Abstract
In science as in mathematics, it is popular to know little and resent much about category theory. Less well known is how common it is to know little and like much about set theory. The set theory of almost all scientists, and even the average mathematician, is fundamentally different from the formal set theory that is contrasted against category theory. The latter two are often opposed by saying one emphasizes Substance, the other Form. However, in all known systems of mathematics throughout history, mathematicians have moved fluidly between ideas conceived of as thing-like, property-like, and process-like. On the other hand one way to advance science is to better distinguish between thing, property, and process. All this constitutes a distracting background for those interested in, or distressed by, the possible application of category theory to science, and to mathematics as well.
Similar content being viewed by others
References
Antonelli GA (2008) Logicism, quantifiers, and abstraction. http://orion.uci.edu/~aldo/papers/FONA.pdf. Cited 1 Sept 2008
Baez J (2008) Categorifying fundamental physics. http://math.ucr.edu/home/baez/diary/fqxi_narrative.pdf. Cited 1 Sept 2008
Baez JC, Dolan J (1998) Categorification. In: E Getzler, M Kapranov (eds) Higher category theory. Contemp. Math. 230, American Mathematical Society, Providence, Rhode Island, pp 1–36. arXiv:math/9802029v1. Cited 1 Sept 2008
Barr M (1999) Functional set theory. ftp://ftp.math.mcgill.ca/pub/barr/pdffiles/varset.pdf. Cited 1 Sept 2008
Benacerraf P (1965) What numbers could not be. Philos Rev 74:47–73
Beran MJ (2008) The evolutionary and developmental foundations of mathematics. PLoS Biol 6(2):e19. doi:10.1371/journal.pbio.0060019 Cited 1 Sept 2008
Boysen ST, Berntson GG (1989) Numerical competence in a chimpanzee (Pan troglodytes). J Comp Psychol 103:23–31
Bunge MA (1977) The furniture of the world (vol 3 of Treatise on basic philosophy). Reidel, Dordrecht
Bunge MA (1997 [2001]) Moderate mathematical fictionism. In: M Mahner (ed) Scientific realism: selected essays of Mario Bunge. Prometheus, Amherst, pp 187–203
Cantlon JF, Brannon EM (2007) Basic math in monkeys and college students. PLoS Biol 5(12):e328. doi:10.1371/journal.pbio.0050328 Cited 1 Sept 2008
Cassirer E (1923 [2004]) Substance and function and Einstein’s theory of relativity. Dover, Mineola, New York
Chow T (1996) A short proof of the rook reciprocity theorem. Electron J Comb 3(1): R10. http://www.combinatorics.org/Volume_3/PDFFiles/v3i1r10.pdf. Cited 1 Sept 2008
Diester I, Nieder A (2007) Semantic associations between signs and numerical categories in the prefrontal cortex. PLoS Biol 5:e294. doi:10.1371/journal.pbio.0050294 Cited 1 Sept 2008
Ehresmann A, Vanbremeersch J-P (2007) Memory evolutive systems: hierarchy, emergence, cognition. Elsevier, Amsterdam
Eilenberg S, Mac Lane S (1945) General theory of natural equivalences. Trans Am Math Soc 58:231–294
Frieden BR (1998) Physics from fisher information: a unification. Cambridge University Press, Cambridge
Gárcía-Sucre M (1990) On the relationship between mathematics and physics. In: Weingartner P, Dorn GJW (eds) Studies on mario bunge’s treatise. Rodopi, Amsterdam
Goldblatt R (1984) Topoi, the categorial analysis of logic. Studies in logic and the foundations of mathematics, vol 98. North-Holland, Amsterdam
Halmos PR (1974) Naive set theory. Springer, Berlin
Kary M, Mahner M (2002) How would you know if you synthesized a thinking thing? Mind Mach 12:61–86
Landry E, Marquis J-P (2005) Categories in context: historical, foundational, and philosophical. Philos Math (III) 13:1–43
Lawvere FW (1963) Functorial semantics of algebraic theories. PhD thesis, Columbia University. http://tac.mta.ca/tac/reprints/articles/5/tr5abs.html. Cited 1 Sept 2008
Lawvere FW (1965 [2005]) An elementary theory of the category of sets (long version) with commentary. Repr Theory Appl Categ 11: 5–35. ftp://ftp.tac.mta.ca/pub/tac/html/tac/reprints/articles/11/tr11. Cited 1 Sept 2008
Lawvere FW (1966) The category of categories as a foundation for mathematics. In: Eilenberg S et al (eds) Proceedings of the conference on categorical algebra, La Jolla, 1965. Springer-Verlag, Berlin-Heidelberg, pp 1–21
Lawvere FW (2003) Foundations and applications: axiomatization and education. Bull Symb Logic 9:213–224
Lawvere FW, Schanuel SH (1998) Conceptual mathematics: a first introduction to categories. Cambridge University Press, Cambridge
Letelier J-C, Soto-Andrade S, Guíñez Abarzúa F, Cornish-Bowden A, Cárdenas ML (2006) Organizational invariance and metabolic closure: analysis in terms of (M, R) systems. J Theor Biol 238(4):949–961
Lui SH (1997) An interview with Vladimir Arnol’d. Notices Am Math Soc 44(4):432–438
Marquis J-P (1995) Category theory and the foundations of mathematics: philosophical excavations. Synthese 103:421–447
Marquis J-P (2002) Categories, sets and the nature of mathematical entities. PILM 2002, Philosophical insights into logic and mathematics: the history and outcome of alternative semantics and syntax. Nancy, France. http://www.univ-nancy2.fr/poincare/colloques/symp02/abstracts/marquis.pdf. Cited 1 Sept 2008
Nieder A, Miller EK (2003) Coding of cognitive magnitude: compressed scaling of numerical information in the primate prefrontal cortex. Neuron 37:149–157
Quian Quiroga R, Reddy L, Kreiman G, Koch C, Fried I (2005) Invariant visual representation by single neurons in the human brain. Nature 435(23):1102–1107
Rosen R (1958a) A relational theory of biological systems. Bull Math Biophys 20:245–316
Rosen R (1958b) The representation of biological systems from the standpoint of the theory of categories. Bull Math Biophys 20:317–341
Rosen R (1959) A relational theory of biological systems II. Bull Math Biophys 21:109–128
Shalizi C (2000) Laboring to bring forth a mouse. (Review of Frieden 1998, physics from fisher information). Bactra Rev 118. http://cscs.umich.edu/~crshalizi/reviews/physics-from-fisher-info/. Cited 1 Sept 2008
Uller C, Jaeger R, Guidry G, Martin C (2003) Salamanders (Plethodon cinereus) go for more: rudiments of number in an amphibian. Animal Cognit 6:105–112
Zdravkovska S (1987) Conversation with Vladimir Igorevich Arnol’d. Math Intell 9(4):28–32
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kary, M. (Math, Science, ?). Axiomathes 19, 321–339 (2009). https://doi.org/10.1007/s10516-009-9064-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10516-009-9064-5