I think writing to an audience has its virtues, so this is also more than a diary. An audience challenges me to commit to truth—especially when it’s vulnerable and so easy to be swayed by the expectations of others, and perhaps even moreso by absurd expectations of myself.

So that’s a little bit of the why a new blog. As for the what, Left Ideals means two things to me:

1. I was raised in the world of left ideals—politically, morally, relationally—and I have been reflecting a bunch lately on how this has shaped my lens on the world.
   
   Politics in America has become so absurd, that it feels to me like something funny is going on. More human-condition funny than foul-play funny.
   
   It feels highly unlikely to me that either the political left or right in America is ultimately “in the right”, but any time I reflect on any particular details of disagreement, I feel compelled by weight of my personal history to believe I am more in the right. For example, that I have loved ones voting for Trump simply perplexes me.
   
   Somehow, it feels more likely to me that we see the shadow sides of each other, shadows that we are blind to or dismissive of in ourselves, and that whatever impossible gap there is between perspectives, it may have more to do with a gap within myself than with a gap between me and my loved ones. If they are anything like me, they are guarding within themselves some inviolable truths. I find it hard to believe those truths don’t also live within me.
   
   Perhaps I have taken something which is inviolable to them as simply a part of the cost of living. Are those I most disagree with protecting gifts of truth that I abandoned? I suspect the more I understand the internal worlds of those most unlike me, I’ll find more of myself.

2. My favorite mathematical place in the world to explore is called a left ideal of a Clifford algebra, and one interesting fact is that a left ideal (or right ideal—doesn’t matter) is generated by a projection operator p which forgets information about a whole, which I’ll call ψ.
   
   For example, if I give you ψ and a friend ψp (multiplying p on the left with ψ), then you have more information than your friend, and if you know the projector p, which generates the ideal that your friend is stuck in, then you can reconstruct what your friend sees ψp. These quantities ψ and ψp are called spinors, and are building blocks of quantum theory.
   
   The projector p also has an orthogonal complement p̄, which together form the identity p + p̄ = 1 and mutually annihilate each other pp̄ = 0 = p̄p.
   
   Since I love an analogy, let’s say I live in a left ideal and you live in a right ideal. The world ψ looks like ψp to me and ψp̄ to you. Neither of us sees the whole ψ—we only see projections, flattenings of it. But if we hold our images side by side, just to see what would happen, then (perhaps) the illusions give way:

I make no promises about what I’m going to write about in this blog, but I want a place where I can write about math and love, and language, and symmetries, and gender, and breaking, and plenty of older brother cringe material!

I expect analogies always to break down somewhere, so my commitment is not to precision first, but to freeing the flow of my thinking where there are knots in me.

If I have to choose between math and love, may I choose love. But, if I need not choose, let me quotient out the differences. Like a good mother loving more than one child—all the same, without forgetting the beautiful differences between. Like a good father—listening & understanding deeply. Up to a flip in gender—I call that Z2 :)

-Luke

Just finished a cross country drive from Philadelphia to Los Angeles! Here are some views from the road.