April 11th, 2017, 06:22 AM  #21 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,400 Thanks: 100  The points are locations on a ball, just as points are locations on a line. The points are not the ball. They are an imaginary ball. If it's not a real ball, it's not a real paradox. 
April 11th, 2017, 01:34 PM  #22 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,400 Thanks: 100 
Consider a set of n "things. "If I remove m things, but don't change their identifying characteristics, and replace them, it's the same set (order not defined). Now consider an infinite set. Infinity is not defined. The set doesn't exist, except as words "an infinite set." On the other hand, if i define a set as a container, the contents don't matter. An egg basket is different than a collection of eggs. A ball (sphere) as an infinite collection of points is meaningless because infinity is undefined. A ball as a container of points is independent of the points. You can't remove points to create another ball. All you can do is move the ball or place some points in another, preexisting, ball. (A basket is independent of the eggs. You can't remove eggs to create another basket. All you can do is place some eggs in another basket.) The mathematical gyrations you go through with the points (eggs) is totally irrelevant. 
April 11th, 2017, 01:38 PM  #23  
Senior Member Joined: Aug 2012 Posts: 1,973 Thanks: 551  Quote:
You're making a really bizarre argument. You're refusing to do math because math doesn't exist. If math doesn't exist, why do you do math in other threads? The concerns you're bringing up apply to all math, not just the BT theorem. Why would you start a thread about BanachTarski just to complain that mathematical 3space isn't real? That doesn't even have anything to do with BanachTarski. Last edited by Maschke; April 11th, 2017 at 01:45 PM.  
April 12th, 2017, 07:11 AM  #24 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,400 Thanks: 100 
It's not a matter of philosophy, it's a matter of mathematical precision. You can do anything you want with an infinite set of points. You can take enough points from a pea and create the universe and still have the pea left over. There is a big difference between is an infinite set of points and contains an infinite set of points. If the pea contains an infinite set of points (locations actually), I can put an infinite set of points (locations) in every preexisting object in the universe, and every imaginable geometric figure, and still have an infinite set of points left in the pea. Now some philosophy. The problem is the continuum. If you did all your thinking in terms of finite number and then let that number become very large, everything would make sense (classical analysis). If you specifically reject making sense, that's fine for you and your little club, but don't take your results to the world and announce a "paradox" as a brilliant intellectual achievement. It isn't. It's just a misconception obscured by smoke. EDIT Short and Sweet You can't create a finite dimension from a 0 dimension. Last edited by zylo; April 12th, 2017 at 07:52 AM. 
April 12th, 2017, 09:39 AM  #25 
Senior Member Joined: Aug 2012 Posts: 1,973 Thanks: 551  
May 8th, 2017, 09:24 AM  #26 
Member Joined: Dec 2016 From: United States Posts: 53 Thanks: 3 Math Focus: Abstract Simulations 
Dude, math is just rules that are consistent with the context you apply it in. There's a lot of religion around math. We keep trying to "discover" more things, when we're just really interpreting constraints from preexisting contexts. For example. Topology is just the study of geometry with a function applied to everything in the system. 

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