
The Unreasonable Effectiveness of Mathematics in the Natural Sciences (1960) - ColinWright
https://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html
======
shas3
A thoughtful counterpoint to Wigner's enthusiasm for Platonistic math is Derek
Abbott's "The Reasonable Ineffectiveness of Mathematics" from IEEE Proceedings
vol. 101, no. 10, 2013
[http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=6...](http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=6600840)

The crux of Abbott's argument is based on Richard Hamming's 1980 critique of
Wigner. It is centered on four propositions, quoting Abbott:

"1) we see what we look for; 2) we select the kind of mathematics we look for;
3) science in fact answers comparatively few problems; and 4) the evolution of
man provided the model."

The first three are fairly straightforward and largely indisputable. The
fourth point is kinda iffy and inexact.

Abbott adds fifth and sixth:

"5) Physical Models as a Compression of Nature (as opposed to exact
representations) 6) Darwinian Struggle for the Survival of Ideas"

(parenthesis mine)

The sixth sounds mystical, but Abbott is really talking about how the 'fittest
ideas survive' and rise to the top, eventually leading to a world where the
mass of mathematical ideas is comprised largely of those that were accurate in
describing natural phenomena, as opposed to those that failed at this. This
reminds me of Brian Greene's tweet today: "How often have you noticed a
coincidence that didn't happen?"

------
tjradcliffe
If math was somehow "deep" in the manner Wigner suggests, you'd expect it
would have a lot fewer rough edges. In particular you'd expect we wouldn't
have to throw away solutions to our differential equations that are perfectly
good mathematically, but unfortunately unphysical:
[http://www.tjradcliffe.com/?p=381](http://www.tjradcliffe.com/?p=381)

The Dirac equation is notable as one of the very, very few mathematical
descriptions of reality that describe only what is, and nothing of what is
not. Pretty much any wave other equation has non-physical (time reversed)
solutions, and the Navier-Stokes equation has mathematically OK solutions (if
you consider the odd singularity "OK") that are unphysical. Even the Dirac
equation has issues with pre-acceleration that are at best physically
ambiguous.

My own take is that math is a language we use to describe reality in a way
that allows us to do precise book-keeping. Like any language, we are able to
express "mimsy were the borogoves" as easily as perfectly meaningful things
like "colourless green ideas sleep furiously" (as in: "Beneath the snows of
winter that blanket the land, colourless green ideas sleep furiously"...
there's actually a minor literature using this phrase in meaningful sentences,
just to prove Chomsky was no poet.)

~~~
dTal
I am not entirely sure I agree with you that reality is the master and
mathematics, the servant - or that it's basically a human construction like
language. For a start, you don't invent new mathematics - you _discover_ it.
You can't discover language. There's a strong sense when you look at things
like the Mandelbrot set that mathematics is in some sense already "out there".
What constrains maths is consistency - it is the study of inevitable
consequences. Assuming that the laws of reality _are_ indeed consistent (see
hackinthebochs's comment for an argument to why they must be), this makes it a
branch of mathematics. At this point considering the branch of mathematics
that we personally can observe somehow more "real", in some vaguely-defined
cosmic sense, is multiplying entities unnecessarily.

I don't think it has "rough edges" either. If your equations don't predict the
world well, it doesn't mean that they are bad mathematics, or that the world
is "wrong". You're just misapplying them.

As an aside, that Chomsky poem is the most beautiful thing I've read all week.

~~~
dllthomas
_" As an aside, that Chomsky poem is the most beautiful thing I've read all
week."_

To clarify (unsure whether or not you understand this):

"Colorless green ideas sleep furiously" was picked by Chomsky as a nonsense
phrase that he asserted had no meaning despite being syntactically correct.
That was an aside from his actual _use_ of the phrase, which was to
demonstrate that there is in fact something that we see as "grammatical" in
even novel sentences that play by the rules - the point of the sentence was
that it had not been uttered before. This was in the space of an argument at
the time of whether we deem things grammatical because they are sensical, or
deem things grammatical because we have encountered them before. I don't know
offhand whether this was used directly as evidence or as pedagogy (certainly,
I first encountered it in pedagogical context) but it makes it pretty clear
that it cannot be the latter and it probably isn't the former, and we're
really recognizing some deeper structure. A lot of this is more obvious to we
who've been playing with compilers for ages - though I am only moderately
confident that this "obvious" result is actually correct.

Coming back... the first half of the quoted sentence is someone else's work,
trying to place Chomsky's "nonsense" sentence in a situation where it has a
meaning. I'm not convinced it was successful - I think it plays with _mood_
rather than _meaning_.

Regardless, that's the context...

~~~
dTal
Thanks - I was aware that the sentence was selected because it was
syntactically valid but semantic nonsense, but not so familiar with the
background debate. For what it's worth, I do think the poem was successful at
extracting meaning from each word, albeit metaphorical. It clearly refers to
hibernating seeds: "Colorless" refers to the embryonic nature of the seeds -
seeds are indeed white or transparent. "Green ideas" are obviously biochemical
ambitions for the future - the seeds do not want to be colorless forever.
"Sleep furiously" \- they may be hibernating, but they are alive and raring to
go!

I was so tickled because even without the background, I thought it was good
enough poetry to stand on its own. For what it's worth, I believe conveying
mood counts as meaning anyway, but in this case I think the poem is
surprisingly coherent and specific.

~~~
dllthomas
Ah, I'd missed that interpretation. Well done.

------
hackinthebochs
I've never found this article persuasive, that one should expect math to not
be effective in such disparate areas. The question is, could it have been any
other way, and what would such a world look like? While its conceivable that
math wouldn't be so effective, such a world would necessarily have different
laws that apply at different scales that are irreducible to more fundamental
laws. The laws of the universe would necessarily be huge, perhaps uncountable.
Considering the extreme end of this spectrum, such a universe would be
_random_ ; that is, each particle and each subset of particles would have
their own individual laws governing their interactions. Order of any kind
would not exist.

Taking the other extreme, where there are a small set of laws that apply to
the fundamental units of the physical world, there is order to the universe
and everything is comprehensible with a finite amount of information. In this
world mathematics is necessarily effective at all scales.

It seems pretty obvious that we exist in a universe with order on most scales,
and so the effectiveness of mathematics is expected.

~~~
pndmnm
_each particle and each subset of particles would have their own individual
laws governing their interactions._

An interesting (philosophical science fiction) read on this topic is Stanislaw
Lem's "The New Cosmogony" (in the book "A Perfect Vacuum").

------
danbruc
I don't follow the last part about insurmountable incompatibilities or
conflicts between physical theories. The universe does not - at least as far
as we can tell - blue screen and present us a speed of light exceeded,
unexpected singularity or unsupported particle interaction error. The laws of
nature seem to work in a flawless and consistent manner.

Maybe we are just to dumb to discover a theory or set of theories that
describe nature in its entirety in a consistent manner using the language of
mathematics. Maybe we are not to dumb, maybe such an theory does not exist
because for some deep reasons there ere just no mathematical expressions
describing (some parts of) nature. Maybe we can only get away with using some
really huge lookup tables to describe nature. Maybe even lookup tables are not
up to the task.

But what seems absolutely impossible to me is that we have different correct
theories, i.e. they describe nature flawlessly where they apply, but they
nonetheless lead to inconsistencies or contradictions.

~~~
DenisM
>Maybe we are not to dumb, maybe such an theory does not exist because for
some deep reasons there ere just no mathematical expressions describing (some
parts of) nature.

I find Godel's incompleteness theorem suggests there are fundamental limits to
"knowledge" as we define it.

[http://en.wikipedia.org/wiki/G%C3%B6del's_incompleteness_the...](http://en.wikipedia.org/wiki/G%C3%B6del's_incompleteness_theorems)

~~~
eli_gottlieb
No, that only puts limits on the provability of particular theorems in formal
proof systems. Inductive reasoners have no such trouble.

~~~
DenisM
I would think I that provability is of critical importance. Could you
elaborate? Is there something I can read on this point of view?

~~~
eli_gottlieb
_Formal_ proof is not actually a part of real epistemology outside of
mathematics.

~~~
DenisM
I see what you mean. I don't feel like I agree but I don't have a position of
my own to retort with, not yet anyway. Thanks for your perspective.

------
leephillips
This version of this famous paper, although bearing all the promise of a PDF,
has horribly messed up typography and is a Unicode train wreck. This HTML
version:

[http://ned.ipac.caltech.edu/level5/March02/Wigner/Wigner.htm...](http://ned.ipac.caltech.edu/level5/March02/Wigner/Wigner.html)

is in much better shape.

------
thisjepisje
Plato.stanford has a very nice article about this as well:

[http://plato.stanford.edu/entries/mathphil-
indis/](http://plato.stanford.edu/entries/mathphil-indis/)

------
Confusion
I'm going to ramble a bit. I haven't thoroughly philosophically investigated
any of this, this is just a condensed version of my train of thought:

There is a very obvious way in which mathematics is related to nature: the
concept of an 'integer' originates directly from nature. The integers
initially allowed our ancestors to count predators, prey, enemies. They did
not originate from some brilliant flash of abstract light. The abstract
definition of the integers is merely an _ad hoc_ construction to make some
things easier. They were already known and used, very effectively, in ancient
times, without that abstraction. They are only known to us, only came into
existence as memes in our memepool, because they are _useful_ for this
purpose.

Integers count multiplicity: how many distinct objects does a group of
objects, considered 'equal' for the purposes for which you are counting them,
contain? Sometimes just knowing they are unequal is enough, making integers
themselves signifiers of a difference. Sometimes you want to know how unequal
they are. Do I need to kill one other bison to have the same amount as we had
yesterday? Twice as many (meaning: the same count once over)?

Everything else is just derivative from those principles: counting things
considered equal and the equality of the countings themselves. Everything else
in mathematics is just convenient representations of Gödel numberings to say
what is equal and how much things aren't.

So _of course_ mathematics applies to nature. Mathematics was born as a way to
abstract over nature, to quantify it.

~~~
yequalsx
I think it is definitely not true that

"<i>Everything else in mathematics is just convenient representations of Gödel
numberings to say what is equal and how much things aren't</i>"

It is true that it appears that non-negative integers are inherent from
nature. Things come in discretized quantities. From non-negative integers one
gets in an isomorphically unique way all integers. From these we get in an
isomorphically unique way the rationals. The completion of the rationals are
the reals. The algebraic closure of the reals are the complex numbers. At each
stage the extension is unique. So in some sense complex numbers are baked into
the universe. What is surprising, at least to some, is that these concepts are
useful in making predictions about the universe and in describing how the
universe works.

~~~
Confusion
Well, suppose the following:

Given a universe finite in extent, with a finite lifetime and a finite mass-
energy content, then for all possible practical purposes only reals up to a
finite precision are needed to calculate and describe observable properties of
the universe. With that constraint, there is e.g. an isomorphism between
integers and all needed reals and as such every observable property in the
universe can be quantified using the integers.

It may seem that that would 'invalidate' a lot of mathematical machinery,
which uses concepts that can not be enumerated by the integers (i.e. the set
of reals), but that need not be the case. The mathematical machinery we have
developed may partly be a very effective 'thought experiment': what if the
reals were uncountable? It turns out that makes things easier to calculate
than to take their countability into account, while giving results
indistinguishable from taking their countability into account.

I lack the mathematical sophistication to prove this is possible. I'm just
proposing a view of mathematics that would make it obvious why it is
effective. I don't believe in miracles or coincidence and I'm not a Platonist
or other sort of idealist, so that doesn't leave me much choice, does it? :)

~~~
yequalsx
Suppose you get into your car and start driving. You drive for 1 hour and
travel 70 miles. At the end of 1 hour your car is at rest since you've reached
your destination.

Now let x be a number less than or equal to 70. Was there a point in time you
were traveling x miles per hour? Is this true for all x in [0, 70]? If so then
there is a need for an uncountable number of reals. If not then which values
of x did you skip? Do these values of x become part of the needed numbers
since they are an excluded set of a numbers of an experiment?

~~~
judk
In the first branch, you are assuming the existence of uncountably continuous
time and space with perfect precision. Scientific observation is not on your
side. (Plancks constant, QM, for example)

For the second, if the universe were somehow rational, there is no need to
drag in irrational numbers to describe it.

~~~
yequalsx
I don't believe I'm assuming anything. I asked specific questions. The answers
to those questions are relevant to what the OP wrote. Your last sentence does
not make sense to me. The universe is not a number and therefore is not a
rational number. It is not rational in the sense of thought either since it
does not think (as far as I can tell). I have not idea what is meant by the
statement, "if the universe were somehow rational...".

------
api
I studied complexity, evolution, emergent behavior, etc. quite a bit, and I
think this is a mirage.

More accurately, it's confirmation bias. Mathematics is only unreasonably
effective at describing the subset of physical systems for which it's
unreasonably effective.

These are systems that are deterministic (or statistically so), linear, and
generally well behaved. Things like the laws of motion, thermodynamics, heat
transfer, and of course basic arithmetic all qualify as well behaved systems
and lend themselves to simple straightforward reductionistic mathematical
reasoning.

These systems however are only a subset of the much larger set of all natural
systems.

This larger set includes living systems, chaotic systems with feedback loops,
and systems like those in QM that exhibit what seems to be non-deterministic
behavior.

Mathematics does have _some_ content around chaotic systems. It's possible to
write equations for systems that exhibit chaotic and emergent behavior. Yet
what you can't do _easily_ with math is write equations that correctly model
real instances of these systems in the same way that you can for, say,
billiard balls on a table or the motion of bodies in the solar system.

You can try, but what you end up with is a system of mathematical
relationships whose complexity approaches that of the source data you are
trying to describe. You do not get the kind of miraculous conceptual "data
compression" you achieve with inorganic physics.

I am not arguing for some kind of supernatural underlying principle or that
these things could never be described mathematically. I'm just saying that
this should be a fertile area for new math. The language as it stands is not
up to the challenge of describing stuff like the N^N^N^N^N^N^N^N^... causal
interaction combinatorics of genetic regulatory networks or the behavior of
real economies at scale.

I'm still very much a fan of Stephen Wolfram's core thesis in A New Kind of
Science. The book as a whole is a mixed bag and I understand why some had a
negative reaction to it, but what he's basically saying if you cut away some
of the hype is that CS may offer new mathematical primitives that can describe
some of these systems. IMHO the biggest problem with the book is the title--
it's not a "new kind of science," just a new domain of math. But Wolfram also
oversells this emerging area of math a little... there is still a _ton_ of
work to be done here. In studying this subject I really got the impression
that there are monumental mathematical (and possibly physical) discoveries
hiding in there.

~~~
thisjepisje
Are you saying that mathematics will never be able to describe certain
systems, or simply that mathematics is not yet complete?

~~~
api
The latter.

Also there is no "complete." See Godel's incompleteness theorem and the
Church-Turing theorem.

I'm also explicitly an anti-platonist, or rather more precisely a "post-
Platonist." The problem with Platonic idealism is that we have found, within
the realm of Platonic forms, theorems that invalidate the central thesis of
Platonic idealism.

------
VLM
"The complex numbers provide a particularly striking example for the
foregoing. Certainly, nothing in our experience suggests the introduction of
these quantities."

Written in an era long before 3d graphics and spatial navigation systems.
Quaternions are just so darn useful that if we stubbornly insisted on not
inventing complex numbers, we'd none the less end up with a magical set of
manipulations that implement them without understanding them, and inevitably
someone would look at those peculiar mechanical arithmetic steps and "invent"
complex numbers.

(edited to add for a good time visit

[http://en.wikipedia.org/wiki/Quaternions_and_spatial_rotatio...](http://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation)

and read all about it)

~~~
theoh
Actually there are alternatives to thinking of them as complex numbers,
specifically geometric algebra. Depending on who you believe, GA may be a
better "design" of math for geometric calculations.
[http://www.mrao.cam.ac.uk/~clifford/introduction/intro/intro...](http://www.mrao.cam.ac.uk/~clifford/introduction/intro/intro.html)

------
quarterwave
I first encountered this wonderful quote in the foreword to one of the
editions of the Feynman Lectures, the foreword in this case being written by
an awesome physics professor in my college (V.Balakrishnan a.k.a Balki): _"...
the combination of physics and mathematics in the Feynman Lectures surely
embodies in an exemplary manner what has been called 'the unreasonable
effectiveness of mathematics in the physical sciences' (Wigner)_"

Balki also introduced us to the mythical student HAROLD - Hypothetical Alert
Reader Of Limitless Dedication, whom he had encountered in Schwinger's
lectures.

------
jordanpg
Love this: "Well, now you are pushing your joke too far," said the classmate,
"surely the population has nothing to do with the circumference of the
circle."

This still fascinates and resonates with me to this day and remains an
impossible barrier for some non-mathematicians to cross.

------
AnimalMuppet
If we are the product of random evolutionary chance, if natural selection
shaped us (which probably means a bias toward fast decisions that are probably
right, rather than for slow decisions that are provably correct[1]), then it's
really hard to see why math (this game we play in our heads) should
necessarily have a deep connection with the rules governing the physical
world.

On the other hand, if there is a God, if God is a He rather than an It, if He
created the world, and if He also created humans in His likeness (sharing
something of His nature), then it becomes much easier to see how there can be
a valid correspondence between how we think in our heads and how the external
universe works.

[1]: Darwinian survival is often a real-time problem: The right answer, too
late, means you still die. A fast answer with a reasonable probability of
being right beats a slow answer that is guaranteed to be wrong simply by being
slow.

~~~
andreasvc
It seems to me you are confusing how humans think and how the physical world
is governed. Humans may make shortcuts that have evolved through evolution,
but the physical laws need not have anything to do with that and may in fact
be better described by abstract mathematics. I do not agree that speculating
about god makes this any easier to reason about.

