Mathematical investigation, when done well, can confer understanding. This bare observation shouldn’t be controversial; where obstacles appear is rather in the effort to engage this observation with epistemology. The complexity of the issue of course precludes addressing it tout court in one paper, and I’ll just be laying some early foundations here. To this end I’ll narrow the field in two ways. First, I’ll address a specific account of explanation and understanding that applies naturally to mathematical reasoning: the view proposed by Philip Kitcher and Michael Friedman of explanation or understanding as involving the unification of theories that had antecedently appeared heterogeneous. For the second narrowing, I’ll take up one specific feature (among many) of theories and their basic concepts that is sometimes taken to make the theories and concepts preferred: in some fields, for some problems, what is counted as understanding a problem may involve finding a way to represent the problem so that it (or some aspect of it) can be visualized. The final section develops a case study which exemplifies the way that this consideration – the potential for visualizability – can rationally inform decisions as to what the proper framework and axioms should be. The discussion of unification (in sections 3 and 4) leads to a mathematical analogue of Goodman’s problem of identifying a principled basis for distinguishing grue and green. Just as there is a philosophical issue about how we arrive at the predicates we should use when making empirical predictions, so too there is an issue about what properties best support many kinds of mathematical reasoning that are especially valuable to us. The issue becomes pressing via an examination of some physical and mathematical cases that make it seem unlikely that treatments of unification can be as straightforward as the philosophical literature has hoped. Though unification accounts have a grain of truth (since a phenomenon (or cluster of phenomena) called “unification” is in fact important in many cases) we are far from an analysis of what “unification” is..
|Keywords||No keywords specified (fix it)|
|Categories||categorize this paper)|
References found in this work BETA
No references found.
Citations of this work BETA
Pi on Earth, or Mathematics in the Real World.Van Kerkhove Bart & Van Bendegem Jean Paul - 2008 - Erkenntnis 68 (3):421-435.
Rota's Philosophy in its Mathematical Context.Sébastien Gandon - 2016 - Philosophia Mathematica 24 (2):145-184.
Similar books and articles
Why Unification is Neither Necessary nor Sufficient for Explanation.Victor Gijsbers - 2007 - Philosophy of Science 74 (4):481-500.
Outline of a Theory of Scientific Understanding.Gerhard Schurz & Karel Lambert - 1994 - Synthese 101 (1):65-120.
Explanatory Unification and Scientific Understanding.Eric Barnes - 1992 - PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association 1992:3 - 12.
Unification: What is It, How Do We Reach and Why Do We Want It?Erik Weber - 1999 - Synthese 118 (3):479-499.
The Causal and Unification Approaches to Explanation Unified—Causally.Michael Strevens - 2004 - Noûs 38 (1):154–176.
Explanatory Unification and Scientific Understanding.Jennifer Wilson Mulnix - 2011 - Acta Philosophica 20 (2):383 - 404.
Unification of Mathematical Theories.Krzysztof Wójtowicz - 1998 - Foundations of Science 3 (2):207-229.
Added to index2009-01-28
Total downloads60 ( #87,744 of 2,171,744 )
Recent downloads (6 months)4 ( #76,305 of 2,171,744 )
How can I increase my downloads?