I am just reading Rebecca Goldstein's Incompleteness, her intellectual biography of Goedel. A great book. She discusses that Wittgenstein's reaction to the incompleteness theorems was almost word for word what you posted. But those theorems did "fall out of mathematical formalisms" and they do have real consequences for us, as Penrose's claims in The Emperor's New Mind emphasize. Thinking of mathematics as just a tool box rather than as an element of reality on its own leads to such restricted views!Chronos said:It is ludicrous to entertain the idea that discoveries drop out of mathematical formalisms. That is about as sensible as claiming houses drop out of tool boxes.
What do you think of Joseph Silk's book on the Big Bang theory compared to Singh's?SpaceTiger said:I think this highlights one of the biggest problems in physics/astrophysics today. Einstein seemed slow relative to his peers simply because he wasn't anything like his peers -- he was a conceptual thinker, not a calculator/database. He wasn't content just to know things and then spit them out, he needed to really understand them and connect the dots of the big picture. Sadly, that kind of physicist is still quite rare, despite the fact that we now have machines to do the computing and compiling for us. We still think someone that can multiply 100 numbers in their head is the epitome of intelligence. I've had colleagues try to convince me that virtually every major discovery in physics "just came out of the math" and that having a conceptual understanding was pointless.
If you ask me, we need to broaden, or perhaps even change, our common understanding of intelligence.
That's very Platonic, but not a logical necessity for the correctness of mathematical theorems. Mathematics is an expression of ideas from which one can derive irrefutable conclusions, but the irrefutability of those conclusions is not inconsistent with the view that mathematics is a "tool box". When I talk about things "falling out of the math", I'm referring to discoveries in which the author did not consider the implications of the mathematical formalism and just stumbled upon a result by blind application of known mathematical procedures. I very much doubt that Goedel was guilty of this. Mathematicians, physicists, and bioligists alike thrive in their respective fields because of an intuition that they've developed for their area of study, an understanding that drives their experiments and derivations. In the absence of this intuition, their work would be little more than fancy guesswork.selfAdjoint said:Thinking of mathematics as just a tool box rather than as an element of reality on its own leads to such restricted views!
Obviously, the universe works just fine, so it is not wrong. Our descriptions of the universe are approximations, whether expressed in conceptual terms or quantified in mathematical terms. Where we run into trouble, I believe, is elevating some past approximations to the level of "unquestioned truths" and thus blocking real progress. If we are not willing to ask hard questions about fundamentals (i.e. How does mass curve space-time? How can the Higgs field and the gravitational field be precisely congruent over all visible space and time?, etc) we will not progress.Chronos said:Not all mathematical predictions have physical counterparts. Which begs the question: is the universe wrong, or just our version of it?
Einstein said:"How does it happen that a properly endowed natural scientist comes to concern himself with epistemology? Is there no more valuable work in his specialty? I hear many of my colleagues saying, and I sense it from many more, that they feel this way. I cannot share this sentiment. ...Concepts that have proven useful in ordering things easily achieve such an authority over us that we forget their earthly origins and accept them as unalterable givens. Thus they come to be stamped as 'necessities of thought,' 'a priori givens,' etc. The path of scientific advance is often made impassable for a long time through such errors. For that reason, it is by no means an idle game if we become practiced in analyzing the long common place concepts and exhibiting those circumstances upon which their justification and usefulness depend, how they have grown up, individually, out of the givens of experience. By this means, their all-too-great authority will be broken."