I don't think AGI is right around the corner

Our silicon machines exist in a countable state space (you can easily assign a unique natural number to any state for a given machine). However, 'standard biological mechanisms' exist in an uncountable state space - you need real numbers to properly describe them. Cantor showed that the uncountable is infinitely more infinite (pardon the word tangle) than the countable. I posit that the 'special sauce' for sentience/intelligence/sapience exists beyond the countable, and so is unreachable with our silicon machines as currently envisaged.

I call this the 'Cardinality Barrier'

That’s an interesting thought. It steps beyond my realm of confidence, but I’ll ask in ignorance: can a biological brain really have infinite state space if there’s a minimum divisible Planck length?

Infinite and “finite but very very big” seem like a meaningful distinction here.

I once wondered if digital intelligences might be possible but would require an entire planet’s precious metals and require whole stars to power. That is: the “finite but very very big” case.

But I think your idea is constrained to if we wanted a digital computer, is it not? Humans can make intelligent life by accident. Surely we could hypothetically construct our own biological computer (or borrow one…) and make it more ideal for digital interface?

That is a really insightful take, thank you for sharing!

> 'standard biological mechanisms' exist in an uncountable state space

Everything in our universe is countable, which naturally includes biology. A bunch of physical laws are predicated on the universe being a countable substrate.

Isn't a Planck length just the minimum for measurability?

Physically speaking, we don’t know that the universe isn’t fundamentally discrete. But the more pertinent question is whether what the brain does couldn’t be approximated well enough with a finite state space. I’d argue that books, music, speech, video, and the like demonstrate that it could, since those don’t seem qualitatively much different from how other, analog inputs stimulate our intellect. Or otherwise you’d have to explain why an uncountable state space would be needed to deal with discrete finite inputs.

Can you explain why you think the state space of the brain is not finite? (Not even taking into account countability of infinities)

It sounds like you are making a distinction between digital (silicon computers) and analog (biological brains).

As far as possible reasons that a computer can’t achieve AGI go, this seems like the best one (assuming computer means digital computer of course).

But in a philosophical sense, a computer obeys the same laws of physics that a brain does, and the transistors are analog devices that are being used to create a digital architecture. So whatever makes you brain have uncountable states would also make a real digital computer have uncountable states. Of course we can claim that only the digital layer on top matters, but why?

Measurability is essentially a synonym for meaningful interaction at some measurement scale. When describing fundamental measurability limits, you're essentially describing what current physical models consider to be the fundamental interaction scale.

Cantor talks about countable and uncountable infinities, both computer chips and human brains are finite spaces. The human brain has roughly 100b neurons, even if each of these had an edge with each other and these edges could individually light up signalling different states of mind, isn't that just `2^100b!`? That's roughly as far away from infinity as 1.

Not quite. Smaller wavelengths mean higher energy, and a photon with Planck wavelength would be energetic enough to form a black hole. So you can’t meaningfully interact electromagnetically with something smaller than the Planck length. Nor can that something have electromagnetic properties.

But since we don’t have a working theory of quantum gravity at such energies, the final verdict remains open.

But this signalling (and connections) may be more complex than connected/unconnected and on/off, such that we cannot completely describe them [digitally/using a countable state space] as we would with silicon.

If you think it can't be done with a countable state space, then you must know some physics that the general establishment doesn't. I'm sure they would love to know what you do.

As far as physicists believe at the moment, there's no way to ever observe a difference below the Planck level. Energy/distance/time/whatever. They all have a lower boundary of measurability. That's not as a practical issue, it's a theoretical one. According to the best models we currently have, there's literally no way to ever observe a difference below those levels.

If a difference smaller than that is relevant to brain function, then brains have a way to observe the difference. So I'm sure the field of physics eagerly awaits your explanation. They would love to see an experiment thoroughly disagree with a current model. That's the sort of thing scientists live for.

Please describe in detail how biological mechanisms are uncountable.

And then you need to show how the same logic cannot apply to non-biological systems.

Absolutely nothing in the real world is truly infinite. Infinity is just a useful mathematical fiction that closely approximate the real world for large enough (or small enough in the case of infinitesimals) things.

But biological brain have significantly greater state space than conventional silicon computers because they're analog. The voltage across a transistor varies approximately continuously, but we only measure a single bit from that (or occasionally 2 for nand).