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Uninstantiatables in Aristotelian Mathematics
Any successful Aristotelian foundations of mathematics needs to account for mathematical objects that are uninstantiated and even uninstantiatable. Examples include (1) positive whole (or “natural”) numbers larger than the number of objects in reality, (2) negative numbers, and (3) infinities. Uninstantiated natural numbers As the Aristotelian sees things, we abstract quantity and structure from reality,… →
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Actualisation of potentiality as such
While we’re on the topic of confusing things Aquinas said, we can talk about his analysis of change, which he in turn gets from Aristotle. We’ve noted before that the first step in analysing change is the realisation that it involves the actualisation of a potential: When a hot cup of coffee gets cold, for example,… →
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Analogy at the foundations of mathematics
Consider the Benacerraf identification problem in philosophy of maths: there are multiple different ways of “defining” natural numbers in terms of sets, so there is no way of determining which definition is the “correct” one. This is not just a problem about natural numbers but they’re a useful notion to introduce the problem with. In fact,… →