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In part 1, I introduced the subject and gave some background metaphysical assumptions I regard as a necessary starting point for philosophical inquiry. In part 2, I summarized the way in which I view quantum mechanics, how it differs from Copenhagen, and how the positivist features of Copenhagen are strongly tied to formalist views of mathematics. In part 3, I'd like to shift the focus more squarely onto the ontological status of mathematical objects which is where I regard the most interesting philosophical questions to lie.
I've stated before that I think of myself as a mathematical realist, but my degree of certainty on that position is a lot weaker than most of the other philosophical positions I hold (which for the record, include: empiricism, representational realism, bayesianism, reductionism, materialism/functionalism (with regard to philosophy of mind), existential nihilism, ethical nihilism/relativism, metaphysical objectivism, atheism, transhumanism, feminism, and social constructivism (with regard to things like gender and race)). I'm a mathematical realist, and yet I feel that many mathematical realists (especially those who choose to call themselves "Platonists" take their beliefs too far and end up rejecting empiricism in spirit if not in practice.
I finished reading a book on Kurt Godel recently, called Incompleteness. It was a biography of his life focusing on the progression of his mathematical and philosophical ideas, his relationship with Einstein and the Vienna Circle. I've since then started reading a book called "A Madman Dreams of Turing Machines" which I was hoping would be about Alan Turing, but unfortunately it's not nearly as good and it's mostly about Godel with only small sections every now and then dealing with Turing. Actually, it's beautifully and poetically written, it's just that I feel like the author of Madman Dreams of Turing Machines (who is a cosmologist I think) was not sophisticated enough to understand the philosophical views of Godel or Turing very well, whereas the author of Incompleteness (a philosopher) came a lot closer. Unfortunately, the Incompleteness author, Rebecca Goldstein, also uses the book to sell Godel's brand of Platonism which is a bit overboard.
Goldstein emphasizes one thing in the book that Godel and Einstein both had in common. According to her, they were "philosophical exiles" because they were both metaphysical objectivists in an age of constructivism... and both of the theories they became famous for ended up being misunderstood by many philosophers and used to support views diametrically opposed to their own. Both Einstein and Godel believed firmly that there was an objective world beyond the senses, one which was not constructed by humans but has an absolute existence outside of all human constructs and inventions. This is also one of the things I liked best about Ayn Rand. Neither she, nor Einstein, nor Godel bought in to the anthropocentric notion that ultimate reality itself is merely a product of human thought or in any way based on it (even though I think the more everyday social reality we live in is very much such a product). Godel knew that no human has ever been smart enough to make up the unending complexity and beauty of the structures which have been discovered in mathematics. His Incompleteness Theorem was intended as the ultimate proof that mathematics has an independent absolute existence which is not dependent on the axioms and formal systems that humans use as tools to study it. Einstein's theory of relativity as well as Godel's Incompleteness Theorem were subsequently used by certain continental philosophers to argue for the possibility that mathematics and physics were themselves socially constructed, yet both of them would have loathed this view. Relativity establishes that lengths and clocks in 3-dimensional space are relative to the observer, but without the existence of absolute 4 dimensional spacetime, and an absolute speed limit, there would be no structure to make such an argument. The interesting thing about relativity is not that it establishes a relational metaphysics (since it does not), but that it moves us a step further away from anthropocentrism by discarding outdated human concepts such as space and time, or energy and momentum, in favor of more absolute universal inhuman 4 dimensional entities such as spacetime and energy-momentum. She postulates that their shared feelings of alienation from the fashionable intellectual climate of the day was the strongest basis for their close friendship during their careers at the Institute for Advanced Study.
Well, it's 2am and I haven't even started to say what I was going to say about mathematics. So I guess there is going to have to be a part 4, and most likely a part 5. But I have been wanting to give my book review of Incompleteness for a while, so I'm glad I got that part out of the way. Till next time...
I've stated before that I think of myself as a mathematical realist, but my degree of certainty on that position is a lot weaker than most of the other philosophical positions I hold (which for the record, include: empiricism, representational realism, bayesianism, reductionism, materialism/functionalism (with regard to philosophy of mind), existential nihilism, ethical nihilism/relativism, metaphysical objectivism, atheism, transhumanism, feminism, and social constructivism (with regard to things like gender and race)). I'm a mathematical realist, and yet I feel that many mathematical realists (especially those who choose to call themselves "Platonists" take their beliefs too far and end up rejecting empiricism in spirit if not in practice.
I finished reading a book on Kurt Godel recently, called Incompleteness. It was a biography of his life focusing on the progression of his mathematical and philosophical ideas, his relationship with Einstein and the Vienna Circle. I've since then started reading a book called "A Madman Dreams of Turing Machines" which I was hoping would be about Alan Turing, but unfortunately it's not nearly as good and it's mostly about Godel with only small sections every now and then dealing with Turing. Actually, it's beautifully and poetically written, it's just that I feel like the author of Madman Dreams of Turing Machines (who is a cosmologist I think) was not sophisticated enough to understand the philosophical views of Godel or Turing very well, whereas the author of Incompleteness (a philosopher) came a lot closer. Unfortunately, the Incompleteness author, Rebecca Goldstein, also uses the book to sell Godel's brand of Platonism which is a bit overboard.
Goldstein emphasizes one thing in the book that Godel and Einstein both had in common. According to her, they were "philosophical exiles" because they were both metaphysical objectivists in an age of constructivism... and both of the theories they became famous for ended up being misunderstood by many philosophers and used to support views diametrically opposed to their own. Both Einstein and Godel believed firmly that there was an objective world beyond the senses, one which was not constructed by humans but has an absolute existence outside of all human constructs and inventions. This is also one of the things I liked best about Ayn Rand. Neither she, nor Einstein, nor Godel bought in to the anthropocentric notion that ultimate reality itself is merely a product of human thought or in any way based on it (even though I think the more everyday social reality we live in is very much such a product). Godel knew that no human has ever been smart enough to make up the unending complexity and beauty of the structures which have been discovered in mathematics. His Incompleteness Theorem was intended as the ultimate proof that mathematics has an independent absolute existence which is not dependent on the axioms and formal systems that humans use as tools to study it. Einstein's theory of relativity as well as Godel's Incompleteness Theorem were subsequently used by certain continental philosophers to argue for the possibility that mathematics and physics were themselves socially constructed, yet both of them would have loathed this view. Relativity establishes that lengths and clocks in 3-dimensional space are relative to the observer, but without the existence of absolute 4 dimensional spacetime, and an absolute speed limit, there would be no structure to make such an argument. The interesting thing about relativity is not that it establishes a relational metaphysics (since it does not), but that it moves us a step further away from anthropocentrism by discarding outdated human concepts such as space and time, or energy and momentum, in favor of more absolute universal inhuman 4 dimensional entities such as spacetime and energy-momentum. She postulates that their shared feelings of alienation from the fashionable intellectual climate of the day was the strongest basis for their close friendship during their careers at the Institute for Advanced Study.
Well, it's 2am and I haven't even started to say what I was going to say about mathematics. So I guess there is going to have to be a part 4, and most likely a part 5. But I have been wanting to give my book review of Incompleteness for a while, so I'm glad I got that part out of the way. Till next time...
