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. (shrink)

Wittgenstein's conception of infinity can be seen as continuing the tradition of the potential infinite that begins with Aristotle. Transfinite cardinals in set theory might seem to render the potential infinite defunct with the actual infinite now given mathematical legitimacy. But Wittgenstein's remarks on set theory argue that the philosophical notion of the actual infinite remains philosophical and is not given a mathematical status as a result of set theory. The philosophical notion of the actual infinite is not to be (...) found in the mathematics of set theory, only in a certain associated philosophy – what Wittgenstein calls a certain kind of “prose”. (shrink)

This paper will evaluate whether Aristotle’s discussion of friendship in the Nicomachean Ethics points towards a plausible account of friendship. We shall evaluate Whiting’s claim that Aristotle provides us with a model of how friendship should be and is at its best, even if most friendships do not live up to this. Whiting’s view centres on a view of friendship as grounded on mutual admiration of ethical character. Whilst there is appeal in the idea, stressed by Whiting, that friendship is (...) grounded upon a kind of impersonal admiration, Aristotelian metaphysics restricts us to a single vision of the human good and prevents us from admiring different people in different ways. It will be argued that supplementation with a more pluralistic notion of human nature can help to bolster the account. Following Telfer , it shall further be argued that the grounds of a friendship must be seen to include elements of a sharing of perspectives and not be limited to mutual admiration of character. Whilst Telfer’s criticism does expose a blindspot in Aristotle, it will be argued that it is a criticism which an Aristotelian account can deal with. (shrink)

It will be argued that Wittgenstein did not outright reject the law of excluded middle for mathematics or the proof-techniques that constructivists reject in connection with the law of excluded middle. Wittgenstein can be seen to be critical of the dogmatic claims of Brouwer and Weyl concerning how proofs should be constructed. Rather than himself laying down a requirement concerning what is and is not a proof, Wittgenstein can be read as exploring the differences between constructive and non-constructive proofs. I (...) will read Wittgenstein as arguing that Brouwer’s rejection of excluded middle is based upon an over-extension from examples that Wittgenstein reads as special cases. Wittgenstein is certainly interested in the difference between constructive and non-constructive proof but only for the purposes of exploring the easily-missed differences between the two, not in order to reject non-constructive approaches. (shrink)

It will be argued that Wittgenstein did not outright reject the law of excluded middle for mathematics or the proof-techniques that constructivists reject in connection with the law of excluded middle. Wittgenstein can be seen to be critical of the dogmatic claims of Brouwer and Weyl concerning how proofs should be constructed. Rather than himself laying down a requirement concerning what is and is not a proof, Wittgenstein can be read as exploring the differences between constructive and non-constructive proofs. I (...) will read Wittgenstein as arguing that Brouwer’s rejection of excluded middle is based upon an over-extension from examples that Wittgenstein reads as special cases. Wittgenstein is certainly interested in the difference between constructive and non-constructive proof but only for the purposes of exploring the easily-missed differences between the two, not in order to reject non-constructive approaches. (shrink)