
The Unreasonable Effectiveness of Mathematics in the Natural Sciences (1960) - mutor
https://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html
======
ethn
Despite the author's argument, for me at least, there are an order of a priori
"hints" which argue why Mathematics ought to be seen in the natural sciences.

In the beginning of mathematics, it was likely, if not definitively created to
enumerate similar objects in a particular given set. In order to enumerate an
object two things must happen. The object must be turned into an object: this
means it must be reduced into some abstract concept which holds certain
properties to belong in a set. Second, it must be mapped with a further
abstract ID, the number. The most significant consequence of this is that the
enumerated ID's and objects now hold the property of logic, because in our
world all objects seem to exhibit properties of logic (most of the time). This
would have begun as simple arithmetic, an early model of a useful description
of our world. From this it becomes evident that you can now map any integer to
objects, and perform operations on those integers which map onto the real
world as if you were performing the operation on those very real objects.

Later, mathematicians would then become less concerned with the objects
mathematics served to represent but more on the relationships in between them.
In the logical manipulations of these objects, mathematicians began really
exploring the very logical relationships between objects (the creation of
variables would coincide here over strict numbers). They once represented an
object, yet still can be used again to represent an object. It follows, that
these objects should now follow these logical relationships.

From these concerns, I reduce mathematics further as a nice description of
logical relationships; a derivative of our own model of the universe.

~~~
akud
> because in our world all objects seem to exhibit properties of logic

This is the thing that needs to be explained. I get that math is just logic,
but why should we expect the natural world to follow logical rules?

~~~
eli_gottlieb
What sort of logical rules would we even consider logical if they didn't
manage to explain anything about the natural world?

~~~
ethn
Exactly.

>I reduce mathematics further as a nice description of logical relationships;
a derivative of our own model of the universe.

The rules of logic are the behaviors of objects that we experience and
observe. In a differing reality, objects may have properties which could be
illogical to the perspective of someone in our own; yet if we move to that
strange reality, our brains would slowly process the information into a new
model of reality with the ~new logic~. However that ability is mostly
pointless as any group of mostly consistent behaviors will have either more
but never differing logical relationships as any other reality with mostly
consistent behavior (this is also why we see something called duality in
mathematics).

The further consequence of this, in our studies of natural sciences, we hold
our brain's model of reality as supreme. We don't consider something to be
~true~ until at least its conclusions can be witnessed and perceived. The
consequence and perhaps also the cause of this, is all of mathematics and
natural sciences are mostly attempts to formalize as well as quantify our own
brain's unified model of reality.

------
dekhn
It's worth re-reading this a few times to understand what Wigner was actually
getting at:

He's specifically saying we have no a priori reason to expect that mathematics
should have predictive ability in the real world, but it seems to do so (I'm
not sure I buy this), and we should be thankful.

~~~
psyc
I can readily understand the perspective, at least as far as the a priori
reason being hard to discern or articulate. But I don't hold the perspective.
Like Tegmark, I believe the reason math works is that the universe is math.
(And it follows that mathematicians who believe their work is unconnected in
principle from the physical world, are being myopic or escapist.)

Working on a rather advanced procedurally generated game world for a long time
has only strengthened this belief. The world looks quite natural, but it is
literally a visualization of an ordinary integer hash function being computed
many millions of times. Exploring the Mandelbrot set intensely, has the same
effect on me, and that math is much simpler. I'm not trying to assume these
analogies are tautologies, but when I'm inside the game world and it looks
both totally natural, and also exactly like my hash function, all I can say is
it's very suggestive.

------
shas3
A counterpoint to this is Derek Abbott's "The Reasomable Ineffectiveness of
Mathematics." Personally, I think domains with low-N (dimensions and sample
sizes) and high complexity remain beyond the optimistic expectations of Eugene
Wigner.
[https://pdfs.semanticscholar.org/462d/7b6b1ee8243b6aa8897be3...](https://pdfs.semanticscholar.org/462d/7b6b1ee8243b6aa8897be3cf306239fb43c6.pdf)

~~~
discreteevent
Also: A naturalist account of the limited, and hence reasonable, effectiveness
of mathematics in physics - Lee Smolin

[https://arxiv.org/abs/1506.03733](https://arxiv.org/abs/1506.03733)

------
imh
>It is true, of course, that physics chooses certain mathematical concepts for
the formulation of the laws of nature, and surely only a fraction of all
mathematical concepts is used in physics. It is true also that the concepts
which were chosen were not selected arbitrarily from a listing of mathematical
terms but were developed, in many if not most cases, independently by the
physicist and recognized then as having been conceived before by the
mathematician. It is not true, however, as is so often stated, that this had
to happen because mathematics uses the simplest possible concepts and these
were bound to occur in any formalism.

That last point is unconvincing. Maybe they aren't the simplest possible
concepts, but it surely must be the case that had we evolved in any universe,
we would have sought to understand it and would have formalized that
understanding. That's not all that math is, but it has led to several popular
branches of it. Surprising? Unreasonable? Not at all.

------
thunk1xxx
No mention of Frege here, and only a cursory quote from Russell. While I can't
comment on the author's perspective based on this article alone, it is
consistently shocking to me how little scientists and mathematicians are
interested in in reading and engaging with philosophy related to their fields.

~~~
Smaug123
I, for one, would be happy to read mathematical philosophy. The problem is
that there is so _much_ of it, and lots of it contradicts lots of other bits
of it. As a mathematician attempting to find out about the philosophy of
mathematics, I want a source of truth and correctness. In order to find it
(assuming it even exists), I also have to wade through reams of falsity.

The additional problem is that philosophy, like any field, has a steep
learning curve. It takes a _lot_ of effort to go from "I want to learn about
the philosophy of my field" to "I know a little bit about the philosophy of my
field".

~~~
black_knight
There is no consensus as to what is the true and correct philosophy of
mathematics. People are going to disagree about that, just like they disagree
about almost every other part of philosophy. But there are books which give
you a survey of different views, such as Shapiro’s “Thinking about
mathematics” — which surely will bring you to “I know a bit about the
philosophy of mathematics”-level.

During my time as a master student in mathematics we used to have a joint
seminar with students from philosophy where we would take turns reading and
presenting topics from philosophy of mathematics. I think it was useful for
everyone, and it wasn’t difficult to organise and didn’t take much time (2
hours per week during one term).

~~~
Smaug123
> There is no consensus as to what is the true and correct philosophy of
> mathematics.

This is part of the problem. I start reading; I see something "obviously
false" which is endorsed by lots of people; I see something "obviously true"
which is endorsed by lots of people but is contradictory to the obviously-
false thing; I see a case which contradicts both of those and isn't obviously
true or obviously false; and I give the whole thing up as non-rigorous and/or
tedious. But I'll look out Shapiro and see if it helps.

------
im3w1l
Well before we tried explaining the world with mathematics - the study of
riddles and games and money.

Before we tried that, we tried explaining the world with gods and spirits -
the study of the _personalities_ of the forces of nature.

I just like to keep that in mind, that mathematics was our second attempt.

------
marcosdumay
I dunno. We have spent a couple of millennia trying things and taking whatever
is effective. It may be unreasonable that something is effective, but I can't
convince myself it is so.

------
EpiMath
For a different take on this, see Dan Shanks' "The case for pythagoreanism"
starting on pg. 130 in his classic text Solved and Unsolved Problems in Number
Theory:

[https://archive.org/stream/SolvedAndUnsolvedProblemsInNumber...](https://archive.org/stream/SolvedAndUnsolvedProblemsInNumberTheory/Shanks-
SolvedAndUnsolvedProblemsInNumberTheory#page/n139/mode/2up)

------
nonbel
"Math" is just a language designed for deductive logic. Lots of science is now
being done in other (programming) languages that can fill the same role. Is
"math" still special?

~~~
goatlover
The physicist Max Tegemark claims that the only real properties are
mathematical properties. Notice how much physics relies on math. What are the
properties of an electron? An electron is understood mathematically.

~~~
aswanson
Yeah, mathematics seems rather meta. If the universe/multiverse we live in
turns out to be a simulation, if every scientific discovery turns out to be
incorrect, etc....set theory remains. There is something incredibly powerful
and beautiful about that.

------
SonOfLilit
I highly recommend Hamming's response, which was very influential to the way I
think about mathematics and physics:

[https://www.dartmouth.edu/~matc/MathDrama/reading/Hamming.ht...](https://www.dartmouth.edu/~matc/MathDrama/reading/Hamming.html)

Edit: a teaser:

"Let us next consider Galileo. Not too long ago I was trying to put myself in
Galileo's shoes, as it were, so that I might feel how he came to discover the
law of falling bodies. I try to do this kind of thing so that I can learn to
think like the masters did-I deliberately try to think as they might have
done.

Well, Galileo was a well-educated man and a master of scholastic arguments. He
well knew how to argue the number of angels on the head of a pin, how to argue
both sides of any question. He was trained in these arts far better than any
of us these days. I picture him sitting one day with a light and a heavy ball,
one in each hand, and tossing them gently. He says, hefting them, "It is
obvious to anyone that heavy objects fall faster than light ones-and, anyway,
Aristotle says so." "But suppose," he says to himself, having that kind of a
mind, "that in falling the body broke into two pieces. Of course the two
pieces would immediately slow down to their appropriate speeds. But suppose
further that one piece happened to touch the other one. Would they now be one
piece and both speed up? Suppose I tied the two pieces together. How tightly
must I do it to make them one piece? A light string? A rope? Glue? When are
two pieces one?"

The more he thought about it-and the more you think about it-the more
unreasonable becomes the question of when two bodies are one. There is simply
no reasonable answer to the question of how a body knows how heavy it is-if it
is one piece, or two, or many. Since falling bodies do something, the only
possible thing is that they all fall at the same speed-unless interfered with
by other forces. There's nothing else they can do. He may have later made some
experiments, but I strongly suspect that something like what I imagined
actually happened. I later found a similar story in a book by Polya [7. G.
Polya, Mathematical Methods in Science, MAA, 1963, pp. 83-85.]. Galileo found
his law not by experimenting but by simple, plain thinking, by scholastic
reasoning.

I know that the textbooks often present the falling body law as an
experimental observation; I am claiming that it is a logical law, a
consequence of how we tend to think."

[..]

"But if you do not like these two examples, let me turn to the most highly
touted law of recent times, the uncertainty principle. It happens that
recently I became involved in writing a book on Digital Filters [8. R. W.
Hamming, Digital Filters, Prentice-Hall, Englewood Cliffs, NJ., 1977.] when I
knew very little about the topic. As a result I early asked the question, "Why
should I do all the analysis in terms of Fourier integrals? Why are they the
natural tools for the problem?" I soon found out, as many of you already know,
that the eigenfunctions of translation are the complex exponentials. If you
want time invariance, and certainly physicists and engineers do (so that an
experiment done today or tomorrow will give the same results), then you are
led to these functions. Similarly, if you believe in linearity then they are
again the eigenfunctions. In quantum mechanics the quantum states are
absolutely additive; they are not just a convenient linear approximation. Thus
the trigonometric functions are the eigenfunctions one needs in both digital
filter theory and quantum mechanics, to name but two places.

Now when you use these eigenfunctions you are naturally led to representing
various functions, first as a countable number and then as a non-countable
number of them-namely, the Fourier series and the Fourier integral. Well, it
is a theorem in the theory of Fourier integrals that the variability of the
function multiplied by the variability of its transform exceeds a fixed
constant, in one notation l/2pi. This says to me that in any linear, time
invariant system you must find an uncertainty principle."

------
vorg
> We now have, in physics, two theories of great power and interest: the
> theory of quantum phenomena and the theory of relativity. [...] The two
> theories operate with different mathematical concepts -- the four
> dimensional Riemann space and the infinite dimensional Hilbert space. So
> far, the two theories could not be united, that is, no mathematical
> formulation exists to which both of these theories are approximations. All
> physicists believe that a union of the two theories is inherently possible
> and that we shall find it. Nevertheless, it is possible also to imagine that
> no union of the two theories can be found.

Yes! Perhaps there's a meta-law which says any Universe in which Consciousness
exists can not be fully explained by one single mathematical formulation.
After all, Math ultimately originates in the conscious Mind. Perhaps only a
Universe that requires two or more distinct Mathematical models to explain it
(for us, GR and QM) can host Consciousness within it.

