> If AI can give us an abstraction that lets us form and combine our ideas for driving the computational processes, why can’t we delegate to AI the thinking effort that goes into writing the actual programs?
Suppose that "AI gives us abstraction", but what is an "abstraction"? A set of symbols and their relations is not sufficient for thinking, esp. in complex problems.
For instance, if I know the abstract definitions of calculus, I am still far away from understanding something like a coupled nonlinear equation (e.g., weather prediction) or even the Newton's equation of motion.
All these are based on a fundamental unproved presumption: hyperscaling.
I feel, perhaps, "abstraction" is misunderstood here. I meant a higher order interface that is expressive enough so humans can formulate their ideas naturally. E.g., an interface that can take a problem statement in natural language and output a code that solves that problem. These are non-rhetorical questions though. I am genuinely interested to know the answers for the questions that I have raised in the last paragraph.
On Math specifically, I recently came across the OpenAI's announcement of planar unit distance problem disproof [1] by a general model. Because I am no mathematician, I cannot tell if it is legitimate, but if it is, I don't know how I should feel about it!
First regarding OpenAI's claim on the disproof of a conjecture in discrete geometry:
Against any rosy framing, this piece of advertorial confirms that AI can be used, as an aide, for exploratory problems, like this one:
1) The groundwork is already done and the stage is set; the statement of the problem was based on this firm basis:
Since Erdős’s original work... the problem is explained in the "2005 book Research Problems in Discrete Geometry" as “possibly the best known (and simplest to explain) problem in combinatorial geometry.”
An internal OpenAI model has disproved this longstanding conjecture, providing an infinite family of examples that yield a polynomial improvement.
2) The proof still required the contribution of group of expert mathematicians (internal and external):
This problem was solved in a completely automated fashion. ... internal human researchers and mathematicians began to examine the solution carefully. After preliminary AI-assisted verification and rewriting, a draft was sent to external mathematicians, including several number theory experts, who confirmed the proof’s correctness (and have already simplified and strengthened the argument). ... They have also written a companion paper explaining the argument and providing further background and context for the significance of the result.
3) The proof connects two mature fields:
The proof came from a new general-purpose reasoning model... [and] brings... ideas from algebraic number theory to bear on an elementary geometric question.
4) Advertorial take-away:
Better mathematical reasoning can make AI a stronger research partner: something that can hold together difficult lines of thought, connect ideas across distant areas of knowledge, surface promising paths experts may not have prioritized, and help researchers make progress on problems that would otherwise be too complex or time-intensive to tackle.
That future still depends on human judgment. Expertise becomes more valuable, not less... .
The bone of contention is the relation between abstraction and thinking.
Current LLMs allow us to use a natural language as well as a symbolic mathematical formalism to describe our ideas.
The main question is whether “we can delegate to AI the thinking effort into writing… programs”.
My sharp answer is negative. The act of “writing programs” or programming can be done in pure symbolics (like pseudo-code); that is syntax does not count much. Yet, writing a programme is far from mere symbolic expression of a conceived idea; it is I) conceiving an algorithm, II) making the algorithm efficient on a certain architecture (e.g. von-Neumann architecture), and finally III) making it efficient within a certain programming paradigm (say, Algol).
LLMs can certainly assist us in all these 3 levels. As we arrive closer to pure symbolics (aka. “code”), LLMs become more efficient and reliable. However, on any of those 3 we cannot “delegate the thinking effort”.
The basic reason is that LLMs (small or large) are merely stochastic predictors of sequences: They sample the elements of a sequence from a high-dimensional non-trivial probability space. As the problem becomes more complex, this high-dimensionality of the probability space becomes a huge unsurmountable obstacle. We know that already from the Monte-Carlo simulation: [The curse of dimensionality](https://en.wikipedia.org/wiki/Curse_of_dimensionality).
That means, finding the “correct” solution(s) becomes extremely unlikely as the dimensionality of the probability space grows. That is also what we face in the real world: finding the correct solution to a problem is like finding a needle in a large unorderly haystack. This is essentially where magical process of “thinking” comes into play.
My sharp answer is negative. The act of “writing programs” or programming can be done in pure symbolics (like pseudo-code); that is syntax does not count much.
I feel, I am still being misunderstood. What the essay says is this: We think to form ideas and combine them; we then use programs (can also be a pseudo-code) to implement those ideas. In other words, we ideate and then implement. The question is: can the thinking effort that is required for the implementation (not for ideation) be delegated to AI.
> If AI can give us an abstraction that lets us form and combine our ideas for driving the computational processes, why can’t we delegate to AI the thinking effort that goes into writing the actual programs?
Suppose that "AI gives us abstraction", but what is an "abstraction"? A set of symbols and their relations is not sufficient for thinking, esp. in complex problems.
For instance, if I know the abstract definitions of calculus, I am still far away from understanding something like a coupled nonlinear equation (e.g., weather prediction) or even the Newton's equation of motion.
All these are based on a fundamental unproved presumption: hyperscaling.
I feel, perhaps, "abstraction" is misunderstood here. I meant a higher order interface that is expressive enough so humans can formulate their ideas naturally. E.g., an interface that can take a problem statement in natural language and output a code that solves that problem. These are non-rhetorical questions though. I am genuinely interested to know the answers for the questions that I have raised in the last paragraph.
On Math specifically, I recently came across the OpenAI's announcement of planar unit distance problem disproof [1] by a general model. Because I am no mathematician, I cannot tell if it is legitimate, but if it is, I don't know how I should feel about it!
[1] https://openai.com/index/model-disproves-discrete-geometry-conjecture/
First regarding OpenAI's claim on the disproof of a conjecture in discrete geometry:
Against any rosy framing, this piece of advertorial confirms that AI can be used, as an aide, for exploratory problems, like this one:
1) The groundwork is already done and the stage is set; the statement of the problem was based on this firm basis:
Since Erdős’s original work... the problem is explained in the "2005 book Research Problems in Discrete Geometry" as “possibly the best known (and simplest to explain) problem in combinatorial geometry.”
An internal OpenAI model has disproved this longstanding conjecture, providing an infinite family of examples that yield a polynomial improvement.
2) The proof still required the contribution of group of expert mathematicians (internal and external):
This problem was solved in a completely automated fashion. ... internal human researchers and mathematicians began to examine the solution carefully. After preliminary AI-assisted verification and rewriting, a draft was sent to external mathematicians, including several number theory experts, who confirmed the proof’s correctness (and have already simplified and strengthened the argument). ... They have also written a companion paper explaining the argument and providing further background and context for the significance of the result.
3) The proof connects two mature fields:
The proof came from a new general-purpose reasoning model... [and] brings... ideas from algebraic number theory to bear on an elementary geometric question.
4) Advertorial take-away:
Better mathematical reasoning can make AI a stronger research partner: something that can hold together difficult lines of thought, connect ideas across distant areas of knowledge, surface promising paths experts may not have prioritized, and help researchers make progress on problems that would otherwise be too complex or time-intensive to tackle.
That future still depends on human judgment. Expertise becomes more valuable, not less... .
The bone of contention is the relation between abstraction and thinking.
Current LLMs allow us to use a natural language as well as a symbolic mathematical formalism to describe our ideas.
The main question is whether “we can delegate to AI the thinking effort into writing… programs”.
My sharp answer is negative. The act of “writing programs” or programming can be done in pure symbolics (like pseudo-code); that is syntax does not count much. Yet, writing a programme is far from mere symbolic expression of a conceived idea; it is I) conceiving an algorithm, II) making the algorithm efficient on a certain architecture (e.g. von-Neumann architecture), and finally III) making it efficient within a certain programming paradigm (say, Algol).
LLMs can certainly assist us in all these 3 levels. As we arrive closer to pure symbolics (aka. “code”), LLMs become more efficient and reliable. However, on any of those 3 we cannot “delegate the thinking effort”.
The basic reason is that LLMs (small or large) are merely stochastic predictors of sequences: They sample the elements of a sequence from a high-dimensional non-trivial probability space. As the problem becomes more complex, this high-dimensionality of the probability space becomes a huge unsurmountable obstacle. We know that already from the Monte-Carlo simulation: [The curse of dimensionality](https://en.wikipedia.org/wiki/Curse_of_dimensionality).
That means, finding the “correct” solution(s) becomes extremely unlikely as the dimensionality of the probability space grows. That is also what we face in the real world: finding the correct solution to a problem is like finding a needle in a large unorderly haystack. This is essentially where magical process of “thinking” comes into play.
My sharp answer is negative. The act of “writing programs” or programming can be done in pure symbolics (like pseudo-code); that is syntax does not count much.
I feel, I am still being misunderstood. What the essay says is this: We think to form ideas and combine them; we then use programs (can also be a pseudo-code) to implement those ideas. In other words, we ideate and then implement. The question is: can the thinking effort that is required for the implementation (not for ideation) be delegated to AI.
Reading the previous response, I see that my question is answered. Here is my take in part 2.