I like this response.

I have no time to craft a nuanced reply so let me just take a shortcut and concede: If you can get the student from "here are a bunch of physical concepts" to "here is a mathematical object, with interesting abstract properties that you can reason about" without introducing formalism you'll have succeeded in teaching math. Excellent.

Is this plan really going to work very often? It is easy to say that you can derive, say, the utility of "zero" without ever doing any arithmetic -- just as it is easy to say that you can be a full-fledged computer scientist without ever touching a computer -- but in practice?

It's true that the presence of a supercomputer in everyone's pocket will change this argument significantly. But simulations go only so far. They, too, are only metaphors, and if you don't know enough to tinker under their covers they are rather inflexible metaphors. Your classical mechanics simulator is not going to discover quantum mechanics for you.

In my experience learning about mathematical abstractions requires all of the above tools -- you tinker with the formalism, you ponder the physical analogies, you draw mental pictures of clouds and colors, you play with a simulator, you build some circuits in the lab, you go for a walk, you tinker with the formalism again, and six years later you finally get it.

And I like this one!

I agree completely that the process of learning mathematics is probably highly multidimensional for... pretty much everyone ever. In particular, it's easy to see how formal descriptions can push mathematical generalization forward far before we have a suitable concept of the mathematical object we're describing.

I think we're all (incl. the op) in some kind of agreement here about the didacticism of math. The op didn't disregard the power and utility of mathematical languages — he came from being trained pretty heavily in engineering math, at least up to playing with higher-order differential equations — but instead was, perhaps not directly, arguing for increased metaphorical/physical descriptions in taught mathematics. He's just responding to the rather eye-opening feeling one gets when one starts to realize that math is so interpretable!

I think that's a perfectly fair argument to have. I know that in my own experience, I never understood the joy of math until the day that linear algebra took an interpretation as linear space transformation.

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So we're just sort of all oscillating here in strong rebuttals of whatever interpretation of the "heart" of math the prior author champions for a while. Which is fun but unproductive.

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I like the challenge of taking someone from physical concepts and metaphors directly to a mathematical object. I think it'd be possible, and maybe even useful when someone first starts to learn real math, but certainly it's not the most efficient way to become well-read. It'd be a lot like explaining the meaning of, I dunno, Día de los Muertos without immersing someone in Mexican language and culture. A single point of contact can be forced, but you lose so much context and fluency.

I would be more comfortable with the original author's goals if he wanted to augment conventional math education with simulation. Perhaps "augment" is too weak a word, perhaps they should be done equally. But I certainly don't want it to replace manipulating abstract symbols. I think that understanding one helps understanding the other, but both are much less meaningful in isolation. In particular, I think that symbolic manipulation is important because it is both simplified and precise; simulation and visualization is important because it appeals to our intuition.

Are you sure you're accurately describing Godel's Incompleteness theorem? It's been a while, but I understood it at a technical level briefly. I'd be interested in knowing more about how you think it applies here.

No, I'm not. I'm interpreting it pretty loosely. Sorry about that. Correct me if I'm wrong here, though.

More directly, what I meant to say is that since there exist true theorems which cannot be proven within any particular choice of mathematical formalism, we need to operate with tools beyond simply symbolic manipulation. That was the death knell of Hilbert's Program and solidly separated the formal specification of math from "that thing which we're studying".

I think you are wrong - even though there are true, unprovable statements, they're unusual and to the best of my knowledge, not the subject of extensive study. More like curiosities (http://en.wikipedia.org/wiki/List_of_statements_undecidable_...). Even if that were not the case, there is no alternate approach to studying them.

I'm not really sure that you mean about the "formal specification" for math vs. the "that thing we're studying". An informal (ie, not expressed in ZFC + 1rst order predicate calculus) proof of something non-trivial can go on for dozens, if not hundreds of dense pages of symbols. If I recall correctly, Whitehead's Principae Mathematica derived arithmetic from ZCF and predicate calculus, and it took the whole book.

I did a little reading to refresh myself on the subject, and this stood out as a good summary of the topic:

"In a sense, the crisis has not been resolved, but faded away: most mathematicians either do not work from axiomatic systems, or if they do, do not doubt the consistency of ZFC, generally their preferred axiomatic system. In most of mathematics as it is practiced, the various logical paradoxes never played a role anyway, and in those branches in which they do (such as logic and category theory), they may be avoided."

(http://en.wikipedia.org/wiki/Foundations_of_mathematics#Foun...)

Your picture of mathematics is quite a bit more robust, so I won't deny that what you wrote here is more accurate than my "Gödel say it won't work, woe is our field!" brimstone version. So I'll explain my reasoning for invoking Gödel.

I mostly wanted to walk around the historical event I mentioned, the breaking of the Hilbert Program. At the time, it seemed that formal specification of math would provide a complete picture of what math was! Once the Program was finished then the job of mathematician would eke out into "computer" (of the abacus sort) or into other fields which interpreted the canon.

I'm not sure which death stroke was stronger, the incredible opaqueness and complexity of proof systems like ZFC or Gödel just saying what he was trying was outright impossible, but Hilbert's Program was killed before it even seriously took off, leaving the study of mathematics and the practical formalisms we use to study it pretty ad-hoc instead of grand and unified.

I'm unifying that with the fact that the way math seems to be practiced never comes from the formal language but instead first comes from imagining some kind of "mathematical object" and then taming its behavior with formalisms. You could consider them to be one and the same and argue that the difference is highly philosophical, and then this is where I'd invoke Gödel and inform you that there definitely exist things we could benefit from reasoning about that your formal language would fail to describe. This existence proof separates the classes of true things and provable things and makes their distinction more than philosophical.

Now, talking about what a "mathematical object" is gets you to the bleeding heart of the philosophy of science and epistemology. It's a tough question!

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As a final note, ZFC is ZF + Axiom of Choice... which, yes, most practicing mathematicians just accept AoC so that they can integrate or whatever. The formal world without AoC is very sparse, but nobody has any sort of idea what the arbitrary decision means. I know that there has been some significant study of ZF-C, though it's been "impractical", I don't know if anyone is willing or capable of stating that ZF-C is in any way worse than ZFC. Impractical is a Mathematicians favorite adjective, so they're just two extant formal systems which disagree quite a lot on important things but we mostly pay attention to ZFC.

He's not.

roughly correct insofar as there are undecidable mathematical statements. I don't understand though, how "we can study, meaningfully" these objects that our formal systems cannot compute. These statements do exist, and we can (interestingly) express them in language and understand them (as a gut feeling), but all we can do is stare awkwardly at them, undecidedly. imvvho, we cannot claim that metaphors (or language) are more powerful than formal systems by that.

I didn't mean to imply that one is stronger than the other. Instead that if they're well-designed, then they overlap (largely). Where they don't overlap is highly interesting.