From A Mathematics Of NecessityTo A Mathematics Of Convention (original) (raw)
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Philosophical Investigations, 2024
The mature Wittgenstein's groundbreaking analyses of sense and the logical must-and the powerful new method that made them possible-were the result of a multi-year process of writing, rearranging , rewriting and one large-scale revision that eventually produced the Philosophical Investigations and RFM I. In contrast, his struggles during the same period with questions of arithmetic and higher mathematics remained largely in first-draft form, and he drops the topic entirely after 1945. In this paper, I argue that Wittgenstein's new method can be applied to the cases of arithmetic and set theory and that the result is innovative, recognizably Wittgensteinian, and independently appealing. I conclude by acknowledging the reasons Wittgenstein himself might have had to resist applying his own proven method to the case of mathematics-particularly to set theory-and by indicating why I think those reasons are ultimately unsound.
Some Remarks on Wittgenstein's Philosophy of Mathematics
Open Journal of Philosophy, vol. 10(1), pp. 45-65, 2020
Drawing mainly from the Tractatus Logico-Philosophicus and his middle period writings, strategic issues and problems arising from Wittgenstein's philosophy of mathematics are discussed. Topics have been so chosen as to assist mediation between the perspective of philosophers and that of mathematicians on their developing discipline. There is consideration of rules within arithmetic and geometry and Wittgenstein's distinctive approach to number systems whether elementary or transfinite. Examples are presented to illuminate the relation between the meaning of an arithmetical generalisation or theorem and its proof. An attempt is made to meet directly some of Witt-genstein's critical comments on the mathematical treatment of infinity and irrational numbers.
Stages of wittgenstein’s philosophy of mathematics
European Journal of Science and Theology, 2018
Wittgenstein‟s philosophy of Mathematics is concerned relatively few philosophers of Mathematics, historians of Philosophy from the XX century and analytically oriented philosophers. The paper deals with the periodization of Wittgenstein‟s philosophy of Mathematics. It is well known that Wittgenstein‟s philosophy of Mathematics does not correspond to the overall periodization of his work. We find the obvious context of Wittgenstein‟s grasping of value and Mathematics. Mathematics of Tractatus Logico -Philosophicus corresponds to the division of the sentences by meaning. Equations as the mathematical assertions are thus only apparent sentences (Scheinsätze) because they try to discuss the logical form. The claims of Mathematics are, in Wittgenstein‟s sense, absurd. According to the Tractatus, we have only one language, not a meta-language. Strict view of Mathematics is related to a rigorous view of values and ethics. Wittgenstein‟s philosophy of Mathematics works in the overall portfolio of opinions in many ways as anarchist. Wittgenstein‟s rejection of some mathematical objects is closely related to his understanding of the syllables: exuberance, meaningfulness. It is undoubtedly related to Wittgenstein‟s ethics.
Leaving Mathematics As It Is - Wittgenstein's Philosophy of Mathematics
Wittgenstein’s later philosophy of mathematics has been widely interpreted to involve Wittgenstein’s making dogmatic requirements of what can and cannot be mathematics, as well as involving Wittgenstein dismissing whole areas (e.g. set theory) as not legitimate mathematics. Given that Wittgenstein promised to ‘leave mathematics as it is’, Wittgenstein is left looking either hypocritical or confused. This thesis will argue that Wittgenstein can be read as true to his promise to ‘leave mathematics as it is’ and that Wittgenstein can be seen to present coherent, careful and non-dogmatic treatments of philosophical problems in relation to mathematics. If Wittgenstein’s conception of philosophy is understood in sufficient detail, then it is possible to lift the appearance of confusion and contradiction in his work on mathematics. Whilst apparently dogmatic and sweeping claims figure in Wittgenstein’s writing, they figure only as pictures to be compared against language-use and not as definitive accounts (which would claim exclusive right to correctness). Wittgenstein emphasises the importance of the applications of mathematics and he feels that our inclination to overlook the connections of mathematics with its applications is a key source of a number of philosophical problems in relation to mathematics. Wittgenstein does not emphasise applications to the exclusion of all else or insist that nothing is mathematics unless it has direct applications. Wittgenstein does question the alleged importance of certain non-applied mathematical systems such as set theory and the logicist systems of Frege and Russell. But his criticism is confined to the aspirations towards philosophical insight that has been attributed to those systems. This is consonant with Wittgenstein’s promises in (PI, §124) to ‘leave mathematics as it is’ and to see ‘leading problems of mathematical logic’ as ‘mathematical problems like any other.’ It is the aim of this thesis to see precisely what Wittgenstein means by these promises and how he goes about keeping them
Wittgenstein and the Turning Point in the Philosophy of Mathematics
1987
for Wittgenstein was the discovery of a notation in terms of which this can be expressed. He thinks that Wittgenstein deduces the contingency of propositions from the 'law of excluded middle'. But in Wittgenstein's notation there is no such 'law'; all tautologies are given the same expression.
Wittgenstein on pure and applied mathematics
Some interpreters have ascribed to Wittgenstein the view that mathematical statements must have an application to extra-mathematical reality in order to have use and so any statements lacking extra-mathematical applicability are not meaningful (and hence not bona fide mathematical statements). Pure mathematics is then a mere signgame of questionable objectivity, undeserving of the name mathematics. These readings bring to light that, on Wittgenstein's offered picture of mathematical statements as rules of description, it can be difficult to see the role of mathematical statements which relate to concepts that are not employed in empirical propositions e.g. set-theoretic concepts. I will argue that Wittgenstein's picture is more flexible than might at first be thought and that Wittgenstein sees such statements as serving purposes not directly related to empirical description. Whilst this might make such systems fringe cases of mathematics, it does not bring their legitimacy as mathematical systems into question.