Alamo-Girl, I am just loving it! It's a workout -- I'm only able to do a chapter a day, because so often I have to get out pen and note pad and work out Penrose's ideas "manually." His treatment of complex numbers is fascinating. I'd heard of the "imaginary" i before; but didn't realize its indispensability for, say, plotting the Mandelbrot set. I anticipate his speculation that certain mathematical ideas are "discoveries" (e.g., the Mandelbrot set) rather than "inventions" (e.g., algorithms) will have direct relevance to the problem of consciousness.
An excerpt for today:
"Is mathematics invention or discovery? When mathematicians come upon their results are they just producing elaborate mental constructions which have no actual reality, but whose power and elegance is sufficient simply to fool even their inventors into believing that these mere mental constructions are 'real'? Or are mathematicians really uncovering truths which are, in fact, already 'there' -- truths whose existence is quite independent of the mathematicians' activities?...
"...These are the cases where much more comes out of the structure than is put into it in the first place. One may take the view that in such cases the mathematicians have stumbled upon 'works of God'. However, there are other cases where the mathematical structure does not have such a compelling uniqueness, such as when, in the midst of a proof of some result, the mathematician finds the need to introduce some contrived and far from unique construction in order to achieve some specific end. In such cases no more is likely to come out of the construction than was put into it in the first place, and the word 'invention' seems more appropriate than 'discovery'. These are indeed just 'works of man'....
"Having made these points, however, I cannot help feeling that, with mathematics, the case for believing in some kind of ethereal, external existence, at least for the more profound mathematical concepts, is a good deal stronger than in those other cases. There is a compelling uniqueness and universality in such mathematical ideas which seems to be of quite a different order from that which one could expect in the arts or engineering...."
Then he notes that this was, indeed, the view of Plato; and that this insight "will have considerable importance" for his further treatment of the issues in this book. So, needless to say, I'm looking forward with enormous interest to the further development of his thesis!
Thank you so much for recommending this astounding work!
I anticipate his speculation that certain mathematical ideas are "discoveries" (e.g., the Mandelbrot set) rather than "inventions" (e.g., algorithms) will have direct relevance to the problem of consciousness.
Indeed, it will have a very great deal to do with our discussion on consciousness. Penrose is Platonist, as am I. And he is a wonderful mathematician/scientist. I believe he is Physicalist (rather than Materialist) and thus will explore every possibility, including the natural.
Hugs!