I think likely to be infinite, per Gödel's incompleteness theorems. Given a corpus of maths called A we can always find new unprovable axioms under A, giving us a new corpus. From there we can prove new theorems.
There may be a finite point at which we've proven and codified everything but the most inane meta-meta-meta-maths though.
That's now how it works. You generally can't know if a statement is provable/unprovable under A. In most cases, there's nothing to identify a new corpus.
I'm not saying we can know for a particular statement. Just that, per the incompleteness theorems, we know A is incomplete. Hence there is always room for our knowledge of mathematics to grow.
It's more of a theoretical corpus for the sake of this discussion. Just think of it as "all math humans have discovered/invented".
> Are all theorems or axioms useful?
Definitively no! Check out the first incompleteness theorem proof by diagonalization. The whole proof is based around generating new "facts" that are completely useless and uninteresting, outside of their utility in the proof itself.
It's actually a great question: to what extent do the incompleteness theorems actually apply to stuff that isn't silly and useless! The same goes for a couple similar proofs in other domains. An important one in Computer Science is Turing's proof that the halting problem can't be solved. There's at least one other important proof that is similar but I can't remember it at the moment.
To massively oversimplify, they can be boiled down to variations of:
> This statement is false.
Which proves that allowing self-referential statements can create a statement that isn't true or false. But they all raise the question: how often does this actually happen with statements we actually care about?
Would these alternatives to backprop make it more feasible to have constant live-training going on in a model? Giving it something akin to neuro-plasticity?
One aspect of how current training and continual learning are somewhat at odds is that the memory required to train a model is often times 2-3x the memory required to just run it (probably not as bad for PEFT, not sure).
DUST does have an advantage specifically along those lines because it doesn't have to save a ton of intermediate state other than each layer's input activations during a single forward pass.
There are many other issues that this algorithm does not address thoigh like catastrophic forgetting. it's still operating on a transformer which contains no inherent mechanism for selecting the relative value of a training step based on current knowledge, nor does it have segmentation of functionalities with specialized areas used for specific things that can be sequestered off and ignore new updates (we do not risk forgetting how to walk as we increase our French vocabulary)
They are candidates for antiferromagnetic semiconductors. Apparently this is invaluable for spintronics and ultra-fast-switching (terahertz) transistors among other things.
In functional programming, there is no "generic list iterator", though. It matters what the output is, whether you need to look at the value of previous iterations, etc.
Because the metaphysical is definitionally outside the realm of science and rationality, and every bit of scientific and technological progress humankind has made has come from discarding superstition and approaching the world as a physical system.
If the brain isn't just a physical system, then we may as well give up. There's certainly no point discussing it, as any assertions will be untestable and one persons elaborate and well thought out theory will be just as valid and predictive as the next persons "consciousness is created by invisible purple unicorns" theory.
So if we are going to discuss it, we should start from the assumption that there is no magic or witchcraft or religion involved.
> If the brain isn't just a physical system, then we may as well give up
Forgive my obtuseness, but if it turns out that the brain is more than just a physical system, what are we giving up? On science as a whole?
Isn’t science predicated on reliably predicting reality? Do we know how to describe reality accurately?
I asked below as well, but how does the observer effect work as a physical system? I get we don’t know so you can’t really answer, but if we can’t accurately describe reality then our science is woefully incomplete! Given that incompleteness I really don’t understand how we can rule out “invisible purple unicorns”?
The issue is that we have yet to prove or find evidence for the existence of anything but the physical. When we do find a new phenomenom in one of these ways, it becomes a part of physics. What would it mean to concretely discover a non-physical phenomenon? It would have to genuinely be magic of some sort, and defy rationality and description, to not simply be a new part of physics. (And if it defies rationality and description, it doesn't advance us scientifically, except to let us know we must abandon science in that domain.)
> I asked below as well, but how does the observer effect work as a physical system?
Just to be sure - the observer effect doesn't refer to a human observer, but rather to any measurement of a quantum system.
It is unsatisfying that quantum mechanics posits that particle states are fundamentally probabilistic. It's possible that in some sense there is an underlying order such as string theory that is impossible to test or verify. In that case, it would make sense to consider string theory as metaphysics. A fun exercise, but not scientific and not useful for understanding the world. By definition it would not be helpful in predicting or modeling the world, otherwise that would serve as a way of testing the theory.
> It is unsatisfying that quantum mechanics posits that particle states are fundamentally probabilistic.
You can alleviate this mostly by accepting the MWI.
Under the MWI, all the possibilities do occur, and the probability manifests because when you split into an infinitude of yous, each of them is just one you, and has no way of predicting which you it will be on the other side.
So the MWI makes the wave equation deterministic when viewed from the outside. The trouble is that we are inside it.
It turns out that God does throw dice - it's just that all the combinations come up at the same time.
All these "AI can't" arguments seem to secretly rely on the assumption that human reasoning/sentience/whatnot is dependent on a soul (or any equivalent metaphysical entity). Everything else is handwaving.
I agree - many are. But there's another camp to watch - LeCun/etc don't believe in a soul but also don't believe that LLMs are capable, because they don't contain the right neural network.
I'd argue that they're still intuitively trying to keep intelligence in the gaps, as humans like to do. Or maybe they're mad they bet on the wrong horse... or maybe both. Hard to say.
> because they don't contain the right neural network.
I'm totally onboard with that as a plausible argument, but I think the architectural limitations with respect to consciousness are very deliberate rather than some lack of technology. We've invested huge sums of money and human effort to create tools with explicit goals that are completely at odds with consciousness or AGI. If we had invested similarly with the clear goal of creating something with agency, self-determination, neuroplasticity, etc. instead of controllability, repeatability, reliability, I think we would be there already.
Maybe. I'm just suspicious of the certainty of the people who are certain of it.
And... I remember (all of six years ago) back when language was considered the pinnacle of the human mind. Sure, animals might be smart, but they don't have language!
The moment LLMs appeared it suddenly took a back seat to physical navigation and child rearing.
Feels like more gap seeking to me. But time will tell.
> they are incapable [...] of independently starting a reasoning task.
This is a simple and intentional design choice though. Biological lifeforms are always on and always receiving sensory input. LLMs don't functionally exist out side of when we decide to run them. An always on agent with a looping prompt of "if you aren't doing anything else, ruminate" overcomes this limitation.
In fact, if your brain were removed from your body, and you were locked in a room that was completely empty and precisely calibrated to your brain's ambient temperature, and you were then ordered on your life to not ruminate. Well -- I would posit that you wouldn't last very long.
There's something else going on with human cognition -- and reasoning by extension -- that LLMs are, to date, not replicating.
With the big caveat of AFAIK. I'm not in one of the labs close to this stuff.
> Not really. Noone has to prompt you to ruminate.
Why do the internal mechanics need to mirror how it works in humans?
I have agents that receive sensory input (video/audio) which run continuously processing it, receiving events as they occur. When something strange happens they notice and take action (i.e. there's an unrecognized person in the living room -> ask who they are.)
Presumably you've seen the agents that play video games - build things to progress in Minecraft, plan routes, avoid adversaries, etc.
Why doesn't that count?
I feel like you're failing to consider anything outside the chatbot UI.
EDIT: The first morning my agents had access to an outside video stream they commented on the sunrise. While I was sleeping through it.
> they are incapable ... of independently starting a reasoning task.
I have provided several examples of them independently starting reasoning tasks.
... and I'm wondering why you seem to be entirely unaware of such examples. The most likely explanation being you are only aware of the chat interface.
... and I have absolutely no idea of what you mean by "... and feel".
Comparisons to how humans reason beg the question: is the way humans do it the only way?
Lots if these arguments are similar to birds saying "Jets don't flap their wings so they aren't even flying."
The arguments about reasoning are even shakier because they usually rely on totally unproven assertions about human reasoning. At least we know birds flap their wings.
There may be a finite point at which we've proven and codified everything but the most inane meta-meta-meta-maths though.
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