Synthesis is harder than analysis • Hacker News Board

Re: Synthesis is harder than analysis

#21

Post by ssivark » Sat, Jul 04, 2026, 9:03 AM UTC

The analysis and synthesis approaches to understanding systems have respectively been the driving forces for two major breakthroughs in 20th century physics: reductionist and emergent phenomena. Reductionism aims to understand a system by reducing it to component parts and assuming that the composition is simple. This attitude drives particle physics. On the other hand, one of the most important themes from condensed matter physics has been how "more is different" and collective phenomena can induce emergent behaviour which is very different from the behaviour of the constituent parts. In this perspective, the precise constituent parts don't really matter too much -- many substrates which like completely different to analysis can end up looking very similar in synthesis. This is the principle behind universality classes in critical phenomena. This patter of thought should also be familiar to folks who advocate for a "systems perspective".

In the language of differential vs integral calculus, you can have perfectly well behaved and physical functions whose derivatives can completely miss the global behaviour of the function i.e. smooth but not analytic eg exp(-1/x) at x=0. Funnily enough, this is exactly the form of the action taken by quantum mechanics and an argument for how the classical limit irrecoverably ignores the physics of quantum systems.

Re: Synthesis is harder than analysis

#22

Post by marouen19 » Sat, Jul 04, 2026, 9:04 AM UTC

ajeet wrote: post #5

I loved reading this article. It was reasonably short to not make the reader lose interest. It jumped around in different domains to make a core point at the end about SREs managing complex systems. This gets even more difficult with the rate of change each of those systems has with coding agents. I don't know if I agree with the terms "synthesis" and "analysis" as the equivalent of global and local respectively, but…

Agree

Re: Synthesis is harder than analysis

#23

Post by ykonstant » Sat, Jul 04, 2026, 10:15 AM UTC

teiferer wrote: post #19

dreamcompiler wrote: post #4

> 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.

Not always. As an exercise, try to find a continuous real function which is differentiable if and only if the Riemann hypothesis is true!

Re: Synthesis is harder than analysis

#24

Post by inigyou » Sat, Jul 04, 2026, 10:53 AM UTC

dreamcompiler wrote: post #15

jonlong wrote: post #10

Earlier quoted context omitted.

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.

On the other hand, integration is numerically well-defined for piecewise continuous functions, while differentiation isn't and may result in nothing useful. Derivative of a square wave is constant zero with some gaps.

Re: Synthesis is harder than analysis

#25

Post by torben-friis » Sat, Jul 04, 2026, 11:45 AM UTC

>incident response is one area where we are frequently confronted with synthesis problems: we have to understand how the pieces normally fit together in order to make sense of what is currently going wrong.

I'm not sure I can accept without further clarification equating that with synthesis. Normally in incident response, you start with a full system where in principle any root cause is possible and gradually reduce scope until finding the solution.

For me integration could be used for say taking three adhoc solutions and designing a general one that fits all, or merging two different services after a company merge. But anything debugging is very clearly analytic.

Re: Synthesis is harder than analysis

#26

Post by hliyan » Sat, Jul 04, 2026, 1:03 PM UTC

If my understanding of Kant is correct, then all analysis (i.e. taking things apart) is synthesis (i.e. putting things together) in reverse, and therefore the latter is a necessary prerequisite of the former. According to him, our minds very first brush with the real world (whatever that might be) is in the form of the "manifold" of raw sensory signals sans any form whatsoever (not even spatial or temporal arrangement), and the only way we can make sense of that manifold of signals or data is to reduce them through a process of combination by commonalities and separation by lack thereof. All objects, categories and concepts ultimately spring from this process of synthesis, some of it involuntary (e.g. the arrangement of certain inputs into three dimensional space) and some of it volitional (e.g. the positing of concepts and categories). Analysis is taking these constructs apart, much like one takes apart a mechanical watch. Except that the exercise tells us more about how our brain puts things together than what the thing actually is.

Re: Synthesis is harder than analysis

#27

Post by nayuki » Sat, Jul 04, 2026, 1:27 PM UTC

> Synthesis is harder than analysis

Taking this statement at face value, it means something in computer science: computing an answer (synthesis) is currently believed to be harder than checking an answer (analysis).

The simplest example to illustrate the claim is that factoring a number is harder than multiplying the two factors to check that it equals the original number, or even to decide whether the original is prime or composite (but without yielding the factors).

This also cuts to the heart of what NP means - it means that the answer to a yes/no problem about binary strings can be checked in polynomial time. It doesn't give a recipe for how to generate the answer, but it is implied that finding an answer can take up to exponential time and no more.

Re: Synthesis is harder than analysis

#28

Post by m_m_carvalho » Sat, Jul 04, 2026, 1:35 PM UTC

AI has made me appreciate synthesis much more.

Understanding existing code or generating possible solutions is getting easier. The hard part is still deciding what to build, what not to build, and how independent ideas fit together into something coherent.

In my experience, AI reduced the cost of implementation far more than it reduced the cost of synthesis.

Re: Synthesis is harder than analysis

#29

Post by efavdb » Sat, Jul 04, 2026, 2:00 PM UTC

dreamcompiler wrote: post #4

> 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

FWIW in numerical analysis stable differentiation is harder than integration.

Re: Synthesis is harder than analysis

#30

Post by adverbly » Sat, Jul 04, 2026, 2:07 PM UTC

ajeet wrote: post #5

I loved reading this article. It was reasonably short to not make the reader lose interest. It jumped around in different domains to make a core point at the end about SREs managing complex systems. This gets even more difficult with the rate of change each of those systems has with coding agents. I don't know if I agree with the terms "synthesis" and "analysis" as the equivalent of global and local respectively, but…

Same.

I thought it was interesting that the author mentioned using AI for the integral - they clearly didn't use it for the prose which is exceptional!

I miss reading high quality stuff like this.