My Math Forum Is the Is definition of a ‘point’ in geometry contradictory?

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September 25th, 2018, 08:28 AM   #11
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Quote:
 Originally Posted by skipjack How do you define "exist"? Aren't you effectively saying that no measurement "exists"? For example, that "noon" doesn't exist?
- Measurements exist but not as part of the system being measured
- 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 non-zero length

Last edited by skipjack; September 25th, 2018 at 09:48 AM.

September 25th, 2018, 08:30 AM   #12
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Quote:
 Originally Posted by Devans99 - Measurement exist but not as part of the system being measured - 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 - Id argue a point is a geometric object so must have non-zero length

You still didn't mathematically define length.

 September 25th, 2018, 08:34 AM #13 Senior Member   Joined: Oct 2009 Posts: 784 Thanks: 280 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, 08:46 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 non-zero 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, 08:46 AM #15 Senior Member   Joined: Jun 2018 From: UK Posts: 103 Thanks: 1 -
 September 25th, 2018, 09:02 AM #16 Senior Member     Joined: Feb 2010 Posts: 706 Thanks: 140 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. 295-300. Thanks from Benit13
 September 25th, 2018, 10:08 AM #17 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,660 Thanks: 2635 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, 10:26 AM #18 Senior Member     Joined: Sep 2015 From: USA Posts: 2,431 Thanks: 1315 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, 10:51 AM #19 Senior Member   Joined: May 2008 Posts: 299 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, 11:15 AM #20 Senior Member   Joined: May 2016 From: USA Posts: 1,310 Thanks: 551 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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