The Unreasonable Effectiveness of Mathematics(dartmouth.edu)
dartmouth.edu
The Unreasonable Effectiveness of Mathematics
http://www.dartmouth.edu/~matc/MathDrama/reading/Hamming.html
43 comments
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File it as a bug. It's in the menu and only a click away.
I did. Should have mentioned that, sorry.
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.
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.
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!
Next up: Unreasonable effectiveness of words in communication. We keep using the same words in different sentences, & they keep making sense.
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.
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.
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.
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.
It's right there on the top: "It is evident from the title that this is a philosophical discussion". In case you aren't interested in philosophy, I don't see why bother reading the article and commenting on it -- unless perhaps you want to say something against doing any philosophy in the first place.
But if you are interested in philosophy, then you should know that the philosopher often starts by wondering about something that most sane people take for granted. Case in point, some people do wonder why the word "wheel" describes wheels. There are tons of papers and books about philosophy of language.
Others ask why wheels exist, what is a wheel, or whether do they really exist at all. You can only be sure that the existence of wheels is an "empirical fact" after you have examined these questions. After all, "empirical fact" is a philosophical term.
Incidentally, "6 sheep + 7 sheep = 13 sheep" is not an empirical fact.
But if you are interested in philosophy, then you should know that the philosopher often starts by wondering about something that most sane people take for granted. Case in point, some people do wonder why the word "wheel" describes wheels. There are tons of papers and books about philosophy of language.
Others ask why wheels exist, what is a wheel, or whether do they really exist at all. You can only be sure that the existence of wheels is an "empirical fact" after you have examined these questions. After all, "empirical fact" is a philosophical term.
Incidentally, "6 sheep + 7 sheep = 13 sheep" is not an empirical fact.
Prefacing an article with "this is a philosophical discussion" is not an excuse to stumble through nonsense like a drunken sailor staggering home. Similarly if I preface this post with "I don't mean to be rude" only to continue to slander and defame, I've not excused my actions.
Counting is defined by objects, and 13 is the sum of 6 and 7. Indeed it is an empirical fact in as much as it has a physical interpretation.
Counting is defined by objects, and 13 is the sum of 6 and 7. Indeed it is an empirical fact in as much as it has a physical interpretation.
> Is it not remarkable that 6 sheep plus 7 sheep make 13 sheep; that 6 stones plus 7 stones make 13 stones?
Personally, I find it rather remarkable that a pattern observed from pebbles, sheep or apples (i.e., 6 + 7 = 13) should hold for any set of discrete objects anywhere in the universe. It could well have been that, say, 1 apple + 1 apple = 2.5 apples, but 1 kitten + 1 kitten = 1.5 kittens.
Personally, I find it rather remarkable that a pattern observed from pebbles, sheep or apples (i.e., 6 + 7 = 13) should hold for any set of discrete objects anywhere in the universe. It could well have been that, say, 1 apple + 1 apple = 2.5 apples, but 1 kitten + 1 kitten = 1.5 kittens.
Not really, because having one something plus one something equal two somethings is pretty much the definition of discrete object.
Yes, I agree and think this is a very smart observation. It doesn't hold for jugs of water, say, if you're allowed to pour water between the jugs.
It doesn't hold for loads of things. 1 heap + 1 heap = 1 heap. 1 bunny + 1 bunny = 12 bunnies, depending on how fast you are with your observing. 1 dl of ethanol + 10 dl of water < 11 dl of diluted alcohol.
But "heaps" and "dl of fluid" are not discrete objects, which was precisely my point.
Nor for sheep if you mix the sexes and leave them alone for a bit.
Cool article, though.
Cool article, though.
Mathematicians still are split into formalists, realists, intuitionists and logicists, debating around the nature of mathematical truth; i. e. does mathematical truth pre-exists (and is just "discovered") or is it a human invention? Good luck sorting this one out!
Well that's what comes of not taking a scientific approach with hypotheses. The invention of math is out there for anyone to see.
Interesting. So are you so down-to-earth that you consider philosophical speculation on the nature of mathematics (or anything else, I suppose) to be nonsensical?
Don't make this about me. Read the history of math & tell me what conclusion you come to when you see people making algorithms, essentially, for dealing with natural processes.
Of course, as long as you stay in the realm of measurable things. However, we still need to explain how and why the perfect abstract mathematical objects are of any use in the real world (as opposed to other abstract objects such as patonician ideas, or more commonly gods).
>(as opposed to other abstract objects such as patonician ideas, or more commonly gods)
Well neither of those two deal with counting... We already know (historically, empirically) that counting & manipulating 'stuff' is what works for making applicable theories. Mathematical abstraction preserves those "traits", & makes the object more general - ie more flexible. Chess for example is about counting, but it's not explicitly written in a form that allows you to drop it in a theory.
I think the important thing is the traits aren't arbitrary. They were forced on people, eg you need to learn counting if you want to keep track of your goats.
Well neither of those two deal with counting... We already know (historically, empirically) that counting & manipulating 'stuff' is what works for making applicable theories. Mathematical abstraction preserves those "traits", & makes the object more general - ie more flexible. Chess for example is about counting, but it's not explicitly written in a form that allows you to drop it in a theory.
I think the important thing is the traits aren't arbitrary. They were forced on people, eg you need to learn counting if you want to keep track of your goats.
Did you read Pinker, Chomsky or both ? :)
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.
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.
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.
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.
I'm not talking about how we build them. I'm talking about what they are, and what they are is as they would be whether they were built by humans or computers or nature. That the theorems follow from their axioms is not a human invention, nor could it be. There is a difference between discovering something and inventing it. Man could not invent mathematics any more than man could invent electricity. When I say physics exists, I mean that the physical world exists and follows rules. If we discover those rules, it does not mean we have invented them.
You're "not talking about how we build them", but you were responding to my post where I am talking about how we build them. I'm gonna say it again: I don't understand this tendency to shift attention away from the "how"--rather insistently, in your case.
Either you're not just talking about how we build them or I misunderstand you, because the first thing you said was, "Mathematics is a product of the mind."
On philosophy of mathematics, have you checked out Reuben Hersh? Or Lakatos? They're much more interesting to me than the usual platonism/formalism.
I'm guessing platonism/formalism were popular in arguing against other ways of understanding the world, like folk science, authoritarianism and mysticism. (I'm not equating the last three.) Maybe also as a foundation myth for professional mathematics.
I'm guessing platonism/formalism were popular in arguing against other ways of understanding the world, like folk science, authoritarianism and mysticism. (I'm not equating the last three.) Maybe also as a foundation myth for professional mathematics.
Thanks--I'll check them out. I hope it's the kind of stuff that is only called philosophy, but actually falls under the purview of science, which doesn't yet have a framework to deal with it. That's the only kind of "philosophy" I've ever found of value.
This guy could easily cut 50% of his words and lose no meaning. He needs an editor.
Richard Hamming needs an editor -- that's a good one...
PS: http://en.wikipedia.org/wiki/Richard_Hamming
PS: http://en.wikipedia.org/wiki/Richard_Hamming
for those who haven't read it, the transcript of his "you and your research" talk is worth repeatedly reading.
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.
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.
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The original article by Wigner: http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html
There is also, more recent, the unreasonable effectiveness of data: http://www.computer.org/portal/web/csdl/doi/10.1109/MIS.2009...
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.
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.
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.
Whereof one cannot speak, thereof one must be silent.
-Wittgenstein
" 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
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
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.