A priori knowledge and the four-colour theorem
The subject of this thesis is mathematical proof involving the use of computers. The proof in 1976 of the Four-Colour Theorem, in which an essential lemma is proved using a computer program which took over 1200 hours of computer time to complete, raises philosophical questions concerning the epistemological status of the proof and the extent to which its acceptance as a proof effects an alteration in the traditional concept of mathematical proof. Section I provides an exposition of this proof and a discussion of the Kantian conception of a priori knowledge to provide a background for the following analysis of the philosophical controversy which immediately developed after the publication of the proof. The unsurveyable length of the proof gave rise to the view that its structure was fundamentally empirical and closer to a scientific experiment than a traditional a priori proof. Objectors to this view claimed that the proof differed from most others only in that its empirical content was greater. No essential qualitative difference was involved. These views are examined, and an analysis of those of Frege and J.S. Mill are used to support the opinion that a detailed reassessment of the a priori/a posteriori distinction
Britain, Daphne C.
Mathematical theorems
A priori knowledge and the four-colour theorem. The subject of this thesis is mathematical proof involving the use of computers. The proof in 1976 of the Four-Colour Theorem, in which an essential lemma is proved using a computer program which took over 1200 hours of computer time to complete, raises philosophical questions concerning the epistemological status of the proof and the extent to which its acceptance as a proof effects an alteration in the traditional concept of mathematical proof. Section I provides an exposition of this proof and a discussion of the Kantian conception of a priori knowledge to provide a background for the following analysis of the philosophical controversy which immediately developed after the publication of the proof. The unsurveyable length of the proof gave rise to the view that its structure was fundamentally empirical and closer to a scientific experiment than a traditional a priori proof. Objectors to this view claimed that the proof differed from most others only in that its empirical content was greater. No essential qualitative difference was involved. These views are examined, and an analysis of those of Frege and J.S. Mill are used to support the opinion that a detailed reassessment of the a priori/a posteriori distinction
priori knowledge colour theorem subject mathematical proof involve use_computer proof colour theorem essential lemma prove use_computer program take hours computer time complete raise philosophical_question concern epistemological status proof extent acceptance proof effect alteration traditional concept mathematical proof section provide exposition proof discussion kantian conception priori knowledge provide_background follow analysis philosophical controversy immediately develop publication proof length proof give_rise view structure fundamentally empirical closer scientific traditional priori proof view claim proof differ others empirical content great essential qualitative difference involve view examine analysis frege mill use_support opinion detail reassessment priori posteriori distinction