Abstract
In their correspondence on the nature of axioms, Frege and Hilbert clashed over the question of how best to understand axiomatic mathematical theories and, in particular, the nonlogical terminology occurring in axioms. According to Frege, axioms are best viewed as attempts to assert fundamental truths about a previously given subject matter. Hilbert disagreed emphatically, holding that axioms contextually define their subject matter; thus, so long as an axiom system implies no contradiction, its axioms are to be thought of as true. This paper considers whether it is possible to extend Hilbert’s “algebraic” view of axioms to preaxiomatic mathematical reasoning, where our mathematical concepts are not yet pinned down by axiomatic definitions. I argue that, even at the preaxiomatic stage, our informal characterizations of mathematical concepts are determinate enough that viewing our mathematical theories as setting well-defined “problems” with mathematical concepts as “solutions” remains illuminating.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
The label ‘preaxiomatic’ may suggest the expectation that such theories will eventually receive an axiomatization. There is, however, scope for important mathematical reasoning that never becomes axiomatized. I take the considerations of this paper to apply to such theories as well as to theories that eventually succumb to axiomatization.
- 2.
As opposed to truth relative to a particular interpretation of the theory’s axioms.
- 3.
For this reason, one should be wary of Hilbert’s rather misleading characterization of consistent axioms as true by definition. According to Hilbert, “if the arbitrarily given axioms do not contradict each other with all their consequences, then they are true and the things defined by the axioms exist. This is for me the criterion of truth and existence.” (Hilbert to Frege, 29/12/1899) The concepts of truth and existence in mathematics are, for Hilbert, quite different from the concepts of truth and existence as we know them elsewhere. So different that I think it is preferable to speak of consistency and existence-according-to-a-consistent-theory, rather than hijacking terminology that to most minds has more substantial implications than this.
- 4.
That is, a Banach space which is also an algebra with norm satisfying ||xy|| ≤ ||x||.||y||.
- 5.
In fact, on seeing the Gelfand-Naimark axiomatization of C*-algebras, Irving E. Segal was able to argue convincingly that C*-algebras provided the perfect basis for quantum field theory. Indeed, according to Segal, C*-algebras “rendered quantum mechanics no less intuitive than classical mechanics once one became familiar with them.” (Segal 1994: 58), while along the same lines, Edward G. Effros tells us that, “Physicists have viewed quantization as a mysterious process that ultimately cannot be explained. Feynman remarked that “it is safe to say that no one understands quantum mechanics”. Owing in large part to the Gelfand-Naimark theorem, this is certainly not the case in mathematics.” (Effros 1994: 100).
References
Benacerraf, P. (1965). What numbers could not be. Philosophical Review, 74: 47–73.
Benacerraf, P. (1973). Mathematical truth. Journal of Philosophy, 70: 661–80.
Doran, R. S. (1994). C*-Algebras: 1943–1993; A fifty year celebration (1993: San Antonio, Texas). Vol. 167 of Contemporary Mathematics. Providence, RI: American Mathematical Society.
Effros, E. G. (1994). Some quantizations and reflections inspired by the Gelfand-Naimark theorem. In Doran (1994): 99–114.
Field, H. (1980). Science without numbers: a defence of nominalism. Princeton, NJ: Princeton University Press.
Field, H. (1989). Realism, mathematics, and modality. Oxford: Blackwell.
Gabriel, G., Hermes, H., Kambertel, F., Thiel, C., & Veraat, A. (eds.) (1980). Gottlob Frege: philosophical and mathematical correspondence. Chicago: University of Chicago Press. Abridged from the German edition by Brian McGuinness.
Gelfand, I. M., & Naimark, M. A. (1943). On the imbedding of normed rings into the ring of operators in Hilbert space. Mat. Sbornik, 12: 197–213. Reprinted in Doran (1994): 3–20.
Hellman, G. (1989). Mathematics without numbers: towards a modal-structural interpretation. Oxford: Clarendon Press.
Hellman, G. (2003). Does category theory provide a framework for mathematical structuralism? Philosophia Mathematica, 11(2): 129–157.
Leng, M. (forthcoming). Mathematics and reality. Oxford: Oxford University Press.
Tait, P. G. (1866). Sir William Rowan Hamilton. North British Review, 14: 35–54.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer Science+Business Media, LLC
About this chapter
Cite this chapter
Leng, M. (2010). Preaxiomatic Mathematical Reasoning: An Algebraic Approach. In: Hanna, G., Jahnke, H., Pulte, H. (eds) Explanation and Proof in Mathematics. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-0576-5_4
Download citation
DOI: https://doi.org/10.1007/978-1-4419-0576-5_4
Published:
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4419-0575-8
Online ISBN: 978-1-4419-0576-5
eBook Packages: Humanities, Social Sciences and LawEducation (R0)