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September 25th, 2018, 08:30 AM   #1
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Is the Is definition of a ‘point’ in geometry contradictory?

“A point is defined only by some properties, called axioms, that it must satisfy. In particular, the geometric points do not have any length, area, volume or any other dimensional attribute”

https://en.m.wikipedia.org/wiki/Point_(geometry)

If a point has no length it does not exist so the definition is surely logically contradictory?

The definition of point in maths leads to confusion. For example:

- people talk about an actually infinite number of points on a line segment
- But point must has length > 0 else it does not exist.
- With this revised definition of a point the number of points on any line segment is always a finite number rather than Actual Infinity.
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September 25th, 2018, 08:40 AM   #2
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Cool.

So how do you define length?
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September 25th, 2018, 08:46 AM   #3
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By length, I mean the a point cannot exist as a 0 dimensional object (that is another contradiction). So the point must have dimensions, one of which I’ve labelled ‘length’. So length might be equivalent to the x dimension of Cartesian space.
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September 25th, 2018, 08:52 AM   #4
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Quote:
Originally Posted by Devans99 View Post
By length, I mean the a point cannot exist as a 0 dimensional object (that is another contradiction). So the point must have dimensions, one of which I’ve labelled ‘length’. So length might be equivalent to the x dimension of Cartesian space.
Please give a rigorous axiom system or definition or anything, that allows you to say what you mean with length.
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September 25th, 2018, 08:57 AM   #5
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https://en.m.wikipedia.org/wiki/Length

“length is the most extended dimension of an object.[1] In the International System of Quantities, length is any quantity with dimension distance”
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September 25th, 2018, 09:01 AM   #6
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Quote:
Originally Posted by Devans99 View Post
https://en.m.wikipedia.org/wiki/Length

“length is the most extended dimension of an object.[1] In the International System of Quantities, length is any quantity with dimension distance”
Not a mathematical definition or axiom, sorry. Try again.
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September 25th, 2018, 09:04 AM   #7
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Well how would you define length then?
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September 25th, 2018, 09:06 AM   #8
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Quote:
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Well how would you define length then?
Read a geometry book.

For a rigorous treatment defining length from scratch, read Moise, "Elementary geometry from an advanced standpoint". That provides into to the two most treated geometries, Birkhoffs system, and Hilbert's system.

Once you digested that, come back and discuss this again.
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September 25th, 2018, 09:07 AM   #9
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Quote:
Originally Posted by Devans99 View Post
. . point must has length > 0 else it does not exist.
How do you define "exist"? Aren't you effectively saying that no measurement "exists"? For example, that "noon" doesn't exist?
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September 25th, 2018, 09:16 AM   #10
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Hey dude, if you want to model the earth rotating around the sun without taking the simplification that the earth is a point without length and with therefore taking into account the entire rigid body dynamics involved, go right ahead, not going to stop you.
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