The Unreasonable Effectiveness of Mathematics in the Natural Sciences (1960)
dartmouth.edu
dartmouth.edu
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?"
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.)
The tools of continuous physics/math are so ingrained into the culture at this point that people seem to have forgotten the distinction between the map and the territory.
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.
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...
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.
In the end Math has a more restrictive syntax, but it's just as valid to prove say X+44.217 > X vs X + 1 > X. So the 'elegance' of math IMO has more to do with us selecting for it vs some innate property.
Or to get back to the parent's argument about the Mandelbrot set, you can draw plenty of pictures with Math. The tiny fraction of equations that are both simple and cool looking are what we select for. Worse, the equation works just fine in black and white we add false color and select for a small subset of the space that's interesting all of which is designed to make pretty pictures, and then say look how many 'simple' math equations make pretty pictures. Despite the fact it takes a fair amount of code for the equation to end up looking like that.
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.
An interesting (philosophical science fiction) read on this topic is Stanislaw Lem's "The New Cosmogony" (in the book "A Perfect Vacuum").
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.
But isn't that exactly where we are with quantum mechanics and relativity? Everyone thinks that if we were smarter or given more time, a unified theory will emerge that looks like one or the other at the appropriate scale. What if that never happens? I think the author is asking "What if that's not even possible?". It seems possible to him because there is no known reason for the mathematics to fit the world in the first place.
But I just read the part again and I probably misread it the first time. He is not talking about having absolutely correct but inconsistent or contradicting theories, but about theories that are correct, consistent and non-contradicting where we can test them and maybe even where we can ever expect to test them, but ultimately lead to inconsistencies or contradictions if we extrapolate them beyond experimental reach. This leaves us in a situation where the theories are in perfect agreement with reality but on purely theoretical grounds it seems impossible that they are correct. This is more similar to the situation of relativity and quantum mechanics but we have not yet reached the experimental limits and the theories are not even in perfect agreement with all available data. (I will read it a third time, I am still not totally convinced that I got it right.)
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://ned.ipac.caltech.edu/level5/March02/Wigner/Wigner.htm...
is in much better shape.
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.
"<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.
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? :)
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?
For the second, if the universe were somehow rational, there is no need to drag in irrational numbers to describe it.
Everything would be discretized, including possible speeds. Note that actually only speed differences need to be discrete, as those are the only thing actually observable. You cannot measure any absolute speed in this universe.
That is, irrational real numbers do exist in nature (diagonal of a square, to start), but only countably many of them exist. The rest are just an intellectual completion to create an elegant-looking symmetry.
It all depends on one's philosophy and existence. When you say, "only countably many of them exists....The rest are just..." you are admitting that at least in some sense they all exist. Indeed, one must be careful when making a statement, "irrational real numbers exist in nature". You can't point to anything and say, "There is the number square root of 2." It does not exist as a physical object.
Even if one were to make a giant philosophical leap and believe that one can make right triangle whose legs have precisely the same length (down to the last particle!) you don't have the existence of an irrational number. You have the hypotenuse of a triangle. The number itself does not exist in the physical sense.
Numbers are concepts and to say that one such concept exists but not another because there is some physical quantity is "equal" to it in former case but not the latter is an absurdity in my opinion. There are errors in all physical measurements. How is one going to distinguish between which numbers exist in your sense and which ones don't when the numbers are close enough together that no measurement can possibly distinguish between them?
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.
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.
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...
and read all about it)
Balki also introduced us to the mythical student HAROLD - Hypothetical Alert Reader Of Limitless Dedication, whom he had encountered in Schwinger's lectures.
This still fascinates and resonates with me to this day and remains an impossible barrier for some non-mathematicians to cross.
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.