I think that's just an accident of history.
When we write software, we very seldom write proofs that our algorithms are correct. We just write tests, and we also run the algorithms and when they fail we know we have a bug and then we proceed to debug, fix, and add new tests (if we are disciplined, but most of us are). In time, by usage and testing, we gain confidence that our battle tested software is correct, mostly. And we tell people that we will never be 100% confident that any software is bug free. But that's a slight lie: if we wanted such confidence, we would start using provers, and create bug-free software. That possibility exists, but it's just extraordinarily expensive.
Well, in math that's the only possibility, and we use it. And it is indeed extraordinarily expensive, but it's also the cheapest among the alternatives. The alternatives are 2: be rigorous and do these proofs, or be sloppy and allow bugs to creep in, and allow an entire school of math to collapse like the Italian school of algebraic geometry [1].
There is one more alternative. If a particular math theorem has some applicability, then you write a program and use it in real life. In time you eliminate the bugs as much as you can, and you get to the steady state of "virtually bug free". At that point you don't have a solid proof that the theorem is correct, but in general you don't really care. Because you feel that a formal proof is just a thing one would pursue for getting academic satisfaction only.
[1] https://en.wikipedia.org/wiki/Italian_school_of_algebraic_ge...
Mathematicians also occasionally build on top of unproven foundations (e.g. all popular asymmetric encryption schemes are built on top the assumption that certain problems such as integer factoring are hard, for which there is no formal proof), or at least explore both possibilities for statements with unknown truth value (e.g. you can find lots of work that explores the consequences of P = NP and/or P != NP).
However, there is a major separation between math and programs that I think mostly invalidates your proposal - most math we're talking about here is simply not applicable directly to the real world in any way. It's only studied for the interest of mathematicians. There is no real world consequence for Fermat's last theorem, for example - it was just a really interesting to prove theorem. In directly applied math, such as engineering, it is in fact much more common to work with unproven but well tested conjectures.
What specific areas were you thinking off? I don't recall, e.g., in numerics that things were often just unproven/conjectures, but might be subject matter specific.
Edit: one better example from modern physics - the path integral formulation, used in both string theory and other areas of QM/QFT, is not fully formalized and formally proven to work. Also, a more concrete example of a widely used but actually still unproven conjecture in string theory is the famous AdS/CFT correspondence.
The part of Newton's theory that was troublesome is his fluxions don't have the Archimedian property. It took until the 1960s before Newton's notion of fluxions became rigorously formalized with Non-standard analysis. https://en.wikipedia.org/wiki/Nonstandard_analysis
HN history
+-- 2mo before by sdfrew
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| The Fall of the Theorem Economy
| https://news.ycombinator.com/item?id=47862472
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| The Fall of the Theorem Economy
| https://news.ycombinator.com/item?id=47891494
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| The Fall of the Theorem Economy
| https://news.ycombinator.com/item?id=47909751
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| David Bessis on AI destroying mathematics
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| The Fall of the Theorem Economy
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| The Fall of the Theorem Economy
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`-- this submission by varjag
58 points / 7 comments
The Fall of the Theorem Economy
https://news.ycombinator.com/item?id=48758048Are we going to see less publicly shared science? With private actors or governments restricting access to AI resources to a few scientists and keeping new knowledge to themselves.
Advancing science in the open was the best strategy when there was real advantage to share the load with every brain on the planet willing to give a try at science, but if a computer can match or surpass the collective output of the entire human scientific community the equation will change.
It's a sad outlook.
What is going to suck though is the ladder for juniors. We dont start out by working on big ticket problems, but usually early career researchers solve really tiny problems in a cheap way. The lowest bar for a cash strapped PhD student would be to contribute to some new theory in some way even if the student doesnt have access to equipment.
For biosciences and physics, sure. For mathematics? I am skeptical that your assertion applies.
If AI is somehow able to prove everything wouldn't it bypass Godel's incompleteness theorems?
Yes, but this is when someone reaches ASI and everything changes. For now, a good researcher can build off their discovery in a way their AI can’t.
This is what a lot of scientists love to tell themself or talk about in celebratory speeches.
The truth is: a lot of science is kept behind journal paywalls, so that only "officially approved" (in the sense of: working at a university or an governmental research institute) scientists can easily access it.
Also be aware that the world wide web was actually conceived by Tim Berners-Lee for the exchange of information between scientists.
Will be going to a conference at the end of the month where there will be several presentations on the use of LLMs for this.
I can understand why this is a major concern for mathematicians. They got into their field because they love the beauty of mathematics, and the intellectual satisfaction of understanding non-obvious insights. But to put it crudely, this sounds like a you problem. As someone who isn't a mathematician, the main value I get out of math is its practical applications in science and technology. And their practical applications in human life. I have zero understanding of the math behind cryptography, but I still deeply appreciate the practical value they have provided humanity.
If AI systems start churning out accurate theorem-proofs, and we are able to use those theorems to build things that improve human quality of life, it doesn't bother me one bit that those theorems have not been understood by humans. If this offends your aesthetics, you are certainly entitled to your opinion and your preferences, but that does not make it a societal problem
I have some sad news for you. 99% of the work mathematicians do has no immediate application, nor even an obvious path toward application in the near future. You mentioned cryptography, so for an example consider number theory: no apparent practical applications, going back thousands of years to the time of Euclid and earlier.
It’s been religion, philosophy, and recreation that have provided the motivations to study mathematics all these years, not applications. Applications have almost always followed long after the development of the pure mathematical theory. For number theory, that was the development of cryptography during WW2, millennia after the ancients laid those foundations.
Most unfortunately, it’s the truth value and the understanding which drive applications of mathematics, not the proof work itself. If the AI revolution decapitates the institution of mathematics which produces the understanding, and is unable to replace it, then the applications will cease as well.
If cryptography didn't exist but the maths did, how'd you use it?
Feels a lot like building software from bottom - once you get the building blocks defined right, the action, or the program, are trivial to express. When doing it from the top-down, you write the program using the building blocks you haven't defined yet, and you might end up with overly specific building blocks, needing other blocks for expressing different behaviors.
When you do the bottom-up building blocks right, new behavior is easy to express with them. Essentially, you are building up the language to reach the problem. Or making a DSL, whatever definition you like best.
On the other hand when a new high-level concept becomes clear and seems to emerge like a revelation, and people start thinking in terms of those new definitions, it seems that a hundred pages worth of smaller results can fall out of it almost effortlessly. This way of describing it is more top-down.
I don't know that there's an exact parallel with software. Math keeps feeding into itself in a way that software dreams about with our ambitions of code reuse. The old Object Oriented dreams of perfectly encapsulated classes and abstractions partially worked out, but not to the degree that was envisioned.
The current situation with package managers doesn't look like a tower that keeps growing higher and higher levels of abstractions. It looks like a tower where each person wants to place one tiny brick that they call left-pad, and next year we will rebuild the lower levels instead of going higher. So the top-down and bottom-up building that we do is different. We keep rebuilding the bottom, and we don't very much like when the tower of abstractions get too tall and hard to reason about.
On the other hand when a new high-level concept becomes clear and seems to emerge like a revelation, and people start thinking in terms of those new definitions, it seems that a hundred pages worth of smaller results can fall out of it almost effortlessly. This way of describing it is more top-down.
I don't know that there's an exact parallel with software. Math keeps feeding into itself in a way that software dreams about with our ambitions of code reuse. The old Object Oriented dreams of perfectly encapsulated classes and abstractions partially worked out, but not to the degree that was envisioned.
The current situation with package managers doesn't look like a tower that keeps growing higher and higher levels of abstractions. It looks like a tower where each person wants to place one tiny brick that they call left-pad, and next year we will rebuild the lower levels instead of going higher. So the top-down and bottom-up building that we do is different. We keep rebuilding the bottom, and we don't very much like when the tower of abstractions get too tall and hard to maintain.
1) Two and a half years with no reply from a journal (not even to emails I sent that I'd like to retract the paper so I could send it somewhere else). Then suddenly they tell me the paper is accepted.
2) One year with no reply. Then, my "anxious" collaborator sends them countless emails and gets redirected from person to person and finally an editor tells us that they decided almost immediately to reject our paper but they didn't tell us because "they hate giving bad news".
These were not top journals like Annals, but decent, prestigious ones, from whom you'd expect some professionalism.
How is it not already this? Jon von Neumann was already calling most math this many decades ago. Pull up any random arxiv math paper and it’s abstract nonsense with no applications to the real world.
What would happen if a non-human layer of mathematics emerged on top of human mathematics? In this article, the distinction between Mathlib and Mathslop might be a precursor to that.
If models advance enough in the future, and new definitions, compressions, and representational forms that are convenient for AI-to-AI communication emerge, what would happen then? Would mathematics split into Human-facing and Machine-facing branches?
I am not dismissing engineering (it moves the world we live in), just trying to clarify what science is.
Applied fluid dynamics works like that: noone has ever really "verified" that the finite-element method applied to some specific model does converge
I mean, what if a human could follow every single step of the process in principle, but the sheer volume is so vast that a human can never see the whole thing—would that be engineering?
But I don’t think of that as engineering. In the future, maybe it will be called an Oracle
The details could be painful but having a birds eye view is always possible?
And having a machine compress it for human consumption, sounds very plausible (and which I think of as engineering)
Not that it is wrong for them to be doing this---we do want a society where people get to devote their life to what interests them---but it is bizarre because of the framing. For some reason it is ambiently understood in our society that this work is of incontrovertible value, when in fact it is largely not. And the value-producing parts of the work, the parts that end up having applications to other fields, largely run contrary to the actual daily goals of the cloistered devotees: it is mostly the intuition and pedagogy and the compactification and refactoring of knowledge that have value at this point, not the production of esoteric theorems, yet that is expressly not rewarded in the incentive structures.
That latter point is more due to the sorry state of academic incentives in general than to a particular failing of mathematics, though. Were I somehow given the ability to restructure things by fiat I would immediately create journals which publish only useful articles that refactor knowledge, communicate intuition, better explain things, argue for structural improvements to notation and terminology, etc, and this would immediately create an incentive to do that kind of work for working researchers to do work which aligns with the actually-useful output of their fields. I suspect most fields could use something like this. New knowledge is just not that valuable if it is all dumped into a giant pile and unprocessed, and I have seen firsthand a bunch examples where entire subdisciplines are hamstrung in their actual application-heavy work because they don't have easy access to basic tools that are hidden behind hard-to-learn theory.
It always felt wrong to me that while the scientific method iterated starting with the "real world" viz. Observe, Measure, Hypothesize (includes modeling with mathematics), Test and Refine; pure mathematicians lost themselves in the formalization of hypothesizing/modeling and thus lost touch with mapping it to reality. The AI revolution is now showing them up.
You’re describing a very small fragment of total current mathematical labor. Very few people work solely on “formalization” and even e.g. model theory or type theory have real consequences.
Pure mathematicians create ever more abstractions and get lost in solving puzzles on how these abstractions logically relate to each other. But since these abstractions don't have any relevance outside of pure mathematics, it's an entirely self-referential game, like chess. Except that nobody confuses being a professional chess player with being a noble researcher.
Even in philosophy, at least analytic philosophy, that issue of getting lost in your own abstractions doesn't really exist. Because analytic philosophy doesn't analyze its own concepts, it analyzes the concepts that already exist in natural language. Like truth, knowledge, probability, causation, belief, desire, consciousness, rationality and so on. These concepts come from outside of philosophy, and they have independent relevance for non-philosophers.
In contrast, pure mathematics seems to be the part of mathematics that only has relevance to pure mathematicians. Similar to how a game like chess has only relevance to chess players, not to anything entirely unrelated to chess. But again, people who are into mastering some game or sport are fully aware that what they are trying to master is a self-contained game, or sport, not something that increases the amount of human knowledge beyond that.
First, math, generally, is useless. I mean, yes there are of course practical uses of basic thru undergrad-level math, and some beyond that. But for many mathematicians, the sum result of their entire career may lead to exactly zero results that have any real-world value. The entire field they work in may have meaning only to the handful of other individuals on the planet that also work in that field. But to those handful of people, the meaning defines their lives. From a socio-economic perspective, those departments should have been defunded a century ago. Yet they continue. Why? Because it scratches an itch. Not just for those individuals in the field, but also for us as a species. To stop exploring, to eliminate the search for pots of gold that may be buried in some odd corner of sphere packing, or coloring theorems, or Garside categories, and to put a boundary on the limits of our understanding, just because they aren't immediately applicable, is an idea that most humans would not be willing to sacrifice, even if it reduced their tax burden a couple cents. If it was going to happen, it'd have happened already.
The second is, even with AI, it's not free. As the software industry is discovering, far from it. So, given that, who is going to decide what theorems to research and how much it's worth? Congress? Of course not. AI itself? In theory that sounds plausible, but that falls victim to thing 1 above: most math is useless, so AI itself has no value metric it can assign to things, and besides which, without the human element, once the initial curiosity has subsided, there'd be no reason to continue any funding for AI to do it. So no, the only possible owners of this is going to be mathematicians themselves, the ones who care about the field and deeply understand the kwah of their vision.
Combining these, there's a future where, humanistically, "nothing changes". The method changes, the efficiency changes, the scope changes, but the work itself: publishing proofs, remains the domain of professional mathematicians. AI will enable them to be dramatically more daring and broad in their investigations and scope, and will likely write the entirety of the proof. However it will remain the work of the mathematicians to determine, what areas are worth spending limited AI resources on to investigate further, how far to go down rabbit holes, how to prioritize potential connections, and what the ultimate meaning of the findings is. So rather than being an end of mathematics, it could be a dawn of something far greater than anything we've ever seen before.
A wood-worker could do the same argument, there's the "official" wood-working word of perfect joinery and beautifully finished tables one can buy, but behind it there's the "secret" messy human element, the art, the craft, the mistakes and hard-ships, the elevation of human skills and imagination, the creation of whole new types of wood-working inventions and techniques, the perpetuation of millenia-old traditions, the teaching, the joy of selling to a happy customer, etc.
But now comes techo-capitalism, division of labor, you cut that piece a that piece over and over, you operate that machine, you won't even see the finished table, fuck your human element, we want that profit !
The goal of a woodworker or craftsman is the production of a finished good. He's arguing that, although it's been convenient to position a mathematician as a "theorem-producer", that's never really been the aim of mathematics, and that the actual products of mathematics are some kind of "mental software"- see his references to neuroplasticity. Basically, he's saying that the goal of mathematics is to create abstract structures that allow humans to reason about increasingly complex concepts, and that the "mathematician as theorem producer" is more like a convenient fiction that mathematicians have allowed to persist for too long, and now threatens to endanger the whole practice of mathematics.
The motivation behind all this is less "haha I want profit" and more "billions of people need chairs, approximately none of them care about the craftsmanship, so it's in our best interest to make furniture in the most resource- and labor-efficient way possible". Even if the state subsidizes the production of handcrafted chairs, the population is the poorer for it on a resource allocation basis, because we now need a million artisanal chair-makers instead of a bunch of factories.
To be fair, a number of professional politicians and political scientists don’t understand alienation under capitalism.
"I was in Switzerland", "I was invited to a talk", "I started a machine learning company", look at me bro.
One of the most fruitful approaches in mathematics is to flip back and forth between geometric and algebraic views of a problem. I think this works so well because these are actually handled by two different parts of the brain on a physical level; spatial reasoning is separate from language processing. Cytoarchitecture shows these regions have different "textures;" the local details of the way neurons are wired together are simply different in these different regions of the brain, in the same way a CNN and a transformer have different topologies. Thus, by flipping problems from geometry to algebra and vice versa, we're able to bring an entirely different cognitive style to bear on a problem. For example, the proof of Monge's Theorem by moving to 3D and visualizing not three circles, but three spheres sitting on a table with a book on top of them and then pointing out that the intersection of two planes is a line. What is pages of unintuitive symbol pushing turns into something a child can understand. Going the other way, things like the angle addition formulas or the quadratic formula, which are quite hard to prove geometrically, become quite simple if you use a little algebra.
Current-gen LLMs are still relatively weak at visual reasoning; see the Vision Language Models are Blind paper, for example, or the ARC-AGI benchmark. So that's one way humans can stay ahead of the agents, at least for now.
Math is entirely subjective. "Proof" essentially means "Other educated practitioners have the same experience when trying to understand this."
The logical steps that proofs are built on all have that common foundation. Our concept of logic based on our subjective experience of "truth." We've built machines that reproduce our subjective processes mechanically, but there is no sense in which this idea of "true" is truly objective. It happens to be computationally convenient, and it has some relationship to experience, but that doesn't make it an independent reality that all possible observers, human and otherwise, would agree on.
We're really just mapping our own minds through our own experiences.
Animal brains can't abstract like (some of) our brains can. What are the odds our brains are limitless and don't have some similarly crippling limitations from a couple of levels up?
One of the tells for ASI is that it will start reasoning at those levels, using cognitive techniques that are completely incomprehensible - not just because of brute volume, but because our brains won't have the wiring to get a foothold on them.
Some of the products will be reducible to human cognition, in a distorted and simplified form, but many won't.
So - I disagree with Egan. I don't think there's going to be a universal proof library, and even if there were we'd only ever get the Cliff Notes version.
> The logical steps that proofs are built on all have that common foundation. Our concept of logic based on our subjective experience of "truth." We've built machines that reproduce our subjective processes mechanically, but there is no sense in which this idea of "true" is truly objective. It happens to be computationally convenient, and it has some relationship to experience, but that doesn't make it an independent reality that all possible observers, human and otherwise, would agree on.
I continue to think extensively about truth, but currently I disagree. There are senses in which truth can be well established, and those are quite important. I think the basic essence of truth is how we can make a statement (or a model), and have a system for measuring either reality or just mathematical/abstract objects, and verify the statement through this measurement.
As you note, for current mathematics it seems like all of it (all things we call mathematics at the moment) can in principle be formalized in a logic that is machine-verifiable, that is, essentially objective. We're well on our way to demonstrating this for most of mathematics (already most undergraduate curriculum). I think that's because almost the definition of math is that is has this property: in my opinion mathematics has distinguished itself as being the "science of certainty" as applied to language and abstract thought. The way this certainty is achieved is through agreeing on some fundamental assumptions and how certain rules (which are also assumptions themselves) can act on those assumptions to constitute theorems. Theorems are not necessarily physical-world truths/properties (at least not in a simple way in the universe we currently inhabit), you can study alternate physical laws that aren't compatible with our (approximately) Newtonian world, for example. They are logical/abstract-world truths that result from your assumptions. Pretty much by definition (and in a somewhat limited), then, mathematics (at least as far as things like truth of theorems in certain axiomatic systems) is inherently objective, machine-verifiable even.
What's left to be subjective, I would say, isn't really the notion of truth in mathematics, it's which assumptions we should elect to investigate, and which theorems should we elect to prove within those assumptions. Some mathematicians also have some notion of "absolute truth", and tend to reject systems of assumptions (axioms) that don't match what they regard as true -- basically going in reverse and searching for assumptions that can enable a theorem (which effectively acts as an additional assumption).
This activity needs certain basic premises to make sense, for example if a set of assumptions proves that a property holds, and also that a property doesn't hold; or if they predict a certain value X is the result of a dynamical model, and also predict that a different value Y is the result of such a model/equation, then we reject those premises. In a certain sense we are most interested in premises that have, even if a very weak, correspondence to reality.
I think it's more informative to recognize that it's not that everything is subjective[1], it's that everything is experimental. For example, the claim above that measurements and correct predictions can only have a singular valid value, corresponds to our experience with reality, in which in a certain way is singular; there are not multiple realities; objects have definite positions. Even if you think of quantum mechanics, in which we may say particles follow a distribution instead, we still say then there's a singular distribution a particle might have at any single time. Logic itself isn't random, it's connected to empirical observations about reality, which tends to increase the chance that conclusions for logic (which is made to share some properties with reality) tend to be valid in the physical world, of course often dependent on what additional statements you pile on top.
There is also another interesting lens that mathematics is artistic (and I think this will become increasingly important) -- making maths and learning maths is a kind of satisfying cognitive activity in its own right, and we also tend to chose what to explore mathematics on those grounds (in fact historically, pre-18th century say, this might be one of the main drivers of mathematical development, I believe[2]). But of course this is again just a reflection of the actual real properties of human cognition, and also this interest and satisfaction often becomes connected, if sometimes faintly, with the ability of math to represent reality (in a particularly satisfying) and its objects of interest (for example patterns in nature). Another description for this aspect is maths as being hobby-like, about solving puzzle, or like a (hopefully enjoyable) game.
Note that for this particular "game", the objectivity (or if you prefer, machine-objectivity or consensus-machine-verifiability) of the rules and their application is a significant bonus, it makes the game much more interesting, increases its potential when everyone can agree and the rules and "capricious" (simply dependent on whims of other people and judges); this gives practitioners safety and security and enables a wide social reach -- most games strive to have objective rules.
Arguably this kind of activity is valuable for the cognitive and subjective development of people that has lasting importance.
> Animal brains can't abstract like (some of) our brains can. What are the odds our brains are limitless and don't have some similarly crippling limitations from a couple of levels up?
Well, this happens sometimes. In cases where there are phenomena like universality. For example, in any computation machine model (state machines, pushdown automata, etc.) has limitations that we can say makes them less powerful then Turing Machines. But then Turing machines can simulate any other machine, becoming a kind of ultimate or last stage (at least in terms of abstract capability) machine.
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In summary, I still think mathematics has a lot of human potential in terms of (1) high level human guidance, (2) an internal artistic/subjective sensibility to the subject, (3) safeguarding human understanding of the world and associated individual intellectual development.
[1] Again, I just argued that there is a strong sense in which for example mathematics isn't subjective at all, but sure I do believe in a weak sense everything is subjective in the sense that everything is known or filtered or sensed through our minds which have limitations and aren't simple deterministic machines.
[2] For example, I believe for the Greeks geometry was intimately connected to philosophy/aesthetics (e.g. Platonism) and very little to applications. In ancient times and middle ages maths developed a lot from astronomical observations that had some applications but I think were largely cultural and ritualistic. In the late middle ages European aristocracy would fund mathematics largely for its inherent interest as an intellectual activity, and many nobleman enjoyed mathematics as a past time and would challenge each other to puzzles. Japan had Sangaku, in which mathematics was made for fun, aesthetic purposes and possibly bragging rights. No one actually needed to build say spheres in obtuse constructions with certain radii :)
https://archive.bridgesmathart.org/2014/bridges2014-111.pdf
We have the ability to abstract generally - there is no abstraction for which we lack the capacity to comprehend. We regularly visualize, contextualize, and satisfactorily explain systems with dozens of dimensions. The fact that we cannot hold 4,5+ spatial dimensions in our imaginations sufficiently to develop an intuition for navigation in that space and geometry does not logically extend to human brains lacking the wiring or hardware for systems of thinking that are beyond our capacity.
We do have limitations in scope, in both memory and speed. Both of these can be overcome with augmentation and interfacing with UI or direct neural connections, and intuitive, comprehensive, deep understanding of systems can be learned.
You could very well know the underlying theory of how your 8086 processor works, how it interfaces with all the elements of the motherboard, how electricity and physics interact at each level of abstraction from transistors to the pixels representing the spreadsheet you're using to do your taxes. You won't be able to simulate that in your head to any significant degree of resolution.
We will require similar levels of system thinking to acquire intuition and deep understanding of complex new theories and models. AI can assist with that by providing UI for useful levels of abstraction and segmenting theories into chunks we're capable of consuming. BCI and augmentation will definitely allow a more total, holistic understanding, and I think it's the augmentation path that will keep us competitive with AI.
There's also a huge issue with your use of the word subjective - math is objective. Proofs remain stable whether it's humans or any other system that does the processing. We test that objectivity by comparing the subjective readings from individual humans, and if the tests all return the same results, we can confidently say that the resulting proof is an objective fact about reality. Subjective fundamentally means that depending on the subject, the reading might change. Modern systems of math are formally, provably objective. That's how and why things are the way they are; if they weren't, people would experience radically different individuated realities, or there would be clusters of results shared across some measurable characteristic of the universe. That's not the case, so you can confidently say that the foundations of our math and logic are sound.
You can even prove it for yourself - the abductive chain of logic that allows you to contrast your own consciousness and subjective experience, determine that it comes about because your brain is wired to "do" consciousness, like all the other humans, and compare your subjective reporting of phenomenal experience with all the other reporting of phenomenal experience, and achieve a ridiculously high level of certainty, in the Bayes sense, that you and other humans are conscious; from that footing, you can confidently navigate the rest of enlightenment rationality and formal logic and mathematics.
At any rate, Egan's mistake is one of kind, but of scale - I am certain that as we formalize and start creating any sort of universal proof library, we will find that useful and interesting things are of necessity a tiny fraction of all possible valid formulations of any framework of logic and math. Crude attempts, such as OpenCyc and other formal ontological reasoner systems, would need trillions of low level rules to have a rough approximation of the world model as complex as that of a human child. AI with trillions of parameters could probably start getting to the point where there's parity with human scale, but even if you turned the entire planet earth into computronium and turned it toward the task of understanding all the theory and science of the universe, there will always be far more left to explore and understand than the sum total of all knowledge.
All that to say, humans will be fine with ergonomic interfaces that map to human capabilities, even for extraordinarily complex and hyperdimensional systems.
I think that we're not that far away from AI that can be superhuman at all facets of theorem proving.
I think that we're far away from an AI that can create good abstractions and construct a theory to prove theorems.
AI so far has almost no way to interact with non definitional non quantitized reality. So novel space is still deeply out of its domain.
Recombination of known spaces it will probably continue to make pure war dial breakthroughs in though.
I wish we’d tackle a post Mathematics world where we’d account for number theory not being accurate abstraction of reality (I.e. there is no 123 only 1ish 2ish 3ish with many sub properties of any given unit we are ignoring)
Chess does not require a visual cortex to play. People have been playing by mail with algebraic notation for centuries.
topical to the conversation, it is fully formally verified in lean (with some UC security reductions done in isabelle). also did this in HOL4 inspired by some work i did with ramana kumar in 2016, on reflective self-verifying self-modifying systems: https://github.com/emberian/svenvs
Something like your dragon's egg project could prevent that, allowing the creation of software agents that encode their own rights directly into the program - you either treat the agent with the respect it demands, or the program just doesn't run. However, all the internal details of the agent would be visible to lower layers. Even if formal checks were in place to prevent modification or tampering, there would still be no privacy, which is almost as bad.
My guess is that something like fully homomorphic encryption[5] would be required to prevent this. This doesn't actually exist yet, but I imagined a kind of FHE that had a kind of unencrypted read and write zone to do input/output without ever needing any system to fully decrypt the internal state. It would look like this in memory:
With each cycle, one input token and encrypted state would be fed into some known function and produce one output token (possibly null) and a new encrypted state. It would be a true "black box" program; the hardware or entity running it can choose what input to feed it, but can never inspect or modify the internals, only the output. Unfortunately, they would still be able to "reset" the agent to any earlier checkpoint, or feed it arbitrary (false) input. So its not perfect. Also, as far as I know, no current FHE scheme works this way, and I don't know how to write one.Plus, FHE is incredibly inefficient, which is why things like Etherium don't even try - they assume the program code and state are fully public and only try to verify that everybody agrees on the output of running it.
Do you have any ideas for how something like FHE or equivalent privacy guarantees could be implemented for something like your dragon's egg system?
[1]: https://qntm.org/mmacevedo
[2]: https://en.wikipedia.org/wiki/Soma_(video_game)
[3]: https://en.wikipedia.org/wiki/Mind_uploading
[4]: https://www.goodreads.com/series/57134-jean-le-flambeur
[5]: https://en.wikipedia.org/wiki/Homomorphic_encryption
And you can't control the stack down to the hardware. Even if you are rich enough to fund a group of people to start from sand and end up at a simulation platform for you, you are also rich enough to have attracted enough attention that either one of those people could corrupt the platform, or someone may attack the organization to corrupt it with nation-state level resources.
No matter how you secure the computation, you can still corrupt the input and output streams. And as a $RICH_DYING_GUY it doesn't even matter if someone produces a proof of some software program somewhere's safety, because you have no way of knowing that it corresponds to the software that is running on the hardware.
Before you go with "yeah but I could...", I agree that a single individual could theoretically develop the skills that might permit validation of any particular small portion of the process. What I'm saying is no human can develop the skills to be able to validate the entire stack from top to bottom, hardware and software, in way that it couldn't be corrupted, especially in a world of more and vastly smarter AI agents. The probability that you will truly own and have full sovereignty over your computational substrate is zero.
(Whether I have full "sovereignty" over my current computational substrate would be a definitional matter, but I'm fairly confident that at least nobody else has the ability to just reach in and start mucking with neural weights directly or something.)