Category: Mathematics
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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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Happy Super π Day!
π day is 3/14. But today is super π day: 3/14/15. To celebrate it properly you must do so at 9:26 A.M. or P. M. Years ago, as a student of electrical engineering, I memorized π this far out: 3.14159. The decimal expansion is non-terminating. But that is not what makes it an irrational number. …
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Closure: Some Mathematical and Philosophical Examples
A reader asks, "What is meant by 'closure' or 'closed under'? I've heard the terms used in epistemic contexts, but I've not been able to completely understand them." Let's start with some mathematical examples. The natural numbers are closed under the operation of addition. This means that the result of adding any two natural numbers…
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What are Numbers? Some Dubious Philosophy of Mathematics Exposed
Here we read: . . . aren't all numbers inventions? It is not like they grow on trees! They live in our heads. We made them all up. The author of the quotation is introducing a discussion of the imaginary number i = the square root of -1. His point is that we are…
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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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Innumeracy in the Check-Out Line
The Sarah Lee frozen pies were on sale, three for $10, at the local supermarket. I bought two, but they rang up as $4.99 each. I pointed out to the check-out girl that this was wrong, and she sent a 'gofer' to confirm my claim. Right I was. But now the lass was perplexed, having…
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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…
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On Potential and Actual Infinity
Some remarks of Peter Lupu in an earlier thread suggest that he does not understand the notions of potential and actual infinity. Peter writes: (ii) An acorn has the potential to become an oak tree. But the acorn has this potential only because there are actual oak trees . . . . If for some…
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Zeno’s Regressive Dichotomy and the ‘Calculus Solution’
The Regressive Dichotomy is one of Zeno's paradoxes of motion. How can I get from point A, where I am, to point B, where I want to be? It seems I can't get started. A_______1/8_______1/4_______________1/2_________________________________ B To get from A to B, I must go halfway. But to travel halfway, I must first traverse half…
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Social Utility and the Life of the Mind: The Example of Complex Numbers
Much as I disagree with Daniel Dennett on most matters, I agree entirely with the following passage: I deplore the narrow pragmatism that demands immediate social utility for any intellectual exercise. Theoretical physicists and cosmologists, for instance, may have more prestige than ontologists, but not because there is any more social utility in the satisfaction of…