Saturday, April 6, 2013
astro picture for the day/ equivalent expressions generalisation of Jacob Bronowski's philosophy of mathematical knowledge
Image Credit: Hubble Legacy Archive, ESA, NASA; Processing - Donald Waid
Quote for the day
"Keep your work and your self-esteem separate." - Israel Moiseevich Gelfand
Israel Gelfand was a pretty major mathematician of the twentieth century. Emil Artin made major generalisations to Galois theory in algebra and reciprocity theorems in number theory; he's the only solver of two of the Hilbert problems. These are two of the major themes of mathematics. Another would be invariant theory. Invariant theory appeared to be solved by David Hilbert in the late 1800s. But, Hermann Weyl found a way out by a representation theory generalisation. Well, Gelfond's perhaps major contribution was taking this clue of Hermann's to the tilt. The central theme of Gelfond's vast representation theory beyond Hermann Weyl's appears to be compact spaces. But, this mathematics of Gelfand's, vast as it was(almost impossible to describe here) is just one contribution. He made contributions to functional analyses - taking clues from Laurant Schwartz's work. His general sum total contribution here was like six volumes. But, he did even more! I just have to give some idea of the vast contribution of this guy!
Some more mathematical thought for the day. While I may have established Jacob Bronowski's philosophy of knowledge, it is kind of trivial. I found some other ideas to continue making more progress on these ideas about mathematical knowledge. The relation between idealisation and generalisation continued to bug me. Something came up about equivalent expressions when dealing with a rather aggresive intellectual who felt that Einstein's General theory of relativity must be wrong and so muct quantum theory! I mean and so does Jacob Bronowski that all theory is provisional even if proven right in the first place. Newton's mechanics was wrong the day he proved it right! Einstein's work 'generalised' it. Quantum theory generalised chemistry and Maxwell's electromagnetism. The same affects happen in mathematics. It's all 'equivalent expressions.' And equivalant expressions are generalisations. I pretty much had this answer in mind for the longest time; but, I never wrote them down till now. I knew all this time(over a year now) that this was a clue to some generalisation of even my understandings of Jacob Bronowski's philosophy of knowledge. Recently, in reviewing how to do deductive proof, I noticed something relevant. Besides but perhaps including the equivalent expression of quantifyers(the words all and some variables in symbolic logic), the fact is that all logical proof has to do with making a cut in terms of the hypotheses and conclusion; you either work from hypothesis to conclusion or the other way around. Either way requires you to define the problem more clearly to find connections. Equivalent expressions enters to find those connections. This is a good stopping point for now.