Wednesday, April 24, 2013

Math and causality

I was reflecting on my old mini-series on causality (from 2010??), and I thought I'd conclude it with a recap of sorts.  But in this recap, I offer a reinterpretation of what I've written--with mathematics.

I am not a serious philosopher.  My discussions on causality are in the category of "personal musings", a tool for me to talk about other things that interest me.  They aren't really informed by a dialogue with previous philosophical writings or anything.  However, what they are informed by is a mathematical method of thinking.  I may not have explained it in mathematical terms, but I was certainly thinking in them.

It's just part of my education, as a physicist.  I see everything in mathematical terms, because that's the best way to understand physics.  When I've explained physics in the past, I generally avoid math because it scares people.  But when I'm talking to math people, physics is just so much easier to explain.  It goes something like this:

"Could you explain String Theory?  What's this about 11 dimensions?"
"Well, you're familiar with N-dimensional differential manifolds, right?"
"Yeah"
"It's like that, with N=11"
"Oooooh"
"And you can decompose it into a Cartesian product of space-time and a 7-dimensional compact manifold."
etc.

Math people: soooo cool.

Causation is easy to think of in mathematical terms.  Causation is a derivative of results with respect to prior conditions.  If the prior conditions are changed, then how much do the results change?

As I pointed out in Colds and causality, experimentally, we can't really measure derivatives.  So instead we take multiple data points and infer the derivative.  Usually there are a lot of random factors, so instead of looking at the results of individual data points, we look at the expected results.

In Women and causality I pointed out other complications with the definition as derivative.  The derivative of a function varies from point to point.  So causal relations may depend on current conditions.

Furthermore, results are not just a function of one variable.  They are functions of many variables.  I identify causation with the partial derivative with respect to one of the variables.  But this definition is ambiguous unless we specify what the other variables are.  For example, if I transform (u,v) to (u,w), this may change the partial derivative with respect to u, even though the value of u was unchanged.

In Nature/nurture and causality, I talked about another limitation of the derivative concept of causality.  If we want to compare the relative effect of two causes (for example, the effect of nature and nurture on human behavior), it seems easy enough to just take the derivative with respect to each of those variables.  However, the two variables have different units, and thus cannot be compared!  The typical solution is to consider the quantity

∂f/∂x * σx

where f is the results as a function of prior conditions, x is one of the prior conditions, and σx is the spread of variable x.  This quantity is no longer affected by the units of x.

In Responsibility and causality, I talk about how causality relates to ethics.  The relationship is often mediated by game theory, which is... math!  Well, okay, maybe ethics doesn't completely reduce to mathematics, but the math is important.

In Physics and causality, I talk about causality on a more fundamental level.  Fundamentally, the state of the universe obeys some differential equations.  By specifying the differential equations and the state of the universe at any given time, we end up specifying the entire history of the universe!  Fun.

Now I am going to go through this post and delete most of the exclamation marks so I don't sound so excited.

Other posts in this mini-series:
Colds and Causality
Women and Causality
Responsibility and Causality
Nature/nurture and Causality
Physics and Causality
Math and causality

Larry, The Barefoot Bum said...

The typical solution is to consider the quantity

∂f/∂x * σx

where f is the results as a function of prior conditions, x is one of the prior conditions, and σx is the spread of variable x. This quantity is no longer affected by the units of x.

Why are you multiplying instead of dividing?

Larry, The Barefoot Bum said...

Er... multiplying by σx instead of dividing by σx.

miller said...

When you differentiate with respect to x, that results In units that are the inverse of x's. You need to multiply by σx to cancel the units again.

Qualitatively, we can say that a greater the variance in, say, genetics, means that genetics contributes a greater proportion of the total variance in a particular trait.

Larry, The Barefoot Bum said...

D'oh! The x's are on the bottom of the differential. Thanks.