Posted by Anon84 11 hours ago
I mean... yes, most people will find it reassuring, but history has proven that's not necessary. People have been using medicines of which the mechanism wasn't understood for a very long time and greatly enjoyed their benefits. Even widely used one (e.g. Paracetamol).
> Or a human mathematician who understands the mathematical model used to locate the cocktail?
If we're being fair to AI - it contains collective knowledge from all fields, which means it's probably less likely to miss something that a human would.
removing some key theories or inputs
(e.g., finding an elementary proof for a result currently only provable by non-elementary means)."
As usual, Tao is brilliant in all that he researches, all that he writes about.
I chose the above statement (which is brilliant, in and of itself!) to comment on, because it leads to the following idea:
There there exists, or should exist, a dependency map in the fields of not only Mathematics, but also of Computer Programs/Software, Engineering, and even a seemingly non-related field: The Law...
In other words, how do we get from the simplest of axioms or foundational things (aka "first principles", aka "self-evident truths") to much more complex entities?
In Law for example, how do we go from the simplest of historical legal constructs to the most complex of the most complex Supreme Court cases?
You see, there is, or should be a map, you could call it a dependency map, you could call it a dependency graph, which shows more and more abstract/complex mechanisms/things/assertions/statements/truths/functions which is mapped back to , that is, dependent on various chains, various stackings, various "stacks" of simpler ones.
In Engineering, for example, how do we get from the simplest of machines to the most complex of machines? What simpler machines and/or sub-components (aka "dependencies", aka "subcomponents") are required to build it, and how do those simpler machines work, and what's the dependency graph or map for their subcomponents?
More generalized, if we have something of complexity, then how do we get there, step by step, from individual subcomponents, individual inputs, individual proofs, individual software systems, step by step?
What is the map of those dependencies?
Note that in some systems, Math proofs, for example, there may be different paths which can be traversed to get to the same destination.
Ablation Studies could be thought of in Travel, in Geography as "if I cannot take one, or a specific set of routes to get to a place, can I still get there?"
A simple example would be in Google Maps, where you'd like to drive somewhere, but you'd like to avoid tolls. Is the route still traversable while avoiding tolls? Well, that's an example of one constraint. In Ablation Studies, you might wish to remove a bunch of routes with whatever criteria or characteristics , i.e. muddy roads, roads that have characteristic X, roads that do not have characteristic Y, etc., etc.
Getting back to Math, specifically proofs, it would be great to create a dependency map/graph of all of them, and then try removing inputs (aka, paths to them, dependencies on other mathematical proofs/objects that they may have) and see if they are still reachable.
In software, when we desire the tightest, cleanest, source code, the above is related to refactoring.
In the future, I'd love to see dependency maps/graphs (call them whatever you will) for not just Mathematical Proofs (although I'd love to see that too!), but also in such diverse subjects as Science, Engineering, Programming/CS, and even the Law!
Because they should exist in all of those subjects!
Anyway, another great piece of work by Terrence Tao!
1. The use of semi-ugly slides to communicate this is just perfect and quite heartwarming, but it does highlight my main criticism of the mathstadon version of this thesis: he's myopically focused on mathematics as he has practiced it, rather than mathematics as a ~2400y old academy. Like, "stable for almost a century" sounds impressive, but should be a pretty obvious red flag in hindsight!
2. Glossing over "objective verifiability" feels like another place where he's ignoring a ton of relevant philosophy for no clear reason -- yes, mathematics is the only academy based in pre-conscious cognitive facts about our processing of time and space, but that's not the end of the story on "objectively verifiable". To say the least! He hedges with "broad consensus" which doesn't need to be absolute, but that seems to be not only dismissing a highly relevant question, but even implying that he might be unaware of it. I doubt he is, but still: not great.
3. Who is this for...? Why is an explanation of Lean needed in a talk given at CalTech? I suppose he's welcoming his role as a bit of an influencer, there?
4. Re:the focus-on/centrality-of 'highly digitizable' as a unique class of task that applies to mathematics in particular, I must sadly trot out the increasingly-common trop: Yudkowsky called it... https://intelligence.org/files/IEM.pdf
5. "the space of mathematical problems remains infinite" is, again, ignoring really important philosophy around academies as social structures, built for human means. Mathematics is only infinite if we decide that all knowledge is useful (the quintessential example being 'counting the grains of sand on a beach'). Not really important in the first place, but another worrying case of the above.
6. Problem solving is the goal of mathematics; he has a completely valid point here (that we shouldn't throw AIs at unsolved problems in bulk and thus lose human expertise), but it's obscured by the use of "[open] problem" being a too-technical one. IMHO. Slide 16 fails to disabuse me of this notion.
7. Slide 18 is describing the differences between functions and systems, and is arguably even talking about assemblages.
8. If we're gonna explain lean up-front, it feels like a baffling choice to throw the "maybe AI will solve cancer but use it to secretly plot to kill us all" slide in there. It's also already lead to misunderstandings and backlash on Reddit, where 'yes we want to not die of cancer!' is a pretty convincing counterpoint (if a ultimately a subtle strawman, ofc). It's also quite distinct from the rest of the talk.
As always, the best part of any Tao publication is his ability to inspire and rally and organize. I think Math 5.0 will indeed be a matter for creativity! Hopefully the IE levels off before we cease to be helpful in that capacity...
https://x.com/RohanArun/status/2109336015814959200?s=20
It seems like OpenAI opened the door for many more people to participate in math discovery process. It's fascinating to follow!
Perhaps, but there is a danger: it is difficult to un-see things. At least for now, AI-generated proofs are likely to be of a somewhat brute-force nature. As each such proof pops into existence, two things happen: (1) it is much harder to maintain an untainted mind and go on to discover alternative, perhaps deeper and more conceptual, proofs that cannot be obtained by massaging and cleaning up a more brute-force approach (think of this as getting 'stuck in a local minimum', if you like), and (2) at a societal level there is thereafter much less incentive to attempt to do so (announcing a proof of a previous unproven result is, for now, far more prestigious than announcing a much better, more elegant, proof of a known result).
One could ask why we prefer conceptual explanations to brute-force proofs; after all, a proof is a proof, isn't it? I suppose one reasonable answer is that conceptual explanations lead more readily to new questions and that there's also intrinsic beauty in such explanations -- though, of course, many outside the field will simply not care.
The popular perception of him (as popular perceptions generally tend to do) reduced his overall position to basically early adopter went sour grapes, and I'm really glad to see that substantively falsified.
Humans have a bias towards assuming that problems have solutions. Mathematicians probably more than average.
This talk is predicated on the theory that the problem of "maintaining the relevance of humans in mathematics" is tractable. Not sure I agree