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Extra resources for Caliban's Problem Book: Mathematical, Inferential and Cryptographic Puzzles
We briefiy take a look at Cantorial set theory and examine how it is used in the explanation of the paradox We wilaloo l look at geometric explanation of the paradox. The Notion of Limif The mathematical explanations of Zeno's paradoxes are nomally based on the notion of limit. Let us look at this notion first. Before the nineteenth century, mathematidans did not have a clear concept of fundions and Iimits. W. Leibniz Although their ideas gave concrete results that yielded definite and precise solutions, their explanations for why their proofs worked were full of unanswered questions.
Of course, this argument had much to do with the belief in God, who is the only tnie infinite. Descartes argued that it is God that has ingrained in our mind the idea of infinite. ) Although we are finite, and aKis unable to grasp the notion of infinitude this should not preclude its existence. Other rationalists, such as Pierre Gassendi, argued that our idea of the infinite was really only based on ouf rejedion of the finitist theory. Spinoza rej8ded this view and held a view that was more in line with Descartes and cwresponded to the Parmenidean principle of the denial of pluality.
Acoording to Bolzano, this leads to the m d u s i o n that the dass of al1 infinite time or moments, and the dass of al1 infinite spaœ or points are not real either. Sinœ intuitions must have an existing objed, we cannot Say that time or space are intuitions but pure concepts. 111)'. To Bolzano, time is a neœssary characteristic for the existence of any real thing and space is defined by the dass of al1 possible locations. Other philosophem such as Kant and Schultz treated spaœ end time not as concepts but as intuitions as they pertained to the ideas of total infinite spaœ and time (Bolzano.