting' to my philosophy. He wrote,

'Did Shakespeare 'discover' his sonnets? Surely all finite sequences of English words 'exist,' and Shakespeare simply chose a few that he liked. I think most people would find the argument incorrect and hold Shakespeare created his sonnets. In the same way, mathematicians create their concepts, theorems, and proofs. Just as not all grammatical sentences are theorems. But theorems are human creations no less than sonnets.'

Similarly, the molecular neurobiologist Jean-Pierre Changeux believes that mathematics is invented: 'For me [mathematical axioms] are expressions of cognitive facilities, which themselves are a function of certain facilities connected with human language.'

[…If you have a favorite math-related passage that might make a nice Sunday morning reflection here let me know ( If I use one submitted by a reader, I'll cite the contributor.]

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