Eric Dane Walker

Even after what he considered the devastation of his logicist program, Gottlob Frege never stopped insisting on the importance of his begriffsschrift, the logico-mathematical writing system he devised to carry out that program. In contrast to most scholarship, this dissertation proceeds under the assumption that Frege had good philosophical reasons for his insistence. Being expansive and articulated along two dimensions, the graphically idiosyncratic signs belonging to Frege’s begriffsschrift have met with everything from polite bemusement to derision. I uncover and explain the rationale Frege had for designing his begriffsschrift to look the way it does, and I show his rationale not only to be plausible and interesting but to be consequential both for understanding Frege’s philosophy as a whole and for understanding the role of written signs in logic and mathematics generally. The architecture of Frege’s rationale is this: Frege designed his writing system with a view to a certain epistemic aim, the pursuit of which calls for certain hermeneutic requirements to be met by certain semiotic means. Specifically, and to put things the other way around, the two-dimensionally configured signs of Frege’s begriffsschrift are designed to express concepts in such a way that their logico-mathematical structure appears for the mind in a form meant to enable and guide our grasp of them, so that we may reason from those concepts and achieve insight into the connections through which new concepts may be developed from others. Frege’s idea implies a picture of the human being’s form of mindedness according to which the purely intelligible order is nothing for us if not capable of being inscribed in the sensible order’s terms. In logico-mathematics, we must design signs that make logico-mathematical concepts appear for the mind in a form specifically addressed to our sensibility, and such suitability for our sensibility is meant precisely to inform and perfect our grasp of those concepts, a perfection which would hardly be possible without signs that were so designed.

One of Gottlob Frege’s most original contributions to logic and philosophy was his logical notation, his ‘Begriffsschrift’. While long criticized, dismissed, or simply ignored, the recent secondary literature contains some helpful re-evaluations and partial defenses of it. These rely largely on technical, pragmatic, or cognitive-psychological considerations. In this paper, we reconsider Frege’s own reasons for valuing his notation highly. We argue that there is a further semiotic dimension, one that matters epistemologically. This dimension becomes evident once one takes seriously, partly also literally, some striking visual metaphors Frege uses. The result is an interpretation highlighting a side of Frege’s position that is not widely known. It involves his views about the indispensable role that language, or sign systems more generally, play for human thought, and especially, for logic and mathematics. More particularly and as we read Frege, a good logical notation allows for expressing conceptual/inferential relations in a ‘visually inscribed’, thus ‘perspicuous’ way, and this leads to a special kind of ‘insight’ in mathematics. Or as we elaborate this point further, it involves a kind of articulation and understanding hardly possible without it. Frege designed his Begriffsschrift with that specific goal in mind.