Monday, August 19, 2019

The Nature Of Proof

     There are days I want to throw up my hands and write off Mankind in its entirety. Too many people think with their wishes – or their ambitions – rather than with their faculty of reason. Too many people substitute their tastes for rational argument. And far too many people are willing to ignore the plain meanings of words if those meanings obstruct their purposes.

     You can’t advance on any goal a man has some chance of reaching by doing such things. Moreover, it doesn’t take a lot of smarts to see that. So why?

     Motivation, according to Robert C. Townsend, is “a door locked from within.” Smith can’t motivate Jones to clean up his ratiocinative act. But it’s possible, on occasion, to jar a man out of his mental habits by asking an unpleasantly pointed question: “How’s that working out for you?”

     So your Curmudgeon is off on another hopeless quest. Blame it on my inner Jeremiah.

     Proof is a difficult concept for many persons, being mathematical in nature. Its definition admits of no compromises; to prove a proposition, one must follow the rules of proof strictly and unbendingly. Unless all the requirements of proof have been satisfied, no assertion can be regarded as proven.

     Every proof, regardless of the proposition at issue, has certain mandatory components:

  1. The established, previously agreed-upon postulates of the formal system within which the proposition is to be proved.
  2. One or more statements of initial conditions.
  3. One or more statements of deduction or induction that couple the initial conditions to the postulates, leading toward the desired conclusion.

     A sound proof establishes that its conclusion is true beyond all possibility of dispute – that if it were untrue, one or more of the postulates of the system would perforce be false! That would contradict the requirements of a formal system.

     (A quick tangent: Induction is a chancy thing outside the realm of purely mathematical reasoning. For non-mathematical purposes, we can omit consideration of it. There have been too many black swans to make use of induction as a method of proof in ordinary, non-mathematical life.)

     But actual life in our spatiotemporal universe, in which time passes and things happen, is not a formal system. There are no postulates upon which everyone agrees. Even the statement that there is such a thing as objective reality, independent of anyone’s opinions about it, is disputed by some – and if that be so, then what prospect is there for proving anything else?

     Proof, therefore, does not apply to propositions about non-mathematical matters. What does apply is another, almost equally misunderstood concept: confidence.

     The investigative technique by which we acquire knowledge, or something akin to it, about the real world is called scientific method. It was first formalized by Francis Bacon some four centuries ago and has been shown to be a reliable – indeed, the only reliable – method for acquiring new knowledge about the spatiotemporal world.

     Scientific method is nicely, if casually, depicted by the graphic below:

     Note that there is no termination – no “escape from the loop” – to the method. Regardless of the proposition(s) at issue, nothing is ever regarded as “proven.” The gathering of relevant data and the comparison thereof to the predictions made by the proposition(s) continue without limit...because a single false prediction is enough to falsify any proposition.

     Predictions that come true increase our confidence in the proposition being investigated. The more such predictions we make, and the more varied our techniques for observation, testing, and measurement, the greater our confidence rises. But it never attains certainty.

     Isaac Newton’s “classical” law of gravity acquired more and more confidence over the three centuries that followed his original formulation. The arrival of unprecedentedly sensitive observational and measurement techniques in the early Twentieth Century called it into dispute. The dispute could not be resolved until a certain young physicist, at that time employed by the German postal service, proposed a new formulation that would:

  1. Replicate the predictions of classical gravity with equal precision;
  2. Outperform the predictions of classical gravity around objects of very large mass.

     General relativity equaled classical gravity in those realms of measurement where classical gravity held firm. However, it produced results classical gravity could not match in the newly explored realms of very large mass where measurement had previously been unable to go. Thus, general relativity superseded classical gravity as the dominant conception of gravitation. At this time it remains dominant, which is to say: scientists’ confidence in it has not been disturbed by contradictory results in any sphere.

     However, no scientist regards general relativity as having been proved. There remain realms to which Mankind’s investigative powers are unequal, at least for now. Perhaps some far-future experiment will reveal that at masses beyond what we can currently investigate, general relativity’s predictions don’t hold true. But for the moment that’s only a hazy possibility.

     There’s more to be said about the concept of proof and why it fails to apply to the spatiotemporal or “real” world. But this is enough for starters. The salient point here is that human ability is limited: far too limited for us to be certain about very much. Indeed, justifiable certainty is confined to personal things: personal observations, personal emotions, and personal experiences. All else is within the domain of the legitimately disputed.

     And so we come back to the virtue it’s currently most fashionable to slough: humility. Or as Josh Billings put it, “It’s better not to know so much than to know so many things that aren’t so.”

     More anon.

1 comment:

Cordolf said...

While I always enjoy your work, Francis, I especially enjoyed reading this piece this morning. Proof is a concept that I'm currently struggling with (along with causation) in political discussions with friends.

Your last paragraph before saying "[t]here's more to be said ..." left me wanting more (as good writing should). I was ready for you to dive into Bayes theorem and clarify it for us all!

Maybe that's part of the 'more' to come anon.

In any event, thanks for the clarity around the formulation of Replicate and Outperform. I think that that helps put some structure around discussions of scientific versus non-scientific theories.

- Rich D