quotes: Babylonian vs Euclidean method, why physicists dread precision

[gusl]

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 connections in between, but you have never fully realized that it could all come up from a bunch of axioms... Even in mathematics you can start in different places... In physics we need the Babylonian method, and not the Euclidian or Greek method.
— Richard Feynman

The physicist rightly dreads precise argument, since an argument which is only convincing if precise loses all its force if the assumptions upon which it is based are slightly changed, while an argument which is convincing though imprecise may well be stable under small perturbations of its underlying axioms.
— Jacob Schwartz, "The Pernicious Influence of Mathematics on Science", 1960, reprinted in Kac, Rota, Schwartz, Discrete Thoughts, 1992.

When I say I'm an advocate of formalization, I'm not saying we need to understand all the precise details of what we're arguing for (although this usually is the case in mathematics, at least more so than in physics). What I want to do is to formalize the partial structure that does exist in these vague ideas. Favoring a dynamic approach, I hold that we must accept formalizing theories in small steps, each adding more structure. We will need "stubs", and multiple, parallel stories to slowly evolve into a formal form. The point is that a vague, general idea *can* be formalized to a point: this is evidenced by the fact that we humans use precise reasoning when talking about them.
Again, the idea is about doing what humans do, formally. If the humans' idea is irremediably vague, we don't hope to do any better, but we do hope to formalize it as far as the ideas are thought out / understood (even if vaguely). To the extent that there exists a systematic (not necessarily "logical", but necessarily normative) in the way we reason and argue, it will be my goal to formalize this in a concrete form.

Regarding the normative aspect, the reason we need one is: not all ideas make sense! For fully-formalized mathematics (i.e. vagueness-free mathematics), it's easy to come up with a normative criterion: a mathematical idea or argument is fully-formalized if it corresponds to a fully-formal definition or a fully-formal proof. One of the challenges of this broader approach is to define what it means for an idea to "make sense": what does it attempt to do? What is its relation with related concepts?

The "natural" medium expression of these ideas is English. The idea is to connect English words to concepts in the formal knowledge system. We say an English sentence makes sense in a given context iff it addresses the goal / there is sound reasoning behind it (not all may be applicable).

Comments

[jcreed.livejournal.com]

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.livejournal.com]

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.livejournal.com]

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.