MathFoundations24: The basic framework for geometry (II)

[samruby82]

Later on in your video series you give a proof of perpendicular in terms of quadrances using pythagoras. I feel like this is circular since that proof depends on right angle triangles (perpendicular lines) in the first place. In this video you just give a definition for perpendicular without explanation @2.20: a1a2 + b1b2 = 0. Is it possible to give a proof using something else? Preferably only using concepts you've discussed up to now and not using the word "angle". ps. these videos are great.

[njwildberger]

Hi samruby82,

The best way to define perpendicularity is via a formula, namely a1a2+b1b2=0. That way it is completely

unambiguous. Later one proves that perpendicular lines satisfy Pythagoras' theorem. You are quite right to be keeping your eye out for possible circular reasoning---this is surprisingly prevalent in modern mathematics. We want to avoid it at all costs. So I hope this clarifies the position: trying to define perpendicularity by mimicking our `physical' intuition doesn't work.

[samruby82]

similarly if we aren't supposed to think of the geometric representation then how would you have divined the definition of line passing through a point [x,y] as

=> ax + by + c = 0

yes the algebra works out but thats because you have a fabulous guess here based on years of looking at the intuitive, geometric/ picture representation. How would you postulate such a theorem/definition without a real world analogue to base it off?

[njwildberger]

Hi samruby82 When we investigate a mathematical theory we use pictures, intuition, perhaps also physical models. However at some point we should try to become more precise and state exactly what we are talking about, and provide careful proofs. So while the geometrical pictures you (and I) are very fond of are important to develop intuition, the mathematical foundations of the subject need to be separate from them. That is why the algebra is so important.

[hugstablebear]

Why are the vertices of a triangle defined as {l sub 1, l sub 2}, when the meet of two lines l sub 1 l sub 2 has already been defined, i.e. if a triangle has lines, aren't its vertices simply the meets of its lines?

I also don't understand why the notation, which normally represents a line segment is being used to represent a set consisting of only two points. It is not explained why a side is considered to consist only of it's endpoints and without length. How can ratios of sides be described?

[erictsfx]

nice