The Unreasonable Effectiveness of Mathematics in the Natural Sciences
Encyclopedia
The Unreasonable Effectiveness of Mathematics in the Natural Sciences is the title of an article published in 1960 by the physicist
Eugene Wigner. In the paper, Wigner observed that the mathematical
structure of a physics theory
often points the way to further advances in that theory and even to empirical
predictions, and argued that this is not just a coincidence
and therefore must reflect some larger and deeper truth
about both mathematics
and physics
.
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 Hertz in 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 mysterious and that there is no rational explanation for it." He concludes his paper with the same question with which he began:
, and has been fairly often cited in the academic literature on the philosophy of physics
and of mathematics. Wigner speculated on the relationship between the philosophy of science
and the foundations of mathematics
as follows:
Later, 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 bias
Wigner cautiously labeled as "not reliable," he went further:
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 cosmology
and the philosophy of mathematics
.
Wigner also laid out the challenge of a cognitive approach to integrating the sciences:
He further proposed that arguments could be found that might...
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
.
in Computer Science, Arthur Lesk
in Molecular Biology from , Peter Norvig
in data mining , Max Tegmark in Physics, Ivor Grattan-Guinness
in Mathematics and Vela Velupillai
in Economics
, an applied mathematician
and a founder of computer science
, reflected on and extended Wigner's Unreasonable Effectiveness in 1980, 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.
2. Humans create and select the mathematics that fit a situation. The mathematics at hand does not always work. For example, when mere scalar
s proved awkward for understanding forces, first vectors, then tensor
s, 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.
4. Evolution
has 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.
in 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.
finds the effectiveness in question eminently reasonable, and explained in terms of analogy, generalisation, metaphor, and similar techniques
How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality? — Albert Einstein
There is only one thing which is more unreasonable than the unreasonable effectiveness of mathematics in physics, and this is the unreasonable ineffectiveness of mathematics in biology. — Israel Gelfand
"... 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." - Pope Benedict XVI
"We should stop acting as if our goal is to author extremely elegant theories, and instead embrace complexity and make use of the best ally we have: the unreasonable effectiveness of data."
Physicist
A physicist is a scientist who studies or practices physics. Physicists study a wide range of physical phenomena in many branches of physics spanning all length scales: from sub-atomic particles of which all ordinary matter is made to the behavior of the material Universe as a whole...
Eugene Wigner. In the paper, Wigner observed that the mathematical
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
structure of a physics theory
Theoretical physics
Theoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...
often points the way to further advances in that theory and even to empirical
Empirical
The word empirical denotes information gained by means of observation or experimentation. Empirical data are data produced by an experiment or observation....
predictions, and argued that this is not just a coincidence
Coincidence
A coincidence is an event notable for its occurring in conjunction with other conditions, e.g. another event. As such, a coincidence occurs when something uncanny, accidental and unexpected happens under conditions named, but not under a defined relationship...
and therefore must reflect some larger and deeper truth
Truth
Truth has a variety of meanings, such as the state of being in accord with fact or reality. It can also mean having fidelity to an original or to a standard or ideal. In a common usage, it also means constancy or sincerity in action or character...
about both mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
and physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
.
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 uncanny number 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
Maxwell's equations
Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.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 Hertz in 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 mysterious and that there is no rational explanation for it." He concludes his paper with the same question with which he began:
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 mathematicsPhilosophy of mathematics
The philosophy of mathematics is the branch of philosophy that studies the philosophical assumptions, foundations, and implications of mathematics. The aim of the philosophy of mathematics is to provide an account of the nature and methodology of mathematics and to understand the place of...
, and has been fairly often cited in the academic literature on the philosophy of physics
Philosophy of physics
In philosophy, the philosophy of physics studies the fundamental philosophical questions underlying modern physics, the study of matter and energy and how they interact. The philosophy of physics begins by reflecting on the basic metaphysical and epistemological questions posed by physics:...
and of mathematics. Wigner speculated on the relationship between the philosophy of science
Philosophy of science
The philosophy of science is concerned with the assumptions, foundations, methods and implications of science. It is also concerned with the use and merit of science and sometimes overlaps metaphysics and epistemology by exploring whether scientific results are actually a study of truth...
and the foundations of mathematics
Foundations of mathematics
Foundations of mathematics is a term sometimes used for certain fields of mathematics, such as mathematical logic, axiomatic set theory, proof theory, model theory, type theory and recursion theory...
as 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.
Later, Hilary Putnam
Hilary Putnam
Hilary Whitehall Putnam is an American philosopher, mathematician and computer scientist, who has been a central figure in analytic philosophy since the 1960s, especially in philosophy of mind, philosophy of language, philosophy of mathematics, and philosophy of science...
(1975) explained these "two miracles" as being necessary consequences of a realist (but not Platonist) view of the philosophy of mathematics
Philosophy of mathematics
The philosophy of mathematics is the branch of philosophy that studies the philosophical assumptions, foundations, and implications of mathematics. The aim of the philosophy of mathematics is to provide an account of the nature and methodology of mathematics and to understand the place of...
. However, in a passage discussing cognitive bias
Cognitive bias
A cognitive bias is a pattern of deviation in judgment that occurs in particular situations. Implicit in the concept of a "pattern of deviation" is a standard of comparison; this may be the judgment of people outside those particular situations, or may be a set of independently verifiable...
Wigner cautiously labeled as "not reliable," he went further:
The writer is convinced that it is useful, in epistemological discussions, 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 cosmology
Cosmology
Cosmology is the discipline that deals with the nature of the Universe as a whole. Cosmologists seek to understand the origin, evolution, structure, and ultimate fate of the Universe at large, as well as the natural laws that keep it in order...
and the philosophy of mathematics
Philosophy of mathematics
The philosophy of mathematics is the branch of philosophy that studies the philosophical assumptions, foundations, and implications of mathematics. The aim of the philosophy of mathematics is to provide an account of the nature and methodology of mathematics and to understand the place of...
.
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
String theory
String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...
, 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
Cognition
In science, cognition refers to mental processes. These processes include attention, remembering, producing and understanding language, solving problems, and making decisions. Cognition is studied in various disciplines such as psychology, philosophy, linguistics, and computer science...
.
Responses to Wigner's original paper
Wigner's original paper has provoked and inspired many responses across a wide range of disciplines. These include Richard HammingRichard Hamming
Richard Wesley Hamming was an American mathematician whose work had many implications for computer science and telecommunications...
in Computer Science, Arthur Lesk
Arthur M. Lesk
Arthur M. Lesk, is an accomplished protein science researcher, who is currently a professor of biochemistry and molecular biology at the Pennsylvania State University, University Park.- Profile :...
in Molecular Biology from , Peter Norvig
Peter Norvig
Peter Norvig is an American computer scientist. He is currently the Director of Research at Google Inc.-Educational Background:...
in data mining , Max Tegmark in Physics, Ivor Grattan-Guinness
Ivor Grattan-Guinness
Ivor Grattan-Guinness, born 23 June 1941, in Bakewell, in England, is a historian of mathematics and logic.He gained his Bachelor degree as a Mathematics Scholar at Wadham College, Oxford, got an M.Sc in Mathematical Logic and the Philosophy of Science at the London School of Economics in 1966...
in Mathematics and Vela Velupillai
Vela Velupillai
Kumaraswamy Velupillai is an academic economist.He is a Professor of Economics - 'Professore di Chiara Fama' - in the department of economics at the University of Trento, Italy. Till recently he was the John E. Cairnes Professor of Economics at the National University of Ireland, Galway and a...
in Economics
Richard Hamming's follow-on
Richard HammingRichard Hamming
Richard Wesley Hamming was an American mathematician whose work had many implications for computer science and telecommunications...
, an applied mathematician
Applied mathematics
Applied mathematics is a branch of mathematics that concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Thus, "applied mathematics" is a mathematical science with specialized knowledge...
and a founder of computer science
Computer science
Computer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...
, reflected on and extended Wigner's Unreasonable Effectiveness in 1980, 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
Arthur Stanley Eddington
Sir Arthur Stanley Eddington, OM, FRS was a British astrophysicist of the early 20th century. He was also a philosopher of science and a popularizer of science...
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 Galileo discovered the law of falling bodies not by experimenting, but by simple, though careful, thinking. Hamming imagines Galileo as having engaged in the following thought experimentThought experimentA thought experiment or Gedankenexperiment considers some hypothesis, theory, or principle for the purpose of thinking through its consequences...
(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 gravitation necessarily follows from the conservation of energyConservation of energyThe nineteenth century law of conservation of energy is a law of physics. It states that the total amount of energy in an isolated system remains constant over time. The total energy is said to be conserved over time...
and of space having three dimensionsSpacetimeIn physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
. Measuring the exponent in the law of universal gravitation is more a test of whether space is EuclideanEuclidean spaceIn mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
than a test of the properties of the gravitational fieldGravitational fieldThe gravitational field is a model used in physics to explain the existence of gravity. In its original concept, gravity was a force between point masses...
.
- The inequality at the heart of the uncertainty principleUncertainty principleIn quantum mechanics, the Heisenberg uncertainty principle states a fundamental limit on the accuracy with which certain pairs of physical properties of a particle, such as position and momentum, can be simultaneously known...
of quantum mechanicsQuantum mechanicsQuantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
follows from the properties of Fourier integrals and from assuming time invariance.
- Hamming argues that Albert EinsteinAlbert EinsteinAlbert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...
's pioneering work on special relativitySpecial relativitySpecial relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...
was 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 ExperimentMichelson-Morley experimentThe Michelson–Morley experiment was performed in 1887 by Albert Michelson and Edward Morley at what is now Case Western Reserve University in Cleveland, Ohio. Its results are generally considered to be the first strong evidence against the theory of a luminiferous ether and in favor of special...
), 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 scalar
Scalar (mathematics)
In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector....
s proved awkward for understanding forces, first vectors, then tensor
Tensor
Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...
s, 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
Value theory
Value theory encompasses a range of approaches to understanding how, why and to what degree people should value things; whether the thing is a person, idea, object, or anything else. This investigation began in ancient philosophy, where it is called axiology or ethics. Early philosophical...
, including ethics
Ethics
Ethics, also known as moral philosophy, is a branch of philosophy that addresses questions about morality—that is, concepts such as good and evil, right and wrong, virtue and vice, justice and crime, etc.Major branches of ethics include:...
, aesthetics
Aesthetics
Aesthetics is a branch of philosophy dealing with the nature of beauty, art, and taste, and with the creation and appreciation of beauty. It is more scientifically defined as the study of sensory or sensori-emotional values, sometimes called judgments of sentiment and taste...
, and political philosophy
Political philosophy
Political philosophy is the study of such topics as liberty, justice, property, rights, law, and the enforcement of a legal code by authority: what they are, why they are needed, what, if anything, makes a government legitimate, what rights and freedoms it should protect and why, what form it...
. To assert that the world can be explained via mathematics amounts to an act of faith.
4. Evolution
Evolution
Evolution is any change across successive generations in the heritable characteristics of biological populations. Evolutionary processes give rise to diversity at every level of biological organisation, including species, individual organisms and molecules such as DNA and proteins.Life on Earth...
has 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.
Max Tegmark's response
A different response, advocated by Physicist Max TegmarkMax Tegmark
Max Tegmark is a Swedish-American cosmologist. Tegmark is a professor at the Massachusetts Institute of Technology and belongs to the scientific directorate of the Foundational Questions Institute.-Early life:...
in 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.
Ivor Grattan-Guinness response
Ivor Grattan-GuinnessIvor Grattan-Guinness
Ivor Grattan-Guinness, born 23 June 1941, in Bakewell, in England, is a historian of mathematics and logic.He gained his Bachelor degree as a Mathematics Scholar at Wadham College, Oxford, got an M.Sc in Mathematical Logic and the Philosophy of Science at the London School of Economics in 1966...
finds the effectiveness in question eminently reasonable, and explained in terms of analogy, generalisation, metaphor, and similar techniques
Related quotes
The most incomprehensible thing about the universe is that it is comprehensible. — Albert EinsteinHow can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality? — Albert Einstein
There is only one thing which is more unreasonable than the unreasonable effectiveness of mathematics in physics, and this is the unreasonable ineffectiveness of mathematics in biology. — Israel Gelfand
Israel Gelfand
Israel Moiseevich Gelfand, also written Israïl Moyseyovich Gel'fand, or Izrail M. Gelfand was a Soviet mathematician who made major contributions to many branches of mathematics, including group theory, representation theory and functional analysis...
"... 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." - Pope Benedict XVI
"We should stop acting as if our goal is to author extremely elegant theories, and instead embrace complexity and make use of the best ally we have: the unreasonable effectiveness of data."
See also
- CosmologyCosmologyCosmology is the discipline that deals with the nature of the Universe as a whole. Cosmologists seek to understand the origin, evolution, structure, and ultimate fate of the Universe at large, as well as the natural laws that keep it in order...
- Foundations of mathematicsFoundations of mathematicsFoundations of mathematics is a term sometimes used for certain fields of mathematics, such as mathematical logic, axiomatic set theory, proof theory, model theory, type theory and recursion theory...
- Mark SteinerMark SteinerMark Steiner is a professor of philosophy at the Hebrew University of Jerusalem, where he specializes in the philosophy of mathematics and physics. He is best known for his book The Applicability of Mathematics as a Philosophical Problem, in which he attempted to explain the historical utility of...
- Philosophy of sciencePhilosophy of scienceThe philosophy of science is concerned with the assumptions, foundations, methods and implications of science. It is also concerned with the use and merit of science and sometimes overlaps metaphysics and epistemology by exploring whether scientific results are actually a study of truth...
- Quasi-empiricism in mathematicsQuasi-empiricism in mathematicsQuasi-empiricism in mathematics is the attempt in the philosophy of mathematics to direct philosophers' attention to mathematical practice, in particular, relations with physics, social sciences, and computational mathematics, rather than solely to issues in the foundations of mathematics...
- Unreasonable ineffectiveness of mathematicsUnreasonable ineffectiveness of mathematicsThe unreasonable ineffectiveness of mathematics is a catchphrase, alluding to the well-known article by physicist Eugene Wigner, "The Unreasonable Effectiveness of Mathematics in the Natural Sciences". This catchphrase is meant to suggest that mathematical analysis has not proved as valuable in...
- Where Mathematics Comes FromWhere Mathematics Comes FromWhere Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being is a book by George Lakoff, a cognitive linguist, and Rafael E. Núñez, a psychologist...
Further reading
, a piece of "mathematical fiction".- http://plato.stanford.edu/entries/mathphil-indis/