18090
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Without infinity time has limits, magnitudes are indivisible, and numbers come to an end [Aristotle]
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Full Idea:
If there is, unqualifiedly, no infinite, it is clear that many impossible things result. For there will be a beginning and an end of time, and magnitudes will not be divisible into magnitudes, and number will not be infinite.
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From:
Aristotle (Physics [c.337 BCE], 206b09), quoted by David Bostock - Philosophy of Mathematics 1.8
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A reaction:
This is a commitment to infinite time, and uncountable real numbers, and infinite ordinals. Dedekind cuts are implied. Nice.
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22929
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Aristotle's infinity is a property of the counting process, that it has no natural limit [Aristotle, by Le Poidevin]
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Full Idea:
For Aristotle infinity is not so much a property of some set of objects - the numbers - as of the process of counting, namely of its not having a natural limit. This is 'potential' infinite
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From:
report of Aristotle (Physics [c.337 BCE]) by Robin Le Poidevin - Travels in Four Dimensions 06 'Illusion'
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A reaction:
I increasingly favour this view. Mathematicians have foisted fictional objects on us, such as real infinities, limits and zero, because it makes their job easier, but it makes discussion of the natural world very obscure.
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16150
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One is, so numbers exist, so endless numbers exist, and each one must partake of being [Plato]
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Full Idea:
If one is, there must also necessarily be number - Necessarily - But if there is number, there would be many, and an unlimited multitude of beings. ..So if all partakes of being, each part of number would also partake of it.
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From:
Plato (Parmenides [c.364 BCE], 144a)
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A reaction:
This seems to commit to numbers having being, then to too many numbers, and hence to too much being - but without backing down and wondering whether numbers had being after all. Aristotle disagreed.
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9974
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Ten sheep and ten dogs are the same numerically, but it is not the same ten [Aristotle]
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Full Idea:
If there are ten sheep and ten dogs, the number is the same (because it does not differ by a numerical difference), but it is not the same ten (because the objects it is predicated of are different - dogs in one instance, horses in the other).
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From:
Aristotle (Physics [c.337 BCE], 224a2-14)
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A reaction:
Mega! Abstract objects are unique, and can't be 'added' to themselves. I think we need 'units' here, because 2+2 adds four units, so each 2 refers to something different. '2' must refer to something other than itself.
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