B Existence of parallels in axiomatic plane geometries

[mathwonk]

I agree with you that ratios preceded numbers. I.e. the ability to compare things of the same kind is the fundamental idea, i.e. equivalence relations, or when are two things to be considered the same in some sense. e.g. if you consider two plane vectors as the same if they have the same length and direction, you have an equivalence relation that can be used to define complex numbers, i.e. to add them you use vector addition and to multiply them you add their angles and multiply their lengths.
It is interesting that again to define multiplication you need a unit vector, i.e. to define addition of angles, as well as multiplication of lengths.

numbers arise when you try to capture the essence of an equvalence class by some one thing. but "numbers" also imply to me some way to have computational operations on the classes that obey some of the usual laws.

 

[bhobba]

Bear in mind that when we write "two lines that intersect intersect in exactly one point" we are defining what a point is;

I thought point was a primitive of the axioms rather than a definition in it based on other primitives:
https://en.wikipedia.org/wiki/Hilbert's_axioms

Or am I missing something?

Thanks
Bill

 

[pbuk]

I thought point was a primitive of the axioms rather than a definition in it based on other primitives:
https://en.wikipedia.org/wiki/Hilbert's_axioms

Maybe (this thread is not using Hilbert's axiomatisation), but I don't think it is wrong to say "we are defining what a point is" rather than "we are adopting a primitive notion of what a point is".

 

[lavinia]

@mathwonk

So starting with two geometrically determined line segments what does it mean to compare them with no unit as a guide?

Is it modular arithmetic?

 

[mathwonk]

Part of the axiom system is the ability to compare two segments and say whether they are equal and if not, which is larger. Euclid's axioms are full of conditions on the notion of "equal", which he seems to apply to lengths, areas, volumes and angles. His theorem statements also include statements about angles being equal, greater than or less than another. See his postulates 4,5 and all 5 common notions, as well as many propositions, e.g. all of the first 20 or so that I have quickly scanned, contain one of these comparison words. Hilbert has a set of axioms for congruence, and one can define less than and greater than in terms of congruence to a subset.

 

[lavinia]

Part of the axiom system is the ability to compare two segments and say whether they are equal and if not, which is larger. Euclid's axioms are full of conditions on the notion of "equal", which he seems to apply to lengths, areas, volumes and angles. His theorem statements also include statements about angles being equal, greater than or less than another. See his postulates 4,5 and all 5 common notions, as well as many propositions, e.g. all of the first 20 or so that I have quickly scanned, contain one of these comparison words. Hilbert has a set of axioms for congruence, and one can define less than and greater than in terms of congruence to a subset.

It would seem that one can lay multiples of one segment down on the other and get remainders. With the remainder one can repeat the process. If eventually one gets only repeats of congruent segments then the process ends in a finite number of steps and the two original segments are rational multiples of each other. If the process continues forever then they are not.

Examples.

The two segments are in a ratio of 5 to 3. Lay the smaller segment on the larger to get a remainder that is in a ratio of 5 to 2. Lay the smaller segment twice on the larger to get a remainder in a ratio of 5 to 1. All segments are multiples of this and the process stops.

If one introduces a unit it seems that one will end of with a segment whose length is the greatest common divisor to the original two.

 

[mathwonk]

yes! this is the famous euclidean algorithm for finding gcd's.