Local and Global and the Terrible Telescope, Part 1

In the comment thread for my previous post, I remarked that

I don’t know of a good theory that bridges the chasm between individual justice (where I think free association is an important fundamental right) and global justice (universal behaviors like [modes of systemic oppression]). I don’t pretend to have one.

What I’d like to do in this series is try to convince you that you probably don’t have one either.

• Local and Global • William Lane Craig is Terrible • Kant is Also Terrible
• the Repugnant Conclusion is Neither • Telescoping Considered Harmful
• You Are Not God, You’re Not Even President of the Galaxy
•  The Can’tegorical Imperative • Further Reading

1. Local and Global

One hard lesson every student of advanced mathematics learns is that a construction that seems easy and obvious locally–for instance, in a small region of a space–can be very difficult, or outright impossible, to make meaningful globally–eg, over the whole space.

For example, think about the direction ‘North’.  Which way is north? Wherever you happen to be reading this, you probably know already; if you don’t, your phone can probably tell you. ‘North’ is an easy local concept–we can all check which way north lies. We can find ‘north’, no matter where we go. Right?

Wrong.

There are two points on the surface of the Earth where you can’t meaningfully define ‘north’: the poles. At the North Pole, you can’t go any further north: every direction is south! At the south pole, every direction is north, so ‘north’ isn’t unique, and so isn’t well-defined.

(East and west are also problematic.)

In fact, you can prove mathematically that there is no possible way to extend ‘north’ in a consistent way to every point on the globe. This fact is called the Hairy Ball Theorem. (Yeah, seriously, that’s what it’s called.)

The local-global problem is ubiquitous in mathematics. Ideas that make sense up close turn out, again and again, to be difficult or impossible to extend everywhere.  There are whole areas of mathematics that focus on the question of whether you can extend an idea from a part to the whole.

What’s any of this got to do with moral philosophy? I hope to persuade you: a whole lot.

Next: Geometric Theology.

Author: strevdrrev

"Then go away!" said Zhuangzi, "and I will drag my tail in the mud." View all posts by strevdrrev