Unreasonable Effectiveness Of Mathematics Research Articles

Does the Differential Structure of Space-Time Follow from Physical Principles?

This article examines Wigner's view on the unreasonable effectiveness of mathematics in the natural sciences, which was based on Cantor's claim that 'mathematics is a free creation of the human mind'. It is contended that Cantor's claim is not relevant to physics because it was based on his power set construction, which does not preserve neighborhoods of geometrical points. It is pointed out that the physical notion of Einstein causality can be defined on a countably infinite point set M with no predefined mathematical structure on it, and this definition endows M with a Tychonoff topology. Under Shirota's theorem, M can therefore be embedded as a closed subspace of RJ for some J. While this suggests that the differentiable structure of RJ may follow from the principle of causality, the argument is constrained by the fact that the completion processes (analyzed here in some detail) required for the passage from QJ to RJ remain empirically untestable.

Fractals: The Secret Code of Creation

Fractals: The Secret Code of Creation

What’s the Use? The Unreasonable Effectiveness of Mathematics by Ian Stewart

What’s the Use? The Unreasonable Effectiveness of Mathematics by Ian Stewart

Борьба сомнения с созерцанием: к ситуации в современной философии математики

The article deals with the concept of intuition (ϑεωρία, intuitio, Anschauung, Wesensschau) and the method of doubt in the philosophy of mathematics from Antiquity to the present day. In the classical philosophy of mathematics, there was no doubt about the attainability of reliable mathematical knowledge. Doubt was always present, but did not claim to deconstruct contemplation. The objects of contemplation are either the higher world of “ideas” (Plato, Neoplatonists), or innate ideas (René Descartes, Gottfried Leibniz), or a priori forms (Immanuel Kant), or something mysterious, perhaps the subconscious (Henri Poincaré). Edmund Husserl was the last classical philosopher to require intuition, and Kurt Gödel was the last mathematician. Modern mathematics is characterized by a departure from intuition, and therefore the philosophy of mathematics has fundamentally changed: now intuition is being questioned. The article provides a brief overview of the history of doubt, from Plato to the triumph of doubt in the second half of the 19th and early 20th centuries. The overthrow of intuition occurs in the works of Hans Hahn, Ludwig Wittgenstein, David Bloor and modern nominalists (Hartry Field). Often, intuition is replaced by a formal conclusion, logic, as was the case with Poincaré and David Hilbert. However, the deconstruction of intuition has deeper implications for the philosophy of mathematics than replacing it with logic. Logic, in turn, also needs to be justified. Why are modus ponens and the substitution rule considered reliable formal operations? Here logical evidence comes into play: Husserl showed that it is also a kind of intuition. However nothing should be obvious, including the simplest formal operations should be considered as purely conventional. What philosophy of mathematics awaits us in the future? The article makes the assumption that mathematics will be considered as a kind of game according to arbitrary rules. This is the current trend of its development. However, this trend is contradicted by the problem of applications of mathematics and Eugene Wigner’s riddle of “the unreasonable effectiveness of mathematics in the natural sciences.”

Vedic Hridaya Chakshu : Why is mathematics so effective in Exploring Nature

Mathematics rightly viewed, wrote Russell in his 'Study of Mathematics', 'possesses not only truth, but supreme beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show." The physicist Eugene Wigner once remarked that [t]he miracle of the appropriateness of the language of mathematics for the formulation of the laws of (Nature) physics is a wonderful gift which we neither understand nor deserve.' Steven Weinberg is another physicist puzzled with it, put, "It is very strange that mathematicians are led by their sense of mathematical beauty to develop formal structures that physicists only later find useful, even where the mathematician had no such goal in mind." The question is , how does the mathematician, who is more like an artist than an explorer, while working in abstract, away from nature, arrive at the most appropriate descriptions of nature? Nearly 82 years ago, Eugene Wigner set the puzzle up in his famous paper to call it "the unreasonable effectiveness of mathematics". It is a puzzle well known to the scientists and philosophers, but caught very little attention in the philosophical literature, which may be due to the presuppositions in the paradigm shift from a formalist philosophy of mathematics, and the rise of anti-realist paradigm in the philosophy of Mathematics. Whatever it may be, but when we look at the issue from our Vedic perspective, we find that the Vedas and Brahmanas provide irrefutable foundations where there can be no place for this so-called puzzle. In this paper we present some of the key ideas from the Vedas and Shatpath Brahmana, which provide the very foundation, a misunderstanding, of which leads to such a puzzle.

Which Number System Is “Best” for Describing Empirical Reality?

Eugene Wigner’s much-discussed notion of the “unreasonable effectiveness of mathematics” as applied to describing the physics of empirical reality is simultaneously both trivial and profound. After all, the relevant mathematics was (in the first instance) originally developed in order to be useful in describing empirical reality. On the other hand, certain aspects of the mathematical superstructure have by now taken on a life of their own, with at least some features of the mathematical superstructure greatly exceeding anything that can be directly probed or verified, or even justified, by empirical experiment. Specifically, I wish to raise the possibility that the real number system (with its nevertheless pragmatically very useful tools of real analysis and mathematically rigorous notions of differentiation and integration) may nevertheless constitute a “wrong turn” (a “sub-optimal” choice) when it comes to modelling empirical reality. Without making any definitive recommendation, I shall discuss several reasonably well-developed alternatives.

A metaphysical foundation for mathematical philosophy

Although mathematical philosophy is flourishing today, it remains subject to criticism, especially from non-analytical philosophers. The main concern is that even if formal tools serve to clarify reasoning, they themselves contribute nothing new or relevant to philosophy. We defend mathematical philosophy against such concerns here by appealing to its metaphysical foundations. Our thesis is that mathematical philosophy can be founded on the phenomenological theory of ideas as developed by Roman Ingarden. From this platonist perspective, the “unreasonable effectiveness of mathematics in philosophy”—to adapt Wigner’s phrase—is analogous to that of mathematical explanations in science. As success-criteria for mathematical philosophy, we propose that it should be correct, responsive, illuminating, promising, relevant, and adequate.

The Unreasonable Effectiveness of Mathematics: From Hamming to Wigner and Back Again

The Unreasonable Effectiveness of Mathematics: From Hamming to Wigner and Back Again

Wigner’s effective mathematics and contradiction

Complex numbers are basic. An inconsistency would question Wigner's unreasonable effectiveness of mathematics. A vehicle to study this question is Kirchoff's scalar diffraction theory. In the paper, an inconsistency in complex phase angle is presented. When this inconsistency is introduced in Kirchoff's theory we can study its influence on the experimental success of this theory. There are no \emph{a priori} reasons to include or exclude phase angles. Referring to Wigner, an experiment can provide more insight. In the experiment a weak intensity, small wavelength source can be employed. When the contradictory phase angle is excluded, a nonzero diffraction amplitude appears physically possible. If it is included, this amplitude vanishes.

Mathematics declaring the glory of God

This article discussed the question ‘Does God speak through the language of mathematics?’ For centuries, mathematicians with different religious backgrounds would have answered this question in the affirmative. Due to changes in mathematics from the 19th century onwards, this question cannot be answered as easily as it used to be. If one regards mathematical concepts as creations of the human mind, it is difficult to argue that mathematical formulae exist in a divine mind. The article argued that there were traces of the divine in mathematics. Six kinds of traces were explained: (1) the existence of indisputable truth, (2) the existence of beauty, (3) the importance of community, (4) rational speaking about infinity, (5) the discovery that speaking about unseen and abstract objects is reasonable and (6) the unreasonable effectiveness of mathematics. In practice, traces (1), (2) and (6) are probably the most convincing.Intradisciplinary and/or interdisciplinary implications: This article is very much interdisciplinary as it combines mathematics and theology, especially the philosophy of mathematics and systematic theology.

Fundamental Physics and Computation: The Computer-Theoretic Framework

The central goal of this manuscript is to survey the relationships between fundamental physics and computer science. We begin by providing a short historical review of how different concepts of computer science have entered the field of fundamental physics, highlighting the claim that the universe is a computer. Following the review, we explain why computational concepts have been embraced to interpret and describe physical phenomena. We then discuss seven arguments against the claim that the universe is a computational system and show that those arguments are wrong because of a misunderstanding of the extension of the concept of computation. Afterwards, we address a proposal to solve Hempel’s dilemma using the computability theory but conclude that it is incorrect. After that, we discuss the relationship between the proposals that the universe is a computational system and that our minds are a simulation. Analysing these issues leads us to proposing a new physical principle, called the principle of computability, which claims that the universe is a computational system (not restricted to digital computers) and that computational power and the computational complexity hierarchy are two fundamental physical constants. On the basis of this new principle, a scientific paradigm emerges to develop fundamental theories of physics: the computer-theoretic framework (CTF). The CTF brings to light different ideas already implicit in the work of several researchers and provides a new view on the universe based on computer theoretic concepts that expands the current view. We address different issues regarding the development of fundamental theories of physics in the new paradigm. Additionally, we discuss how the CTF brings new perspectives to different issues, such as the unreasonable effectiveness of mathematics and the foundations of cognitive science.

Science and faith: Friendly allies, not hostile enemies

Contemporary science is a wonderfully collaborative activity. It knows no barriers of geography, race, or creed. At its best, it enables us to wrestle with the problems that beset humanity, and we rightly celebrate when an advance is made that brings relief to millions. I have spent my life as a pure mathematician, and I often reflect on what physics Nobel Prize–winner Eugene Wigner called “the unreasonable effectiveness of mathematics.” How is it that equations created in the head of a mathematician can relate to the universe outside that head? This question prompted Albert Einstein to say, “The only incomprehensible thing about the universe is that it is comprehensible.” The very fact that we believe that science can be done is a thing to be wondered at.

The role of nature and brain in demystifying the “unreasonable effectiveness of mathematics”

ABSTRACTIn 1960, Eugene P. Wigner shared his observation of the unreasonable effectiveness of mathematics in formulating regularities in nature. Later, Jean Piaget recognized the functioning of the living organism as a source of harmony between mathematics and nature. However, not only did Piaget not demonstrate how more advanced mathematics could be linked to natural behavior, the unique functional feature of human behavior was absent in his explanation of its ultimate outcome (i.e. mathematics). The present analysis proposes that the effectiveness of mathematics seems puzzling because of the apparent disconnection between nature, the characteristics of the human brain and behavior, and the properties of mathematical constructs. The behavior of humans and other animals (i.e. rodents) will be the basis of comparison to meet the objective of the analysis in demystifying the effectiveness of mathematics in natural sciences by (1) showing the potential natural roots of some mathematical constructs in the organization of behavior with some specific examples in mathematics, (2) demonstrating a mechanism for the construction of mathematical reasoning as well as for the invention of mathematical constructs, and (3) discussing how mathematics seems disconnected from nature as a result of mathematical inventions guided by the symmetry principle.

The Utterly Prosaic Connection between Physics and Mathematics

Eugene Wigner famously argued for the “unreasonable effectiveness of mathematics” as applied to describing physics and other natural sciences in his 1960 essay. That essay has now led to some 58 years of (sometimes anguished) philosophical soul searching—responses range from “So what? Why do you think we developed mathematics in the first place?”, through to extremely speculative ruminations on the existence of the universe (multiverse) as a purely mathematical entity—the Mathematical Universe Hypothesis. In the current essay I will steer an utterly prosaic middle course: Much of the mathematics we develop is informed by physics questions we are trying to solve; and those physics questions for which the most utilitarian mathematics has successfully been developed are typically those where the best physics progress has been made.

Foreword to BIOMATH 2017 Proceedings: Some comments on mathematical modelling and biomathematics.

Both biology and mathematics have existed as well established branches of science for hundreds of years and both, maybe not in a well defined way, have been with the humankind for a couple of thousands of years. Though nature was studied by the ancient civilizations of Mesopotamia, Egypt, the Indian subcontinent and China, the origins of modern biology are typically traced back to the ancient Greece, where Aristotle (384-322 BC) contributed most extensively to its development. Similarly, the ancient Babylonians were able to solve quadratic equation over four millennia ago and we can see the development of mathematical methods in all ancient civilisations, notably in China and on the Indian subcontinent. However, possibly again the Greeks were the first who studied mathematics for its own sake, as a collection of abstract objects and relations between them. Nevertheless, despite the fact that the development of such a mathematics has not required any external stimuli, an amazing feature of the human mind is that a large number of abstract mathematical constructs has proved to be very well suited for describing natural phenomena.This prompted Eugene Wigner to write his famous article The Unreasonable Effectiveness of Mathematics in the Natural Sciences, ...

The unreasonable effectiveness of theory based instrument calibration in the natural sciences: What can the social sciences learn?

In his classic paper entitled “The Unreasonable Effectiveness of Mathematics in the Natural Sciences,” Eugene Wigner addresses the question of why the language of Mathematics should prove so remarkably effective in the physical [natural] sciences. He marvels 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.” We have been similarly struck by the outsized benefits that theory based instrument calibrations convey on the natural sciences, in contrast with the almost universal practice in the social sciences of using data to calibrate instrumentation.

Studying the Winger’s “Enigma” about the Unreasonable Effectiveness of Mathematics in the Natural Sciences

The effectiveness of mathematics in the natural sciences was characterized by the famous Nobel prize holder E. P. Winger as being unreasonable. It is not difficult for one to understand that this characterization is related to a question that has occupied the interest of philosophers, mathematicians and other scientists at least from the Plato’s era in ancient , until today: “Is mathematics discovered or invented by humans”? In the present work in an effort to obtain a convincing explanation of the above Winger’s “enigma”, the existing philosophical views about the above question are critically examined and discussed in connection with the advances in the history of mathematics that affected the human beliefs about them.

Eficácia sem razão [da matemática

http://dx.doi.org/10.5007/2175-7941.2016v33n2p399 Em 2003, o matemático Americano Alex Kasman (2003) escreveu um belo conto de “ficção matemática” intitulado “Unreasonable Effectiveness”, uma das várias respostas existentes ao clássico artigo de Wigner (1960) “The unreasonable effectiveness of mathematics in the natural sciences”. Na versão de Kasman, uma pesquisadora acaba, acidentalmente, descobrindo a resposta para a questão de como uma nova teoria encontra, em algum momento, uma utilidade prática na ciência. Isto é, como resultados abstratos, construídos sem quaisquer alicerces no “mundo real”, acabam se tornando tão úteis, mesmo em áreas completamente diversas? Neste artigo é proposta uma tradução comentada deste delicioso ensaio, tanto como proposta não convencional de experiência didática quanto uma reflexão sobre os rumos do desenvolvimento científico, propiciados pela Matemática e Física. Como objetivo secundário, procura-se trabalhar a motivação do estudante na busca por soluções não triviais para problemas científicos e filosóficos.

The Role of Symmetry in Mathematics

Over the past few decades the notion of symmetry has played a major role in physics and in the philosophy of physics. Philosophers have used symmetry to discuss the ontology and seeming objectivity of the laws of physics. We introduce several notions of symmetry in mathematics and explain how they can also be used in resolving different problems in the philosophy of mathematics. We use symmetry to discuss the objectivity of mathematics, the role of mathematical objects, the unreasonable effectiveness of mathematics and the relationship of mathematics to physics.

The Unreasonable Accuracy of Moore's Law [Historical

In 1960, three years before receiving the Nobel Prize in Physics, Hungarian-born American theoretical physicist and mathematician Eugene Wigner (1902?1995) published a paper, ?The Unreasonable Effectiveness of Mathematics in the Natural Sciences.? In it, he argued about the surprising precision with which mathematical equations can describe physical phenomena [1]. He considered Newton?s second law, which states that the gravitational force acting on a falling body is proportional to its mass and no other parameter of that body. Newton brought this law into relation with the motion of the moon, noting that the parabola of a thrown rock and the circle of the moon's path in the sky are particular cases of the same mathematical object, the ellipse, deducing the universal law of gravitation. Initially, this law was supported by a single and rough coincidence with an accuracy of a mere 4%, but subsequent observations proved that it held to an accuracy of 0.001%. Wigner also examined other laws, such as quantum mechanics that, when formulated in the form of matrix mechanics to calculate the lowest energy level of helium, matched experiments to a relative error of 10?7.