Lakatos' mathematical Hegelianism

1 In the preface to the Phenomenology of Spirit, Hegel claims that the knowledge found in the various special disciplines is (at the time of his writing) in one way or another defective. History, mathematics and the natural sciences are all limited in their methods and therefore leave fundamental questions unanswered (and indeed unasked). Consequently, Hegel thought, these specialisms must be completed, explained and in some manner subsumed by philosophy. He was especially scathing about mathematics. He took the subject matter of pure mathematics to be space and number (not an unnatural assumption for the time) and described this subject matter as " inert and lifeless " (§45; p. 26). Mathematical thought, for Hegel, " moves forward along the line of equality " (ibid.). In other words, mathematics consists of equations. There is nothing unstable or incomplete about an equation so there is no dialectical impetus to conceptual revision. Therefore equations, for Hegel, do not get at the essence of anything. Philosophical thought, on the other hand, does consider essences. Any particular essence (i.e. the essence of anything smaller than the entire universe) is in some way incomplete and cries out to be included in some larger scheme. Thus philosophy is goaded onwards and upwards towards ever more comprehensive and self-subsistent conceptions of the world. In consequence, philosophers do not need ingenuity or creativity. All they need do is steep themselves in the subject matter and thus become efficient vehicles for its internal necessity. This does not happen with the " rigid, dead propositions " (ibid.) of mathematics. Of these Hegel says, " We can stop at any one of them; the next one starts afresh on its own account, without the first having moved itself on to the next, and without any necessary connection arising through the nature of the thing itself " (ibid.). In short, mathematics does not have the power of self-movement. What this means is that, having got halfway through a proof, there is nothing in the nature of the case to determine what the next line should be. Hegel (in common with his contemporaries) regarded Euclid's Elements as the paradigmatic mathematical work. In Euclidian geometry it is normal, in the course of a proof, to draw lines in addition to the original figure. As the proof unfolds these lines turn out to have important roles to play, though a student seeing the proof for the …