The beauty of maths — metric spaces

I’m not a maths major. Yet I do feel the built-in elegance of mathematics while I’m learning Analysis this semester.

I. Metric Space: Interpersonal Distance

A metric space X must satisfy:

(M1) d(x,y)>=0 and d(x,y)=0 iff x=y;

(M2) (symmetry) d(x,y)=d(y,x);

(M3) (triangle inequality) d(x,y)<=d(x,z)+d(z,y).

The definition “d” here is like the distance between people. x, y, and z are different individuals. The distance is always positive, even if we are faced with parents or closed friends. We want some private space to store our deepest emotions and secrets. However, there is one person that has no distance from us–ourselves.

The second condition seems to be a bit optimistic. It suggests that my feeling about our distance is the same as your feeling about the distance between us. This is not necessarily the case. Sometimes you trust someone deeply, yet she is not aware of that and just treats you as a “say-hi” friend. Our parents feel close to us, but we always want to avoid being too close to them to keep our independence.

The third equation is more interesting. Friend’s friend is somehow closer than the two distance added together! This makes sense because by introducing his friend to you, your friend saves your efforts. The two of you, though just know each other, are likely to develop longlasting relationship as well.

II. Discrete metric space: Loneliness

X is a non-empty set, d(x,y):=1-corr(x,y). (X,d) is called a discrete metric space. Indeed, the distance from x are all 1 except for x itself (which is 0). Some people keep others out from their heart. They stick to a stubborn distance in their relationships. The world collapses into a point, and the point is the individual life. His social network is like a circle: he is in the middle and everyone else is on the margin of his world.

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