September 25th, 2018, 09:28 AM  #11  
Senior Member Joined: Jun 2018 From: UK Posts: 103 Thanks: 1  Quote:
 A measurement is the process of assigning a numerical quantity to a physical property  So a measurement is a number rather than a geometric object  So measurements do not have a length property  Whereas I’d argue a point is a geometric object so must have nonzero length Last edited by skipjack; September 25th, 2018 at 10:48 AM.  
September 25th, 2018, 09:30 AM  #12  
Senior Member Joined: Oct 2009 Posts: 628 Thanks: 190  Quote:
You still didn't mathematically define length.  
September 25th, 2018, 09:34 AM  #13 
Senior Member Joined: Oct 2009 Posts: 628 Thanks: 190 
So in mathematics, a plane geometry is a set of points and a set of lines such that Axiom 1: Through every two points is exactly one line. Axiom 2: For every line, there is a point not lying on the line. etc etc etc Now, I understand you find all of this garbage, since you made it clear you don't accept it. So, what do you replace it with? What are the axioms you will replace this by? 
September 25th, 2018, 09:46 AM  #14 
Senior Member Joined: Jun 2018 From: UK Posts: 103 Thanks: 1 
Before working out new axioms, we need a clear definition of a point: ‘A point is a one dimensional object with an infinitesimal but nonzero length.’ Where an infinitesimal is a quantity approaching but never reaching 0. I hope with a definition like this most of the existing axioms are still ok? 
September 25th, 2018, 09:46 AM  #15 
Senior Member Joined: Jun 2018 From: UK Posts: 103 Thanks: 1 


September 25th, 2018, 10:02 AM  #16 
Senior Member Joined: Feb 2010 Posts: 697 Thanks: 135 
Most standard developments for geometry start with point, line, plane as undefined terms. Then axioms (postulates) are stated using these undefined terms. Although Euclid starts with a definition of point, I think that this leads to a lack of rigor in the development. You might want to read the article "How Big is a Point", The College Mathematics Journal, Vol. 14, No. 4, (1983), pp. 295300. 
September 25th, 2018, 11:08 AM  #17 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,511 Thanks: 2514 Math Focus: Mainly analysis and algebra 
He doesn't understand that mathematics' primary use is as a model, not reality. I don't know how well he'd accept the idea that a model that doesn't make simplifications is useless.

September 25th, 2018, 11:26 AM  #18 
Senior Member Joined: Sep 2015 From: USA Posts: 2,201 Thanks: 1156 
This whole refusal to understand what a model is and claim that no model has value because it's not a perfect representation of reality is just silly really. No one claims you can do anything with a point of 0 length. I believe it's well established that length's shorter than the Planck length can never be measured. But looking around I'd say we've done pretty well using the model of a point we have. 
September 25th, 2018, 11:51 AM  #19 
Senior Member Joined: May 2008 Posts: 297 Thanks: 81 
Beer soaked ramblings follow. You could try "defining" the point as a cirle whose radius is equal to zero. With the circle defined as a set of points equidistant from a fixed point, it's a good bet that you won't be going in squares any time soon. 
September 25th, 2018, 12:15 PM  #20 
Senior Member Joined: May 2016 From: USA Posts: 1,206 Thanks: 494 
As far as I can tell, the OP insists on physical validation of mathematical axioms. He does not seem to get that axioms and definitions are free creations of the human mind. It is one thing to say you personally do not accept an axiom because it cannot be observed in the physical world. Then develop your mathematics without that axiom. It is something else entirely to legislate for others. The idealizations of mathematics have resulted in mathematical tools that have been found practically useful. I do not believe myself in the physical observability of irrational numbers, but working with the real number system is how I do mathematics nevertheless. 

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‘point’, contradictory, definition, geometry 
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