On the Reasonable and Unreasonable Effectiveness of Mathematics in Classical and Quantum Physics

The point of departure for this article is Werner Heisenberg’s remark, made in 1929: “It is not surprising that our language [or conceptuality] should be incapable of describing processes occurring within atoms, for … it was invented to describe the experiences of daily life, and these consist only of processes involving exceedingly large numbers of atoms. … Fortunately, mathematics is not subject to this limitation, and it has been possible to invent a mathematical scheme—the quantum theory [quantum mechanics]—which seems entirely adequate for the treatment of atomic processes.” The cost of this discovery, at least in Heisenberg’s and related interpretations of quantum mechanics (such as that of Niels Bohr), is that, in contrast to classical mechanics, the mathematical scheme in question no longer offers a description, even an idealized one, of quantum objects and processes. This scheme only enables predictions, in general, probabilistic in character, of the outcomes of quantum experiments. As a result, a new type of the relationships between mathematics and physics is established, which, in the language of Eugene Wigner adopted in my title, indeed makes the effectiveness of mathematics unreasonable in quantum but, as I shall explain, not in classical physics. The article discusses these new relationships between mathematics and physics in quantum theory and their implications for theoretical physics—past, present, and future.

[1]  N. David Mermin,et al.  Boojums All The Way Through , 1990 .

[2]  J. Stillwell,et al.  Symmetry , 2000, Am. Math. Mon..

[3]  William K. Wootters,et al.  Quantum mechanics without probability amplitudes , 1986 .

[4]  B. Inwood.  The Art and Thought of Heraclitus , 1984 .

[5]  B G Sidharth,et al.  Law without Law , 2007, 0710.3900.

[6]  J. Wheeler,et al.  Quantum theory and measurement , 1983 .

[7]  L. Brown.  :The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory , 2004 .

[8]  W. Heisenberg.  A quantum-theoretical reinterpretation of kinematic and mechanical relations , 1925 .

[9]  L. Loeb.  Autobiographical Notes , 2015, Perspectives in biology and medicine.

[10]  Arkady Plotnitsky,et al.  Epistemology and Probability: Bohr, Heisenberg, Schrödinger, and the Nature of Quantum-Theoretical Thinking , 2009 .

[11]  Charles H. Kahn,et al.  The Art and Thought of Heraclitus , 1979 .

[12]  S. Segal.  Plato's Ghost: The Modernist Transformation of Mathematics , 2010 .

[13]  Anton Zeilinger,et al.  Wave–particle duality of C60 molecules , 1999, Nature.

[14]  H. Weyl,et al.  The Continuum: A Critical Examination of the Foundation of Analysis , 1987 .

[15]  Helge Kragh,et al.  Dirac: A Scientific Biography , 1991 .

[16]  Albert Einstein,et al.  Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? , 1935 .

[17]  H. Weyl.  The Theory Of Groups And Quantum Mechanics , 1931 .

[18]  E. Condon.  The Theory of Groups and Quantum Mechanics , 1932 .

[19]  W. Heisenberg.  The Physical Principles of the Quantum Theory , 1930 .

[20]  Aristotle.  The Complete Works Of Aristotle , 1954 .

[21]  W. Heisenberg,et al.  Encounters with Einstein: And Other Essays on People, Places, and Particles , 1989 .

[22]  N. Bohr.  II - Can Quantum-Mechanical Description of Physical Reality be Considered Complete? , 1935 .

[23]  Drew McDermott,et al.  A critique of pure reason 1 , 1987, The Philosophy of Artificial Intelligence.

[24]  Abraham Pais,et al.  Einstein and the quantum theory , 1979 .

[25]  W. Bean.  Physics and Philosophy, the Revolution in Modern Science. , 1959 .

[26]  Robert L. Weber,et al.  Sources of Quantum Mechanics , 1968 .

[27]  E. Wigner.  The Unreasonable Effectiveness of Mathematics in the Natural Sciences (reprint) , 1960 .