- The Unreasonable Effectiveness of Mathematics in the Natural Sciences
In 1960, the
physicist Eugene Wignerpublished an article titled "The Unreasonable Effectiveness of Mathematics in the Natural Sciences", arguing that the way in which the mathematicalstructure of a physical theory often points the way to further advances in that theory and even to empirical predictions, is not a coincidence but must reflect some larger and deeper truth about both mathematicsand physics.
The miracle of mathematics in the natural sciences
Wigner begins his paper with the belief, common to all those familiar with mathematics, that mathematical concepts have applicability far beyond the context in which they were originally developed. Based on his experience, he says "it is important to point out that the mathematical formulation of the physicist’s often crude experience leads in an
uncannynumber of cases to an amazingly accurate description of a large class of phenomena." He then invokes the fundamental law of gravitation as an example. Originally used to model freely falling bodies on the surface of the earth, this law was extended on the basis of what Wigner terms "very scanty observations" to describe the motion of the planets, where it "has proved accurate beyond all reasonable expectations."
Another oft-cited example is
Maxwell's equations, derived to model the elementary electrical and magnetic phenomena known as of the mid 19th century. These equations also describe radio waves, discovered by Heinrich Hertzin 1887 a few years after Maxwell's death. Wigner sums up his argument by saying that "the enormous usefulness of mathematics in the natural sciences is something bordering on the mysteriouscite web | url = http://en.wiktionary.org/wiki/mysterious] and that there is no rational explanation for it." He concludes his paper with the same question he began with:
The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning.
The deep connection between science and mathematics
Wigner's work provided a fresh insight into both physics and the
philosophy of mathematics, and has been fairly often cited in the academic literature on the philosophy of physicsand of mathematics. Wigner speculated on the relationship between the philosophy of scienceand the foundations of mathematicsas follows:
It is difficult to avoid the impression that a miracle confronts us here, quite comparable in its striking nature to the miracle that the human mind can string a thousand arguments together without getting itself into contradictions, or to the two miracles of laws of nature and of the human mind's capacity to divine them.
Hilary Putnam(1975) explained these "two miracles" as being necessary consequences of a realist (but not Platonist) view of the philosophy of mathematics. However, in a passage discussing cognitive biasWigner cautiously labeled as "not reliable," he went further:
The writer is convinced that it is useful, in
epistemologicaldiscussions, to abandon the idealization that the level of human intelligence has a singular position on an absolute scale. In some cases it may even be useful to consider the attainment which is possible at the level of the intelligence of some other species.
Whether humans checking the results of humans can be considered an objective basis for observation of the known (to humans) universe is an interesting question, one followed up in both
cosmologyand the philosophy of mathematics.
Wigner also laid out the challenge of a cognitive approach to integrating the sciences:
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.
He further proposed that arguments could be found that might...
...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.
Some believe that this conflict exists in
string theory, where very abstract models are impossible to test given existent experimental apparatus. While this remains the case, the "string" must be thought of either as real but untestable, or simply as an illusion or artifact of either mathematics or cognition.
Hamming's follow-on to Wigner
Richard Hamming(1980), an applied mathematician and a founder of computer science, reflects on and extends Wigner's "Unreasonable Effectiveness", mulling over four "partial explanations" for it. Hamming concluded that the four explanations he gave were unsatisfactory. They were:
1. "Humans see what they look for". The belief that science is experimentally grounded is only partially true. Rather, our intellectual apparatus is such that much of what we see comes from the glasses we put on. Eddington went so far as to claim that a sufficiently wise mind could deduce all of physics, illustrating his point with the following joke: "Some men went fishing in the sea with a net, and upon examining what they caught they concluded that there was a minimum size to the fish in the sea."
Hamming gives four examples of nontrivial physical phenomena he believes arose from the mathematical tools employed and not from the intrinsic properties of physical reality.
* Hamming proposes that
Galileodiscovered the law of falling bodies not by experimenting, but by simple but careful thinking. Hamming imagines Galileo as having engaged in the following thought experiment(Hamming calls it "scholastic reasoning"):
Suppose that a falling 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 tie 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?"
There is simply no way a falling body can "answer" such hypothetical "questions." Hence Galileo would have concluded that "falling bodies need not know anything if they all fall with the same velocity, unless interfered with by another force." After coming up with this argument, Hamming found a related discussion in Polya (1963: 83-85). Hamming's account does not reveal an awareness of the 20th century scholarly debate over just what Galileo did.
*The inverse square
law of universal gravitationnecessarily follows from the conservation of energyand of space having three dimensions. Measuring the exponent in the law of universal gravitation is more a test of whether space is Euclideanthan a test of the properties of the gravitational field.
*The inequality at the heart of the
uncertainty principleof quantum mechanicsfollows from the properties of Fourier integrals and from assuming time invariance.
*Hamming argues that
Albert Einstein's pioneering work on special relativitywas largely "scholastic" in its approach. He knew from the outset what the theory should look like (although he only knew this because of the Michelson-Morley Experiment), and explored candidate theories with mathematical tools, not actual experiments. Hamming alleges that Einstein was so confident that his relativity theories were correct that the outcomes of observations designed to test them did not much interest him. If the observations were inconsistent with his theories, it would be the observations that were at fault.
2. "Humans create and select the mathematics that fit a situation". The mathematics at hand does not always work. For example, when mere
scalars proved awkward for understanding forces, first vectors, then tensors, were invented.
3. "Mathematics addresses only a part of human experience". Much of human experience does not fall under science or mathematics but under the philosophy of value, including
ethics, aesthetics, and political philosophy. To assert that the world can be explained via mathematics amounts to an act of faith.
Evolutionhas primed humans to think mathematically". The earliest lifeforms must have contained the seeds of the human ability to create and follow long chains of close reasoning. Hamming, whose expertise is far from biology, otherwise says little to flesh out this contention.
A different response, advocated by Physicist
Max Tegmark(2007), is that physics is so successfully described by mathematics because the physical world "is" completely mathematical, isomorphic to a mathematical structure, and that we are simply uncovering this bit by bit.In this interpretation, the various approximations that constitute our current physics theories are successful because simple mathematical structures can provide good approximations of certain aspects of more complex mathematical structures.In other words, our successful theories are not mathematics approximating physics, but mathematics approximating mathematics.
"The most incomprehensible thing about the universe is that it is comprehensible." -- Albert Einstein
"How can it be that mathematics, being after all product of human thought which is independent of experience, is so admirably appropriate to the objects of reality?" -- Albert Einstein
foundations of mathematics
philosophy of science
quasi-empiricism in mathematics
Unreasonable Ineffectiveness of Mathematics
Where Mathematics Comes From"
* [http://www.ias.ac.in/currsci/feb102005/415.pdf "Revisiting the ‘unreasonable effectiveness’ of mathematics", Sundar Sarukkai, CURRENT SCIENCE, VOL. 88, NO. 3, 10 February 2005]
* [http://math.cofc.edu/faculty/kasman/MATHFICT/mfview.php?callnumber=mf334 "Unreasonable Effectiveness", Alex Kasman, Math Horizons magazine, April 2003 (pp. 29--31)] , a piece of "mathematical fiction".
*"The Unreasonable Effectiveness of Mathematics in Molecular Biology", Artuhur Lesk, The Mathematical Intelligencer, Vol. 22, No. 2, pp. 28-36, 2000.
Eugene Wigner, 1960, " [http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html The Unreasonable Effectiveness of Mathematics in the Natural Sciences,] " " Communications on Pure and Applied Mathematics" 13(1): 1–14.
Richard Hamming, 1980, " [http://www.dartmouth.edu/~matc/MathDrama/reading/Hamming.html The Unreasonable Effectiveness of Mathematics,] " "The American Mathematical Monthly87":
George Polya, 1963. "Mathematical Methods in Science". Mathematical Association of America.
Hilary Putnam, 1975, "What is Mathematical Truth?" "Historia Mathematica 2": 529-543. Reprinted in his (1975) "Mathematics, Matter and Method: Philosophical Papers, Vol. 1". Cambridge Univ. Press: 60-78
Max Tegmark, 2007, " [http://arxiv.org/pdf/0704.0646 The Mathematical Universe] ", arXiv 0704.0646
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