Unreasonable ineffectiveness of mathematics

Last updated

The unreasonable ineffectiveness of mathematics is a phrase that alludes to the article by physicist Eugene Wigner, "The Unreasonable Effectiveness of Mathematics in the Natural Sciences". This phrase is meant to suggest that mathematical analysis has not proved as valuable in other fields as it has in physics.

Contents

Life sciences

I. M. Gelfand, a mathematician who worked in biomathematics and molecular biology, as well as many other fields in applied mathematics, is quoted as stating,

Eugene Wigner wrote a famous essay on the unreasonable effectiveness of mathematics in natural sciences. He meant physics, of course. 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. [1]

An opposing view is given by Leonard Adleman, a theoretical computer scientist who pioneered the field of DNA computing. In Adleman's view, "Sciences reach a point where they become mathematized," starting at the fringes but eventually "the central issues in the field become sufficiently understood that they can be thought about mathematically. It occurred in physics about the time of the Renaissance; it began in chemistry after John Dalton developed atomic theory" and by the 1990s was taking place in biology. [2] By the early 1990s, "Biology was no longer the science of things that smelled funny in refrigerators (my view from undergraduate days in the 1960s). The field was undergoing a revolution and was rapidly acquiring the depth and power previously associated exclusively with the physical sciences. Biology was now the study of information stored in DNA - strings of four letters: A, T, G, and C and the transformations that information undergoes in the cell. There was mathematics here!" [3]

Economics and finance

K. Vela Velupillai wrote of The unreasonable ineffectiveness of mathematics in economics . [4] [5] To him "the headlong rush with which economists have equipped themselves with a half-baked knowledge of mathematical traditions has led to an un-natural mathematical economics and a non-numerical economic theory." His argument is built on the claim that

mathematical economics is unreasonably ineffective. Unreasonable, because the mathematical assumptions are economically unwarranted; ineffective because the mathematical formalisations imply non-constructive and uncomputable structures. A reasonable and effective mathematisation of economics entails Diophantine formalisms. These come with natural undecidabilities and uncomputabilities. In the face of this, [the] conjecture [is] that an economics for the future will be freer to explore experimental methodologies underpinned by alternative mathematical structures. [6]

Sergio M. Focardi and Frank J. Fabozzi, on the other hand, have acknowledged that "economic science is generally considered less viable than the physical sciences" and that "sophisticated mathematical models of the economy have been developed but their accuracy is questionable to the point that the 2007–08 economic crisis is often blamed on an unwarranted faith in faulty mathematical models" [7] (see also: [8] ). They nevertheless claim that

the mathematical handling of economics has actually been reasonably successful and that models are not the cause behind the present crisis. The science of economics does not study immutable laws of nature but the complex human artefacts that are our economies and our financial markets, artefacts that are designed to be largely uncertain.... and therefore models can only be moderately accurate. Still, our mathematical models offer a valuable design tool to engineer our economic systems. But the mathematics of economics and finance cannot be that of physics. The mathematics of economics and finance is the mathematics of learning and complexity, similar to the mathematics used in studying biological or ecological systems. [9]

A more general comment by Irving Fisher is that:

The contention often met with that the mathematical formulation of economic problems gives a picture of theoretical exactitude untrue to actual life is absolutely correct. But, to my mind, this is not an objection but a very definite advantage, for it brings out the principles in such sharp relief that it enables us to put our finger definitely on the points where the picture is untrue to real life. [10]

Cognitive sciences

Roberto Poli of McGill University delivered a number of lectures entitled The unreasonable ineffectiveness of mathematics in cognitive sciences in 1999. The abstract is:

My argument is that it is possible to gain better understanding of the "unreasonable effectiveness" of mathematics in study of the physical world only when we have understood the equally "unreasonable ineffectiveness" of mathematics in the cognitive sciences (and, more generally, in all the forms of knowledge that cannot be reduced to knowledge about physical phenomena. Biology, psychology, economics, ethics, and history are all cases in which it has hitherto proved impossible to undertake an intrinsic mathematicization even remotely comparable to the analysis that has been so fruitful in physics.) I will consider some conceptual issues that might prove important for framing the problem of cognitive mathematics (= mathematics for the cognitive sciences), namely the problem of n-dynamics, of identity, of timing, and of the specious present. The above analyses will be conducted from a partly unusual perspective regarding the problem of the foundations of mathematics. [11]

See also

Related Research Articles

<span class="mw-page-title-main">Eugene Wigner</span> Hungarian-American physicist and mathematician (1902–1995)

Eugene Paul "E. P." Wigner was a Hungarian-American theoretical physicist who also contributed to mathematical physics. He received the Nobel Prize in Physics in 1963 "for his contributions to the theory of the atomic nucleus and the elementary particles, particularly through the discovery and application of fundamental symmetry principles".

<i>Where Mathematics Comes From</i> 2000 mathematics book by Lakoff & Núñez

Where 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. Published in 2000, WMCF seeks to found a cognitive science of mathematics, a theory of embodied mathematics based on conceptual metaphor.

The philosophy of mathematics is the branch of philosophy that studies the assumptions, foundations, and implications of mathematics. It aims to understand the nature and methods of mathematics, and find out the place of mathematics in people's lives. The logical and structural nature of mathematics makes this branch of philosophy broad and unique.

Quasi-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. Of concern to this discussion are several topics: the relationship of empiricism with mathematics, issues related to realism, the importance of culture, necessity of application, etc.

"The Unreasonable Effectiveness of Mathematics in the Natural Sciences" is a 1960 article by the physicist Eugene Wigner, published in Communication in Pure and Applied Mathematics. In it, Wigner observes that a physical theory's mathematical structure often points the way to further advances in that theory and even to empirical predictions. Mathematical theories often have predictive power in describing nature.

<span class="mw-page-title-main">Behavioral economics</span> Academic discipline

Behavioral economics is the study of the psychological, cognitive, emotional, cultural and social factors involved in the decisions of individuals or institutions, and how these decisions deviate from those implied by classical economic theory.

<span class="mw-page-title-main">Outline of academic disciplines</span> Overviews of and topical guides to academic disciplines

The following outline is provided as an overview of and topical guide to academic disciplines:

Rigour or rigor describes a condition of stiffness or strictness. These constraints may be environmentally imposed, such as "the rigours of famine"; logically imposed, such as mathematical proofs which must maintain consistent answers; or socially imposed, such as the process of defining ethics and law.

<span class="mw-page-title-main">Dynamical systems theory</span> Area of mathematics used to describe the behavior of complex dynamical systems

Dynamical systems theory is an area of mathematics used to describe the behavior of complex dynamical systems, usually by employing differential equations or difference equations. When differential equations are employed, the theory is called continuous dynamical systems. From a physical point of view, continuous dynamical systems is a generalization of classical mechanics, a generalization where the equations of motion are postulated directly and are not constrained to be Euler–Lagrange equations of a least action principle. When difference equations are employed, the theory is called discrete dynamical systems. When the time variable runs over a set that is discrete over some intervals and continuous over other intervals or is any arbitrary time-set such as a Cantor set, one gets dynamic equations on time scales. Some situations may also be modeled by mixed operators, such as differential-difference equations.

In physics and cosmology, the mathematical universe hypothesis (MUH), also known as the ultimate ensemble theory, is a speculative "theory of everything" (TOE) proposed by cosmologist Max Tegmark.

The argument from beauty is an argument for the existence of a realm of immaterial ideas or, most commonly, for the existence of God, that roughly states that the elegance of the laws of physics or the elegant laws of mathematics is evidence of a creator deity who has arranged these things to be beautiful and not ugly.

The following outline is provided as a topical overview of science; the discipline of science is defined as both the systematic effort of acquiring knowledge through observation, experimentation and reasoning, and the body of knowledge thus acquired, the word "science" derives from the Latin word scientia meaning knowledge. A practitioner of science is called a "scientist". Modern science respects objective logical reasoning, and follows a set of core procedures or rules to determine the nature and underlying natural laws of all things, with a scope encompassing the entire universe. These procedures, or rules, are known as the scientific method.

The term physics envy is used to criticize modern writing and research of academics working in areas such as "softer sciences", liberal arts, business administration education, humanities, and social sciences. The term argues that writing and working practices in these disciplines have overused confusing jargon and complicated mathematics to seem more 'rigorous' and like heavily mathematics-based natural science subjects like physics.

<span class="mw-page-title-main">Applied mathematics</span> Application of mathematical methods to other fields

Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical science and specialized knowledge. The term "applied mathematics" also describes the professional specialty in which mathematicians work on practical problems by formulating and studying mathematical models.

Kumaraswamy (Vela) Velupillai is an academic economist and a Senior Visiting Professor at the Madras School of Economics and was, formerly, (Distinguished) Professor of Economics at the New School for Social Research in New York City and Professore di Chiara Fama in the Department of Economics at the University of Trento, Italy.

The following outline is provided as an overview of and topical guide to social science:

<span class="mw-page-title-main">Relationship between mathematics and physics</span> Study of how mathematics and physics relate to each other

The relationship between mathematics and physics has been a subject of study of philosophers, mathematicians and physicists since Antiquity, and more recently also by historians and educators. Generally considered a relationship of great intimacy, mathematics has been described as "an essential tool for physics" and physics has been described as "a rich source of inspiration and insight in mathematics".

Emmanuel Haven is an academic, author and researcher. He previously held a personal Chair at the University of Leicester (UK) and is currently full professor and the Dr. Alex Faseruk Chair in Financial Management at the Faculty of Business Administration, Memorial University.

References

  1. Borovik, Alexandre (November 2006). Mathematics Under the Microscope.
  2. Gene Genie
  3. Computing with DNA (Scientific American) 1998
  4. Velupillai, Vela (November 2005). "The unreasonable ineffectiveness of mathematics in economics". Cambridge Journal of Economics. 29 (6): 849–872. doi:10.1093/cje/bei084. hdl: 10379/1108 . SSRN   904709.
  5. Velupillai, K. Vela (2004). "The Unreasonable Ineffectiveness of Mathematics in Economics". Technical Report 6, Economia. University of Trento.
  6. Abstract
  7. Focardi, S. & Fabozzi, F. (Spring 2010). "The reasonable effectiveness of mathematics in economics". American Economist. 49 (1): 3–15.
  8. López de Prado, M. and Fabozzi, F. (2018). Who Needs a Newtonian Finance? Journal of Portfolio Management , Vol. 44, No. 1, 2017
  9. Abstract.
  10. Irving Fisher (1930). The Theory of Interest: As Determined by Impatience to Spend Income and Opportunity to Invest It; p. 315
  11. "Poli seminar abstract". Category Theory Research Center, McGill University. 1999.

Bibliography