Abstract
Hans Vaihinger tried to explain how mathematical theories can be useful without being true or even coherent, arguing that mathematicians employ a special kind of fictional or “as if” reasoning that reliably extracts truths from absurdities. Moritz Pasch insisted that Vaihinger was wrong about the incoherence of core mathematical theories, but right about the utility of fictional discourse in mathematics. This essay explores this area of agreement between Pasch and Vaihinger. Pasch’s position raises questions about structuralist interpretations of mathematics.
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Notes
My translation of Dehn (1934, pp. 130, 136). Max Dehn is not to be lightly dismissed: he had the distinction of solving Hilbert’s third problem the very year Hilbert proposed it.
Engel (1934) is a good source of information about Pasch’s later years.
Vaihinger (1935, p. 66). I have relied entirely on C. K. Ogden’s admirable translation. Subsequent page references in this section will be to this volume.
For a taste of the literature on this topic, see Mares (1997).
This journal was the precursor of Erkenntnis. For some of the history, see Hegselmann and Siegwart (1991).
All translations of Pasch are by me.
For a discussion of the effect of Pasch’s empiricism on his geometric work, see Gandon (2005).
My translation of Dehn (1934, p. 138).
As we have seen, Pasch was not of one mind about this, elsewhere locating his tall tales within “pliable mathematics.” Pliable mathematics would be a form of mathematics, not pre-mathematics. The issue is terminological, however, not substantive.
See Pasch’s remarks (1914, pp. 142–143) about the duality between points and lines in projective geometry. Although the terms “point” and “line” are interchangeable, it is part of our mathematical understanding of projective geometry that the corresponding concepts are distinct. If we leave the content terms uninterpreted, then, Pasch concedes, the projective axioms will still characterize a system of relations. He denies, though, that they provide a definition of the fundamental concepts.
Hans Freudenthal (1962, p. 617) is quite wrong to say that, “Pasch was anxious … to postulate no more than experience seems to grant.” Every time Pasch appeals to “as if” reasoning he is postulating more than experience grants.
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Pollard, S. ‘As if’ Reasoning in Vaihinger and Pasch. Erkenn 73, 83–95 (2010). https://doi.org/10.1007/s10670-009-9205-7
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DOI: https://doi.org/10.1007/s10670-009-9205-7