The Unreasonable Effectiveness of Mathematics

http://www.dartmouth.edu/~matc/MathDrama/reading/Hamming.html

by Arun2009 • 16 years ago

59 42 16 years ago

Herring5• 16 years ago

Next up: Unreasonable effectiveness of words in communication. We keep using the same words in different sentences, & they keep making sense.

Groxx• 16 years ago

To get a similar result, it'd be as if "Colorless green ideas sleep furiously" turned out to accurately describe ideas, without modifying the terms.

i.e., if it turns out ideas and "green" operate in the same locations in the brain, and if you exhibit anger + sleep patterns of thought while brainstorming.

Herring5• 16 years ago

Hm, it's not productive to argue over analogies. They're just used to illustrate a point. I was saying math was explicitly designed to work & to be general. Likewise words were designed for communication (not just for describing the real world), & I'd say that sentence communicates your meaning very well.

eg later he says:

>"Is it not remarkable that 6 sheep plus 7 sheep make 13 sheep; that 6 stones plus 7 stones make 13 stones? Is it not a miracle that the universe is so constructed that such a simple abstraction as a number is possible? To me this is one of the strongest examples of the unreasonable effectiveness of mathematics. Indeed, l find it both strange and unexplainable. "

That makes absolutely no sense. You might as well wonder why the word "wheel" describes wheels. Or wonder why wheels exist. These are empirical facts.

wazoox4• 16 years ago

Did you read Pinker, Chomsky or both ? :)

rayder• 16 years ago

The discussion of beauty in mathematics resonates deeply with me and I seem to always relate such discussions with music. As a musician turned math enthusiast, I think the thought processes involved in the creation of music parallel those involved in the creation of proofs quite nicely. My obsession with music seems to complement my obsession with mathematics on some level that I cannot quite define. As far as I know, there is not a distinctly important practical relationship between math and music (please correct me if I am wrong), but I still feel as though there is some connection. I do not know if this is a personal thing or not, but I cannot stop relating the two!

ComputerGuru3• 16 years ago

old proofs of theorems may become false proofs. The old proofs no longer cover the newly defined things. The miracle is that almost always the theorems are still true; it is merely a matter of fixing up the proofs. It is claimed that an ex-editor of Mathematical Reviews once said that over half of the new theorems published these days are essentially true though the published proofs are false.

I'm only around 55% of the way through (according to my scrollbar --- and thank you Readability!) but it's an incredibly interesting read. It's very lengthy, but I really advise taking the time to look it over. It really tickles those brain cells. Thus far, I'm not seeing anything "new" in this article, but seeing all these incredible things expressed and summarized up close is amazing.

jonsen3• 16 years ago

(Safari Reader doesn't show the full article)

ComputerGuru3• 16 years ago

File it as a bug. It's in the menu and only a click away.

saint-loup• 16 years ago

The original article by Wigner: http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html

snippyhollow• 16 years ago

There is also, more recent, the unreasonable effectiveness of data: http://www.computer.org/portal/web/csdl/doi/10.1109/MIS.2009...

pmichaud• 16 years ago

This guy could easily cut 50% of his words and lose no meaning. He needs an editor.

sb• 16 years ago

Richard Hamming needs an editor -- that's a good one...

PS: http://en.wikipedia.org/wiki/Richard_Hamming

carterschonwald• 16 years ago

for those who haven't read it, the transcript of his "you and your research" talk is worth repeatedly reading.

jonsen3• 16 years ago

I hope that I have shown that mathematics is not the thing it is often assumed to be, that mathematics is constantly changing and hence even if I did succeed in defining it today the definition would not be appropriate tomorrow.

I think this kind of intent requires some seemingly superfluous amount of wording to carry a nontrivial flow of narrative to be persuasive if to avoid dullness in expression.

dspeyer• 16 years ago

So after all that verbiage, he concludes:

Some math was designed to be useful.

Science is by definition those practical problems to which math can be applied.

He still doesn't know.

Very disappointing.

ComputerGuru3• 16 years ago

Sometimes it's important to realize just how little we know and clearly define it. Knowing that you don't know, knowing why you should, and understanding the inherent difficulties standing in the way of an explanation is 99% of getting there.

asdflkj4• 16 years ago

Mathematics is a product of the mind; the mind is a product of evolution; evolution is a product of natural laws. If you want answers to these questions, it helps to look at what we are, exactly, and how we came to be.

In fact "simple" mathematics are not simple at all by any objective measure. Starting with any truly formal system, you need a stupendous number of deductions to get to things like elementary laws of arithmetic, or basic plane geometry. Mathematical proofs are not formal proofs--they are instructions for our brains. Evolution made the relevant parts of our brains the same, so same instructions lead to same results. That's why there's never any argument over whether a proof is correct, once a few people got to study it in detail. This also explains Hamming's observation that when proofs turn out to be "wrong" after math has evolved a bit, theorems are still usually correct. We find a new, better route to the same place in our brain, and recognize the hazards of the old route, now deprecated.

Okay, here is the key bit: if evolution made the relevant parts of our brains the same, that means it has arrived at a maximum, or at least a local maximum. What is the nature of this maximum? Physiologically, there are constraints on the amount of brain circuity our body can maintain. Brains consume a lot of energy, take up space, etc. So naturally, evolution ended up with a design where the same circuity can serve the greatest possible number of functions.

Of course, evolution only concerns itself with those functions relevant to our survival and reproduction. But there is nothing niche about those goals. If some general pattern occurs often in our quest for survival, then it likely occurs often in other quests that evolution never knew about--like building airplanes.

calcnerd2563• 16 years ago

Mathematics is not a product of the mind any more than physics is. All theorems are true (or, more precisely, all theorems follow from their axioms), even the ones we haven't discovered yet. That a mind can choose an axiomatic system to explore does not mean the relationships between those axioms and their theorems are created by that mind.

asdflkj4• 16 years ago

Mathematics and physics are products of the mind, obviously. Mathematicians and physicists do their work by using their minds. They don't channel some divine truth--they merely filter what their mind makes through certain criteria. I don't understand this common tendency, exemplified by your comment, to shift attention away from how mind makes things, to the criteria according to which we filter them before we call them "science" or "mathematics". I've studied mathematical logic, and it has been of little use to me in mathematics. Philosophy of mathematics has been of no use at all. The practice of mathematics can get by perfectly well without that stuff. So maybe it's time to put aside the mysticism, and start looking at how our brains actually make what they make. Especially since we are just now starting to understand what brains are and how they've evolved.

jayruy• 16 years ago

Whereof one cannot speak, thereof one must be silent. -Wittgenstein

10ren• 16 years ago

I think our sense of geometry comes from hunting (not body decoration).

It's a little bit like mathematicians invent little "chains of reasoning" rather than "mathematics", and that these chains are interesting and useful; even if their original assumptions turns out to be incorrect, the reasoning is still valid. In the marketplace/ecosystem of mathematics, people then choose the ones that they find most useful and/or interesting.

I love the thought that when we meet aliens, they have utterly different mathematics from us, so it reveals how parochial our particular toolbox is. This has actually happened, in a sense, with Chinese mathematics. Apparently, their approach to "proof" was algorithmic rather than declarative - not just a different toolbox, but a different kind of toolbox.

zeynel1• 16 years ago

" Is it not remarkable that 6 sheep plus 7 sheep make 13 sheep; that 6 stones plus 7 stones make 13 stones? Is it not a miracle that the universe is so constructed that such a simple abstraction as a number is possible? To me this is one of the strongest examples of the unreasonable effectiveness of mathematics. Indeed, l find it both strange and unexplainable."

this is confusing because what he is talking about is -counting- not mathematics -mathematics- is an academic field that may include -counting- as one of its areas of study -but- it is confusing to reduce mathematics to counting

The Unreasonable Effectiveness of Mathematics

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