Philosophia Mathematica 1 (1):24-49 (1993)
In the early years of this century, Poincaré and Russell engaged in a debate concerning the nature of mathematical reasoning. Siding with Kant, Poincaré argued that mathematical reasoning is characteristically non-logical in character. Russell urged the contrary view, maintaining that (i) the plausibility originally enjoyed by Kant's view was due primarily to the underdeveloped state of logic in his (i.e., Kant's) time, and that (ii) with the aid of recent developments in logic, it is possible to demonstrate its falsity. This refutation of Kant's views consists in showing that every known theorem of mathematics can be proven by purely logical means from a basic set of axioms. In our view, Russell's alleged refutation of Poincaré's Kantian viewpoint is mistaken. Poincaré's aim (as Kant's before him) was not to deny the possibility of finding a logical ‘proof’ for each theorem. Rather, it was to point out that such purely logical derivations fail to preserve certain of the important and distinctive features of mathematical proof. Against such a view, programs such as Russell's, whose main aim was to demonstrate the existence of a logical counterpart for each mathematical proof, can have but little force. For what is at issue is not whether each mathematical theorem can be fitted with a logical ‘proof’, but rather whether the latter has the epistemic features that a genuine mathematical proof has.
|Keywords||Poincaré Russell Kant proof role of logical reasoning in proof intuition intuitionism constructivism non-logical character of mathematical proof|
|Categories||categorize this paper)|
References found in this work BETA
No references found.
Citations of this work BETA
Towards a Theory of Mathematical Argument.Ian J. Dove - 2009 - Foundations of Science 14 (1-2):136-152.
Towards a Theory of Mathematical Argument.Ian J. Dove - 2013 - In Andrew Aberdein & Ian J. Dove (eds.), Foundations of Science. Springer. pp. 291--308.
Toward a Topic-Specific Logicism? Russell's Theory of Geometry in the Principles of Mathematics.Sébastien Gandon - 2008 - Philosophia Mathematica 17 (1):35-72.
On Formalism Freeness: Implementing Gödel's 1946 Princeton Bicentennial Lecture.Juliette Kennedy - 2013 - Bulletin of Symbolic Logic 19 (3):351-393.
Similar books and articles
Poincaré's Thesis of the Translatability of Euclidean and Non-Euclidean Geometries.David Stump - 1991 - Noûs 25 (5):639-657.
The Mathematical Philosophy of Giuseppe Peano.Hubert C. Kennedy - 1963 - Philosophy of Science 30 (3):262-266.
Poincaré, Kant, and the Scope of Mathematical Intuition.Terry F. Godlove - 2009 - Review of Metaphysics 62 (4):779-801.
Added to index2009-01-28
Total downloads48 ( #109,891 of 2,177,988 )
Recent downloads (6 months)1 ( #317,698 of 2,177,988 )
How can I increase my downloads?