Abstract
The relationship between physics and mathematics is reviewed upgrading the common in physics classes’ perspective of mathematics as a toolkit for physics. The nature of the physics-mathematics relationship is considered along a certain historical path. The triadic hierarchical structure of discipline-culture helps to identify different ways in which mathematics is used in physics and to appreciate its contribution, to recognize the difference between mathematics and physics as disciplines in approaches, values, methods, and forms. We mentioned certain forms of mathematical knowledge important for physics but often missing in school curricula. The geometrical mode of codification of mathematical knowledge is compared with the analytical one in context of teaching school physics and mathematics; their complementarity is exemplified. Teaching may adopt the examples facilitating the claims of the study to reach science literacy and meaningful learning.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
Heidegger pointed to the etymology of mathematics coming from Greek and mathematical as being learned but refined this by the condition essential for science: awareness of fundamental assumptions which actually makes something worth to be learned as reliable and different from opinion, the true knowledge (Heidegger, 1967, pp. 247–282). This is, however, not the meaning of mathematical commonly held.
It was a highly spiritual trend of natural philosophy of the middle ages to ascribe God the status of the great architect, designer of the universe (Fig. 1). One may get this spirit from the study reflected in the book with a romantic title God’s Philosophers (Hannam, 2009). The representative image of God holding compasses became iconic and was used by Blake in his picture of Newton, shown as revealing of that same design.
This understanding brought Hume to state “demarcation of the necessary statements of mathematics from the contingent statements of empirical science” (Losee, 2001, p. 93).
It is illustrative to see this feature of mathematical modeling as a fundamental difference between the works of Leonardo and Galileo (Galili, 2015).
Salviati expressed Galileo’s view on that: “I am at your service if only I can call to mind what I learned from our Academician who had thought much upon this subject and according to his custom had demonstrated everything by geometrical methods so that one might fairly call this a new science. For, although some of his conclusions had been reached by others, first of all by Aristotle, these are not the most beautiful and, what is more important, they had not been proven in a rigid manner from fundamental principles.” (Galilei, 1638/1914, p. 6)
For Newton, the central law of mechanics was his First Law (Galili & Tseitlin, 2003). However, after conversion of the originally integral statement of the Second Law (Δp ∝ FΔt) to its differential form by Euler in 1738 (F = ma), the Second Law became the classical equation of motion.
Though very different in their methods and distanced by 2000 years, both Aristotle and Faraday never used any mathematics, in the sense of formalism, in their account of reality.
The explanations provided by Faraday, Ampere, Biot, and Savart for the observed phenomena were essentially different (Assis and Chaib, 2015).
Worth to mention that another theory of electromagnetism, which draws on the kind of action at a distance introduced by Ampere and Weber (Assis, 1994), does not use ether. Yet, it ignores the question how interaction is performed exactly as Newton did.
Thus, for instance, from many mathematical options, the quantum theory has only Hilbert space for state vectors and only Hermitian operators to represent observables (Wigner, 1995, p. 541).
Engels (1877, p. 39) wrote: “… in the philosophy of nature it [mathematics] is something completely empirical, taken from the external world and then divorced from it.” Boniolo and Budinich (2005, p. 76) mentioned five accounts for mathematics effectiveness of which three bare metaphysical nature. Here, we are interested in the rest two ascribed to Kant. They explained by the fact that (i) we cognize the world by constructing mathematics and (ii) by means of mathematics we construct concepts of reality to which we have no other access. The two point to the empirical origin of mathematics as rooted in human evolutionary activity of iterative nature. Missing this dynamical perspective creates the enigma of learning possible in principle. Thus, trying to explain the learning success brought Socrates to the alternative explanation, the exoteric model of learning by recollection as a solution of Meno’s paradox (Plato, 2005, p. 113).
Many years later, this image of mathematics depicted by Maxwell was matched in fact by the passionate opinion of the celebrated mathematician Arnold (1998).
Besides the space-time revolution, Einstein performed a cardinal epistemological shift in his theory of special relativity which we do not address here (e.g., Howard, 2005; Goldberg, 1983). This was the major aspect in eyes of those philosophers who were “unwilling to differentiate that which was sensed from that which was deduced mentally” (Canales, 2015, p. 340). This impact of Einstein theory can be illustrated by his debate with Bergson about time, local and distant, and simultaneity. Thus, Bergson, among others, claimed no fixed boundary between physical and mental events, distinction between here and there, now and later (ibid., p. 343)
This trend of thought, seeking the all-inclusive universal harmony across form, number, sound never stopped in culture. In the seventeenth century, Kepler seriously elaborated his perception of the regularity of planets’ speeds and positions as organized in sets as the musical octave. His Chapter 5 in Book 5 in The Harmony of the World (Kepler, 1619/1997, p. 431) was titled “That the Positions in the System, or the Notes of the Musical Scale, and the Kinds of Melody, Hard and Soft, Have Been Expressed in the Apparent (to observers on the sun, so to speak) Planetary Motions.”
This impact, which is about logical rather than numerical proof drawing on the postulated assumptions, seemingly matches Heidegger’s understanding of “mathematical” as that to be learned and represented the essence of being scientific (Heidegger, 1967, pp. 66–108). This aspect was often lacking in the knowledge reached prior to Greek civilization (e.g., Montgomery and Kumar, 2016, pp. 157–158).
One may illustrate the mentioned important transition using an appealing artistic representation (Appendix).
In fact, basing on this result, he could claim the principle of equivalence—the gravitational force is equivalent to the centrifugal force in a rotational frame of reference. Yet, for that, the mathematical result by itself was insufficient. One needs the conceptual frame as reached only by Einstein. This result of Huygens pointed to the operational definition of weight concept hinted by him but actually introduced only in the twentieth century (Galili, 2001, 2012).
It was further elaborated by later development of the new theory of generalized functions (Schwarz, 1950).
Paul Dirac relativistic equation for electron in 1928 provided solutions with negative energy. They were interpreted as corresponding to antielectron—a positron—after Carl D. Anderson in 1932 (Nobel Prize in 1936) reported the detection of a positive particle possessing the same mass as the electron. Murray Gell-Mann and previously Yuval Ne’eman predicted the omega baryon particle in 1961 drawing on the suggested new symmetry in quarks classification.
Yet, some mathematicians (Courant and Robbins, 1996) and physicists (Feynman, 1965) prefer to stress that mathematical objects remained undefined in another sense: they are vacant to represent any real object. In that sense, Feynman (1965) mentioned: “Mathematicians are only dealing with the structure of reasoning, and they do not really care what they are talking about.” Inside their world, the mathematical objects are precisely defined; triangles cannot be confused with squares.
Moreover, Feynman (1965, p. 40) defined this way mathematics, saying: “Mathematics, then, is a way of going from one set of statements to another”. Mathematicians, however, would seemingly not agree with this claim as too simplified, that is, neglecting just the other part—the construction of complex albeit abstract objects and determining their features.
Mathematicians may see a more complex picture. In What is mathematics? Courant and Robbins (1996) say: “Mathematics is nothing but a system of conclusions drawn from definitions and postulates that must be consistent but otherwise may be created by the free will of the mathematicians.” But then, they proceeded: “…mathematics hovers uneasily between the real and the not-real; it’s meaning does not reside in formal abstraction, but neither is it tangible. This may cause problems to philosophers who like tidy categories, but it is the great strength of mathematics—what I have elsewhere called its “unreal reality.” Mathematics links the mental world of mental concepts to the real world of physical things without being located completely in either.” This confession may be related to the understanding of mathematics by Galileo as mentioned above (cf. footnote 9).
This aspect essentially moderates the quoted claim of Wigner (1995) regarding the effectiveness of mathematics in depicting reality.
In reviewing a study regarding teaching physics in some country, I found physics defined as the area of knowledge to produce atomic bomb… This case illustrates the danger of identifying physics with its products.
A somewhat similar structure was suggested by Lakatos (1978a) to depict Scientific Research Program in physics. The components of the structures do not coincide. In particular, Lakatos considered physical theory in a strictly disciplinary perspective, while the Discipline-Culture includes conceptually contrasting elements in the nucleus-periphery confrontation. This feature helps here in the comparison of physics with mathematics.
In case of comparison between classical and quantum mechanics, this would be the limit of h → 0
An example of a still not settled in mathematics object but in common use in physics (area III) is the path integrals introduced by Feynman. Manin (1981, p. 15) wrote: “It would hardly be possible to construct a consistent and applicable mathematical theory of Feynman integrals without progress in the understanding of physics.” (p. 93). Johnson and Lapidus (2000, p. 696) added anticipating: “… it seems that a suitable geometrization (yet to be discovered) of some notion of Feynman path integral will play a significant role.”
Irrational numbers are those that cannot be expressed as a ratio of integers, e.g., \( \sqrt{2} \) . Transcendental numbers are those which cannot be a root of a polynomial equation with integer (or rational) coefficients. π and e are such numbers
Other problems of this kind were calculation of π, squaring a circle, doubling a cube, comprehending incommensurability of certain segments demonstrated specific values of mathematics (Beckman, 1971; Katz, 2009). Pythagoreans kept it in secret and considered as a major threat to their worldview of number harmony making this knowledge public was allegedly punished by death penalty (Hearth, 1921, p. 65)
The famous example could be the proof of the Big Theorem of Fermat in the theory of numbers which was reached using the knowledge from a very distant area of mathematical knowledge dealing with elliptic curves.
It is more precise to say that in some cases, an algebraic account is more inclusive than a geometrical one, especially when the latter addresses a special case of a certain subject. For instance, the analytical form of quadratic equation to represent a parabola is more inclusive than any single case of parabola graph. The idea of geometrical representation of algebraic/arithmetic operations in general led Descartes in his introduction of analytical geometry (e.g., Boyer, 1991, p. 337) and the so-called “Cartesian spirit” of extinction of graphs from mathematics (Arnold, 1998).
In fact, such redundancy provides stability of knowledge justification, attained through multiple and many faceted justification of the same contents.
It was at the time of the scientific revolution of the seventeenth century that the mathematical proof and in particular by means of geometry entered the method of science often replacing the requirement of mechanical (“substantialist”) explanation of natural phenomena (Gingras, 2001).
This dichotomy echoed in confrontation between the proponents of exclusively algebraic approach in mathematical instruction, as being a more advanced (inclusive), with those defending geometrical thinking in the debate of Serre and Arnold (Arnold, 1998, 2002). In the following, we address it supporting the stance of Arnold.
Radical changes and hot debates have being observed regarding mathematical curricula in France and Russia corresponding to polar opposed understanding of the role and status of geometry in mathematics (Arnold, 2002).
Courant and Robbins (1996, p. 59) ascribed this way to Eudoxus, Pythagorean, and stated it to be a masterpiece of mathematics often omitted from the diluted high-school texts. We found this proof in the old textbook of school mathematics by Kiselov (1892/2004, pp. 111–112) preserved in use in Russian schools for more than a hundred years. In mathematical literature, it was addressed by Apostol (2000).
In the following instruction, the discipline-cultural approach suggests for teaching physics to mention the complexity of the process in eyes of mathematics thus providing deeper meaning in comparison.
With respect to involving geometrical, graphical approach in addition to algebraic, one may mention the study which informed about missing graphical representation of kinematics in Palestinian schools (El-Mimi, 2005). The researcher introduced such graphical account and reported about significant beneficial impact, especially on behalf of the students of middle and low level of achievements.
According to Einstein’s comments, he imagined to himself the time-space continuum in terms of “rods and clocks,” spacious images but not elaborated in geometric claims (e.g., Einstein and Infeld, 1938, pp. 186–202): “Time is determined by clocks, space co-ordinates by rods, and the result of their determination may depend on the behavior of these clocks and rods when in motion.” (p. 196)
This perception may remind the perspective of “the two cultures” suggested by C.P. Snow in 1959 with regard to science and humanities (Snow, 1962). The discipline-culture approach removes the oversimplified perspective of separation with respect to knowledge representation (Tseitlin and Galili, 2005).
References
Abiko, S. (2005a). The origins and concepts of special relativity. In M. M. Capria (Ed.), Physics before and after Einstein, Ch. 4. Amsterdam: IOS Press.
Abiko, S. (2005b). The light-velocity postulate the essential difference between the theories of Lorentz-Poincaré and Einstein. Science & Education, 14, 353–365.
Al Khalili, J. (2012). Pathfinders: the golden age of Arabic science. London: Penguin.
Al Biruni (Abu Rayhan) (1966). Geodezia. Tashkent, UzSSR: Academy of Science (in Russian).
Allen, D. H. (2014). How mechanics shaped the modern world. New York: Springer.
Alonso, M., & Finn, E. J. (1967). Fields & Waves. Fundamental University Physics (Vol. Vol. 2). Reading, Massachusetts: Addison-Wesley.
Apostol, T. M. (2000). Irrationality of the square root of two—a geometric proof. The American Mathematical Monthly, 107(9), 841–842.
Archimedes. (1952a). Quadrature of the parabola. The Works of Archimedes. Chicago: The University of Chicago.
Archimedes. (1952b). The method treating of mechanical problems. The Works of Archimedes: Chicago, The University of Chicago.
Arnold, V. (1998). On teaching mathematics. Russian Math. Surveys, 53, 229–236.
Arnold, V. I. (2002). Математическая Дуэль вокруг Бурбаки. Вестник Российской Академии Наук, 72(3), 245–250.
Assis, A. K. T. (1994). Weber’s electrodynamics. Dordrecht: Kluwer Academic Publishers.
Assis, A. K. T., & Chaib, J. P. M. C. (2015). Ampère’s electrodynamics. Aperion: Roy Keys.
Babb, J. (2005). Mathematical concepts and proofs from Nicole Oresme Using History Calculus Teach Mathematics. Science & Education, 14, 443–456.
Beckmann, P. (1971). A history of PI. New York: St. Martin's Press.
Bergmann, P. G. (1942). Introduction to the theory of relativity. New York: Prentice-Hall.
Berry, A. (1961). A short history of astronomy. New York: Dover.
Bohr, N. (1949/1959). Discussion with Einstein on epistemological problems in atomic physics. In P. A. Schilpp (Ed.), Albert Einstein: Philosopher-Scientist (pp. 199–241). New York: Harper Torchbooks.
Boniolo, G., & Budinich, P. (2005). The role of mathematics in physical sciences and Dirac’s methodological revolution. In G. Boniolo, P. Budinich, & M. Trobok (Eds.), The Role of Mathematics in Physical Sciences. Interdisciplinary and Philosophical Aspects (pp. 75–96). Dordrecht, The Netherlands: Springer.
Born, M. (1962). Einstein’s theory of relativity. New York: Dover.
Bridgman, P. W. (1931). Dimensional Analysis. New Haven. London: Yale University Press.
Brunschwig, J., & Lloyd, G. E. R. (2000). Greek thought. Cambridge, Massachusetts: The Belknap Press of Harvard University Press.
Boyer, C. B. (1991). A history of mathematics. New York: Wiley.
Canales, J. (2015). The Physicist & The Philosopher. Einstein, Bergson, and the debate that changed our understanding of time. Princeton and Oxford: Princeton University Press.
Chabay, R., & Scherwood, B. (1995). Electric and magnetic interactions. New York: Wiley.
Clegg, B. (2003). The first scientist: a life of Roger Bacon. London: Constable.
Cohen, M. R., & Drabkin, I. E. (1966). A source book in Greek science. Cambridge: Harvard University Press.
Corless, R. M. & Fillion, N. (2013). A graduate introduction to numerical methods. New York: Springer.
Courant, R., & Robbins, H. (1996). What is mathematics? An elementary approach to ideas and methods. Oxford: Oxford University Press.
Crombie, A. C. (1996). Science, art and nature in medieval and modern thought. London: The Hambledon Press.
Crowe, M. J. (1967). A history of vector analysis. New York: Dover.
D’Alembert, J. (1743/1950). Traité de dynamique./Dinamika. Moscow: Fiziko-Teoreticheskaya Literatura.
Darrigol, O. (2000). Electrodynamics from ampere to Einstein. New York: Oxford University Press.
de Broglie, L. (1949). A general survey of the scientific work of Albert Einstein. In P. A. Schilpp (Ed.), Albert Einstein: Philosopher-scientist (pp. 106–128). New York: MJF Books.
Descartes, R. (1644/1998). The world and other writings. New York: Cambridge University Press.
Dijksterhuis, E. J. (1986). The mechanization of the world picture. Princeton: Princeton University Press.
Dirac, P. A. M. (1958). The principles of quantum mechanics. Oxford: Oxford University Press.
Dreyer, J. L. E. (1953). A history of Astronomy from Thales to Kepler. New York: Dover.
Einstein, A. (1905/1923). On the Electrodynamics of Moving Bodies. In The Principle of Relativity. London: Methuen and Company.
Einstein, A. (1921/2008). Geometry and experience. The lecture at 27 January 1921 at the Prussian Academy of Sciences in Berlin. Published by Methuen & Co. Ltd, London. In A. Diem & D. Lane (Eds.) Illuminated geometry. Walnut: Mt. San Antonio College.
Einstein, A. (1949/1970). Autobiographical notes. In Schilpp, P. A. (Ed.) Albert Einstein: Philosopher-Scientist. La Salle, Ill: Open Court, pp. 2–94.
Einstein, A. (1973). Methods of theoretical physics. In: Ideas and Opinions. New York: Dell.
Einstein, A., & Infeld, L. (1938). The evolution of physics. Cambridge: Cambridge University Press.
El-Mimi O. (2005). Physics instruction contents in perspective of implementations of physics education research - cross national perspective. Ph.D. Thesis. The Hebrew University of Jerusalem. Unpublished.
Engels, F. (1877/1987). Anti-Dühring. Herr Eugen Dühring’s revolution in science. New York: International Publishers.
Euler, L. (1736/1938). Mechanica. Moscow: Technico-Teoreticheskaya Literatura.
Faraday, M. (1952). Experimental researches in electricity. In Great Books of the Western World. Chicago: Encyclopaedia Britannica.
Feynman, R. (1965). The character of physical law. Cambridge: The MIT Press.
Feynman, R. (1966). Lectures on Physics. With R. Leighton and M. Sands. Addison-Wesley, Reading, MA, vol. II.
Finocchiaro, M. A. (2010). Defending Copernicus and Galileo: critical reasoning in the two affairs. Dordrecht: Springer.
Fleming, J. A. (1902). Magnets and electric currents. London: E. & F. N. Spon.
French, A. (1968). Special relativity. The MIT introductory series. New York: Norton.
Galilei, G. (1623/1957). The assayer. In Discoveries and Opinions of Galileo. New York: Anchor Books.
Galilei, G. (1632/1967). Dialogue concerning the two chief world systems, Ptolemaic and Copernican. Berkeley: University of California Press.
Galilei, G. (1638/1914). Dialogues concerning two new sciences. New York: Dover.
Galili, I. (2001). Weight versus gravitational force: historical and educational perspectives. International Journal of Science Education, 23(10), 1073–1093.
Galili, I. (2011). James Hannam: God's Philosophers. How the Medieval World Laid the Foundations of Modern Science. Science & Education, 21(3), 415–422.
Galili, I. (2012). Cultural content knowledge—the case of physics education. International Journal of Innovation in Science and Mathematics Education, 20(2), 1–13.
Galili, I. (2014). Teaching optics: a historico-philosophical perspective. In M. R. Matthews (Ed.). International Handbook of Research in History and Philosophy for Science and Mathematics Education, pp. 97-128, Springer.
Galili, I. (2015). From comparison between scientists to gaining cultural scientific knowledge: Leonardo and Galileo. Science & Education, 25(1), 115–145.
Galili, I., & Tseitlin, M. (2003). Newton’s first law: text, translations, interpretations, and physics education. Science and Education, 12(1), 45–73.
Gingras, Y. (2001). What did mathematics do to physics? History of Science, 39(4), 383–416.
Goldberg, S. (1983). Albert Einstein and the creative act: the case of special relativity. In R. Aris, H. T. Davis, & R. H. Stuewer (Eds.), Springs of scientific creativity: essays on founders of modern science. Minneapolis: The University of Minnesota Press.
Goldreigh, P., Mahajan, S., & Plinney, S. (1999). Order of Magnitude Physics: Understanding the World with Dimensional Analysis, Educated Guesswork, and white Lies. www.inference.phy.cam.ac.uk/sanjoy/oom/book-letter.pdf
Goldenbaum, U. (2016). The geometrical method as a new standard of truth, based on the mathematization of nature. In G. Gorham, B. Hill, E. Slowik, & C. Waters (Eds.), The Language of Nature: Reassessing the Mathematization of Natural Philosophy in the Seventeenth Century. Minneapolis; London: University of Minnesota Press.
Gorelik, G. S. (1959). Oscillations and waves. Moscow: State-Publishing House of Physics-Mathematics Literature.
Gutfreund, H., & Renn, J. (2015). The road to relativity. Princeton: Princeton University Press.
Handscomb, D. C. (1966). Methods of numerical approximation. Oxford: Pergamon Press.
Hannam, J. (2009). God’s philosophers. In How the medieval world laid the foundations of modern science? London: Icon Books Ltd..
Hearth, T. Sir. (1921). A History of Greek Mathematics (Vol. 1). Oxford: The Clarendon Press.
Harvey, S. (2000). The medieval Hebrew encyclopedias of science and philosophy. Dordrecht: Springer Science.
Hecht, E. (1994). Physics. Pacific Grove, CA: Brooks/Cole.
Heidegger, M. (1967). Modern science, metaphysics, and mathematics. In What is a Thing? Chicago: Henry Regnery.
Hobbes, Th. (1660). Leviathan. Of Reason and Science (Ch. 5).
Holton, G. (1973). Thematic origins of scientific thought: Kepler to Einstein. Part II, "On relativity theory". Cambridge, Mass: Harvard University Press.
Holton, G. (2008). Who was Einstein? Why is he still so alive? In P. L. Galison, G. Holton, & S. S. Schweber (Eds.), Einstein for the 21st century: his legacy in science, art, and modern culture. Princeton: Princeton University Press.
Howard, D. A. (2005). Albert Einstein as a philosopher of science. Physics Today, 58(12), 34–40.
Huygens, Ch. (1659/1703). On centrifugal force. From De vis Centrifuga, in Oeuvres Complètes, Vol. XVI, pp. 255-301, Translated by M. S. Mahoney. Online: http://www.princeton.edu/~hos/mike/texts/huygens/centriforce/huyforce.htm.
Huygens, Ch. (1690/1912). Treatise on light: In which are explained the causes of that which occurs in reflection & in refraction, and particularly in the strange refraction of Iceland crystal. London: McMillan.
Johnson, G. W., & Lapidus, M. L. (2000). The Feynman integral and Feynman’s operational calculus. Oxford: Oxford Science Publication, Clarendon Press.
Inwood, M. (1999). A Heidegger Dictionary. Bodmin, Cornwall: Blackwell.
Kapitza, P. L. (1966). Physics problems. Moscow: Znanie.
Karam, R., & Krey, O. (2015). Quod erat demonstrandum: understanding and explaining equations in physics teacher education. Science & Education, 24, 661–698.
Katz, V. J. (2009). A history of mathematics: an introduction. Boston: Addison-Wesley.
Kepler, J. (1619/1997). The harmony of the world. Philadelphia, USA: the American Philosophical Society.
Kepler, J. (1621/1972). Epitome of Copernican Astronomy. In Great Books of the Western World, Vol. 15, p. 845. Chicago: Britannica.
Kipnis, N. (1991). History of the principle of interference of light. Basel: Birkhauser Verlag.
Kiselov, A. P. (1892/2004). Geometry. Textbook for schools. Moscow: Fismatlit.
Kogan, B. Y. (1968). Dimension of the physical quantity. Moscow: Nauka.
Krieger, M. H. (1987). The Physicist’s toolkit. American Journal of Physics, 55, 1033–1038.
Lagrange, J. L. (1788/1997). Mecanique analytique. Paris: Veuve Desaint/Springer.
Lakatos, I. (1978a). The methodology of scientific research programs. Cambridge: Cambridge University Press.
Lakatos, I. (1978b). What does a mathematical proof prove? In Mathematics, science and epistemology. Philosophical papers 2. Cambridge: Cambridge University Press.
Landau, L. D. & Lifshitz, E. M. (1976). Mechanics. Course of Theoretical Physics, Vol. 1. Amsterdam: Elsevier
Liebscher, D. E. (1977). Relativitatstheorie mit Zirkel und Lineal. Berlin: Academie-Verlag.
Lindberg, D. (1992). The beginning of western science. Chicago: University of Chicago Press.
Linton, C. M. (2004). From Eudoxus to Einstein: a history of mathematical astronomy. Cambridge: Cambridge University Press.
Losee, J. (2001). A historical introduction to the philosophy of science. New York: Oxford University Press.
Mahajan, S. (2010). Street-fighting mathematics: the art of educated guessing and opportunistic problem solving. Boston: MIT Press.
Manin, Y. I. (1981). Mathematics and physics. Boston: Birkhauser.
Martins, R. A. (2005). Mechanics and electromagnetism in the late nineteenth century: the dynamics of Maxwell’s ether. In M. M. Capria (Ed.), Physics before and after Einstein, Ch.2. Amsterdam: IOS Press.
Maxwell, J. C. (1873). Treatise on electricity and magnetism. Oxford: Clarendon Press.
Merzbach, U. C., & Boyer, C. B. (2011). A history of mathematics. New York: Wiley.
Migdal, A. B. (1990). Physics and Philosophy. Voprosi Filosopfii, 1, 5–32. Moscow: Nauka (in Russian).
Miller, A. M. (1981). Albert Einstein’s special theory of relativity. Emergence (1905) and early interpretation (1905–1911). Reading: Addison-Wesley.
Montgomery, S. L., & Kumar, A. (2016). A history of science in world cultures: voices of knowledge. London and New York: Routledge.
Nersessian, N. J. (1984). Faraday to Einstein: constructing meaning in scientific theories. Dordrecht, The Netherland: Kluwer Academic Publishers.
Neugebauer, O. (1993). The exact sciences in antiquity. New York: Barrens & Noble.
Newton, I. (1678/1999). The Principia. Mathematical Principles of Natural Philosophy. Berkeley: University of California Press.
Palmerino, C. R. (2016). Reading the book of nature: the ontological and epistemological underpinnings of Galileo’s mathematical realism. In G. Gorham, B. Hill, E. Slowik, & C. Waters (Eds.), The language of nature: reassessing the mathematization of natural philosophy in the seventeenth century. Minneapolis: University of Minnesota Press.
Pedersen, O., & Pihl, M. (1974). Early physics and astronomy. London: McDonald & Janes.
Pines, S. (2000). Studies in Abul Barakat Al Baghdadi. Physics and Metaphysics. Jerusalem: Magness Press.
Plato. (2005). Meno and other dialogues. Oxford: Oxford University Press.
Poincare, H. (2003). Science and method. New York: Dover.
Pospiech, G., Geyer, M., Böhm, U., Lehavi, Y., Bagno, E., & Eylon, B.-S. (2015). The role of mathematics for physics teaching and understanding. Il Nuovo Cimento, 38(C), 1–10.
Redish, E. F., & Kuo, E. (2015). Language of physics, language of math: disciplinary culture and dynamic epistemology. Science & Education, 24, 561–590.
Ross, D. Sir (1923/1995). Aristotle. New York: Routledge.
Russo, L. (2004). The forgotten revolution: how science was born in 300 B.C. and why it had to be reborn. Berlin: Springer.
Sartori, L. (1996). Understanding relativity. Berkeley: University of California Press.
Schiefsky, M. J. (2007). Theory and practice in Heron's mechanics. In Mechanics and Natural Philosophy before the Scientific Revolution, W. R. Laird & S. Roux (eds). Boston Studies in the Philosophy of Science 254. New York: Springer.
Schwarz, L. (1950). Theorie des distributions. Paris: Herman.
Shadowitz, A. (1968). Special relativity. New York: Dover.
Shapiro, A. E. (1984). Experiment and mathematics in Newton’s theory of color. Physics Today, 37(9), 34–42.
Snow, C. P. (1962). The two cultures (pp. 1-20). In The Two Cultures and the Scientific Revolution. Cambridge at the University Press.
Szirtes, T. (2006). Applied dimensional analysis and modeling. Elsevier Science & Technology Books.
Taylor, J. R. (1982). An introduction to error analysis: the study of uncertainties if physical measurements. Susalito, CA: University Science Books.
Taylor, E. F., & Wheeler, J. A. (1997). Space-time physics. New York: Freedman & Co.
Tseitlin, M., & Galili, I. (2005). Teaching physics in looking for its self: from a physics-discipline to a physics-culture. Science & Education, 14(3–5), 235–261.
Tweney, R. D. (2011). Representing the electromagnetic field: how Maxwell’s mathematics empowered Faraday’s field theory. Science & Education, 20, 687–700.
Tzanakis, C. (2000). Presenting the relation between mathematics and physics on the basis of their history: a genetic approach. In V. Katz (Ed.), Using history to teach mathematics. Washington, DC: The Mathematical Association of America.
Vinitsky, L. (2015). On the role of mathematics in the physics curriculum: towards understanding and compatibility. Ph.D. Thesis. The Hebrew University of Jerusalem, Israel.
Vinitsky, L., & Galili, I. (2013). Refinement of logico-mathematical intelligence in the context of physics education. US-China Educational Review, 8A(3), 604–609.
Vinitsky, L., & Galili, I. (2014). The need to clarify the relationship between physics and mathematics in science curriculum: cultural knowledge as possible framework. Procedia-Social and Behavioral Sciences, 116, 611–616.
Von Neumann, J. (1955). Mathematical foundations of quantum mechanics. Princeton: Princeton University Press.
Walker, J. (2008). Fundamentals of Physics (8th ed.). Danvers, Massachusetts: Wiley.
Weinstein, G. (2012). Genesis of general relativity—discovery of general relativity. https://arxiv.org/ftp/arxiv/papers/1204/1204.3386.pdf
Wigner, E. (1995). The unreasonable effectiveness of mathematics in natural science. In J. Mehra & A. S. Whightman (Eds.), The collected works of Eugene Paul Wigner (Vol. 6). Berlin: Springer-Verlag.
Whittaker, E. (1960). A history of the theories of aether and electricity. New York: Harper.
Young, H. D., & Freedman, R. A. (2012). Sears and Zemansky’s university physics. Boston: Pearson, Addison-Wesley.
Zeldovich, Y. B. (2010). Advanced mathematics for beginners and it application to physics. Moscow: Fizmatlit.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
No potential conflict of interest is reported by the author.
Appendix
Appendix
With regard to Greek science, one may illustrate the mentioned changes of epistemological preferences using the imagery appearing in Rafael’s fresco The School of Athens (1504) (Fig. 9). The first swing was from the Pythagorean number paradigm to Platonian theory first paradigm (the claim of initial ideal forms design). Still within the theoretical account, Aristotle changed the game and removed mathematics from the natural philosophy. He drew on the self-evident qualitative principles based on experience. The next change was the refinement of the Aristotelian empiricism returning mathematics to the theory of cosmos. That change facilitated the Hellenistic paradigm of the natural philosophy.
a Using Rafael’s fresco The school of Athens (1504) to depict the dynamics of changes in the relationship of science (natural philosophy) and mathematics. Arrows and numbers point to the changes in time from left to right. 1—from Pythagoras and his “all are numbers” paradigm (its celebrating application—the harmonic intervals of musical octave (epogdoon b)—is shown on the table held for Pythagoras) to Plato’s theory first (rationalism); 2—from Plato to Aristotelian experience first (empiricism); and 3—from Aristotle to Hellenistic account by Euclid in the form of hierarchical deductive mathematical theory, geometry (a geometrical construction is shown on the tablet c)
During the scientific revolution of the seventeenth century, the depicted three epistemological preferences were still considered representative and were addressed by Hobbes (1660, Ch. 5) as three alternative ways to represent the knowledge of science:
For as arithmeticians teach to add and subtract in numbers, so the geometricians teach the same in lines, figures (solid and superficial), angles, proportions, times, degrees of swiftness, force, power, and the like; the logicians teach the same in consequences of words, adding together two names to make an affirmation, and two affirmations to make a syllogism, and many syllogisms to make a demonstration; and from the sum, or conclusion of a syllogism, they subtract one proposition to find the other. (Emphasis added)
Rights and permissions
About this article
Cite this article
Galili, I. Physics and Mathematics as Interwoven Disciplines in Science Education. Sci & Educ 27, 7–37 (2018). https://doi.org/10.1007/s11191-018-9958-y
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11191-018-9958-y