What truths are worth seeking?

by Natália Mendonça2 min read27th Apr 20209 comments

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World Modeling
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Cross-posted from AxiomaticDoubts.

[Epistemic status: extremely uncertain. These ideas came to my mind recently and I am not entirely sure of how to judge them. I’m posting this here in hopes that I gain some feedback.]

If we accept David Hilbert’s dictum that “mathematical existence is merely freedom from contradiction,” and accept Max Tegmark’s hypothesis that all mathematical structures are just as real as the physical world in which we live, we reach an amusing conclusion: that everything that is free from contradiction exists — and, in fact, must exist.

In a sense, if you want to understand reality at a fundamental level, that is all you need to know. That is the “rule” reality follows; that is what determines if any scenario you imagine in your head is factual or not and if any belief you hold is true. Freedom from contradiction is the only game in town.

I have accepted that idea for a while now, and yet, that hasn’t quite quenched my curiosity about the fundamental nature of reality. After some reflection on the matter, I concluded that the only reason that is the case is that I am not infinitely intelligent. In principle, analogously to how Laplace’s demon would be able to perfectly predict the future and retrodict the past by knowing the position and momentum of all particles in the universe, an infinitely intelligent agent would be able to correctly answer any question — even, say, questions about the molecular structure of a perfectly effective and safe treatment for COVID-19 — pretty much by reasoning from the law of non-contradiction.

But to the extent that you’re not infinitely intelligent, I realized, understanding reality requires being explicitly aware of more than just the one rule that governs it. What more, then, do we need to be explicitly aware of?

It’s probably not the case that it is necessary to know a lot of random mere facts, such as, for example, the exact number of grains of sand on Earth or the fact that 64345675432 + 976567898765 = 1040913574197. Those are very different kinds of facts — the former is different in different universes, the latter cannot possibly be — but they are equally irrelevant to people in the vast majority of situations.  There is an infinite number of true facts that would not satisfy anyone’s curiosity.

It seems to me that whether knowing a certain fact can improve your understanding of the world or not depends on what beliefs and ideas are already present in your mind, or are likely to arise in the future.

Galileo’s discovery that any two objects dropped in a vacuum will fall at the same rate was important not only because it was true — so is 64345675432 + 976567898765 = 1040913574197, after all — but also because our experiences in the world lead us to incorrectly develop the belief that they ought to fall at different rates while in a vacuum. If we are prone to accumulating beliefs, whether true or false, as a natural byproduct of existing in and navigating through the world, then in order to understand the world better we ought to seek knowledge specifically of facts that insulate us against holding incorrect beliefs.

Perhaps there is such a thing as the most succinct set of facts the knowledge of which insulates someone from holding incorrect beliefs. And, perhaps, much of curiosity consists of a desire to know such a set. This set is different from person to person — it varies according to the cognitive resources one has available to use and to the incorrect beliefs one is likely to develop — although the difference between what such a set would be for different humans must not be nearly as large as the difference between what it would be for an average human and for a superintelligent AGI.

Perhaps, when David Deutsch claimed in his book The Fabric of Reality that the principles of quantum physics, evolution, computation, and epistemology are “the four main strands of which our current understanding of the fabric of reality is composed,” what he meant was that those principles approximate the most succinct set of facts that could prevent him from holding incorrect beliefs.

I think his choices are very reasonable: there is a very real sense in which knowledge of the principle of evolution, for example, helps a human make sense of the world. Life is constantly presenting us with situations that can only be explained well by referring to it — the emergence and spread of diseases, the popularity of different memes, the behavior of others, the appearance of emotions, fears and desires in our own consciousness, etc. Someone without knowledge of it would be liable to accumulating a wide range of incorrect beliefs.

The principle of evolution is very much derivable from the law of non-contradiction, and therefore it would not be useful to an infinitely intelligent AI. But, as an ordinary human being, it is arguably one of the most essential pieces of knowledge you could have — along with the principles of quantum physics, computation and epistemology, the law of non-contradiction itself, and all other ideas that would insulate you against holding incorrect beliefs, it is more than merely true. Only truths of that same stature, it seems to me, are actually worth seeking.

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We don't know that all possible worlds are actual. This could be the only one. Also, non-contradiction doesn't tell you what's possible, only what's impossible. How were you first informed of the existence of numbers, colors, space, time, or people? It wasn't by non-contradiction.

If contradiction tells you what's impossible, non-contradiction tells you what's possible..for some value of "possible"...perhaps not the most interesting one.

We don't know that all possible worlds are actual. This could be the only one.

Indeed. This entire post assumes all possible worlds are actual and reasons from there; I didn't mean to argue for their existence.

How were you first informed of the existence of numbers, colors, space, time, or people? It wasn't by non-contradiction.

Correct. But we are quite bad at actually reasoning from the law of non-contradiction; we often tend to act as if we believed contradictory things (as is shown by how frequently we make math errors). I conjecture that that is the reason why we need observation to figure things out (assuming all possible worlds exist), although I am not completely sure.

Start with the truths applicable to biological aging, or you'll never get to the rest of them.

In principle, analogously to how Laplace’s demon would be able to perfectly predict the future and retrodict the past by knowing the position and momentum of all particles in the universe, an infinitely intelligent agent would be able to correctly answer any question — even, say, questions about the molecular structure of a perfectly effective and safe treatment for COVID-19 — pretty much by reasoning from the law of non-contradiction.

Even an infinitely intelligent agent is limited by the amount of empirical data it has. Sample complexity is a limitation even if computational complexity is unbounded. From the perspective of the Tegmark multiverse, there are different universes in which the molecular structure is different, and you don't know in which one you are. By making observations, you can rule some of them out, but others remain.

Moreover, we need to assume a probability measure over universes to be able to conclude anything, otherwise you can always imagine a universe matching our observations so far and producing anything whatsoever after that. There is nothing logically inconsistent about a universe which is just like our own, except that a pink unicorn materializes inside my house after I click "submit". It's just that the unicorn universe much higher description complexity than the universe in which no such thing will occur. Having assumed such a probability measure, we get a probability measure over predictions: but these probabilities are not 0 and 1, there is still uncertainty!

but also because our experiences in the world lead us to incorrectly develop the belief that they ought to fall at different rates

-while in a vacuum.

Thanks for pointing this out! I fixed it.

Ironically I read this after the correction and still thought it scanned a little oddly; as if were suggesting that the world naturally leads us to believe that things ought to fall at different rates specifically or only while in a vacuum.

For my contribution to the bikeshedding, maybe "ought to fall at different rates, even while in a vacuum".

an infinitely intelligent agent would be able to correctly answer any question — even, say, questions about the molecular structure of a perfectly effective and safe treatment for COVID-19 — pretty much by reasoning from the law of non-contradiction.

That seems like a free lunch. Is it a free lunch? The Infinite Intelligence can't figure out where in the multiverse you are from the law of non contradiction, so you might have to ask a very long question.