The Crucial Role of Proof: A Classical Defense Against Mathematical Empiricism

Dissertation, Massachusetts Institute of Technology (1993)
Catherine Womack
Bridgewater State University
Mathematical knowledge seems to enjoy special status not accorded to scientific knowledge: it is considered a priori and necessary. We attribute this status to mathematics largely because of the way we come to know it--through following proofs. Mathematics has come under attack from sceptics who reject the idea that mathematical knowledge is a priori. Many sceptics consider it to be a posteriori knowledge, subject to possible empirical refutation. In a series of three papers I defend the a priori status of mathematical knowledge by showing that rigorous methods of proof are sufficient to convey a priori knowledge of the theorem proved. ;My first paper addresses Philip Kitcher's argument in his book The Nature of Mathematical Knowledge that mathematics is empirical. Kitcher develops a view of a priori knowledge according to which mathematics is not a priori. I show that his requirements for knowledge in general as well as a priori knowledge in particular are far too strong. On Kitcher's view, some correct proofs may not even convey knowledge, much less a priori knowledge. This consequence suggests that Kitcher's conception of the a priori does not respond to properties of mathematics that have been responsible for the view that it is non-empirical. ;In my second paper I examine Imre Lakatos' fallibilism in the philosophy of mathematics. Lakatos argued that some mathematical propositions are subject to what he calls "refutations", by which he means to include falsification on extra-logical grounds. Lakatos cites Kalmar's scepticism about Church's Thesis as a case in point. I examine this case in detail, concluding that the failure of Lakatos' thesis in this prima facie favorable case casts doubt upon the thesis generally. ;My third paper is a defense of the classical conception of proof against Thomas Tymoczko's thesis that only arguments that are surveyable by us can count as proofs. Tymoczko concluded from his thesis that the computer-assisted proof of the Four Color Theorem involves an extension of the concept of proof hitherto available in mathematics. The classical conception regards the computer-assisted proof as a real proof, which we are unable to survey. Tymoczko recognizes that formalizability is a criterion for whether an argument is a proof, but he does not, in published work, note that formalizability and surveyability are often conflicting ideals. The classical theory recognizes both ideals because it regards the question whether something is a proof as distinct from the question of whether we can recognize it as such, or how confident we can be that it is one
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