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• W103275182 abstract "In this paper we try to convert the mathematician who calls himself, or herself, “a formalist” to a position we call “methodological pluralism”. We show how the actual practice of mathematics fits methodological pluralism better than formalism while preserving the attractive aspects of formalism of freedom and creativity. Methodological pluralism is part of a larger, more general, pluralism, which is currently being developed as a position in the philosophy of mathematics in its own right. 1 Having said that, henceforth, in this paper, we abbreviate “methodological pluralism” with “pluralism”. 1. Characterization of Formalism We shall begin with Delefsen’s characterisation of formalism, modify it to better fit the modern mathematician’s conception, use Hilbert’s two principles to set parameters on the notion of rigor, and end with a general description of For example see (Friend (2012)) ANDREA PEDEFERRI, MICHELE FRIEND 174 formalism. Formalism is a philosophy of mathematics which was developed in the late nineteenth century and at the beginning of the twentieth. To be more precise, it is not simply one philosophy. Following Detlefsen’s careful characterization of formalism, 2 we can think of formalism as a family of positions, each member of which has some of the five following characteristics. For our characterisation of formalism we shall ignore (i). We shall then add Hilbert’s two principles. The following are the characteristics: i. geometry no longer sets the standard for rigor. Instead, the standard is set by arithmetic 3 . ii. Formalists have a particular conception of what rigor is. For them rigor follows from an act of abstraction away from intuition instead of from an embedding into or onto a previously accepted theory (which is what we often find in pre-Hilbert and pre-Tarski presentations of geometry). This is a methodological constraint on the practice and presentation of mathematics. iii. Formalists reject the idea that mathematical proof should be based on a “genetic” model of proof, because they do not believe that we have knowledge of mathematical truths 4 by having knowledge of their origins and causes (although this might, of course, be useful for some learning purposes). They replace the genetic model with axiomatic theories. Stipulating a set of axioms, or rules of inference, is what we need to have a mathematical theory. iv. Formalism advocates “a nonrepresentational role for language in mathematical reasoning”. 5 v. Mathematicians have the freedom to create and work with different reasoning tools in order to get genuine knowledge. The idea that mathematical knowledge can be reached only by the use of contentual (in the sense of interpreted (semantically or ontologically)) reasoning is essentially alien to formalism. Hilbert’s principles 1 and 2 6 are: 2 See Frege (1980, pp. 236-237). 3 Detlefsen characterizes formalism as a position with all five charactersistics. We are less stringent in our definition The word “truth” is not to be taken too seriously here. It could be replaced by “content”. 5 Frege (1980, p. 237). Who best represents these characteristics is a separate historical question. ARE MATHEMATICIANS BETTER DESCRIBED AS FORMALISTS OR PLURALISTS? 175 1. Mathematicians should aim to construct concepts and inferential methods, which are fruitful in mathematical practice. 2. It is necessary to make inferences as reliable as possible and to search on which basis this can be done. We interpret (iv) and 2 to imply that we need a unique, specifed and tight proof theory for all our proofs. We ignore characteristic (i) (the one about arithmetic setting the standard for rigor above geometry) because we think that this has largely been eclipsed in present day mathematics namely by the arithmetization of geometry by Hilbert and Tarski. However, since we also use Hilbert’s two principles as characterising formalism, they will serve as parameters on what counts as rigor for the formalist. Characteristics (iv) and (v) are important for us because the mathematicians who call themselves formalist tell us that they feel that as formalists they enjoy freedom and creativity. They are free to interpret the symbols as they choose – give them any interpretation which fits the formal constraints of rigor (characteristic (iv)). For Hilbert, and for our formalist, mathematics is only to be thought of as the collection of all mathematical theorems ever given in history, where the theorems are generated in axiomatic theories, and that" @default.
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• W103275182 date "2011-01-01" @default.
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• W103275182 title "Are Mathematicians Better Described as Formalists or Pluralists" @default.
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