# B Banach-Tarski

#### thinkandmull

I am an avid student of philosophy, not mathematics. I was more interested in these topics for there philosophical implications. It seems to me that Zeno showed the uncountable infinity of space long before Cantor, and that that infinity alone leads directly to Banach-Tarski. I don't know why one's a paradox but not the other. Pierre Bayle (his 1696 article on Zeno), Kant, Hegel and others thought these arguments show that reality is contradictory in the sense that Escher's "Ascending, Descending" picture is. Mathematicians don't always understand the shock waves they send through the field of philosophy. The Zenonian box: half of it stacked progressively on top of the other half, one fraction at a time, first blue then green, then blue again ect. When the box is whole again, will the top be blue or green? If traditional philosophy is true, there must be one color left on top. Period. But this is definitely a discussion for a different forum. Thanks

#### Mark44

Mentor
I am an avid student of philosophy, not mathematics.
We don't discuss philosophy at PF, so I'm closing this thread.
thinkandmull said:
I was more interested in these topics for there philosophical implications. It seems to me that Zeno showed the uncountable infinity of space long before Cantor, and that that infinity alone leads directly to Banach-Tarski.
I don't think so. What Zeno's Paradox shows is a lack of understanding about how an infinite sum can add up to a finite number.
thinkandmull said:
I don't know why one's a paradox but not the other. Pierre Bayle (his 1696 article on Zeno), Kant, Hegel and others thought these arguments show that reality is contradictory in the sense that Escher's "Ascending, Descending" picture is.
If you really want to understand mathematics, you would be better off reading articles newer than 1696, written by mathematicians.
thinkandmull said:
Mathematicians don't always understand the shock waves they send through the field of philosophy.
I don't think this keeps them up at night.
thinkandmull said:
The Zenonian box: half of it stacked progressively on top of the other half, one fraction at a time, first blue then green, then blue again ect.
Et cetera is abbreviated as etc., not ect.
thinkandmull said:
When the box is whole again, will the top be blue or green?
If traditional philosophy is true, there must be one color left on top. Period.
Since there can be no last box in an infinite sequence of ever smaller boxes, one can conclude only that traditional philosophy is not true, using the principles of logical arguments. That is, $p \Rightarrow q$ is equivalent to $\neg q \Rightarrow \neg p$.
thinkandmull said:
But this is definitely a discussion for a different forum. Thanks

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