Category: Infinity
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Topics in Current Technical Threads
1) Potential versus actual infinity. 2) Are there mathematical sets? 3) Does mathematics need a foundation in set theory? 4) Is there irreducibly plural reference, predication, and quantification? If yes, does plural quantification allow us to avoid ontological commitment to sets? 5) Discreteness, density, and continuity at the level of number theory, geometry, and nature…
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Notes on Infinite Series
The resident nominalist writes, Your post generated a lot of interest. What I have to say now is better put as a separate post, rather than a long comment. Feel free to post. 1) Plural reference provides a means of dealing with numbers-of-things without introducing extra unwanted entities such as sets. Even realists agree that…
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On Potential and Actual Infinity, and a Puzzle
Consider the natural numbers (0, 1, 2, 3, . . . n, n +1, . . . ). If these numbers form a set, call it N, then N will of course be actually infinite. This because a set in the sense of set theory is a single, definite object, a one-over-many, distinct from each…
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World + God = God: A Mathematical Analogy
The Big Henry offers the following comment on my post, World + God = God? "World + God = God" is (mathematically) analogous to "number + infinity = infinity", where "number" is finite. If God embodies all existence, then God is "existential infinity", and, therefore, no amount of existence can be added to or subtracted…
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Does Potential Infinity Presuppose Actual Infinity?
Returning to a discussion we were having back in August of 2010, I want to see if I can get Peter Lupu to agree with me on one point: It is not obvious or compellingly arguable (arguable in a 'knock-down' way) that there are infinite sets. Given my aporetic concerns, which Peter thoroughly understands, I…
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Infinity and Mathematics Education
A reader writes, Regarding your post about Cantor, Morris Kline, and potentially vs. actually infinite sets: I was a math major in college, so I do know a little about math (unlike philosophy where I'm a rank newbie); on the other hand, I didn't pursue math beyond my bachelor's degree so I don't claim to…
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Kline on Cantor on the Square Root of 2
Morris Kline, Mathematics: The Loss of Certainty, Oxford 1980, p. 200: . . . when Cantor introduced actually infinite sets, he had to advance his creation against conceptions held by the greatest mathematicians of the past. He argued that the potentially infinite in fact depends on a logically prior actually infinite. He also gave the…
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Does Potential Infinity Rule Out Mathematical Induction?
In an earlier thread David Brightly states that "The position on potential infinity that he [BV] is defending is equivalent to the denial of the principle of mathematical induction." Well, let's see. 1. To avoid lupine controversy over 'potential' and 'actual,' let us see if we can avoid these words. And to keep it simple, let's…
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Doron Zeilberger’s Ultrafinitism
This is wild stuff; I cannot say whether it is mathematically respectable but the man does teach at Rutgers. It is certainly not mainstream. Excerpt: It is utter nonsense to say that sqrt 2 is irrational, because this presupposes that it exists, as a number or distance. The truth is that there is no such number…