To the memory of Ludovico Geymonat, for his genuine encouragement and generous intellectual support.
The problems of Mathematics are not problems in a vacuum. There pulses in them the life of ideas which realize themselves in concreto through our human endeavors in our historical existence, but forming an indissoluble whole transcending any particular science (H. Weyl).
Abstract
The history and philosophy of science are destined to play a fundamental role in an epoch marked by a major scientific revolution. This ongoing revolution, principally affecting mathematics and physics, entails a profound upheaval of our conception of space, space–time, and, consequently, of natural laws themselves. Briefly, this revolution can be summarized by the following two trends: (1) by the search for a unified theory of the four fundamental forces of nature, which are known, as of now, as gravity, electromagnetism, and strong and weak nuclear forces; (2) by the search for new mathematical concepts capable of elucidating and therefore explaining such a relationship. In fact, the first search is essentially dependent on the second; that is to say, that in order for a new theory of physics to come to light, the development of a deeper geometric theory capable of explaining the structure of space–time on a quantum scale appears to be necessary. On careful consideration, we notice that both of these developments converge in the direction of a unitary and fundamental tendency of modern science—which is the geometrization of theoretical physics and of natural sciences. This new emergent situation carries within it a profound conceptual change, affecting the way in which relations are conceived of, first and foremost, between mathematics and physics. This new paradigm can be summed up by the intimately interdependent points: (1) the immense variety of physical phenomena and of natural forms follows from the equally infinite variety of geometric and topological objects that can be made out in space and from which space is made up; (2) the second point, which ensues from the former one and which is of great historical and epistemological significance, is that mathematics is involved in rather than applied to phenomena. In other words, phenomena are effects that emerge from the geometrical structure of space–time. There is no doubt that this new conception of the relationship between the universe of mathematical ideas and objects and the world of natural phenomena is the true scientific revolution of our century, of great conceptual importance, and consequently, capable of changing our view of science and of nature at one and the same time. It is all at once of a scientific, philosophical and aesthetic order.
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Notes
For all that, I am not saying that it will have important practical consequences on our lives nor on the economical and social destiny of our societies, although one shouldn't hasten to entirely exclude the possibility.
Within the same framework, we should recall the very interesting attempt made by David Bohm to give a non-conventional interpretation of quantum mechanics, and furthermore to provide it a more intelligible and essentially ontological meaning. He was thus led to a number of new concepts, whose the most important are these of non-locality and objective wholeness. That is to say, particles may be strongly connected even when they are far apart, and this arises in a way which implies that the whole cannot be reduced to an analysis in terms of its constituent parts. In his important last book, The undivided universe (London 1927), Bohm introduced a radically new overall framework which he called the implicate or enfolded order. First, he showed that the failure of quantum theory and relativity to cohere conceptually already begins to point to the need for such a new order for physics as a whole. He then introduced the implicate order and explain it in terms of a number of examples which illustrates the enfoldment of a whole structure into each region of space, e.g. as happens in a hologram. One of the main new ideas implied by this approach is that the geometry and the dynamics have to be in the same framework, i.e. that of the implicate order. In this way one come to a deep unity between quantum theory and geometry in which each is seen to be inherently conformable to the other. To that purpose, one need not to begin with traditional Cartesian notions of order and then try to impose the dynamics of quantum theory on this order by using the algorithm of ‘quantization’. Rather quantum theory and geometry are united from the very outset and are seen to emerge together from what may be called pre-space.
Weil (1979).
Abraham Flexner has stressed this point in a very appropriate way in the paper “The usefulness of useless knowledge” published in the Harper's Magazine, October 1939. Talking about the equations of magnetism and electricity introduced by Maxwell in his famous treatise published in 1873, and about the problem of the detection and demonstration of the electromagnetic waves, which are the carriers of wireless signals, solved by Heinrich Hertz in 1888, he wrote: “Hertz and Maxwell could invent nothing, but it was their useless theoretical work which was seized upon by a clever technician and which has created new means for communication, utility, and amusement… Who were the useful men? Not Marconi, but Clerk Maxwell and Heinrich Hertz. Hertz and Maxwell were geniuses without thought of use. Marconi was a clever inventor with no thought but use (…) Hertz and Maxwell had done their work without thought of use and that throughout the whole history of science most of the really great discoveries which had ultimately proved to be beneficial to mankind had been made by men and women who were driven not by the desire to be useful but merely the desire to satisfy their curiosity.” Still more interesting appear what he add further: “Institutions of learning should be devoted to the cultivation of curiosity and the less they are deflected by considerations of immediacy of application, the more likely they are to contribute not only to human welfare but to the equally important satisfaction of intellectual interest which may indeed be said to have become the ruling passion of intellectual life in modern times. (…) I am not for a moment suggesting that everything that goes on in laboratories will ultimately turn to some unexpected practical use or that an ultimate practical use is its actual justification. Much more am I pleading for the abolition of the word “use”, and for the freeing of the human spirit. (…) The considerations upon which I have touched emphasize the overwhelming importance of spiritual and intellectual freedom. I have spoken of experimental science; I have spoken of mathematics; but what I say is equally true of music and art and of every other expression of the untrammeled human spirit. The mere fact that they bring satisfaction to an individual soul bent upon its own purification and elevation is all the justification that they need. And in justifying these without any reference whatsoever, implied or actual, to usefulness we justify colleges, universities, and institutes of research. An institution, which sets free successive generations of human souls, is amply justified whether or not this graduate or that makes a so-called useful contribution to human knowledge. A poem, a symphony, a painting, a mathematical truth, a new scientific fact, all bear in themselves all the justification that universities, colleges, and institutes of research need or require.”
An interesting view on this question, which concerns more specifically physics, although different from ours, has been expressed recently by Piet Hut. He asked himself “what does the notion that physics is a complete system, covering all of reality, imply, what is the 'matter' that materialism refers to? According to relativity theory, it includes energy as well, and according to quantum theory, we can no longer talk about uniquely defined states of being.” Further, Hut lays stress on the fact “that materialism cannot hold on to determinism, cannot talk about the existence of anything, although it can talk, and very precisely so, about interactions… Interactions act. They are actions 'inter' other actions. There is a web of actions, but the 'is' does not mean 'existing' in a way that defines any type of state or endurance. Actions act, without anything extra required. (…) it is a fascinating outcome of twentieth-century physics that materialism has tended to converge to phenomenology. Phenomena are understood in ever greater detail, while previous explanations of properties in terms of things that have those properties are being abolished. Properties is all we deal with. Not only is there no need to postulate 'things' underneath, to somehow hold up those properties, but what is more, there is no simple consistent way to do so. Starting with a world given in terms of nouns and verbs, particles and interactions, in classical physics, we are now forced to face a world in terms of verbs only: a world of interactions interacting with interactions. Phenomena have become more concrete, matter has become more fluid, and the two have lost their sharp separation” (I quote an unpublished manuscript). I thank Piet Hut for having drawn my attention on this point and for the interesting discussions we have had at the Institute for Advanced Study in Princeton.
For a deeper analysis of this question, see L. Boi, Geometry and Perception. Mathematical modeling and philosophical interpretations of spatial perception, to be published in 2018.
Poincaré (1902).
Chandrasekhar (1987).
See, for example, on this subject his pioneering work “Topological models in biology,” Topology, 8 (1969), 313–335.
Ibid., pp. 313–314.
See on this important subject L. H. Kauffman (Editor), Knots and applications, World Scientific, Singapore, 1995, especially Chapters 9, 10 and 11.
In this respect, it is interesting to quote what Yang wrote: “… when Mills and I worked on non-Abelian gauge fields, our motivation was completely divorced from general relativity and we did not appreciate that gauge fields and general relativity are somehow related. Only in the late 1960s did I recognize the structural similarity mathematically of non-Abelian gauge fields with general relativity and understand that they both were connections mathematically.”, in “Hermann Weyl’s Contribution to Physics”, Hermann Weyl Centenary Lectures, Ed. by K. Chandrasekharan, Springer-Verlag, 1989, p. 17. For a very good historical overview of the birth and early developments of gauge theory, see L. O’Raifeartaigh, The Dawning of Gauge Theory, Princeton, Princeton University Press, 1997.
In Commentationes societatis regiae scientiarum Gottingensis recentiores, Vol. VI, 99–146. Göttingen, 1828. Reprinted in C. F. Gauss, Werke, Vol. IV, pp. 217–258. Göttingen, 1873.
“Über die Hypothesen, welche die Geometrie zu Grunde liegen,” Abh. Konigl. Gesell. Wiss. Gött., Vol. XIII, 1867 (also in Collected Works, new edition ed. by R. Narasimhan, Heidelberg–Berlin, Springer-Verlag, 1990, pp. 304–319). Bernhard Riemann was one of the outstanding mathematicians of the 19th century. His work revolutioned complex analysis and geometry—Riemann surfaces and the still unsettled Riemann hypothesis, Riemannian geometry and the Riemann integral testify to his lasting influence. One of the main novelties of Riemann’s approach was the way he replaced algorithmic calculations by conceptual reasoning wherever he could. A leitmotif of his was to find results with as little calculation as possible, fast ohne Rechnung. In mathematics as well in physics, Riemann had one underlying principle: He attempted to understand geometry, analytic functions, and physical phenomena from behavior in infinitely small regions. In physics he aimed at what we now call a unified field theory of matter, electricity, magnetism, and gravity. He was also a great philosopher, who wrote some fragmentary, yet very interesting notes on the body-mind problem and about other philosophical and scientific topics.
Published in the Royal Society Transaction, Vol. CLV, 1864. Reprinted in The Scientific Papers of J. C. Maxwell, Vol. 1, Cambridge, 1890.
It should here be mentioned the important fact the Yang-Mills theory retains its gauge symmetry with respect to rotations of the isotopic-spin arrow, but the objects described—protons and neutrons—do not express the symmetry. In despite all this difficulties, the Yang-Mills theory had begun as a model of the strong interactions, but by the time it had been renormalized interest in it centered on applications to weak interactions. In 1967 S. Weinberg, A. Salam and C. Ward have proposed a model of the weak interactions based on a version of the Yang-Mills theory in which the gauge quanta take on mass through the Higgs mechanism. The Weinberg-Salam-Ward model actually embraces both the weak force and electromagnetism. The conjecture on which the model is ultimately founded is a postulate of local invariance with respect to isotopic spin; in order to preserve that invariance four photon-like fields are introduced, rather than the three of the original Yang-Mills theory. The fourth photon could be identified with some primordial form of electromagnetism. It corresponds to a separate force, which had to be added to the theory without explanation. For this reason the model should not be called a unified field theory.
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Acknowledgements
The author was supported by the John Simon Guggenheim Memorial Foundation (New York) and the Canadian Council for Social Sciences and the Humanities (Ottawa), to whom he would like to express his deep gratitude. The author also warmly acknowledges the suggestions, comments and criticisms of Professors Piet Hut, Chiara Nappi and Irving Lavin of the IAS in Princeton.
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Boi, L. Some Mathematical, Epistemological, and Historical Reflections on the Relationship Between Geometry and Reality, Space–Time Theory and the Geometrization of Theoretical Physics, from Riemann to Weyl and Beyond. Found Sci 24, 1–38 (2019). https://doi.org/10.1007/s10699-018-9550-6
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DOI: https://doi.org/10.1007/s10699-018-9550-6