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.
I agree. Thanks for the link, turbo. And well put, SpaceTiger.
I second the motion by ST. 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.
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!
What do you think of Joseph Silk's book on the Big Bang theory compared to Singh's?
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.
Sorry, I've not read either.
I agree with SA in spirit, but disagree in principle. Not all mathematical predictions have physical counterparts. Which begs the question: is the universe wrong, or just our version of it? It takes intuition and observation to separate physical from unphysical solutions. Ultimately, however, I believe math and intuition will converge in agreement with observation.
I believe mathematics IS an element of reality because it is born of the same fabric: The human brain evolved as a survival strategy to cope in a non-linear world. From such a non-linear brain emerges another coping strategy: a non-linear concept we call mathematics that works so well to describe nature precisely because it mirrors the fractal, nested, non-linear geometry which bore it.
I dare say that if nature was linear in its geometry, then live would have evolved likewise,and if intelligence were to have arisen, it would have created a linear mathematics.
I suspect discoveries fall out of mathematics because at a fundamental level, both math and nature are equilivent because they have equivalent geometries: fractal.
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.
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