The Scientific Method, Pt. 2 – Godel’s Incompleteness Theorems

Previously I discussed my misgivings about totally subscribing to the scientific method. I was unnerved by many epistemologically unsound underpinnings that seem to show that the scientific method on offers a small light to the corpus of truth. These weren’t criticisms against practitioners or assumed malice at scientists, but actual structural flaws in the system itself. I now want to discuss what caused most of that line of thought – Godel’s Incompleteness Theorem.

This is dense, but try to follow along. It is tremendously rewarding and pretty much the biggest mind-f*** I’ve ever dealt with. Consider a computer that when prompted will tell you the correct answer to any question – super-Google.

That’s super-Google’s schtick, that it is a complete system and can answer any question correctly.

Say you pull out the source code for this super-Google and you splice in a sentence. Let’s call this sentence “S”, for sentence. “S” actually says “Any algorithm run on this program will never say this sentence is true”

Think about this for a minute.

Think about it deeply.

Can you say whether that statement is true or not?

Let’s say you are lazy and just ask super-Google – Super-Google poops itself. Why?

If super-Google says “S” is true, then “S” becomes false because originally “S” said that no algorithm will say that “S” is true. Since super-Google’s whole point is to be correct about everything and it is now wrong it is no longer complete. This means that super-Google must change its answer.

Ok, so the sentence “S” is false. But that isn’t right either. Because now super-Google must eventually say that “S” is true. And we already saw the trouble that occurs when super-Google claims “S” as true.

So, the best thing for super-Google is to not answer. Then, “S” remains true. But now super-Google is no longer complete. The whole point was to be correct about everything. You know “S” is true, but super-Google can never say it and thus is no longer able to provide a correct answer to any question.

This is Godel’s Incompleteness Theorem.

Godel generalized this thought experiment and was able to show that any system must eventually become either incomplete or accept that any “complete” system will inevitably breed paradox. Douglas Hofstadter, author of Godel, Escher, Bach: An Eternal Golden Braid, states that Godel showed that provability is a weaker notion than truth. And of course, the Math community has run with this idea, as it forms the underpinnings of Axiom of Choice, the Church-Turing thesis, Peano Arithmetic, the Halting Problem…

There’s a lot to unpackage with this thought. That book, the Eternal Golden Braid? 825 pages. I can’t even get through the first ten without having to set it aside just to digest what I’ve read. There has been discussion on the proof of divinity and the role of faith, of human consciousness and machine intelligence, of the formal underpinnings of logic itself…

It’s heavy.

At least for me. Maybe you can digest this more quickly than I can.

While there are a lot of ways you can extend this idea, some of which I mentioned in the last paragraph, I specifically wrote this regarding the scientific method. Godel’s theorem works because it relies on self-referential statements. Statements like “This statement is false” refer to themselves and inevitable introduce paradox. Any system will eventually create self-reference. It may not be as concise as the “A liar say’s he’s a liar, is he lying?” variety. It may be more like:

“The statement X lines below is true

the statement X lines above is false”

From a systems point of view, this raises serious questions about sustainability. But putting that aside, this is why I started to question the scientific method. The scientific method is, after all, simply another system.

I wrote long ago once about how I believed that the heart of definition is self-referential. Consider the definition for mass, a property of matter. And what is matter? Something with mass. In the framework of physics, this is the self-referential statement that espouses a field of study. Now, I understand there have been advances in our understanding of both of these phenomenon, Higgs fields and all, but my point is this is how the field developed. More importantly, this offers a new way to look at self-reference and paradox.

Instead of paradox as a catastrophic inevitability that will render systems as obsolete and incomplete, paradox is the starting point, the definitions to accept in order to begin study. And any paradox encountered is now the discovery of a new definition. A new definition is the start of a new system or field of study. Continue ad infinitum.

One thing Godel’s Theorem doesn’t discuss (as far as I know anyways) is whether we can predict when a system designed for completeness will create a self-referential statement. And, going off of my idea of paradox as definition, the length of time/output of complexity until the discovery of a new definition.

Is it also possible that the collapse of systemic complexity and length of output between definitions could be view in a chain?

Could this idea be extended to be viewed in conjunction with evolutionary algorithms?

These are all high-level, grandiose ideas. But I do think that this dialetheistic method is a counterpoint to the scientific method that may offer another way of relating man to the universe. Instead of careful study, repetitive trials, consensus based attempts at objectivity, a chaotic embrace of dialethesia and feverish attempts to scale and collapse systems plays well with our natural human inclinations.

So what would this dialetheistic method look like? Somewhat esoteric actually. In my view the steps would center around microcosms, just as the scientific method centers around experimentation. The “Steps” of the dialetheistic method to me would be to

  • First identify a non-trivial paradox. Sure, you can create artificial paradoxes, like the Barber Paradox, but naturally occurring or seemingly evident paradoxical phenomenon are better areas of study.
  • Pear down any seemingly relevant factors to construct a minimal microcosm – a small world that can function as a self-sustaining ecosystem.
  • Then, add variables to increase complexity and observe growth trajectories. Eventually, complexity will increase to the extent that some self-reference occurs.
  • Isolate that self-referential variable and begin again.

As a form of systems thinking or systems analysis, this may help address our seemingly feeble attempts to rein in large systems like pollution in the environment or systemic collapse in the economy.

I’m just trying to provide a different way of interrogating our surroundings and our place in the universe beyond the historical or scientific methods. I could be entirely off base. Maybe this sounds too close to the scientific method as I’ve described it now. Maybe it sounds too Hegelian with a thesis-antithesis-synthesis sort of feel.

And who knows what issues this brings. Several of the biases that I claim cast doubt on the scientific method don’t apply to the dialetheistic method but some do, and I’m sure careful thinkers will discover other reason to doubt this heuristic. And I should mention that there wasn’t a single moment in history everyone said, “Hey, let’s use science” – it kind of just developed.

But maybe this will develop.

Or maybe I’m wrong and science will solve all our problems.

3 thoughts on “The Scientific Method, Pt. 2 – Godel’s Incompleteness Theorems

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