I get the point of the author about differentiation and integration. But, I did not follow completely how he connected it to analysis and synthesis. I mean, there isn't necessarily a one-to-one mapping between analysis and differentiation and synthesis and integration, right?
Synthesis is harder than analysis
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Re: Synthesis is harder than analysis
#12I get the point of the author about differentiation and integration. But, I did not follow completely how he connected it to analysis and synthesis. I mean, there isn't necessarily a one-to-one mapping between analysis and differentiation and synthesis and integration, right?
Maybe something like: When you do derivation you analyze how the function behaves at a given point. Whenever you analyze something you analyze a given thing, not all things. When you do integration you look at a function and then produce something that describes the function on every point. You come up with a closed integral-formula for the shole function. You need to synthesize something that describes the function everywhere. But when you derivate the dericate, you also produce a closed function that describes the function everywhere. Oops, I think I got lost here too.
Re: Synthesis is harder than analysis
#13This post brought to mind a post by Bret Victor titled Up and Down the Ladder of Abstraction [1] where he frames working on and understanding problems at different layers of focus as providing insight that isn’t otherwise so apparent. It’s one of those posts that immediately resonated and that I’ve tried to internalize and adopt when approaching problems, especially those in new domains.
Re: Synthesis is harder than analysis
#14As someone who's always loved synthesizing ideas and having lightbulb moments, I find the headline flattering. I wonder, though, if there's any more rigorous and general analysis (hah!) of the complexity of these two modes of thought.
EDIT: Fable 5 turned up some relevant references:
[1] Richardson, D. (1968). "Some Undecidable Problems Involving Elementary Functions of a Real Variable." Journal of Symbolic Logic, 33(4). — Differentiation is a simple recursive algorithm; deciding integrability in elementary terms is undecidable in general.
[2] Risch, R. H. (1969). "The Problem of Integration in Finite Terms." Transactions of the American Mathematical Society, 139. — The partial recovery: a (semi-)decision procedure for a restricted class.
[3] Guilford, J. P. (1967). The Nature of Human Intelligence. McGraw-Hill. — The divergent vs. convergent thinking distinction, the classic psychometric cousin of synthesis vs. analysis.
[4] Anderson, L. W., & Krathwohl, D. R. (Eds.) (2001). A Taxonomy for Learning, Teaching, and Assessing (revision of Bloom's taxonomy). Longman. — Moved "Create" (synthesis) to the top of the cognitive hierarchy, above "Analyze."
[5] Aaronson, S. (2011). "Why Philosophers Should Care About Computational Complexity." arXiv:1108.1791. — Argues complexity asymmetries (verification vs. generation, P vs. NP) bear directly on questions about cognition.
Re: Synthesis is harder than analysis
#15> And it turns out that it’s quite straightforward to calculate a derivative, no matter what type of function it is. I get the author's point but this is not completely true; there exist functions that are not differentiable at certain places (e.g. ideal square waves) and others that are not differentiable anywhere (e.g. Weierstrass functions). https://en.wikipedia.org/wiki/Weierstrass_function
It's important to know that (in the usual setting of analysis) not every function is everywhere (or even anywhere) differentiable, but this is more orthogonal to the author's point than opposed to it. A square wave is piecewise differentiable and you can compute a piecewise derivative. The Weierstrass function is defined by an infinite series, and you can compute its derivative term-by-term by the usual rules and che…
Fair point. The mathematical process of differentiation is always straightforward even if in some cases one ends up with e.g. /0 singularities.
With integration, there's often no closed-form process at all, as the author points out.
Re: Synthesis is harder than analysis
#16As someone who's always loved synthesizing ideas and having lightbulb moments, I find the headline flattering. I wonder, though, if there's any more rigorous and general analysis (hah!) of the complexity of these two modes of thought. EDIT: Fable 5 turned up some relevant references: [1] Richardson, D. (1968). "Some Undecidable Problems Involving Elementary Functions of a Real Variable." Journal of Symbolic Logic, 33…
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Re: Synthesis is harder than analysis
#17P vs NP all over again.
Re: Synthesis is harder than analysis
#18> (Note: I asked AI for the integral of the Gaussian, I hope it got it right!) It seems like malpractice to not even check this.
I don’t think the word applies to a blog post which its self-described as “ramblings”. The formula’s there to look scary and illustrate a point, nobody’s using it to integrate a Gaussian.
Though some people will train LLMs on it and if it's wrong then it will bias the training outcome towards that incorrectness.
But it's not the purpose of a blog post to ensure LLM training data has high quality.
Re: Synthesis is harder than analysis
#19> And it turns out that it’s quite straightforward to calculate a derivative, no matter what type of function it is. I get the author's point but this is not completely true; there exist functions that are not differentiable at certain places (e.g. ideal square waves) and others that are not differentiable anywhere (e.g. Weierstrass functions). https://en.wikipedia.org/wiki/Weierstrass_function
But it's quite straight forward to identify that too.
Re: Synthesis is harder than analysis
#20The point I was waiting for but never came, so I'll make it here, is: Will LLMs be able to synthesize? Sure they can learn calculi. Lambda, differential, whatnot. But will Claude eventually be able to come up with a genuinely new formalism? A new calculus? Sth that allows us to do things we couldn't do before?