January 22nd, 2019, 08:17 AM  #41  
Senior Member Joined: Jun 2014 From: USA Posts: 564 Thanks: 43  Quote:
 
January 22nd, 2019, 08:44 AM  #42 
Senior Member Joined: Oct 2009 Posts: 863 Thanks: 328  
January 22nd, 2019, 09:13 AM  #43 
Senior Member Joined: Jun 2014 From: USA Posts: 564 Thanks: 43  
January 22nd, 2019, 09:19 AM  #44 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,681 Thanks: 2659 Math Focus: Mainly analysis and algebra 
Also, because maths happens in the mind rather than the physical world, mathematicians are able to work with (some) infinite objects such as infinite decimals and infinite sequences.

January 22nd, 2019, 09:36 AM  #45 
Senior Member Joined: Oct 2009 Posts: 863 Thanks: 328  
January 22nd, 2019, 10:53 AM  #46 
Senior Member Joined: Jun 2014 From: USA Posts: 564 Thanks: 43  Let each line segment between 0 and 1 inch in length represent 1/3. We can always come up with more arbitrary ways to represent 1/3... Edit: PS  Just in case you don't have a magnifying glass that powerful, for each real number x (of which there are uncountably many), let 1/3 = x/3x. Last edited by AplanisTophet; January 22nd, 2019 at 11:04 AM. 
January 22nd, 2019, 11:11 AM  #47  
Senior Member Joined: Aug 2012 Posts: 2,386 Thanks: 746  Quote:
Last edited by skipjack; January 22nd, 2019 at 12:57 PM.  
January 22nd, 2019, 11:27 AM  #48  
Senior Member Joined: Jun 2014 From: USA Posts: 564 Thanks: 43  Quote:
My only point is that a representation of a number ought not be confused as being the number itself. There certainly are uncountably many representations that we could conjure. The line segment 'thing' above demonstrates that, as there are uncountably many line segments that have lengths of between 0 and 1 inches long. Last edited by skipjack; January 22nd, 2019 at 12:58 PM.  
January 22nd, 2019, 12:01 PM  #49  
Senior Member Joined: Aug 2012 Posts: 2,386 Thanks: 746  Quote:
That couldn't be more false, as it contradict known physics and profoundly misrepresents the nature of physical measurement. Last edited by skipjack; January 22nd, 2019 at 12:59 PM.  
January 22nd, 2019, 12:06 PM  #50  
Senior Member Joined: Oct 2009 Posts: 863 Thanks: 328  Quote:
 