Science's Kernel

Science can mean a lot of things to a lot of people.

The main working definition I’ve settled on after spanning science, engineering, and medicine: science is the discovery of dynamics.

Science studies $y$

If we want to study how our dependent variable $y \in \mathbf{R}$ changes, an implicit thing we start with is “over time”: $y(t) \in \mathbf{R}$. Since we’re so often frequentist, we tend to think of $y$ as a single static thing.

So either we just measure $y$ once and assume it’s always like that $y(t) = y(t=t_0)$ or we measure $y$ several times and collapse all of them into just a single $y$ - often by treating all observed ${y(t)}, t = t_0, t_1, …, t_n$ as just instances of that single static thing.

In any case, this sort of description of $y$ is often where we stop before we jump over to independent variables that we seek to link directly to these descriptions. But we shouldn’t stop here…

What are Dynamics?

To say something is dynamic is to say that it changes. That it changes is almost trivially true - name one thing in the universe that doesn’t change. How it changes is the key part of dynamics of interest to scientific inquiry.

Dynamical Systems is the study of the rules by which things change. How it changes over time is a function of things.

So if we care about a dependent variable $y$ and think that it’s influenced by some independent variable $x$ then we could construct a TSM ¹ machine:

And our null hypothesis is that $f(x) = 0 \times x$, but that any variation we see in $y$ is purely noise $\eta$. This is garbage and silly ².

What makes more sense is that some independent variable $x$ affects how $y$ changes over time.

So:

$y(t) \rightarrow \dot{y} = \frac{dy}{dt} = f(x) + \eta$

Here, $x$ doesn’t directly influence $y$ - it influences how $y$ changes over time. This is subtle in text, but seismic in effect.

Science is Inference of Dynamics

There’s much more to say about all of this, and I will, but suffice it to say, I think “science” is often just estimating the dynamical system of interest. In other words, science (even “causal” science) is effectively real-world differential equation estimation (O/PDE estimation).

So any/all AI efforts in science should anchor themselves in O/PDE estimation… we sure as hell will at nform.ai.