In this article, we examine the origins of the dialectical approach to the philosophy of mathematics. While this approach is commonly taken to begin with Imre Lakatos’s Proofs and Refutations, first published as a series of articles in 1963–64, it was preempted by the British logician G. T. Kneebone in a pair of forgotten articles in 1955 and 1957 and a chapter of his 1963 book. We introduce Kneebone’s dialectical approach to mathematics and compare it with Lakatos’s. Furthermore, we give (...) evidence from the Lakatos archives at the London School of Economics that Kneebone and Lakatos were acquainted, in correspondence, and that Lakatos read and annotated Kneebone’s papers. Nonetheless, Kneebone is nowhere mentioned in Lakatos’s work. We finish with the question of why this might be. (shrink)

Far from being unwelcome or impossible in a mathematical setting, indeterminacy in various forms can be seen as playing an important role in driving mathematical research forward by providing “sources of newness” in the sense of Hutter and Farías :434–449, 2017). I argue here that mathematical coincidences, phenomena recently under discussion in the philosophy of mathematics, are usefully seen as inducers of indeterminacy and as put to work in guiding mathematical research. I suggest that to call a pair of mathematical (...) facts a coincidence is roughly to suggest that the investigation of connections between these facts isn’t worthwhile. To say of this pair, “That’s no coincidence!” is to suggest just the opposite. I further argue that this perspective on mathematical coincidence, which pays special attention to what mathematical coincidences do, may provide us with a better view of what mathematical coincidences are than extant accounts. I close by reflecting on how understanding mathematical coincidences as generating indeterminacy accords with a conception of mathematical research as ultimately aiming to reduce indeterminacy and complexity to triviality as proposed in Rota Indiscrete thoughts, Birkhäuser, 1997). (shrink)

Beyond the obvious technical difficulties, human attempts to communicate with hypothetical Extra-Terrestrial Intelligences also present a number of philosophical puzzles. After all, an alien intelligence is likely the closest thing to a Wittgensteinian lion humanity could ever encounter. In this paper I advance a new challenge for the feasibility of communication with extra-terrestrials. The problem I raise is a practical problem that falls out of the history and philosophy of mathematics and the implementation of METI projects—specifically, the semiprime self-decryption schema (...) of the Drake Pictures message strategy. The Drake Pictures strategy presumes that aliens share the concept ‘prime number’ with us, as understanding that concept is necessary to decrypt our message. However, if the concept ‘prime number’ exhibited open texture at any point in its history, it could have developed in a different direction than it did in our history. If that is so, an alien could have the concept ‘prime number’ and still not be capable of decrypting our messages. I argue that this new problem is more trenchant than previous arguments in both the philosophy and SETI literature, such as applications of the aforementioned Lion argument and other concerns about the possibility of long-range communication without the assumption of shared concepts. (shrink)