quotes: Babylonian vs Euclidean method, why physicists dread precision

[gustavolacerda]

There are two kinds of ways of looking at mathematics... the Babylonian tradition and the Greek tradition... Euclid discovered that there was a way in which all the theorems of geometry could be ordered from a set of axioms that were particularly simple... The Babylonian attitude... is that you know all of the various theorems and many of the

Comments

[pbrane]

The physicist can exhibit a thousand proofs that were fallacious in the formal system and that are still fallacious to her informal intuition, but this does not convince me that there is no informal argument yet to be discovered that passes her intuitive test, but cannot be correctly formalized.

I think the point for physics is that we don't rely on "proofs" at all, really. We often 'prove' things to some degree of satisfaction using formal systems (renormalization of the Standard Model being a good example - Physics Nobel 2000), but we know that the real final arbiter of correctness for a theory is *experiment*: regardless how hokey and seemingly inconsistent a theory seems (mathematicians are pretty sure that QFT as a whole is inconsistent on numerous levels: each term in the perturbation theory is ill defined, even once regulated, the sum of the series diverges, and the defining generating function of the whole system is an integral with a completely undefined measure over an infinite dimensional space), if it produces predictions which are verified by precision experiments (like the 14 decimal places of accuracy that the standard model agrees to), we call it "right".

Continuing the analogy, mathematicians have been trying to formalize QFT for 50 years, without much success, and without any real contribution to finding new "truths" - at best, they've found slightly more reasonable proofs that some little piece of the edifice can be given another (more rigorous) definition which reproduces the same results as the old ill-defined one.

Formal systems are great - for mathematics, linguistics, computer science. But not for the physical sciences.

[jcreed]

So, I don't dispute that consistency with experiment is the right thing to be looking for. However, isn't there still a process that takes the a proposed model and derives what experimental predictions it actually makes? I think that process could (or, really, already does) benefit from improving and deploying more formal understanding. Just because QFT produces formal objects that are foreign to how mathematicians usually think about calculus doesn't mean that there isn't some consistent formal system that encapsulates our current physical theory in that domain.

If the mathematics is irreparably inconsistent (which I doubt) then it falls to mere intuition and the community social dynamics of physicists to even decide what the theory is predictably and what it isn't. But I bet instead that it's a very well-behaved formal system that just hasn't been exactly delineated yet.

[pbrane]

But I bet instead that it's a very well-behaved formal system that just hasn't been exactly delineated yet.

Certainly, I agree with you on this. My point is just that for all the work that mathematicians have done on trying to make advances in finding this well-behaved formal system, we've gained pretty much *no* new insights about physics, and have made no new predictions, and don't really have any new way of seeing what's going on.

There may be a perfectly consistent and well-defined formal system for QFT, but it may also not be as *useful* as the kludgey shorthand set of mnemonics we have now. I'm not saying that formal systems aren't interesting, or that physical law can't be encapsulated in one, I'm just saying that it may not be terribly efficient or productive (in discovering new physics or understanding old).

This is not to say, of course, that *abstraction* isn't useful in physics - it's *immensely* useful, as it allows the condensing of messy systems into more basic components. But making sure that these basic components are all well defined and interact consistently has historically not been that necessary for the gains that are made by the initial ill-defined abstraction process.

[gustavolacerda]

There may be a perfectly consistent and well-defined formal system for QFT, but it may also not be as *useful* as the kludgey shorthand set of mnemonics we have now.

We can, of course, have formal systems that are less mathematically elegant (for example, using redundant axioms)... my dream is precisely to make such systems are natural as possible to use in practice: make them correspond to intuition.

Once formal systems match the way physicists do their thing (i.e. when formal language is close to natural language), then the machines can be used to support scientists in their reasoning... and maybe eventually take over their job.