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Despite positional number systems, Aristotle's argument is quite convincing. Of relevance to our discussion is the remarkable advance made by the Babylonians who introduced the idea of a positional number system which, for the first time, allowed a concise representation of numbers without limit to their size. However they are potentially infinite in the sense that given any finite collection we can always find a larger finite collection. His idea was that we can never conceive of the natural numbers as a whole. Aristotle argued against the actual infinite and, in its place, he considered the potential infinite. He introduced an idea which would dominate thinking for two thousand years and is still a persuasive argument to some people today. Aristotle did not seem to have fully appreciated the significance of Zeno's arguments but the infinite did worry him nevertheless. Of course these paradoxes arise from the infinite. However Zeno's paradoxes show that both the belief that matter is continuously divisible and the belief in an atomic theory both led to apparent contradictions. Parmenides and the Eleatic School, which included Zeno, argued against the atomists. Then there were Atomists who believed that matter was composed of an infinite number of indivisibles. Pythagoras had argued that "all is number" and his universe was made up of finite natural numbers. In their study of matter they realised the fundamental question: can one continue to divide matter into smaller and smaller pieces or will one reach a tiny piece which cannot be divided further. The early Greeks had come across the problem of infinity at an early stage in their development of mathematics and science. We should begin our account of infinity with the "fifth-century Eleatic" Zeno. What happened if one cut a piece of wood into two pieces, then again cut one of the pieces into two and continued to do this. There were more subtle questions about infinity which were also asked at a stage when people began to think deeply about the world. The questions above are very fundamental and must have troubled thinkers long before recorded history.
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What happened if one kept travelling in a particular direction? Would one reach the end of the world or could one travel for ever? Again above the earth one could see stars, planets, the sun and moon, but was this space finite or did it go on for ever? Did the world come into existence at a particular instant or had it always existed? Would the world go on for ever or was there a finite end? Then there were questions about space. Of course from the time people began to think about the world they lived in, questions about infinity arose. The dialectical puzzles of the fifth-century Eleatics, sharpened by Plato and Aristotle in the fourth century, are complemented by the invention of precise methods of limits, as applied by Eudoxus in the fourth century and Euclid and Archimedes in the third. The interaction of philosophy and mathematics is seldom revealed so clearly as in the study of the infinite among the ancient Greeks. This is particularly true in ancient Greek times, as Knorr writes in :. Does one concentrate purely on the mathematical aspects of the topic or does one consider the philosophical and even religious aspects? In this article we take the view that historically one cannot separate the philosophical and religious aspects from mathematical ones since they play an important role in how ideas developed.
Ac infinity archive#
An article on infinity in a History of Mathematics Archive presents special problems.