David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
Learn more about PhilPapers
Erkenntnis 73 (2):265–292 (2010)
Charles Parsons’ book “Mathematical Thought and Its Objects” of 2008 (Cambridge University Press, New York) is critically discussed by concentrating on one of Parsons’ main themes: the role of intuition in our understanding of arithmetic (“intuition” in the specific sense of Kant and Hilbert). Parsons argues for a version of structuralism which is restricted by the condition that some paradigmatic structure should be presented that makes clear the actual existence of structures of the necessary sort. Parsons’ paradigmatic structure is the so-called ‘intuitive model’ of arithmetic realized by Hilbert’s strings of strokes. This paper argues that Hilbert’s strings, considered as given in intuition, cannot play the role Parsons assigns to them: the criteria of identity of these strings do not have the sharpness that Parsons wants to see in them, and Parsons inadvertently projects abstract structures into his ‘intuitive model’. This diagnosis is exemplified with respect to (a) Parsons’ distinction between addition and multiplication on the one hand and exponentiation on the other and (b) his analysis of arithmetical knowledge in simple cases like “7 + 5 = 12”. All in all, it is claimed that Parsons book contains many important insights with respect to, for example, different versions structuralism, the notion of “natural number” and its uniqueness, induction, predicativity and other things, for which he is rightly famous, but that his way of drawing on the notion of intuition leaves too many questions unanswered.
|Keywords||Philosophy Logic Ethics Ontology Epistemology Philosophy|
|Categories||categorize this paper)|
Setup an account with your affiliations in order to access resources via your University's proxy server
Configure custom proxy (use this if your affiliation does not provide a proxy)
|Through your library|
References found in this work BETA
George Boolos, John Burgess, Richard P. & C. Jeffrey (2007). Computability and Logic. Cambridge University Press.
Tyler Burge (2003). Logic and Analyticity. Grazer Philosophische Studien 66 (1):199-249.
Simon Friederich (2010). Structuralism and Meta-Mathematics. Erkenntnis 73 (1):67 - 81.
Andrzej Grzegorczyk (2005). Undecidability Without Arithmetization. Studia Logica 79 (2):163 - 230.
Hannes Leitgeb & James Ladyman (2008). Criteria of Identity and Structuralist Ontology. Philosophia Mathematica 16 (3):388-396.
Citations of this work BETA
No citations found.
Similar books and articles
Colin Cheyne (1997). Getting in Touch with Numbers: Intuition and Mathematical Platonism. Philosophy and Phenomenological Research 57 (1):111-125.
Dennis Lomas (2002). What Perception is Doing, and What It is Not Doing, in Mathematical Reasoning. British Journal for the Philosophy of Science 53 (2):205-223.
Paul Sludds (2003). The Importance of Being Actual: Some Reasons for and Against Procreation. Australasian Journal of Philosophy 81 (4):561 – 568.
Gila Sher & Richard L. Tieszen (eds.) (2000). Between Logic and Intuition: Essays in Honor of Charles Parsons. Cambridge University Press.
Dale E. Miller (2003). Axiological Actualism and the Converse Intuition. Australasian Journal of Philosophy 81 (1):123 – 125.
Charles Parsons (2008). Mathematical Thought and its Objects. Cambridge University Press.
Pierre Cassou-Nogués (2006). Signs, Figures and Time: Cavaillès on “Intuition” in Mathematics. Theoria 21 (1):89-104.
Jaakko Hintikka (1972). III. Kantian Intuitions. Inquiry 15 (1-4):341 – 345.
David Galloway (1999). Seeing Sequences. Philosophy and Phenomenological Research 59 (1):93-112.
Peter Smith (2009). Critical Notice of C. Parsons, Mathematical Thought and its Objects. [REVIEW] Analysis 69 (3):549-557.
Added to index2010-06-09
Total downloads73 ( #28,157 of 1,696,464 )
Recent downloads (6 months)6 ( #93,751 of 1,696,464 )
How can I increase my downloads?