

The Unreasonable Effectiveness of Mathematics - Arun2009
http://www.dartmouth.edu/~matc/MathDrama/reading/Hamming.html

======
ComputerGuru
_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.

~~~
jonsen
(Safari Reader doesn't show the full article)

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

~~~
jonsen
I did. Should have mentioned that, sorry.

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10ren
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.

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rayder
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!

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

~~~
Groxx
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.

~~~
Herring
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.

~~~
nandemo
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.

~~~
brg
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.

------
asdflkj
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.

~~~
calcnerd256
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.

~~~
asdflkj
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.

~~~
calcnerd256
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.

~~~
asdflkj
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.

~~~
calcnerd256
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."

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

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

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

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

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saint-loup
The original article by Wigner:
<http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html>

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snippyhollow
There is also, more recent, the unreasonable effectiveness of data:
[http://www.computer.org/portal/web/csdl/doi/10.1109/MIS.2009...](http://www.computer.org/portal/web/csdl/doi/10.1109/MIS.2009.36)

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dspeyer
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.

~~~
ComputerGuru
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.

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

------
zeynel1
" 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

