The Unreasonable Effectiveness of Mathematics in the Natural Sciences
January 31, 2010 10:30 PM Subscribe
The Unreasonable Effectiveness of Mathematics in the Natural Sciences is a 1960 essay by Eugene Wigner. Via Steve Strogatz.
If you liked this you should read a followup: The Unreasonable Effectiveness of Mathematics by R. W. Hamming
posted by pseudonick at 10:43 PM on January 31, 2010 [2 favorites]
posted by pseudonick at 10:43 PM on January 31, 2010 [2 favorites]
As opposed to the unreasonable ineffectiveness of mathematics in, say, economics or cognitive science.
posted by twoleftfeet at 10:54 PM on January 31, 2010
posted by twoleftfeet at 10:54 PM on January 31, 2010
The book, The Martians Of Science: Five Physicists Who Changed the Twentieth Century, is well worth a read for the immense impact the five Hungarians, Kármán, Szilard, Wigner, von Neumann and Teller had on our world. All were born in and around the turn of last century in Budapest and I don't think it is possible to over-estimate the impact these men had on our world today. Add in another Budapest-born Hungarian, Paul Erdős, and you begin to wonder what alignment of the universe occurred to create such titans.
posted by vac2003 at 11:46 PM on January 31, 2010
posted by vac2003 at 11:46 PM on January 31, 2010
Also worth checking out:
The impact of the incompleteness theorems on mathematics by Solomon Feferman
posted by ageispolis at 11:59 PM on January 31, 2010 [1 favorite]
The impact of the incompleteness theorems on mathematics by Solomon Feferman
posted by ageispolis at 11:59 PM on January 31, 2010 [1 favorite]
I am often surprised at the usefulness of assuming a simple linear relationship between two parameters when I have no physical model to apply to the situation.
But I cannot wrap my head around what Wigner means about finding a fusion of theories at a lower level of intelligence in the following passage:
"In order to obtain an indication as to which alternative to expect ultimately, we can pretend to be a little more ignorant than we are and place ourselves at a lower level of knowledge than we actually possess. If we can find a fusion of our theories on this lower level of intelligence, we can confidently expect that we will find a fusion of our theories also at our real level of intelligence. On the other hand, if we would arrive at mutually contradictory theories at a somewhat lower level of knowledge, the possibility of the permanence of conflicting theories cannot be excluded for ourselves either. The level of knowledge and ingenuity is a continuous variable and it is unlikely that a relatively small variation of this continuous variable changes the attainable picture of the world from inconsistent to consistent."
posted by wigner3j at 12:00 AM on February 1, 2010
But I cannot wrap my head around what Wigner means about finding a fusion of theories at a lower level of intelligence in the following passage:
"In order to obtain an indication as to which alternative to expect ultimately, we can pretend to be a little more ignorant than we are and place ourselves at a lower level of knowledge than we actually possess. If we can find a fusion of our theories on this lower level of intelligence, we can confidently expect that we will find a fusion of our theories also at our real level of intelligence. On the other hand, if we would arrive at mutually contradictory theories at a somewhat lower level of knowledge, the possibility of the permanence of conflicting theories cannot be excluded for ourselves either. The level of knowledge and ingenuity is a continuous variable and it is unlikely that a relatively small variation of this continuous variable changes the attainable picture of the world from inconsistent to consistent."
posted by wigner3j at 12:00 AM on February 1, 2010
Or you can look at this way: Is 'the theory of everything' merely the ultimate ensemble theory?
posted by overyield at 1:54 AM on February 1, 2010 [1 favorite]
posted by overyield at 1:54 AM on February 1, 2010 [1 favorite]
wigner3j,
When he talks about the level of knowledge being a continuous variable, he means that a new theory should be able to increase the precision of an old theory, but that it shouldn't conflict with it in the domains where the old theory has been well studied.
Think of dimensional analysis, taking the classical limit, or thermodynamic considerations as ways that some problems can be naively approached.
Dimensional analysis - We can pretend that we only know the units of the answer and the inputs, and that we don't have a clue as to the physical relationships between them, we should get an expression that is at least consistent with the expression that we derive using our theoretical knowledge of the problem.
Classical Limit - If you've used a lot of clever differential geometry to produce general relativity, you can pretend for a minute that you don't understand any of the derivation, only the end result. If you plug numbers for well studied systems in, you should get numbers that match those that you've observed in the past.
posted by atrazine at 3:02 AM on February 1, 2010
When he talks about the level of knowledge being a continuous variable, he means that a new theory should be able to increase the precision of an old theory, but that it shouldn't conflict with it in the domains where the old theory has been well studied.
Think of dimensional analysis, taking the classical limit, or thermodynamic considerations as ways that some problems can be naively approached.
Dimensional analysis - We can pretend that we only know the units of the answer and the inputs, and that we don't have a clue as to the physical relationships between them, we should get an expression that is at least consistent with the expression that we derive using our theoretical knowledge of the problem.
Classical Limit - If you've used a lot of clever differential geometry to produce general relativity, you can pretend for a minute that you don't understand any of the derivation, only the end result. If you plug numbers for well studied systems in, you should get numbers that match those that you've observed in the past.
posted by atrazine at 3:02 AM on February 1, 2010
The effectiveness of mathematics in describing the universe is a feature of the universe that could perhaps use some explaining. If the universe were "made of" mathematics in some sense, then it would be less surprising.
posted by DU at 5:03 AM on February 1, 2010 [1 favorite]
posted by DU at 5:03 AM on February 1, 2010 [1 favorite]
If the universe were "made of" mathematics in some sense, then it would be less surprising.
See also the Mathematical Universe Hypothesis, mentioned briefly on the wiki page above.
My take on all this is that mathematics is ultimately a human endeavor which is predicated on intuition we derive from making observations of the world around us. So, instead of math discovering more math, it's really physics discovering more physics. Or maybe they're fundamentally the same thing. I'm surprised to hear myself say this, but I actually tend to agree with Pope Benedict XVI here: "... if nature is really structured with a mathematical language and mathematics invented by man can manage to understand it, this demonstrates something extraordinary. The objective structure of the universe and the intellectual structure of the human being coincide."
posted by albrecht at 7:09 AM on February 1, 2010
See also the Mathematical Universe Hypothesis, mentioned briefly on the wiki page above.
My take on all this is that mathematics is ultimately a human endeavor which is predicated on intuition we derive from making observations of the world around us. So, instead of math discovering more math, it's really physics discovering more physics. Or maybe they're fundamentally the same thing. I'm surprised to hear myself say this, but I actually tend to agree with Pope Benedict XVI here: "... if nature is really structured with a mathematical language and mathematics invented by man can manage to understand it, this demonstrates something extraordinary. The objective structure of the universe and the intellectual structure of the human being coincide."
posted by albrecht at 7:09 AM on February 1, 2010
atrazine,
If I understand, his fusion of knowledge occurs in quantum mechanics when we note that the expectation value of position and momentum obey Newton's laws of motion, or that Special relativity reduces to the classical equations of motion at the limit of small velocities.
So if our current theories can be shown to encompass our previous theories, then we should feel more comfortable that the current theories won't contradict each other. But if the current theories do not include the results of the theories they have replaced then we may expect them to conflict with each other.
posted by wigner3j at 7:17 AM on February 1, 2010
If I understand, his fusion of knowledge occurs in quantum mechanics when we note that the expectation value of position and momentum obey Newton's laws of motion, or that Special relativity reduces to the classical equations of motion at the limit of small velocities.
So if our current theories can be shown to encompass our previous theories, then we should feel more comfortable that the current theories won't contradict each other. But if the current theories do not include the results of the theories they have replaced then we may expect them to conflict with each other.
posted by wigner3j at 7:17 AM on February 1, 2010
"...The objective structure of the universe and the intellectual structure of the human being coincide."
I'm always surprised that this idea generates so much excitement. Shouldn't it just be obvious that since "the intellectual structure" of the human mind is governed by the same physical laws that operate on everything else there has to be, at some level, a direct relationship between the laws that operate on the gross physical structures of the brain and the mental processes those structures give rise to? In other words, of course we think in ways that ultimately mirror the structure of an underlying physical reality: Our thinking, too, is a fact of physical reality.
posted by saulgoodman at 7:43 AM on February 1, 2010 [1 favorite]
I'm always surprised that this idea generates so much excitement. Shouldn't it just be obvious that since "the intellectual structure" of the human mind is governed by the same physical laws that operate on everything else there has to be, at some level, a direct relationship between the laws that operate on the gross physical structures of the brain and the mental processes those structures give rise to? In other words, of course we think in ways that ultimately mirror the structure of an underlying physical reality: Our thinking, too, is a fact of physical reality.
posted by saulgoodman at 7:43 AM on February 1, 2010 [1 favorite]
Well, saulgoodman, your assumption is that the human mind is governed by physical laws. So if you take that as a given, then of course the other follows. But I suspect that the Pope, and the audience he is speaking to, doesn't necessarily take that view.
posted by wigner3j at 8:12 AM on February 1, 2010
posted by wigner3j at 8:12 AM on February 1, 2010
And for me, what is exciting about the Pope's statement is that he is getting tantalizingly close to exposing logical inconsistencies in his own beliefs.
posted by wigner3j at 8:15 AM on February 1, 2010
posted by wigner3j at 8:15 AM on February 1, 2010
>>>your assumption is that the human mind is governed by physical laws. So if you take that as a given, then of course the other follows. But I suspect that the Pope, and the audience he is speaking to, doesn't necessarily take that view....
>>>what is exciting about the Pope's statement is that he is getting tantalizingly close to exposing logical inconsistencies in his own beliefs...
Really? Speaking as one who became a Catholic pretty much exactly the time I became an MSci in Physics, I'd say the fact that the human mind is [at least] a physical system has been a Catholic given for centuries.
posted by KMH at 8:49 AM on February 1, 2010
>>>what is exciting about the Pope's statement is that he is getting tantalizingly close to exposing logical inconsistencies in his own beliefs...
Really? Speaking as one who became a Catholic pretty much exactly the time I became an MSci in Physics, I'd say the fact that the human mind is [at least] a physical system has been a Catholic given for centuries.
posted by KMH at 8:49 AM on February 1, 2010
True, wigner3j, but the question gets a fair amount of play in philosophy of mind discussions too.
posted by saulgoodman at 8:52 AM on February 1, 2010
posted by saulgoodman at 8:52 AM on February 1, 2010
To clarify, I personally always find it, philosophically, a pretty pedestrian thought that math and physics are fundamentally similar.
After all, how did we invent counting except by counting some physical stuff, discover geometry except by making physical measurements etc? We then extrapolated from there beyond the physical evidence but have never [and will never] abandoned the physical roots.
The level of self-awareness required to apply this reasoning to more abstract bits of math is not a huge jump i.e. my mind which makes algebraic operations on physically-derived axioms is itself [at least] a physical process.
Any religious reasoning comes on top of the fundamental realism of this humble admission, I think.
posted by KMH at 9:02 AM on February 1, 2010
After all, how did we invent counting except by counting some physical stuff, discover geometry except by making physical measurements etc? We then extrapolated from there beyond the physical evidence but have never [and will never] abandoned the physical roots.
The level of self-awareness required to apply this reasoning to more abstract bits of math is not a huge jump i.e. my mind which makes algebraic operations on physically-derived axioms is itself [at least] a physical process.
Any religious reasoning comes on top of the fundamental realism of this humble admission, I think.
posted by KMH at 9:02 AM on February 1, 2010
Okay, I'll stop pretending I know anything about Catholic beliefs. So if in this belief system the mind is part of a physical system, the brain, then what part of the mind is going to heaven or hell?
posted by wigner3j at 9:19 AM on February 1, 2010
posted by wigner3j at 9:19 AM on February 1, 2010
>>>So if in this belief system the mind is part of a physical system, the brain, then what part of the mind is going to heaven or hell?
Not "is part of" - "is [at least]". Big diff.
The body and soul are distinct but overlap totally (form/matter) - that's Thomism.
They are both going to heaven (or hell) together. Please don't ask me how ;) as far as I know, not even the Pope knows that :D or pretends to.
posted by KMH at 9:23 AM on February 1, 2010
Not "is part of" - "is [at least]". Big diff.
The body and soul are distinct but overlap totally (form/matter) - that's Thomism.
They are both going to heaven (or hell) together. Please don't ask me how ;) as far as I know, not even the Pope knows that :D or pretends to.
posted by KMH at 9:23 AM on February 1, 2010
>>>Please don't ask me how...
Well, possibly "naturally" or "personally" - but I'm not a master of these concepts. C.f. Luigi Giussani, "The Religious Sense".
Catholic philosophy tends to predicate on conscious and studious ignorance and agnosticism (c.f. G K Chesterton, "...I'm an agnostic on most things...") That's why I like it so much :D
posted by KMH at 9:25 AM on February 1, 2010
Well, possibly "naturally" or "personally" - but I'm not a master of these concepts. C.f. Luigi Giussani, "The Religious Sense".
Catholic philosophy tends to predicate on conscious and studious ignorance and agnosticism (c.f. G K Chesterton, "...I'm an agnostic on most things...") That's why I like it so much :D
posted by KMH at 9:25 AM on February 1, 2010
Shouldn't it just be obvious that since "the intellectual structure" of the human mind is governed by the same physical laws that operate on everything else there has to be, at some level, a direct relationship between the laws that operate on the gross physical structures of the brain and the mental processes those structures give rise to?
"At some level," yes, but it's not the right level to be relevant. The substrate of our thoughts (the mechanism that produces them) is obviously operated on by physical laws, but the content of our thoughts may have nothing to do with physical laws. Just because thought is produced by neurons, that does not mean that our theories are of neurons. It took us a long time to discover their existence. And it's similarly surprising that our best physical theories are of the physical world.
posted by painquale at 9:52 AM on February 1, 2010
"At some level," yes, but it's not the right level to be relevant. The substrate of our thoughts (the mechanism that produces them) is obviously operated on by physical laws, but the content of our thoughts may have nothing to do with physical laws. Just because thought is produced by neurons, that does not mean that our theories are of neurons. It took us a long time to discover their existence. And it's similarly surprising that our best physical theories are of the physical world.
posted by painquale at 9:52 AM on February 1, 2010
is obviously operated on by physical laws, but the content of our thoughts may have nothing to do with physical laws.
I'm not talking about the contents of our thoughts here, only about the structure of our thought processes (including our formal rules of reasoning, but likely going even deeper). And I wouldn't even argue you can map those structures in any direct way to the physical laws that constrain them. Still, there must be some consistent relationship between physical reality and the physical operations of the brain. (Neurons, for example, can't fire in ways that aren't constrained by the same rules that constrain all other physical systems. The physical limits that constrain brain processes also necessarily constrain subjective mental processes in certain ways; though as you point out, the content of thoughts may be to a great extent arbitrary, the rules for how thoughts relate to one another, how one thought gives rise to another, etc., must still be constrained by certain physical limits on brain processes, and those physical limits--assuming that the laws constraining physical reality are in some sense consistent--must also form the basis for a consistent relationship/series of determinate correspondences between mental processes and the broader structural framework of the rest of reality).
posted by saulgoodman at 10:35 AM on February 1, 2010
I'm not talking about the contents of our thoughts here, only about the structure of our thought processes (including our formal rules of reasoning, but likely going even deeper). And I wouldn't even argue you can map those structures in any direct way to the physical laws that constrain them. Still, there must be some consistent relationship between physical reality and the physical operations of the brain. (Neurons, for example, can't fire in ways that aren't constrained by the same rules that constrain all other physical systems. The physical limits that constrain brain processes also necessarily constrain subjective mental processes in certain ways; though as you point out, the content of thoughts may be to a great extent arbitrary, the rules for how thoughts relate to one another, how one thought gives rise to another, etc., must still be constrained by certain physical limits on brain processes, and those physical limits--assuming that the laws constraining physical reality are in some sense consistent--must also form the basis for a consistent relationship/series of determinate correspondences between mental processes and the broader structural framework of the rest of reality).
posted by saulgoodman at 10:35 AM on February 1, 2010
Why the particular concepts of mathematics "work" in physics reduces to the mystery of why the physical universe is "regular" / "well-behaved" -- similar to the mystery of why the universe supports intelligent life, when (seemingly) any number of minor differences in underlying physical parameters could have made it impossible. (Not sure there is any reason to believe there are "parameters," but I leave that aside.) Much has been said about this, but I don't think it's the more interesting part of the puzzle.
The harder part of the question could be rephrased: "Why do generalizations work?" Mathematical physics being just a very rigorous discipline of generalization. Why can you abstract properties from any particular thing and reason logically from their application to other things. This in turn I think reduces to basic questions in the philosophy of logic / philosophy of math. Where do things get their truth values etc.
I guess what I'm saying is, I think there are two separate questions: 1) Why are some generalizations physically true? This is "merely" a question of physical cosmology, albeit an open and somewhat hard one. 2) Why are generalizations logically possible? This is a question of philosophy of math / logic.
posted by grobstein at 11:41 AM on February 1, 2010
The harder part of the question could be rephrased: "Why do generalizations work?" Mathematical physics being just a very rigorous discipline of generalization. Why can you abstract properties from any particular thing and reason logically from their application to other things. This in turn I think reduces to basic questions in the philosophy of logic / philosophy of math. Where do things get their truth values etc.
I guess what I'm saying is, I think there are two separate questions: 1) Why are some generalizations physically true? This is "merely" a question of physical cosmology, albeit an open and somewhat hard one. 2) Why are generalizations logically possible? This is a question of philosophy of math / logic.
posted by grobstein at 11:41 AM on February 1, 2010
Oh for goodness sake, I hear about this essay in reverent tones for decades and nod my head in assent, but when I finally get around to reading it, it turns out to be a manifesto of vitalism, if not outright theism!
From the next to last paragraph:
A much more difficult and confusing situation would arise if we could, some day, establish a theory of the phenomena of consciousness, or of biology, which would be as coherent and convincing as our present theories of the inanimate world. Mendel’s laws of inheritance and the subsequent work on genes may well form the beginning of such a theory as far as biology is concerned. Furthermore, it is quite possible that an abstract argument can be found which shows that there is a conflict between such a theory and the accepted principles of physics. The argument could be of such abstract nature that it might not be possible to resolve the conflict, in favor of one or of the other theory, by an experiment. Such a situation would put a heavy strain on our faith in our theories and on our belief in the reality of the concepts which we form. It would give us a deep sense of frustration in our search for what I called "the ultimate truth." The reason that such a situation is conceivable is that, fundamentally, we do not know why our theories work so well. Hence, their accuracy may not prove their truth and consistency. Indeed, it is this writer’s belief that something rather akin to the situation which was described above exists if the present laws of heredity and of physics are confronted. [My emphasis]
posted by jamjam at 4:03 PM on February 1, 2010
From the next to last paragraph:
A much more difficult and confusing situation would arise if we could, some day, establish a theory of the phenomena of consciousness, or of biology, which would be as coherent and convincing as our present theories of the inanimate world. Mendel’s laws of inheritance and the subsequent work on genes may well form the beginning of such a theory as far as biology is concerned. Furthermore, it is quite possible that an abstract argument can be found which shows that there is a conflict between such a theory and the accepted principles of physics. The argument could be of such abstract nature that it might not be possible to resolve the conflict, in favor of one or of the other theory, by an experiment. Such a situation would put a heavy strain on our faith in our theories and on our belief in the reality of the concepts which we form. It would give us a deep sense of frustration in our search for what I called "the ultimate truth." The reason that such a situation is conceivable is that, fundamentally, we do not know why our theories work so well. Hence, their accuracy may not prove their truth and consistency. Indeed, it is this writer’s belief that something rather akin to the situation which was described above exists if the present laws of heredity and of physics are confronted. [My emphasis]
posted by jamjam at 4:03 PM on February 1, 2010
Wrong in important ways, but still frames an interesting / important question, no?
I mean, I agree: it sounds stupid to me to even imagine that we could find an irresolvable contradiction between the laws of physics and of biological evolution. But that is separable from the main argument of the piece.
posted by grobstein at 4:57 PM on February 1, 2010
I mean, I agree: it sounds stupid to me to even imagine that we could find an irresolvable contradiction between the laws of physics and of biological evolution. But that is separable from the main argument of the piece.
posted by grobstein at 4:57 PM on February 1, 2010
jamjam, grobstein: it's simply pointing out that there are complex systems where predictive experiments based on ansatz reasoning will predict, but not in any conclusive. Rather the ansatz represents a point on the manifold of theoretical continuity. Introducing error into our space of possible conception will only produce divergent, contradictory, models. Consider research into drugs that modify brain chemistry and the variety of conflicting scientific conception surrounding them.
He is suggesting that the "Natural Laws", the essence of Truth one might suppose, are intruding upon Actuality. More to the point, these Laws represent one half of a balancing act..
posted by kuatto at 9:32 PM on February 1, 2010
He is suggesting that the "Natural Laws", the essence of Truth one might suppose, are intruding upon Actuality. More to the point, these Laws represent one half of a balancing act..
posted by kuatto at 9:32 PM on February 1, 2010
Wigner means what he says in the title: the effectiveness of mathematics in the physical sciences is "unreasonable."
That is, it cannot be accounted for by rational means (as he says in so many words in the body of the essay). As such, it is an irreducible "mystery"; a mystery which points beyond itself.
I think Wigner means to say in this essay that the mystery of "the unreasonable effectiveness of mathematics" points to and can only be understood in terms of a manifestation of the influence of a transcendent consciousness.
This is at odds with the way the essay is usually spoken of, so I looked around very briefly to try to find some support for my view. Here are a couple of sentences from his Wikipedia page:
He became interested in the Vedanta philosophy of Hinduism, particularly its ideas of the universe as an all pervading consciousness. ...
In his collection of essays Symmetries and Reflections - Scientific Essays, he commented "It was not possible to formulate the laws (of quantum theory) in a fully consistent way without reference to consciousness."
For Wigner, in brief, "the unreasonable effectiveness of mathematics" appears to have been due to intelligent design all the way down.
posted by jamjam at 8:23 AM on February 2, 2010
That is, it cannot be accounted for by rational means (as he says in so many words in the body of the essay). As such, it is an irreducible "mystery"; a mystery which points beyond itself.
I think Wigner means to say in this essay that the mystery of "the unreasonable effectiveness of mathematics" points to and can only be understood in terms of a manifestation of the influence of a transcendent consciousness.
This is at odds with the way the essay is usually spoken of, so I looked around very briefly to try to find some support for my view. Here are a couple of sentences from his Wikipedia page:
He became interested in the Vedanta philosophy of Hinduism, particularly its ideas of the universe as an all pervading consciousness. ...
In his collection of essays Symmetries and Reflections - Scientific Essays, he commented "It was not possible to formulate the laws (of quantum theory) in a fully consistent way without reference to consciousness."
For Wigner, in brief, "the unreasonable effectiveness of mathematics" appears to have been due to intelligent design all the way down.
posted by jamjam at 8:23 AM on February 2, 2010
Sure, sure. But "the unreasonable effectiveness" is still a problem if you're not interested in mysticism and think the "mystery" is not necessarily irreducible.
posted by grobstein at 9:13 AM on February 2, 2010
posted by grobstein at 9:13 AM on February 2, 2010
Yes, I think you are absolutely right.
One of the reasons I put reading Wigner off was that I wanted to have at least a few nascent thoughts of my own as a counterpoise to what I assumed would be his solution, and it's brought me up a bit short to realize that his solution is a higher power (which, as I attempted to allude to with "all the way down", founders on the reef of infinite regress).
posted by jamjam at 9:33 AM on February 2, 2010
One of the reasons I put reading Wigner off was that I wanted to have at least a few nascent thoughts of my own as a counterpoise to what I assumed would be his solution, and it's brought me up a bit short to realize that his solution is a higher power (which, as I attempted to allude to with "all the way down", founders on the reef of infinite regress).
posted by jamjam at 9:33 AM on February 2, 2010
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