Abstract
The 19th-century polymath Bernard Bolzano is widely regarded as a figure who decisively showed that even in Kant’s time, there were philosophical and mathematical reasons to think that an intuition-based philosophy of mathematics was untenable. This paper identifies and clarifies two of Bolzano’s longstanding objections to Kant’s philosophy of mathematics. Once this is done, I defend Kant from these objections. To conclude, I discuss a deeper disagreement between Bolzano and Kant’s philosophical approaches. Namely, Bolzano rejected that a “critique of reason” (i.e., an examination of the principles and limits of human cognition) was necessary in order to secure and explain reliable forms of inference and claims to a priori knowledge.
This is a preview of subscription content, log in via an institution to check access.
Notes
- 1.
“Contributions to a better-grounded Presentation of Mathematics.”
- 2.
“Theory of Science.”
- 3.
“New Anti-Kant.”
- 4.
More specifically, NAK was instigated by Bolzano in 1837, the same year WL was published. Příhonský was responsible for writing and publishing NAK, and Bolzano was actively involved in its development, providing Příhonský with input and eventually, his approval of the finished product. Thus, there is good reason to think that NAK reflects Bolzano’s own views.
- 5.
The critics I have in mind are those who ascribe to what might be suitably called a “Wolffian” or “Leibnizian” philosophy of mathematics according to which a mathematical proof of a proposition P (which is necessary and sufficient to furnish mathematical knowledge of P) proceeds and “depends on the principle of contradiction and in general on the doctrine of syllogisms” (Lambert 1761, §92.14). See Heis (2014) for more on this debate between the Wolffian and the Kantian.
Secondly, it is important to point out that, according to Bolzano, there is a distinction between the actual reasons why a proposition P is true and what merely convinces us that P is true. The former tracks objective ground-consequence relations mentioned in the previous paragraph. These relations hold regardless of whether there are any minds or epistemic agents.
Bolzano was not the first to make this distinction. In fact, the distinction between the metaphysical ordering of truths and the doxastic order by which one comes to believe a truth can be traced back to Aristotle (1993) in Book I of the Posterior Analytics.
- 6.
In this paper, I understand a proposition P to be “general” or “universal” if P applies to a class of objects such as all triangles or all natural numbers and not just particular instances. I take “necessity” to mean something similar except that it is a property that applies primarily to arguments and inferences.
- 7.
To give a mundane mathematical example, the concept < triangle > (I will use angle brackets to denote concepts) has among its marks the concept < three-sided >.
- 8.
Classically understood, genetic definitions are definitions that reveal the “genesis” or “cause” of the thing being defined. What this means is often spelled out in terms of a finite procedure that specifies how to construct the object being defined presumably from a given or privileged set of entities. For instance, a genetic definition for a triangle provides a procedure of constructing a triangle from more basic kinds of objects (e.g., points and lines) by certain rules and methods of (perhaps, straightedge and compass) construction. No claim here is made as to whether for Kant, genetic definitions have to be unique. Friedman (2010) construes schemata as functions of a certain type. They are functions whose inputs (e.g., three arbitrary lines) are a given set of entities and whose output is a representation (e.g., a triangle constructed from those three lines). Precisifying “schemata” in terms of genetic definitions or functions (in Friedman’s sense) works equally well for the purposes of the paper.
- 9.
See, for example, Kant (1781, A303) and (Ibid., A732/B760) for clear articulations of Kant’s conception of axioms, especially as they compare with knowledge that is inferred.
- 10.
Kant formulates the distinction between analytic and synthetic judgments in different possibly inequivalent ways. Nonetheless, in this paper, the distinction will be understood in terms of concept-containment relations. As already touched on, this distinction hinges on a certain view of concepts, namely that concepts have marks, which are just further concepts, and that concepts admit of decomposition in terms of these marks. This view of concepts was somewhat standard and shared among thinkers such as Lambert and even Bolzano.
- 11.
That intuition can furnish knowledge of axioms is taken to be evident for Kant in light of the epistemic legitimacy of Euclidean diagrammatic geometry. See Friedman (2010) for more on the relationship between Kant and Euclidean geometry.
- 12.
It might be helpful to recall Russell’s criticism of “the method postulating” by which one postulates mathematical axioms and definitions without verifying their truth and consistency. According to Russell, without doing the hard philosophical and mathematical work of verifying the axioms and definitions that one has laid down, the method of postulation amounts to nothing more than “theft over honest toil” (Russell 1919, p. 71). Thus, Bolzano cannot appeal to stipulated definitions and axioms in order to undermine Kant’s philosophy of mathematics since the validity of such definitions and axioms are the very thing intuitions are supposed to secure, according to Kant. I thank one of the anonymous referees for pointing this out to me.
- 13.
This is admittedly a delicate issue since Bolzano’s conception of analyticity and syntheticity evolved throughout his writings. As noted, Bolzano in BD defines “analytic” in terms of concept-containment, and defines “synthetic” as being not analytic. However, in WL, Bolzano defines “analytic” according to a variable term criterion. See Chapter 5 of Lapointe (2011) for more on Bolzano’s mature conception of analyticity.
- 14.
See Kant (1781, B182–3).
- 15.
- 16.
This is not to say that serious complications will not arise if Kant’s critical philosophy is adapted to a different set of logical laws.
- 17.
One can give the following counterexample. Given any proposition P, P logically implies itself. However, not every proposition grounds itself.
- 18.
At the end of the day, I argue that the tension between Kant’s and Bolzano’s respective definitions of terms such as “intuition” and “a priori” is merely apparent. I see no reason why a Kantian has to reject Bolzano’s definitions as long as he could supplement them with her own. For Kant, a mathematical inference is composed of a grounding cognition and an a priori construction that allows for the possibility of genuine cognition from the ground to the inferred proposition in a way that goes beyond the laws of traditional logic. Kant’s account leaves room for what Bolzano would consider to be an objective ground of a mathematical truth as its objective ground. As I see it, Kant’s account does not necessarily impose prohibitions on mathematical practice in a way that would preclude any instances of a ground-consequence from being genuine cases of ground-consequence.
- 19.
See Bolzano (1851, §14).
- 20.
“Paradoxes of the Infinite.”
References
Aristotle (1993) Posterior Analytics, 2nd edn. Clarendon Press, Oxford
Bolzano B (1810) Contributions to a Better-Grounded Presentation of Mathematics. In: Russ S (ed) The Mathematical Works of Bernard Bolzano, Oxford University Press, pp 83–139
Bolzano B (1837) Theory of Science. Oxford University Press, Oxford
Bolzano B (1851) Paradoxes of the Infinite. In: Russ S (ed) The Mathematical Works of Bernard Bolzano, Oxford University Press, pp 591–670
Coffa A (1991) The Semantic Tradition from Kant to Carnap: To the Vienna Station. Cambridge University Press, Cambridge
Friedman M (2010) Kant on geometry and spatial intuition. Synthese 186(1):231–255
Heis J (2014) Kant (vs. Leibniz, Wolff and Lambert) on real definitions in geometry. Canadian Journal of Philosophy 44(5–6):605–630, DOI 10.1080/ 00455091.2014.971689
Kant I (1781) Critique of Pure Reason. Cambridge University Press, Cambridge
Lambert JH (1761) Treatise on the Criterion of Truth. In: Watkins E (ed) Kant’s Critique of Pure Reason: Background Source Material, Cambridge University Press, Cambridge, pp 233–257
Lapointe S (2011) Bolzano’s Theoretical Philosophy An Introduction. Palgrave Macmillan, Basingstoke
Příhonský F (1850) New Anti-Kant. Palgrave Macmillan, Basingstoke
Rusnock P (1999) Philosophy of mathematics: Bolzano’s responses to Kant and Lagrange. Revue d’histoire des sciences 52(3/4):pp. 399–427
Russell B (1919) Introduction to Mathematical Philosophy. George Allen & Unwin, Ltd., London
Acknowledgements
Thanks are owed to Karl Ameriks, Mic Detlefsen, Matteo Bianchetti, Curtis Franks, Eric Watkins, two anonymous referees, and the organizers and participants at CSHPM 2016 for their helpful comments and discussion.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing Switzerland
About this paper
Cite this paper
Anh McEldowney, P. (2017). Bolzano Against Kant’s Pure Intuition. In: Zack, M., Schlimm, D. (eds) Research in History and Philosophy of Mathematics. CSHPM 2016. Proceedings of the Canadian Society for History and Philosophy of Mathematics/La Société Canadienne d’Histoire et de Philosophie des Mathématiques. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-64551-3_1
Download citation
DOI: https://doi.org/10.1007/978-3-319-64551-3_1
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-319-64092-1
Online ISBN: 978-3-319-64551-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)