The second, concerns the nominalism-platonism debate. However, the discussion on mathematical explanation in the empirical sciences has fruitfully interacted with some of the proposals on offer.
Baker starts from a debate between Colyvanand Meliathat saw both authors agreeing that the prospects for a successful platonist use of the indispensability argument rests on examples from scientific practice in which the postulation of mathematical objects results in an increase of those theoretical virtues which are provided by the postulation of theoretical entities.
By contrast, mathematics has no such explanatory value: Pincock and Batterman are criticized in Reutlinger and Andersen on account of relying on the assumption that abstract explanations, defined as those that leave out most of the microphysical causal details of the target phenomenon to be explained, have to be non-causal.
Thus Steiner arrives at two criteria for explanatory proofs, i.
We suppose ourselves to possess unqualified scientific knowledge of a thing, as opposed to knowing it in the accidental way in which the sophist knows, when we think that we know the cause on which the fact depends as the cause of the fact and of no other, and further, that the fact could not be other than it is.
There are many ways this stick could be horizontal spin it around in the horizontal planebut only two ways it could be vertical up or down. As the domains to which mathematics could be applied grew, so also did the resistance to it. Consider the following example from evolutionary biology introduced in Baker and discussed extensively in the philosophical literature.
The mapping account emphasizes the centrality of the existence of a mapping between the mathematical structures and the empirical reality that is the target of the scientific modeling.
The explanatory store over K, E Kis the best systematization of K Kitcher here makes an idealization by claiming that E K is unique. Cariani see Other Internet Resources also focuses on the explanatory asymmetry between the two types of induction but with considerations different from those of Baker.
But, second premise, we ought to believe our best theories. Baron revamps the deductive-nomological model to account for mathematical explanations of scientific facts by using an informational account of relevance logic to capture the dependence relation between mathematical facts and physics facts.
Hold the tennis racket horizontally by its handle with face S facing up. Bees that use less wax and thus spend less energy have a better chance at evolving via natural selection. Lange section 3, chapters 7 to 9 brings together several investigations on mathematical explanation by the same author see Lange a, b,b, c; in what follows, we mainly reference the book of In general this is an argument in favor of Platonism, as our best science quantifies over mathematical entities.
Much of the interest in this area has been the analysis of whether, and if so how, such explanations differ from mechanistic explanations see HunemanBrigandtWoodwardJonesKosticHunemanDarrason for case studies and further references.
Indeed, they claim that inference to the best explanation is used differently in arguments for scientific realism from the way it is used in arguments for mathematical realism see also Colyvanand that this undermines the use of the enhanced indispensability argument as establishing Platonism.
As we have seen, much attention has been devoted recently to structural explanations in physics, namely explanations of specific phenomena given in terms of mathematical properties of the systems in which they occur.
Apart from a single dissenting voice Zelcersee also the reply Weber and Franscontributors in this area all accept mathematical explanation of mathematical facts as a datum about mathematical practice that philosophers of mathematics have to investigate. Having established that mathematics seems to play an important role in giving explanations in the natural sciences, we now move to a few historical remarks on how this problem has emerged in the history of philosophy and science.
Marcus distinguishes epistemic and metaphysical readings of the explanatory indispensability argument and claims that once one disambiguates the two readings, the explanatory indispensability argument is no improvement on the standard Quinean indispensability argument.
Kitcher was the first to read Bolzano as propounding a theory of mathematical explanations see Betti and Roski for recent contributions.
What explains this symmetry. Much of the discussion in this area, which addresses whether the mathematics present in the formulation of scientific theories or mathematical explanations offered in science is eliminable, is deeply related to so-called indispensability arguments.
Let y be the intermediate principal axis. We thus see that the problem of the explanatory role of mathematics in science is intimately related to problems of modeling and idealization in science see Morrison In such situations the major conceptual recasting will also produce new proofs but the explanatoriness of the new proofs is derivative on the conceptual recasting.
We can only provide here a very short summary of the main positions.
But this does not mean that the mathematical explanation is restricted to pure mathematics. As we shall see, alleged counterexamples to the causal theory of explanation as well as to the causal theory of modeling come from areas as diverse as mathematics, physics, biology, and economics.
Group work Give the children, in groups or in pairs, a selection of different text types on a consistent theme eg snow: In this book we absolutely and unequivocally refuse to give proofs of this second type.
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Entry Navigation. Entry Contents; Bibliography; Academic Tools; Friends PDF Preview we should also point out that there is another tradition of thinking of explanation in mathematics that includes Mill, Lakatos, Russell and Gödel. and J. Tabery,“Mechanisms in Science”, The Stanford Encyclopedia of Philosophy (Winter How to write a Diary Entry.
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