Wednesday, August 14, 2019

Disjunctions and differential equations

It is plausible that:

  1. Some of the fundamental dynamic laws of nature are given by differential equations.

  2. All fundamental dynamic laws of nature provide fundamental causal explanations.

  3. Facts that involve disjunction do not enter into fundamental causal explanations.

But one cannot believe (1)–(3). For:

  1. Facts about derivatives are facts about limits.


  1. Facts about limits are infinite conjunctions of infinite disjunctions of infinite conjunctions.

For the limit of f(x) as x → y equals z if and only if every neighborhood N of z there is a neighborhood M of x such that for all u ∈ M we have f(u)∈N. Universal quantification is a kind of conjunction and existential quantification is a kind of disjunction.

I am inclined to reject (1).


Alexander R Pruss said...

Maybe this is an argument in favor of infinitesimals in the fundamental laws. For with infinitesimals we can define the limit with two conjunction, at least in a Euclidean space: f(x)->z as x->y iff for every infinitesimal a, |f(x+a)-f(x)| is infinitesimal. And being an infinitesimal is itself a conjuntive property: a is infinitesimal iff for all real b>0, |a|<b.

The downside of that is that deterministic laws seem to cease to be deterministic, because an infinitesimal perturbation will still satisfy the same differential equations.

Michael Gonzalez said...

What motivation is there for (3)?

Heath White said...

I am inclined to reject (5). The definition in terms of infinitesimals is much more natural, and perhaps there are other ways to define limits also. (Or: we could just take them as a primitive, a form of vagueness!)

IanS said...

On (2), I see ‘causal explanations’ as stories we tell ourselves for our own purposes. But nature is not obliged to satisfy our desire for a good story, or to conform to our intuitions.

As a schoolboy, I wondered how a Newtonian particle following f = ma was supposed to ‘know’ its acceleration. Later I learned about the stationary (Lagrangian) action formulation. This avoids the derivatives, but raises other questions. How is the particle is supposed to find the stationary value? Does it really check out all possible paths? (The story continues in QM: Schrödinger’s PDEs, Heisenberg’s matrices and Feynman’s path integrals all describe the same behaviour.)

The moral? It’s not so easy to infer the ‘fundamental dynamical laws’ from the form of the mathematics.

Alexander R Pruss said...

For reasons like these, I doubt the Lagrangian formulation is the fundamental law. There are equivalent formulations, but not all of them -- maybe none of them -- give the *fundamental* law. Sometimes we can at least talk of gradations of fundamentality. For instance F=Gmm'/r^2 is very likely more fundamental than F=rGmm'/r^3.

That it is simpler is evidence of the greater fundamentality, but I do not think the simplicity *defines* the fundamentality.

I think the fundamental laws of nature need to be grounded in something hyperintensional. I think causal powers are hyperintensional.

IanS said...

If you reject derivatives and variational principles, and demand manifest causality, what would you accept? Cellular automata and other discrete systems with local interactions? Or am I thinking too narrowly?

Note that some introductions to QFT start with a discrete spacetime array of operators linked by local difference equations. So discreteness at fundamental scales may not be so silly.

Alexander R Pruss said...

Yeah, I was thinking of discrete systems. Not necessarily with *local* interactions, though.