Abstract
Dirac’s treatment of his well known Delta function was apparently inconsistent. We show how to reconstruct his reasoning using the inconsistency-tolerant technique of Chunk and Permeate. In passing we take note of limitations and developments of that technique.
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To specify things with full rigor, we would need a formal specification of what sort of information is allowed to flow between \(S\) and \(T\). We forego this here, as appropriate for a more technical paper. In this paper, we are concerned only with the heuristic of the procedure.
At this point, Mortensen wishes to register divergence of opinion from Brown and Priest. The point of C&P is to isolate premisses which are inconsistent with one another, to avoid taking such premisses fully collectively, by not allowing all consequences of \(S\) to flow to \(T\). This contrasts with the traditional view of reasoning (deductive, inductive, abductive, consistent or inconsistent), where one believes the conclusion on the basis of believing the premisses used to get there. C&P is something less than full agreement with all the premisses and therefore the conclusion. It thus appears more like a useful heuristic device than a rigorous mathematical proof technique, part of the context of discovery rather than the context of justification, at least for the apriori justifications of pure mathematics. In the case of the delta function, this is apparent from the fact that following theorists strove to discover a (consistent) set of definitions and proofs. It took 30 years before this was found in Laurent Schwartz ’ theory of distributions. This is a rigorous and apparently consistent mathematical theory, which has the main drawback of considerably increased complexity, deriving from its treatment of delta not as a function but as a functional. Of course, heuristics are a useful part of empirical science too (CM).
Mortensen and Priest wish to record with sadness the recent death of their co-author, the talented logician and wit Richard Benham.
References
Brown, B., & Priest, G. (2004). Chunk and permeate: A paraconsistent inference strategy. part 1: The infinitesimal calculus. The Journal of Philosophical Logic, 33, 379–388.
Brown, B., & Priest, G. (2008). Chunk and permeate II: Weak aggregation, permeation and old quantum theory, Fourth World Congress on Paraconsistency, Melbourne.
Dirac, P. A. M. (1930). Principles of quantum mechanics, 3rd (ed) 1947. Oxford: Oxford University Press.
Hardy, G. H. (1908). Pure mathematics, 10 (ed) 1960. Cambridge: Cambridge University Press.
Mortensen, C. (1995). Inconsistent mathematics. Dordrecht: Kluwer.
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R. Benham—deceased.
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Benham, R., Mortensen, C. & Priest, G. Chunk and permeate III: the Dirac delta function. Synthese 191, 3057–3062 (2014). https://doi.org/10.1007/s11229-014-0473-7
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DOI: https://doi.org/10.1007/s11229-014-0473-7