Before discussing them, it should also be pointed out that other models of scientific explanation can be thought to extend to mathematical explanation. Marc Lange a has argued that a whole class of arguments fail to be explanatory, namely arguments by mathematical induction.
Baker offers a version of the indispensability argument that does not depend on holism. See also Pincock b for a detailed working out of the mathematical explanation of the rainbow that extends this approach. Rizza is also deflationary about the alleged ontological consequence of the role of mathematics in the cicadas case; however he does not deny the centrality of mathematical concepts in constructing explanations.
First, a theory of explanation should account for how science advances our understanding of the world. It is not possible here to treat this topic in its full complexity but a few remarks will help the reader appreciate the historical importance of the question.
Saatsi emphasizes the representational role of mathematics and criticizes the alleged explanatory role of mathematics in the explanation of empirical phenomena such as the honeycomb case see below and the cicadas case.
In Newton, there is none of this kind.
We now turn to mathematical explanations within mathematics. One such move would deny that mathematical explanations have any bearing on physical explanations and that some bridge principles linking the mathematics to the physical system, are required.
Thus, the hexagonal tiling is optimal with respect to dividing the plane into equal areas and minimizing the perimeter. In this book we absolutely and unequivocally refuse to give proofs of this second type.
Consider for instance the case of Gregory Brumfiel, a real algebraic geometer. As the domains to which mathematics could be applied grew, so also did the resistance to it. Friedman15 Friedman tried to make this intuition more precise by substituting linguistic descriptions in place of an appeal to phenomena and laws.
This has been beautifully described in an article by Y. Science advances our understanding of nature by showing us how to derive descriptions of many phenomena, using the same patterns of derivation again and again, and, in demonstrating this, it teaches us how to reduce the number of types of facts that we have to accept as ultimate or brute.
Bakerstill relying on the periodical cicada explanation, argues that mathematics can sometimes reduce the ontological commitment to concrete entities and poses a challenge to nominalism, especially the piecemeal nominalism described above, based on that observation.
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Sotahistoria Ensimmäinen Maailmansota Maailmanhistoria Kuntosali Säiliöt Maailmanhistoria. 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. About this resource. Info. Created: May 3, Updated: Sep 4, Worksheet.
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