14 Responses to “Why is Mathematics so Unreasonably Effective?”

1. Stephen Says:
   May 28th, 2008 at 9:48 pm

   One has to note that mathematics has little to say about ethics, felt beauty in the broader sense, philosophy or spirituality.

Comment by Stephen — May 28, 2008 @ 9:48 pm

2. The Pixie Says:
   May 29th, 2008 at 7:10 am

   The "unreasonable effectiveness of mathematics" is one of several rational pillars of intelligent design.

   It is also an excellent argument for the (not unrelated) claim we are all living inside a computer simulation.

Comment by The Pixie — May 29, 2008 @ 7:10 am

3. Salvador T. Cordova Says:
   May 29th, 2008 at 8:30 am

   The original essay by Wigner is Unreasonable Effectiveness of Mathematics.

   Wigner says:

   it is hard to believe that our reasoning power was brought, by Darwin's process of natural selection, to the perfection which it seems to possess.

   and

   physics as we know it today would not be possible without a constant recurrence of miracles similar to the one of the helium atom,

   Along those lines, an essay that should not be forgotten by William Dembski The Last Magic

   Consider, for in stance, the physicist Paul Dirac's discovery of the positron and antiparticles more generally. The positron is a particle just like an electron, only with a positive charge. Yet when Dirac proposed the positron, there was no experimental evidence for it. Indeed, there was no reason even to expect its existence. Why, then, did Dirac propose such a particle?

   Dirac was at the time trying to understand the Klein-Gordon field equation and the energy levels it assigned to certain quantum systems. He wanted to extend this equation relativistically to the electron, but he found that the only way to do so was by factoring it. Unfortunately, the equation resisted factoring over the real and complex numbers. Dirac therefore "brute-forced" the factorization by introducing higher dimensional "number-like" objects (the property where these objects differed from ordinary numbers was commutativity of multiplication).

   The factoring worked and gave Dirac the relativistic solution he wanted for the electron. But because the "number-like" objects he introduced also had a higher dimension than the ordinary numbers, Dirac's solution to the Klein-Gordon equation also yielded extra solutions"”solutions corresponding to the extra dimensions of his "number-like" objects. One of the solutions suggested a positively charged particle that in every other way was identical to the electron. What started as a mathematical trick designed to factor an equation and yield insight into the electron therefore yielded an entirely new particle and, indeed, an entirely new type of matter"”antimatter, the discovery of which fundamentally altered our understanding of the physical universe.

   Dirac's mathematical manipulations and physical speculations would have remained just that except for two facts: (1) In 1932 Carl Anderson experimentally confirmed the existence of the positron. (2) In the nineteenth century mathematicians had already constructed the "number-like" objects that Dirac needed to factor the Klein-Gordon equation. They are known today collectively as the Clifford algebra, and Dirac had to reinvent it to get a relativistic equation for the electron.

   Where is the philosophical problem for naturalism in examples like this (and Steiner makes clear that such examples are wide spread throughout mathematics in its application to physics)? The problem is that mathematics is a thoroughly human enterprise. Nature may condition us to see patterns that are readily perceived"”that, as it were, ride on the surface structure of nature. At the same time, nature should be indifferent to human idiosyncrasies. Thus, the problem for naturalism posed by Dirac's reinvention of the Clifford algebra and subsequent discovery of antimatter is that it occurred entirely through the manipulation of humanly constructed notations, and with attention not to physical reality but to human convenience.

   Equations that are factorable are much easier for us to deal with than those that are not. Factorability, however, has no physical significance. A world indifferent to us has no stake in rendering itself intelligible to us by making the equations that describe it factorable through some mathematical device (like the Clifford algebra). And yet precisely such idiosyncratic manipulations of humanly constructed notations result in genuine and previously unsuspected physical insights.

   There really is a problem here for naturalism. As Steiner notes, in every other area where human constructions are manipulated according to human convenience, naturalism expects and indeed confirms no profound in sight into the structure of the world. The rules of chess, for instance, do not yield insight into the structure of the atom. The study of palindromes (sentences that read the same backward as forward; e.g., "Madam, I'm Adam") tells us nothing about the first three minutes after the Big Bang.

   Indeed, the claim that human constructions manipulated according to human convenience supply insights into reality belongs to what traditionally has been called magic"”the view that what humans do in the purely human world (i.e., the microcosm) mirrors the deep structure of the world at large (i.e., the macrocosm). Naturalism has no place for magic. And yet the applicability of mathematics to physics is magic. Ac cording to Steiner, mathematics is the last redoubt of magic, but one that stands se cure and is in no danger of naturalistic debunking. This is a user-friendly world where we humans are the users, and where the tool of discovery that renders the natural world friendly is mathematics.

   In place of naturalism, Steiner therefore opts for an anthropocentrism which affirms that humans do have a privileged place in the scheme of things. Steiner's anthropocentrism falls short of a full-blown metaphysical position like Judeo-Christian theism, Platonism, or Pythagoreanism. But it stands sharply against the widely held evolutionary view that humans are mere accidents of natural history.

   Physicist Paul Davies won the 1.4 million dollar Templeton Prize for Religion because of his book, The Mind of God which explored these ideas in more detail — namely the relationship of physics, mathematics, computer science and religion.

Comment by Salvador T. Cordova — May 29, 2008 @ 8:30 am

4. Stephen Says:
   May 29th, 2008 at 10:12 am

   One should also note that Brouwer's intuitionist based mathematic finds a better platform to depict mathematical structure as a constructive enterprise. But intuitionist mathematics carries the insight of needing a "creating subject," as Brouwer noted. And this departs from David Herbert's formalistic mathematics that is found depending on contrived axioms that carry the agenda of proof derivation from axioms. To get at deep reality we need something better than formalistic mathematics that forgets the creating subject that is found so important in the discovery of truth. Mathematics is unable to ignore its feelings.

Comment by Stephen — May 29, 2008 @ 10:12 am

5. Todd Berkebile Says:
   May 29th, 2008 at 10:51 am

   Mike: The belief that our universe was intelligently designed and is best understood through a design paradigm can be derived from many things among which is the "enormous usefulness of mathematics in the natural sciences."

   Sure that subject observation might enforce someones preconceived notions. I, however, fail to see why this is anything put expected. Mathematics was not created independent of nature. Our brains do not work in some magical manner that is independent of nature. Why would we expect to develop mental models that do not mimic nature? I find it more amazing when our mental models turn out to be insufficient for the task of understanding nature. This is not to say we should treat instinct as a true reflection of nature, it is to say that our brains would not function at all if they were incapable of modeling nature in some rational way.

   This reminds me of the "utility of abstraction" thread, its really another aspect of the exact same thing.

Comment by Todd Berkebile — May 29, 2008 @ 10:51 am

6. Salvador T. Cordova Says:
   May 29th, 2008 at 11:12 am

   . Mathematics was not created independent of nature. Our brains do not work in some magical manner that is independent of nature. Why would we expect to develop mental models that do not mimic nature?

   Paul Davies points out that if the laws of physics were not algorithmically compressible the universe would not be compehensible even if our minds were more powerful than the ones we have today.

   "Algorithmically compressible" means we can express physical laws (approximations) with simple statements such as:

   F=ma

   The universe is thus highly amenable to computational analysis….

   A computationally sensible universe is highly improbable when viewed in the space of possible universes — the cardinality of computable numbers is smaller than that of non-computable numbers…..

   It is highly probable the universe would have been arranged in such a way that it would resist description by simple mathematical statements. Davies points out that only a narrow set of conditions in quantum cosmology would allow computable universes such as the one we live in.

   We are able to apply data compression to our description of large numbers of natural phenomenon. If the datapoints were not amenable to this sort of compression, we couldn't have laws of physics, just a never ending catalogue of disconnected measurements……

   If the universe were not fine tuned, the approxmately classical behavior on many levels would not be in evidence, and physics would be impossible. We would be unable to make inferences about anything…..

   Without fine tuning, we would not have the approximately classical notions of motion and time, and the orbits of the planets (assuming they even existed) would be more like smeared out probability distributions that we see at the atomic level. Such world's would not be amenable to physical inquiry.

   Davies book explains the details….I've only sketched out his ideas in these posts….Davies refutes the hypothesis that human minds in any universe would have been able to create the appropriate mathematics to explore such a universe. Only in certain possible universes will there be a convergence between mental processes and laws of physics…such a universe must be amenable to computation and algorithmic compression…

   The idea that humans would adapt to whatever their circumstances is a Darwinian idea, it is not a scientific nor mathematically defensible idea

Comment by Salvador T. Cordova — May 29, 2008 @ 11:12 am

7. aiguy Says:
   May 29th, 2008 at 12:25 pm

   The effectiveness of mathematics is surely mysterious, and I also think our ability to understand mathematics can't be explained by evolutionary theory (since our prehistoric ancestors' fitness was not enhanced by their ability to do mathematical physics).

   One could posit that some unknown, unintelligent process caused the universe, and this unintelligent process was mathematically describable, so the resulting universe was mathematically describable too. But this approach fails to identify what this process was, or say why it was mathematically describable in the first place.

   Alternatively, one could posit that some unknown intelligent entity caused the universe, and this entity was capable of understanding mathematics, and so the universe it created was mathematically describable too. But this approach fails to identify what this intelligent entity was, or say why it was capable of understanding mathematics in the first place.

   I don't think there is any reason compelling anyone to prefer one of the alternatives over the other, but at least the former option leaves a simpler construct unexplained.

Comment by aiguy — May 29, 2008 @ 12:25 pm

8. Bradford Says:
   May 29th, 2008 at 12:32 pm

   Todd:

   Our brains do not work in some magical manner that is independent of nature. Why would we expect to develop mental models that do not mimic nature?

   But we do develop mental constructs that do not mimick nature. Salvador Cordova quoting Dembski:

   Equations that are factorable are much easier for us to deal with than those that are not. Factorability, however, has no physical significance. A world indifferent to us has no stake in rendering itself intelligible to us by making the equations that describe it factorable through some mathematical device (like the Clifford algebra). And yet precisely such idiosyncratic manipulations of humanly constructed notations result in genuine and previously unsuspected physical insights.

   There really is a problem here for naturalism. As Steiner notes, in every other area where human constructions are manipulated according to human convenience, naturalism expects and indeed confirms no profound in sight into the structure of the world. The rules of chess, for instance, do not yield insight into the structure of the atom. The study of palindromes (sentences that read the same backward as forward; e.g., "Madam, I'm Adam") tells us nothing about the first three minutes after the Big Bang.

Comment by Bradford — May 29, 2008 @ 12:32 pm

9. Zachriel Says:
   May 29th, 2008 at 1:24 pm

   Dembski: Factorability, however, has no physical significance.

   That's not quite correct. Factoring is an outgrowth of counting and organizing. Factorization of quadratics can reveal underlying geometric relationships, e.g. a² + 2ab + b².

Comment by Zachriel — May 29, 2008 @ 1:24 pm

10. Bilbo Says:
    May 29th, 2008 at 4:39 pm

    aiguy:One could posit that some unknown, unintelligent process caused the universe, and this unintelligent process was mathematically describable, so the resulting universe was mathematically describable too. But this approach fails to identify what this process was, or say why it was mathematically describable in the first place.

    This sounds very much like Pythagoras.

    Alternatively, one could posit that some unknown intelligent entity caused the universe, and this entity was capable of understanding mathematics, and so the universe it created was mathematically describable too. But this approach fails to identify what this intelligent entity was, or say why it was capable of understanding mathematics in the first place.

    I think Plato chose this approach.

Comment by Bilbo — May 29, 2008 @ 4:39 pm

11. aiguy Says:
    May 29th, 2008 at 6:53 pm

    Bilbo,
    I think you're right about Pythagorian cause, but not sure about the Platonic one. Plato held that ideal forms (including mathematical constructs) were real, but did he consider them to be the result of "intelligent" creation?

    Anyway, the point I was making was that neither "unspecified unintelligent cause" nor "unspecified intelligent cause" (ID) goes any distance at all in explaining why the universe is describable with mathematics. "Unspecified unintelligent cause" could be anything, of course – any sort of combination of causes that we currently understand, or a completely new sort of cause that we have never thought of. Likewise, "unspecified intelligent cause" could be anything too, from a "transcendent immaterial being" to an "unconscious algorithmic search mechanism naturally arising from the intrinsic interactive properties of dark matter" or something. Obviously, neither of these ideas go any distance to explaining anything about anything whatsoever, since we really can't say what we're talking about in either case.

Comment by aiguy — May 29, 2008 @ 6:53 pm

12. Salvador T. Cordova Says:
    May 29th, 2008 at 10:14 pm

    Engineers Discover In Nature Exotic Structures Envisioned By Mathematicians

    By the way, Wigner was a physics re-tread….he was an engineer by training who eventually became a Nobel prize winning physicist.

    I'm glad to say, various schools are offering opportunities for engineers to get a retread as physics students…:cool:

Comment by Salvador T. Cordova — May 29, 2008 @ 10:14 pm

13. Rock Says:
    May 29th, 2008 at 10:49 pm

    Is this another "disconfirmation bias" thing?

    Mathematics is unreasonably effective when one ignores how many errors mathematicians make.

    Are mathematicians all that "effective" and what makes them so effective?

Comment by Rock — May 29, 2008 @ 10:49 pm

14. Bilbo Says:
    May 30th, 2008 at 4:24 pm

    aiguy lato held that ideal forms (including mathematical constructs) were real, but did he consider them to be the result of "intelligent" creation?

    Plato was trying to deal with how the forms interact with the physical world. His answer seemed to be (at least in The Timaeus) a demiurge that was conscious of the forms, and then created the physical world after their patterns.
    Then later in The Laws, Plato seemed to say that Mind was the most fundamental thing. Did he mean more fundamental than the Forms? Did the Forms reside in Mind? Not sure. I haven't read all of The Laws. I'll have to do that someday.

Comment by Bilbo — May 30, 2008 @ 4:24 pm