248 research outputs found
Aristotle's Actual Infinities
Aristotle is said to have held that any kind of actual infinity is impossible. I argue that he was a finitist (or "potentialist") about _magnitude_, but not about _plurality_. He did not deny that there are, or can be, infinitely many things in actuality. If this is right, then it has implications for Aristotle's views about the metaphysics of parts and points
Radical anti-realism and substructural logics
We first provide the outline of an argument in favour of a radical form of anti-realism premised on the need to comply with two principles, implicitness and immanence, when trying to frame assertability-conditions. It follows from the first principle that one ought to avoid explicit bounding of the length of computations, as is the case for some strict finitists, and look for structural weakening instead. In order to comply with the principle of immanence, one ought to take into account the difference between being able to recognize a proof when presented with one and being able to produce one and thus avoid the idealization of our cognitive capacities that arise within Hilbert-style calculi. We then explore the possibility of weakening structural rules in order to comply with radical anti-realist strictures
Wittgenstein And Labyrinth Of âActual Infinityâ: The Critique Of Transfinite Set Theory
In order to explain Wittgensteinâs account of the reality of completed infinity in mathematics, a brief overview of Cantorâs initial injection of the idea into set- theory, its trajectory (including the Diagonal Argument, the Continuum Hypothesis and Cantorâs Theorem) and the philosophic implications he attributed to it will be presented. Subsequently, we will first expound Wittgensteinâs grammatical critique of the use of the term âinfinityâ in common parlance and its conversion into a notion of an actually existing (completed) infinite âsetâ. Secondly, we will delve into Wittgensteinâs technical critique of the concept of âdenumerabilityâ as it is presented in set theory as well as his philosophic refutation of Cantorâs Diagonal Argument and the implications of such a refutation onto the problems of the Continuum Hypothesis and Cantorâs Theorem. Throughout, the discussion will be placed within the historical and philosophical framework of the Grundlagenkrise der Mathematik and Hilbertâs problems
Hypatia's silence. Truth, justification, and entitlement.
Hartry Field distinguished two concepts of type-free truth: scientific truth and disquotational truth. We argue that scientific type-free truth cannot do justificatory work in the foundations of mathematics. We also present an argument, based on Crispin Wright's theory of cognitive projects and entitlement, that disquotational truth can do justificatory work in the foundations of mathematics. The price to pay for this is that the concept of disquotational truth requires non-classical logical treatment
On the Reality of the Continuum Discussion Note: A Reply to Ormell, âRussell's Moment of Candourâ, Philosophy: Anne Newstead and James Franklin
In a recent article, Christopher Ormell argues against the traditional mathematical view that the real numbers form an uncountably inïŹnite set. He rejects the conclusion of Cantorâs diagonal argument for the higher, non-denumerable inïŹnity of the real numbers. He does so on the basis that the classical conception of a real number is mys- terious, ineffable, and epistemically suspect. Instead, he urges that mathematics should admit only âwell-deïŹnedâ real numbers as proper objects of study. In practice, this means excluding as inadmissible all those real numbers whose decimal expansions cannot be calculated in as much detail as one would like by some rule. We argue against Ormell that the classical realist account of the continuum has explanatory power in mathematics and should be accepted, much in the same way that "dark matter" is posited by physicists to explain observations in cosmology. In effect, the indefinable real numbers are like the "dark matter" of real analysi
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