I can use AI for coding after decades of coding. I can't use it for theoretical physics because I can't evaluate the responses.

https://larsfaye.com/articles/ai-coding-will-prevent-experti...

If someone asks about AI I tell them step 1 is ask it how to do something you know all about. Step 2 is consider everything else you ask will be that inaccurate.

I wonder if this key point actually holds though:

>The skills to do so, however, are a function of someone who has experienced the friction and challenges over time that culminate in "good taste".

It's certainly sometimes true, but I don't think it's a general rule. Sometimes friction is just friction and sometimes you spend 1000 hours learning something that disappears and becomes obsolete or at least irrelevant to the goal.

Programmers used to need to know the instruction set of the CPU, assembly language and so on. Some still do but for most developers today that's not useful. Everything you know about 6800 assembly will not make your note-taking app any better.

I think we are in a state where AI tools so easily mimic what we used to do by hand that we think the friction is gone, but that's because we haven't raised the bar yet. One day we'll look at the Fable one-shot that's better than anything we ever made by hand ourselves and say "It could be even better".

And then the friction is back, just on a new level.

It could. It could make a simple note-taking app not take gigabytes of memory and take visible delay on each click. Most people don't bother of course because simple note-taking app is not worth the effort. It's possible to do better, it's just often not practical.

Similarly, sometimes I have to manually validate the output of an ai tool, analyzed over a large text I can’t practically read and understand, and best way I’ve found is to ask the tool (or another ai) to ‘show your work,’ ie make it help me make the determination by showing places in the text I have to read to follow its reasoning. We can never trust another ai directly to assess the validity of another ai.

A lot of times that will take physical tests. Or in the case of math/logic, tests to validate each line or validated sources of previously proved theorems

But they did get good and this seems like a non-issue.

> If these tools demand expertise, yet the tools can actively circumvent the friction that cultivates expertise, then what is the path for one to become an expert so they can effectively use these tools?

But I don't think a beginner would have the same experience. The AI still makes A LOT of stupid mistakes and decisions, but I catch them early enough (sometimes while it's still showing it's reasoning steps), stop the prompt, guide it on the right path, rinse and repeat.

Sometimes I am lazy and give the AI a broader prompt, let it do its thing, and then I come back to see that it spent 90% of the time working on some part/feature/implementation that was not really needed and that it over-engineered the solution.

I rarely write any line of code know or manually change any code, I tell the AI how to do it and what to watch out for. Many times it catches some edge-cases before I even haven to think about them. I do still feel like both me and the AI could miss some edge-cases now, because I'm thinking less about the implementation and what problems can arise, but I feel like 90% of "gotchas" are already engrained in my planning after so many years of coding and problem solving.

You can't use it for theoretical physics because you have no meaningful question and there is no result that you can do anything with.

In contrast you don't need to be a coder to understand if your to-do list for cats works: You have an idea of what you want. You know the rough shape of what an app is and what it can do. You can put it in front of your can and look at it go. Or not.

It turned out it was using LocalStorage, which is not entirely unreasonable, but there are obvious drawbacks (e.g if you move the file it might become inaccessible, if you switch machines there's no convenient way to transfer the data, there are all kinds of ways to lose it, etc..).

In this case "how is this storing my data" is a fundamental question with plenty of implications for your app over its lifetime, but most non-computer people don't think to ask it at all. These days many users enter CS programs without knowing how to manage files and folders on their computer, because even that is often abstracted away.

But if you are building some complex data science statistical model and you don’t have any domain knowledge you won’t even know what to ask for.

I think pretty much the opposite. I can ask it to explain to me in ways I understand it. Even drill down the simplest of equations. Since llms have infinite patience. All I need to learn anything is patience.

And if we (LLM tool operators) don’t know the subject matter in question, we can’t easily distinguish what they are right or wrong about.

Professor Tao also put out some YouTubes of him working with an older LLM to do Lean proofs, and his intelligence matters - things where I would be stuck for hours trying to understand what was failing in the model proof were just instantly clear to him and fixed in thirty seconds.

And I am an ok coder (rather than a bad maths grad student), but the LLM will happily thrash around the edges of a problem with me with no clear convergence when I don’t have that clear insight and the problem is weirdly presented enough; I still find the trick to walk around the block and disengage and then return knowing exactly what to do (now prompting it to the right thing) to be a super power for getting what I want out of the coding system.

More broadly, LLMs or anything at all, needs a verifier. If the task can be automatically verified, great, then anyone can use them. If you can't automate the verification, then you need to be able to verify it using your knowledge. Knowledge required to verify is lower than the knowledge required to create in very few cases. This is why you still need a fully trained human verifier. We are yet to reorganise the overall "tasks" in the economy such that verification can be done with much lesser knowledge, for no reason other than that there was never demand for this until creation became automatic few years ago. It is possible and is slowly being done, there are many many startups working on automating verification in different fields and in many cases we will see fields reorganise themselves to be more amenable to automatic verification. Note that _effort_ required to verify is much lower than what is required to create, for almost anybody, and LLMs have economic use just due to that alone, albeit in the hands of a knowledgeable human.

Edit: do give counter examples if you have any in maths, physics, chemistry, biology etc

I asked Opus 4.8 to critique my algebra notes (these are definitely not masters level- just undergrad second year). It hallucinated an error it claimed I made in the notes and then put in a correction I didn't need because what I had written was correct.

What I said in my notes was:

   Notice that a cyclic group is a degenerate (in the sense of "smallest
   non-trivial") case of a finitely generated group where the generating set is
   a singleton.

The point is I see the LLMs as a "smart friend"/colleague I can work with but I do think critically about what I get told and don't just take it as face value because it's not always correct for sure even in relatively basic cases like this.

I think you guys are misunderstanding me, you can still talk to a llm to learn everything about physics or maths.

Any actual physicist would probably be able to tell me why that's a category error. I don't know why because I'm not one. But there are actual mathematics underlying a statement like that and I'm 99% sure the maths don't work like that.

By the way I think that's why everyone thinks of so many weird physics ideas more than other fields. It's because things are explained in words that hide math, and you can make hypotheses in words that would be obviously nonsense at the level of maths. Like your boss asking why you don't just recompile the cloud.

Let’s not pretend you need to be programmer to use AI for programming like mathematicians need to solve math problems. Programming (most of the time)solves real problems rather than abstract constructs.

AI is already good enough to create the next “Facebook”(v1 and maybe v2 as well) without any real programmer. This makes the AI a big enabler and reduces the need for programmers vastly in the early stages of any business.

That being said as more online businesses will flourish these will require actual programmers after they get enough traction so the debate is still on if it will lead to massive layoffs in our industry.

Facebooks early years were dominated by a concern for users per engineer ratio; the rather florid style of LLMs suggest that they will generate so many systems of such a high complexity you will get Hadoop levels of non-application support needed - forget self healing, it’ll need constant LLM spend just to keep running at scale.

Most people posting on the internet, especially people who know about more complex subjects, are terrible teachers. Teaching is it's own skill

That is what will happen though to future generations: they won't be able to use it for anything because none of them will have the "decades of coding" experience that you have had the privelege to have without AI.

to which betteridge's law of headlines says: NO.

A tool, even if it is a chisel, in the hands of a master sculptor would obviously result in a wildly different outcome.

  > I’m also surprised to see that even Terrence Tao seems to use it in a way that resembles, in progression, how I use llms in my area of expertise

I've mostly seen people trying to oneshot a result, while I'll quickly experienced that going through steps/discovery was more effective and more satisfying, since you can always steer it back in the right direction; while oneshotting is hit (and it kind feel like magic) or miss (and you'll have to rework your prompt).

It is still ultimately Terence that is steering things.

What is crazy to me is how few of other people's conversations like this I have actually read.

Tao is really great for this because the anti-AI crowd can't really chime in and take the thread in a pointless direction. It is hard to think of another human alive who can carry the weight of unassailable authority in the same way.

What's somewhat disturbing is just how much Fable's code really does benefit from the review. It tends to leave a lot of low-hanging fruit, and you can see it getting kind of impatient when repeatedly called on it.

This is basically how Fable told me to get therapy.

It's still what I do 90% of the case until I feel it's good enough for my usage.

I think it's not even about the ability to steer the AI. Just the ability to ask the right questions

The first one was someone proving another conjecture false by just repeatedly saying "keep going" to ChatGPT: https://x.com/DmitryRybin1/status/2079904005652893709

What a world we live in.

  > What a world we live in.

Scroll... wait for render... scroll... wait for render... repeat...

We live in a world where there's so much crazy technology but few people use it to make products better or to improve people's lives. Most people use it to just make more money. It's funny too, because there's a million things we could use that tech for that actually reduce costs. Hell, what would be the economic impact of putting ML systems into streetlights so they properly coordinate. Don't even need LLMs for that, and I'm sure it'd save billions of dollars a year. Just a lack of will. I wonder if this will ever change. Is this how we create the high tech low life future?

(FWIW, no problems if I jump into the app. It's purely a web thing, but my point more illustrative than specific)

Did Tao do that?

We recently had some bugs fixed in the geometry kernel of solvespace. Not much conversation, but the analysis from the AI was amazing:

https://github.com/solvespace/solvespace/pull/1729

https://github.com/solvespace/solvespace/pull/1730

https://github.com/solvespace/solvespace/pull/1731

From the Validation section of PR 1730:

"The model family was reconstructed programmatically (parameterized cuboid stack) and swept over 2,304 configurations — extrusion directions, workplane-normal orientations, sketch windings, D's plane/height/depth/extent, including all the exact-coincidence heights. Zero failures with the fix; 576 failing configurations without it. The generator is available on request."

It looks like it wrote a python script to generate test cases in our file format for testing. Just... you know, as a side quest.

On the one hand, agents have done this sort of thing for a year+, if you pushed them to check their work. On the other, I absolutely can feel Fable and Sol have crossed a threshold where they can be trusted far more than before. Huge difference between plans written by Opus or Fable.

Accumulated AI slop can simply be cleaned up by better models. Real cost of technical debt is shrinking due to the the inflationary devaluation of code!

For posterity, this indeed works for most problems where an agent might give up. LLMs don't inherently know something is impossible.

The phrase I tend to use in my harder prompts to automate this with a sane loop breaker:

> **REPEAT THIS PROCESS UNTIL CONVERGENCE AND YOU ARE OUT OF OPTIMIZATION IDEAS.** You have permission to keep iterating.

"Famous mathematician uses ChatGPT to solve famous math problem" is equally technically impressive, but now you're telling those very same knowledge workers "that famous mathematician could've been you". It puts you in the driver's seat, and provides a clear path forward — subscribe, use our product, and reap the rewards.

My guess is they don't do this because they don't have time. They're all trying to build a company that makes them generationally wealthy before the music stops.

EDIT: oh, they do offer grants of $1000 of API credits to researchers https://help.openai.com/en/articles/10139500-researcher-acce...

As an aside, there's this idea in math that when you create a new field you shouldn't solve all the easy problems - you need to entice other people to learn about the field!

I expect there's some element of that here. It's much better for OpenAI and Anthropic if their users are the ones discovering and writing up the results of the AI solving hard math. Look at the high school and college age students who have become ai power users and potentially learned how to use git to contribute ai generated solutions

(Related: I believe Terry has also gotten all of the subscriptions gifted to him)

https://xenaproject.wordpress.com/2026/07/20/human-mathemati...

Depends. The headline 'some rando solved a famous problem with a consumer grade subscription' might be a really good way to sell these subscriptions.

"Some already famous guy with special access solved a problem" might not be as inspiring.

Could they do it? Sure, but to what end? It would make more people hate them and feel even more "take our interesting work." Pitching it as a useful tool just makes more sense on all levels

Although if I were a mathematical quantum physicist I might be trying to prompt one into a % mathematical field formalized quantum field theory. Weirdly this sort of obvious step has not been accomplished in the last 100 years.

That's why the free market works, millions of agents in parallel beats any planned economy (by humans)

https://openai.com/index/model-disproves-discrete-geometry-c...

Marketing for huge bucks sounds like this.

You will own nothing and will be happy (that you are still alive). Probably.

The interesting part of this: while some leading implementations use the same LLM and context for the evaluator, some call out to a different context, some to a tuned LLM and different context; so which is better? many blog-scale benchmarks are calling it a toss-up that is highly dependent on the primary model.

(Of course LLMs can help there, get it right etc, just a caveat that people have to keep in mind)

I would love any tips for other folks who have successfully used similar approaches.

LLM: This package hasn't made it to production.

ME: are you sure? i see it right here!

LLM: You're right to push back. I inferred that based on weak data. I see now that the package has been deployed!

If the above conversation is typical for me, how could one expect to achieve a sound result by repeatedly prompting an LLM to simply "keep going" in dense mathematical proofs? Perhaps the user in this case had actually checked the LLM's work before issuing the prompt, but I think you see my point anyway.

I'll have a third for you soon, here's the obligatory result in a tweet. A detailed post about it is in the works.

For instance, would it be affordable for a research lab to not rely on OpenAI?

[1]: https://en.wikipedia.org/wiki/Dinitz_theorem

"Worked for 88m 24s... >"

"<h1>Complete finite counterexample</h1>"

...

Hearing "here's what I've done, here's the completely unambiguous next steps, I'll wait for you to send a pointless message before I continue" over and over again is a real pain.

> You should do a breakthrough

This is just as funny and ridiculous as those "make no mistake" prompts.

I am talking about someone jokingly asking AI `HOW TO ACHIEVE COLD FUSION` (or `A UNIVERSAL CANCER VACCINE`, or `AN AI FRAMEWORK SUPERIOR TO THE TRANSFORMER`), and getting a usable answer.

Eventually there will be an AI that will be able solve those sorts of questions as simply stated, like "cure all human diseases. also, make no mistakes!".

Don't get me wrong, the way we collectively treat animals is evil, but the idea that somehow we know enough about biology even today to reliably simulate drug behaviors seems unlikely.

> Construct a counterexample to the Collatz conjecture. You should do a breakthrough and find a structured counterexample.

without someone independently verifying it, it just dangles there

...

I started using "[leave-open" for those.

It lasted for a couple of years, until someone went through and "fixed" them all.

>What a world we live in.

Not sure, it sounds pretty boring to me...

1. The counter example wasn't just a brute force selection, the polynomial is structured in a very specific way that ends up getting the result.

2. Terry Tao's questions are very specific and prompts the AI in a useful way, that without high math training you are not going to get the same information out of it. Terry seems to see some aspects of the problem and counter example and uses AI to brute force some parts of it.

"Von Neumann would carry on a conversation with my 3-year-old son, and the two of them would talk as equals, and I sometimes wondered if he used the same principle when he talked to the rest of us."

>I sometimes wondered if he used the same principle when he talked to the rest of us.

I can come pretty close to a humbler example of this, as the more-extroverted "bad twin," of an identical set: my own genetic equal is bored to tears interacting with 90% of daily interactions... we both started with good brains/IQ, then went to the same college; but while twin spent the next decades solving EE problems (and co-founding startups), I "kicked back" and smoked myself sillyretarded, electricianing, relying upon bullshit and sheer luck to get to where I [barely maintain] am.

Watching my twin deal with any normal-intelligence persons is such an unpredictable shitshow, but I pretty much always know he's "dealing with us toddlers" who mostly never grew up. His patience is limited but willing.

Glad he can get by among us mere mortals. I have other similarly-minded brothers, but only this one identical twin.

I think lots of people have heard of him. I agree, that he is probably not as famous as some of the other scientists of the era though.

huh? He's pretty famous to anyone who has studied either the history of computing or of the development of the atomic bomb.

I mean that.

But I think your comment drives at some authentic take on this. Skill with AI is not only crafting iterative prompts the agent will understand, but also very high domain-specific knowledge of what the prompts explore.

One without the other can result in frustration or worse.

It is perplexing how many developers lack this basic skill, some of them borderline lack theory of mind and are incapable to understand that other people can't see the unspoken part in their heads.

One of my first jobs out of undergrad was technical writing on EDA tools for Mentor Graphics Calibre product. For many years, I did not appreciate that experience--it seemed orthogonal to what I'd gone to school.

Once I did become a professional SWE, something that took me too long to realize was that writing documentation and tutorials for people and looking after the quality of onboarding materials was almost never rewarded.

I came to see it more of a liability with ~all emphasis being on human code gen.

I listened to an interview with Steve Yegge earlier this year, one of his concerns was that many people just don't type very fast or read very quickly. IIRC, he expressed concern this would continue to be a big barrier to successful use of AI.

I previously commented on annotation for coding agents that an important skill was knowing the beat of a conversation.

I think that extends into critical reading. For example, it is not that unusual for a Washington Post article to put some ~explosive detail deep in the article--this is called "burying the lede."

I see this regularly reading AI output. That is the AI will lead with the strongest claim it can state cleanly, but not the most explosive implication.

With Claude Code and Codex, you're looking essentially neverending walls of text dense with technical information.

So now you have to have high domain expertise, a good understanding of written communication, a sense for the beat of a conversation (which assists in identifying understatements or overstatements), and the ability to dense text outputs quickly and then highlight critical statements for further exploration.

I'd almost describe part of this work as that of a skilled interviewer.

Turns out that understanding the technical domain is important to getting good results, even if the LLM is more capable at producing output. -- It's still a case of "garbage in, garbage out".

I think for many newbies, they see the LLM is so good at programming and so figure they don't need to learn anything. Apparently not so.

*GIT GUD* (at prompting)

Until they are embedded into automatons that can interact with the world.

(To be clear, I'm both on board with this, and I think it's the natural progression. Currently they are bounded by their training set, but interaction with the world is the imperative for theory -> test -> analysis -> update that is essential for growth and creativity.)

Yes and no. They can extrapolate and build upon the training data, as was the case with the last dozens of math proofs

LLMs giving you something novel would be like if you let a model play chess against itself and become the best player in the world this way. Totally impossible.

/sacrasm

To put it differently, if you have some idealized model in front of you that can do anything a team of humans can do, what do you say to it? It's still just a model that takes an input and provides an output.

Someone will still need to ask for something (or maybe it will just run autonomously making things it thinks we'll like) but it won't require any expertise in the field.

For things like sickness, old age and death, generally the lacks are not knowledge but the consistent will of the group. Like global warming, there is no real profound knowledge gap except how to get everyone to agree to mitigating the carbon burn. Perhaps it is likely LLMs would be able to persuade the group to stop burning the carbon before it’s [more] uncomfortable, but the small group of people committed to accelerating the carbon burn can also pay for LLM persuasiveness.

Just awesome to see new knowledge hit an incredible mind like this. Having these "what if" discussions is what I miss most from JPL and academia.

This "symbiosis" (for lack of better word) of human with AI seems to be an emergent value proposition of AI. In the process of doing stuff with AI, producing artefacts like code diffs, we are continuously able to decide how strong the mental map is of the current stage of the production process.

I could probably have worded this better but I'm sure it's something others have noticed... this choice we are able to make of how high fidelity our own understanding needs to be of the current working problem, and how that choice never really existed prior to AI.

Sorry it's a bit of an aside, but I imagine many other otherwise "technical" folks feel the same unfamiliar sense of total loss like when encountering hard mathematics.

Cache, stack, heap, process, thread, socket, file, tcp, http, tls, websocks, socks, soc2???, deadlock, stack, queue, race, atomic, event loop, coroutine, async, database, transaction, index, replication, sharding, consistency, serialization, DNS, load balancer, container, namespace, and so on.

Every sub fields (web/kernel/backend/etc.) has a million/bazillion weird words used in a dozen different contexts and if you read a paragraph of even semi technical software text you will feel like an over stuffed turkey.

Even cache could mean the CPU caches, the page cache, a browser cache, a CDN cache, a Redis cache, or imagine the flurry of words we have that have real world meaning. Session, handle, pool, buffer, stream, channel, event, task, worker, or queue. Generally there is some overlapping meaning but often there isn't.

compare to

>https://en.wikipedia.org/wiki/Rees_algebra

Most people, especially non-tech technical people, could crash through the TCP article and come out the other side with at least a high level understanding of it.

Most people, even technical ones, could not even get through the first line of the rees article, heck the first statement of the article. And then if they try, they need to know about algebraic rings. And digging into rings becomes totally intractable. None of the words or symbols in any of the articles track to anything even many technical people can grab onto. And this pattern is all over the place in mathematics.

It's not about mastering the difficulty of a topic or it's relative depth, it's about how abstract and removed from anything tangible it is. Anything with math it is always seemingly impossible to get a foothold on the idea anywhere within 10 degrees of explanation. Hell you cannot even clearly understand the problem that is being solved, or anything within 10 degrees of that.

For instance, most people are familiar with polynomials. Take a polynomial in x (with integer coefficients) and substitute x for 5t everywhere. So for instance, 2x^3 - 8x + 3 becomes 250t^3 - 40t + 3.

The Rees algebra Z[5t] (here Z is the integers) is then just the set of all polynomials you can get this way.

If Wikipedia introduced it like this, I don't think most people would have a problem understanding.

The concept is not (at least not always) actually that complicated, like others are implying. It's our communication that is lacking.

I sometimes think it would be good to rename some of the things that have historical names to mnemonics more descriptive than a proper name. But that would be difficult.

The truth is that math is hard: few other subjects have anything close to its crazy conceptual breadth and depth, with hardish concepts being built upon hardish concepts in many layers.

For example, Fourier is more clear to me as: F{t -> sin(t)} = omega -> (...) instead of F(sin(t)) = (...). Or D{x -> x^2}(x) = 2x instead of (x^2)' = 2x

Abuse of notation is very common, like using f(x) both as the function and as the return value at some input x etc. For example, the chain rule is often notated in a way that hides a lot. Math uses so many single letter variables, uses huge formulas instead of factoring out parts and using multiple lines, math people don't seem to appreciate namespaces and dislike nested variable scoping etc.

It's not magic that makes everything super easy, but it helps.

I can tease apart the Rees Algebra article one bit of half remembered terminology at a time and come out of it feeling like I just barely understand what the topic even is.

I can read the TCP article and feel like I have a thorough overview of the topic and could explain it at a high level to someone else.

To understand the definition of the Rees algebra, you would need to define, mostly in order: sets, groups, abelian groups, rings, ideals of rings, algebras over rings, direct sums of rings, adjoining things to rings, etc.

This is just to understand the definition; to understand its significance in algebraic geometry (which I have no idea of), there are a thousand more definitions.

The issue with trying to understand a concept in math is there is a massive directed acyclic graph of prerequisites leading to these concepts, and one needs to traverse this graph in the right order. Unfortunately, knowing the right order is almost tantamount to understanding the concept itself.

Careful, I think you might be committing an https://xkcd.com/2501/ error.

What even is a protocol? What is a host? What is a ‘stream of octets’? Wiki helpfully tells you octets are also known as ‘bytes’.

Nothing in the ChatGPT conversation is tangible. It's all in the realm of concepts.

Now days of course the chips are small so you have to point to where the multiple gigabyte chips are at.

But they are still quite physically.

Heck a C pointer points to an actual physical location on your machine, if you ignore the MMU.

I’m not sure this gets us any closer to a jargon-free explanation of what TCP does.

Such a small sentence and yet it means very little to me. I understand some constituent pieces, but I don't understand what Z is here other than a 'ring' and I don't really grasp how t^-1 converts this into a generalized family of algebra. It would take me a lot of effort to understand this and use it practically. I find that fascinating because it really is such a small statement that seems perfectly cromulent, but there's a lot packed in there that someone like me is totally missing.

I suppose there may be similar concepts in computer science, but nothing comes to mind that ever stumped me. To be frank, the field has been relatively accessible to me because it hasn't been too challenging. Not sure if that's a personal aptitude thing or it is genuinely simpler.

An algebra over a ring (call it S so we don’t confuse it with R from the previous paragraph) is a like a vector space over S, with the added structure that you can multiply elements of the algebra together (vector spaces only let you add their elements together). So for example the collection of even integers 2Z is an algebra over the ring of all integers Z. The collection of all polynomials with integer coefficients, Z[t], is another algebra over Z.

This is a great example of how dense language gets in math. There are tons of concepts hiding in the unstated background. Many are quite simple to explain individually, but there are so many of them that an outsider won’t know where to start to tease them apart. There’s a good reason to do it this way though; it would take a very long time to say anything in math without ever increasing levels of information density.

His point is the terms are dense too

As for your specific questions, I believe Wikipedia does a great job of answering two of them for a layperson:

https://en.wikipedia.org/wiki/Ring_(mathematics)

https://en.wikipedia.org/wiki/Vector_space

For the others, I’ll say that a formal variable is just a symbol (literally, like the letter t). With such a symbol, we can construct polynomials like 2t^2 - t + 3. Also, there’s no need to only use integers as the allowed coefficients; you can use any ring you like instead.

An “algebra over the ring R” is what I was attempting to define in my comment above. The algebra is “over” R if we can multiply an element of the algebra by an element of R. The useful analogy here is scalar multiplication in a vector space: you can multiply a vector by 2 to double it or -1/2 to reflect and shorten it. More generally, it makes perfect sense to consider some more general version of vectors which can be scalar multiplied by elements of any ring R.

Not like a drink with jam and bread.

I'm glad you answered them.

It finally makes sense to me, and now I realize I didn't even understand "over" in that context. That Ring wiki page though, um, nope... :D

Fair enough! At a super high level, a ring is just a collection that has a similar structure to what you’re used to “numbers” having. That is, you can add, subtract, and multiply them. Not divide! If we restrict ourselves to just whole numbers then 2/3 is not allowed. We also require that something like 0 and 1 have to be there. “Like zero” means 0 + x = x for every x in your collection, and “like one” means 1x = x for every x. And lastly, we require that the distributive property holds.

Examples include the set of whole numbers (Z), the rationals aka fractions (Q), the reals (R), complex numbers (C). These are all infinite rings, but there are also finite rings such as the set of whole numbers modulo a fixed number n, denoted Z/nZ. For instance, Z/2Z has only two elements, namely 0 and 1, with rules like 1 + 1 = 0. There are also polynomial rings, like Z[t], whose elements are all polynomials with integer coefficients (e.g. 3t^3 - t - 2). You can add, subtract, and multiply such polynomials and the result is more polynomials, so this collection is indeed a ring.

I studied computer science (and mathematics) in Germany. I am very certain that this was taught to computer science students, even though (compared to the lectures for math students) the lecturer did not get very deep into these topics.

> most CS students would have been terrified of that.

This is a feature, not a bug. :-)

Seriously: In Germany, the "math for ..." lectures often are intended to be "weed-out lectures" so that students who simply are not qualified for their major get to quit their degree course fast (either by realizing that the degree course is too hard for them, or by (typically) failing math exams so that they get exmatriculated), so that they don't waste many semesters on a degree course which they simply are not suited for.

You clearly write from the perspective of the US-American university system.

In Germany, basically everybody can enroll into a computer science program at a university, assuming the person has a Abitur (Allgemeine Hochschulreife) certificate (these terms are difficult to translate into English) from the grammar school [1]. So, "have it made to college" is like "not having been a complete failure in school". [2]

So, making it to the university is no achievement in Germany, and also no sign of motivation either.

> If these courses are so important, they should be taught in a way that students can understand.

These courses are taught in a way that students can understand, but not in a way where you can afford to slack off.

It is basically a consensus in Germany that a university is clearly a wrong place for you if you are incapable of closing knowledge gaps on your own (for example by reading books from the library), and you don't have the self-motivation to sit over the lecture material for hours to finally understand it.

So yes, I would say that among the possible options, weed-out courses in mathematics are in my opinion likely the least bad one.

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[1] In years where there was an insane demand for places at the university to study computer science such as during the dot-com bubble, there were some restrictions (numerus clausus), but for computer science, this was always the exception to the rule.

[2] There exist good reasons for puns like "Abitur: nichts gerafft und doch geschafft" (Abitur: Didn't get a thing, yet still passed) or "A-bier-tur" (a portmenteau of "Abitur" and "beer", which suggests that even pupils who are more into drinking than learning typically get their Abitur certificate).

For example in search algorithms where you want to search a space without visiting state nodes twice. Each state in the search space is produced by the sequence (a product of) of operators from the start state: elements of a monoid (or group if actions are invertible) which define the primitive steps. Trivial example being generating all permutations of a list. More interesting, enumerate all graphs with some property with pathwidth at most k, by adding one edge or vertex at a time. So now you want to know the structure of this group so you know which sequences of elements simplify and don't need to be tried, and you want to canonicalise each state to throw out duplicates.

And you can think in terms of orbits: if there are some symmetries then you might want to factor by the symmetry group and only visit one node in each orbit, grouping states into orbits with a single representative state. See eg. Pochter, Zohar and Rosenschein, Exploiting Problem Symmetries in State-Based Planners.

> https://en.wikipedia.org/w/index.php?title=Algebra_over_a_fi...

> https://en.wikipedia.org/w/index.php?title=Algebra_over_a_fi...

> https://en.wikipedia.org/w/index.php?title=Associative_algeb...

The latter is what aground asked for in https://news.ycombinator.com/user?id=agrounds