Skip to main content
SpringerLink
Log in

Applying mathematics to empirical sciences: flashback to a puzzling disciplinary interaction

  • Published:
Synthese Aims and scope Submit manuscript

Abstract

This paper aims to reassess the philosophical puzzle of the “applicability of mathematics to physical sciences” as a misunderstood disciplinary interplay. If the border isolating mathematics from the empirical world is based on appropriate criteria, how does one explain the fruitfulness of its systematic crossings in recent centuries? An analysis of the evolution of the criteria used to separate mathematics from experimental sciences will shed some light on this question. In this respect, we will highlight the historical influence of three major disciplinary paradigms. According to the Aristotelian classification of the sciences, the separation of mathematics from physics is based on their respective objects of study. The Baconian system distinguishes these sciences by the type of knowledge involved in each field. Finally, the Whewellian disciplinary layout categorises these disciplines by their respective methods. In this paper, we argue that the cascading effect of such disciplinary categorisations—based successively on ontological, epistemological, and heuristic criteria—resulted in a profound redefinition of the border between mathematics and physics, with the consequence of obscuring the foundations of the interplay between these fields. Our approach intends to put forward such a mechanism as a constitutive piece of the puzzle of the applicability of mathematics.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

Notes

  1. The word “discipline” is understood here in the most general sense of a “unit of internal differentiation of science” (Stichweh 2001). Let us note that the conceptual recognition of an autonomous field of research does not necessarily imply its academic institutionalisation (as is especially apparent from a historical perspective).

  2. It should be noted that our main concern in such a task is to support our tripartite disciplinary scheme along the lines of philosophical purposes, rather than targeting an absolute historical accuracy.

  3. See De Caelo 290b1 for a claim on the permanence of celestial bodies, and De Anima II. 7–8 for discussions about light and sound, which are described as incorporeal forms distinguished from sight and hearing.

  4. As Whewell (1837, III.II) has pointed out, Greek’s “geometry” would have been better named “astrometry”.

  5. During the sixteenth century, several scholars, especially Ramus, van Roomen and Snellius, already referred to such practices as “mathematica mixta” (Oki 2013).

  6. The word “idea” actually stands for “principle”, as stated here: “Ideas are not synonymous with Notions; they are Principles which give to our Notions whatever they contain of truth” (Whewell 1840, I.II.6).

  7. Therefore, it is crucial to distinguish “physical objects” from “material objects”: while the former expression is disciplinarily committed, the later remains neutral in this respect.

References

  • Ariew, R. (1990). Christopher Clavius and the classification of sciences. Synthese, 83(2), 293–300.

    Article  Google Scholar 

  • Ariew, R. (1992). Descartes and the tree of knowledge. Synthese, 92(1), 101–116.

    Article  Google Scholar 

  • Bacon, F. (1970). Of the proficience and the advancement of learning. New York: Da Capo Press.

    Google Scholar 

  • Bacon, F. (1612). A description of the intellectual globe. In J. M. Robertson. (1905), The Philosophical Works of Francis Bacon (pp. 670–701). London: Routledge.

  • Bacon, F. (1620). The great instauration. In J. M. Robertson (1905), The philosophical works of Francis Bacon (pp. 243–254). London: Routledge.

  • Bacon, F. (1623). De dignitate et augmentis scientiarum. In J. M. Robertson (1905), The philosophical works of Francis Bacon (pp. 413–638). London: Routledge.

  • Barnes, J. (Ed.). (2014). The complete works of Aristotle: The revised Oxford translation. Princeton: Princeton University Press.

  • Boethius, A. M. (\(\sim \)510) Commentaries on Isagoge. (S. Brandt (1906) Trans.). Leipzig: Freitag.

  • Boethius, A. M. (\(\sim \)520) Theological tractates the consolation of philosophy. (H. F. Stewart, E. K. Rand & S. J. Tester (Eds.) (1973) Trans.). Cambridge, MA: Harvard University Press.

  • Bridges, D., & Richman, F. (1987). Varieties of constructive mathematics. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  • Bridgman, P. W. (1927). The logic of modern physics. New York: Macmillan.

    Google Scholar 

  • Brown, G. I. (1991). The evolution of the term ‘mixed mathematics’. Journal of the History of Ideas, 52(1), 81–102.

    Article  Google Scholar 

  • Buchdahl, G. (1971). Inductivist versus deductivist approaches in the philosophy of science as illustrated by some controversies between whewell and mill. The Monist, 55(3), 343–367.

    Article  Google Scholar 

  • Bueno, O., & Colyvan, M. (2011). An inferential conception of the application of mathematics. Noûs, 45(2), 345–374.

    Article  Google Scholar 

  • Causey, R. (1977). The unity of science. Dordrecht: Reidl.

    Book  Google Scholar 

  • Chabot, H. & Roux, S. (2011). La mathématisation comme problème. Archives contemporaines.

  • Cleary, J. J. (1994). Emending Aristotle’s division of theoretical sciences. The Review of Metaphysics, 48(1), 33–70.

    Google Scholar 

  • Colyvan, M. (2001). The miracle of applied mathematics. Synthese, 127(3), 265–278.

    Article  Google Scholar 

  • Dabars, W. B. (2008). Disciplinarity and Interdisciplinarity: Rhetoric and Context in the American Research University. ProQuest.

  • Darnton, R. (1984). Philosophers trim the tree of knowledge: The epistemological strategy of the encyclopédie. In The Great Cat Massacre: And Other Episodes in French Cultural History (pp. 191–213). Harmondsworth: Penguin.

  • Dear, P. (1995). Discipline and experience: The mathematical way in the scientific revolution. Chicago: The University of Chicago Press.

    Book  Google Scholar 

  • de Chasseneuz, B. (1603). Catalogus Gloriae Mundi. Repr. (2015), ReInk Books.

  • Descartes, R. (1628). Regulae Ad Directionem Ingenii. (G. Heffernan (1998) Transl.) Amsterdam: Rodopi.

  • Distelzweig, P. M. (2013). The intersection of the mathematical and natural sciences: The subordinate sciences in aristotle. Apeiron, 46(2), 85–105.

    Article  Google Scholar 

  • Dorato, M. (2005). The laws of nature and the effectiveness of mathematics. In G. Boniolo, P. Budinich, & M. Trobok (Eds.), The role of mathematics in physical sciences: interdisciplinary and philosophical aspects (pp. 131–144). Dordrecht: Springer.

    Chapter  Google Scholar 

  • Einstein, A. (1921). Geometry and experience. Scientiae Studia, 3(4), 665–675.

    Article  Google Scholar 

  • Fisch, M. (1991). William Whewell, philosopher of science. Oxford: Oxford University Press.

    Google Scholar 

  • Fisher, N. (1990). The classification of the sciences. In R. C. Olby (Ed.), Companion to the history of modern science (pp. 853–868). London: Routledge.

    Google Scholar 

  • French, S. (2000). The reasonable effectiveness of mathematics: partial structures and the application of group theory to physics. Synthese, 125(1/2), 103–120.

    Article  Google Scholar 

  • Gaukroger, S. (1980). Aristotle on intelligible matter. Phronesis, 25(2), 187–197.

    Article  Google Scholar 

  • Gaukroger, S. (2001). Francis Bacon and the transformation of early-modern philosophy. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  • Gingras, Y. (2001). What did mathematics do to physics? History of Science, 39(4), 383–416.

    Article  Google Scholar 

  • Grantham, T. A. (2004). Conceptualizing the (dis)unity of science. Philosophy of Science, 71(2), 133–155.

    Article  Google Scholar 

  • Grosseteste, R. Commentarius In Posteriorum Analyticorum Libros, A Cura Di Pietro Rossi. Reimpr. (1981), Olschki.

  • Haines, V. A. (1992). Spencer’s Philosophy of Science. The British Journal of Sociology, 43(2), 155–172.

    Article  Google Scholar 

  • Hacking, I. (1996). The disunities of the sciences. In P. Galison & D. J. Stump (Eds.), The disunity of science (pp. 37–74). Stanford: Stanford University Press.

    Google Scholar 

  • Holbrook, J. B. (2013). What is interdisciplinary communication? Reflections on the very idea of disciplinary integration. Synthese, 190(11), 1865–1879.

    Article  Google Scholar 

  • Hughes, T. P. (1986). The seamless Web: technology, science, etcetera, etcetera. Social Studies of Science, 16(2), 281–292.

    Article  Google Scholar 

  • Katayama, E. G. (1999). Aristotle on artifacts: A metaphysical puzzle. Albany: SUNY Press.

    Google Scholar 

  • Kawajiri, N. (1980). Francis Bacon’s view of mathematics?: Bacon’s concept of mixed mathematics. Proceedings of the Faculty of Science of Tokai University, 15, 7–21.

    Google Scholar 

  • Klein, J. T. (1990). Interdisciplinarity: History, theory, and practice. Detroit: Wayne State University Press.

    Google Scholar 

  • Kusukawa, S. (1996). Bacon’s classification of knowledge. In M. Peltonen (Ed.), The Cambridge companion to Bacon (pp. 47–74). Cambridge: Cambridge University Press.

    Chapter  Google Scholar 

  • Laird, W. R. (1987). Robert grosseteste on the subalternate sciences. Traditio, 43, 147–169.

    Article  Google Scholar 

  • Lear, J. (1982). Aristotle’s philosophy of mathematics. The Philosophical Review, 91(2), 161–192.

    Article  Google Scholar 

  • Lennox, J. G. (1986). Aristotle, Galileo, and “Mixed Sciences”. In William A. Wallace (Ed.), Reinterpreting Galileo (pp. 29–51). Washington, D.C.: Catholic University of America Press.

    Google Scholar 

  • Lévy-Leblond, J.-M. (1992). Why does physics need mathematics? In E. Ullmann-Margalit (Ed.), The scientific enterprise (pp. 145–161). New York: Springer.

    Chapter  Google Scholar 

  • Mancosu, P. (1991). On the status of proofs by contradiction in the seventeenth century. Synthese, 88(1), 15–41.

    Article  Google Scholar 

  • Mancosu, P. (1996). Philosophy of mathematics and mathematical practice in the seventeenth century. Oxford: Oxford University Press.

    Google Scholar 

  • Maull, N. (1977). Unifying science without reduction. Studies in the History and Philosophy of Science, 8, 143–162.

    Article  Google Scholar 

  • McKirahan, R. D. (1978). Aristotle’s subordinate sciences. The British Journal for the History of Science, 11, 197–220.

    Article  Google Scholar 

  • Mill, J. S. (1843/1882). A system of logic, ratiocinative and inductive: Being a connected view of the principles of evidence and the methods of scientific investigation. New York: Harper & Brothers.

  • Morrison, M. (2000). Unifying physical theories. Physical concepts and mathematical structures. New York: Cambridge University Press.

    Google Scholar 

  • Mueller, I. (1970). Aristotle on geometrical objects. Archiv für Geschichte der Philosophie, 52(2), 156–171.

    Article  Google Scholar 

  • Oki, S. (2013). The establishment of “mixed mathematics” and its decline 1600–1800. Historia Scientiarum, 23(2), 82–91.

    Google Scholar 

  • Pereyra, B. (1576). De communibus omnium rerum naturalium principiis et affectionibus. Rome.

  • Porphyry. Isagoge. (E. W. Warren (1975) Trans.). Pontifical Institute of Mediaeval Studies.

  • Shea, W. R. (1970). Galileo, Scheiner, and the interpretation of sunspots. Isis, 61(4), 498–519.

    Article  Google Scholar 

  • St. Victor, H. (1130). The Didascalicon of Hugh of St. Victor: A medieval guide to the arts (J. Taylor (1961) Trans.). New York: Columbia University Press.

  • Stehr, N., & Weingart, P. (2000). Practising interdisciplinarity. Toronto: University of Toronto Press.

    Google Scholar 

  • Steiner, M. (1998). The applicability of mathematics as a philosophical problem. Cambridge, MA: Harvard University Press.

    Google Scholar 

  • Stichweh, R. (1992). The sociology of scientific disciplines: On the genesis and stability of the disciplinary structure of modern science. Science in Context, 5(1), 3–15.

    Article  Google Scholar 

  • Stichweh, R. (2001), History of Scientific Disciplines. In International Encyclopedia of the Social & Behavioral Sciences, pp. 13727–13731.

  • Suppes, P. (1978). The plurality of science. PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association, 2, 3–16.

    Google Scholar 

  • Thorén, H., & Persson, J. (2013). The philosophy of interdisciplinarity: Sustainability science and problem-feeding. Journal for General Philosophy of Science, 44(2), 337–355.

    Article  Google Scholar 

  • Vert, F. (1994). Note sur la distribution Baconienne des sciences et L’Encyclopédie. Les Études Philosophiques, 3, 357–370.

    Google Scholar 

  • Wettersten, J. (1994). William Whewell: Problems of induction vs. problems of rationality. The British Journal for the Philosophy of Science, 45(2), 716–742.

    Article  Google Scholar 

  • Whewell, W. (1837). The Mechanical Euclid. Cambridge: Cambridge University Press.

  • Whewell, W. (1840). The Philosophy of the Inductive Sciences. London: J. W. Parker.

    Google Scholar 

  • Whewell, W. (1860). On the philosophy of discovery. London: J. W. Parker.

    Google Scholar 

  • Whitney, E. (1990). Paradise restored. The mechanical arts from antiquity through the thirteenth century. Transactions of the American Philosophical Society, 80(1), 1–169.

    Article  Google Scholar 

  • Wigner, E. (1960). The unreasonable effectiveness of mathematics in the natural sciences. Communications in Pure and Applied Mathematics, 13(1), 1–14.

    Article  Google Scholar 

  • Yeo, R. (1993). Defining science: William Whewell, natural knowledge and public debate in early Victorian Britain. Cambridge: Cambridge University Press.

    Book  Google Scholar 

Download references

Acknowledgements

The research carried out in this paper was supported by the Swiss National Science Foundation, through a postdoctoral fellowship at the University of Oxford (grant P2GEP1-155682).

Author information

Authors and Affiliations

  1. Faculty of History, University of Oxford, George Street, Oxford, OX1 2RL, UK

    Raphaël Sandoz

Authors
  1. Raphaël Sandoz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Raphaël Sandoz.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sandoz, R. Applying mathematics to empirical sciences: flashback to a puzzling disciplinary interaction. Synthese 195, 875–898 (2018). https://doi.org/10.1007/s11229-016-1251-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11229-016-1251-5

Keywords

Search

Navigation