By Augustus De Morgan; Sophia Elizabeth De Morgan; David Eugene Smith
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Additional resources for A Budget of Paradoxes [Vol I]
In explaining this idea he chose to represent uncertainty in mathematical knowledge using a kind of Bayesian degree-of-belief framework, where qualitative changes of Introduction 33 direction of degrees of belief occur on discovering new knowledge. Now, while Bayesian reconstructions of rational practice are extremely common in philosophy of science, very little has been done by way of following up P´olya on their suitability for mathematical practice. I discuss the question of which variety of Bayesianism is most appropriate for this task, and proceed to examine the use of analogy, the choice of strategy in proof planning, and large-scale enumerative induction.
The great sticking point, though there are several, concerns the sporadic groups. I find it aesthetically repugnant to accept that these groups are mere anomalies . . Possibly . . The Origin of Groups remains to be written, along lines foreign to those of Linnean outlook’ (quoted in Solomon 2001, 345). Hacking (1992: 5) remarks that ‘A great many inquiries use several styles. ’ 20 Towards a Philosophy of Real Mathematics mathematical entities, as when, for instance, K-theory was constructed to probe topological spaces.
Much closer to our times, when textbooks had to be typeset the expense to be incurred in their production forced publishers to guarantee that they would find a substantial readership and so be cautious. With the advent of Xerox machines, ideas at the frontiers of research could be conveyed rapidly around the world. However, when you see the ghastly typewritten monographs of the 1970s, you can appreciate the limitations of the available notational devices. Now we have the opportunity to read for free papers submitted to preprint servers, written with the notational resources of Latex, and illustrated by diagrams constructed using clever software packages, and we are able to watch remotely lectures given by illustrious mathematicians and view moving graphics.