[JoshCole]

> One abstraction that is very useful is linear algebra which is an abstraction without errors.

Bullshit. Your claim that there is no abstraction error in linear algebra (when run on computers - we're in subdiscussion related to programming) is false.

Computers can't represent all numbers [1]. They can only represent the computable numbers in theory and even then only a subset of computable numbers can actually be computed. Therefore, there is abstraction error in linear algebra on computers. This isn't at all a theoretical thing. This regularly has implication on algorithm design. You ought to have known this, because the truth about floating point numbers has been well publicized [2].

> Same goes for category theory. Grothendiecks work wasn't about tolerating errors in abstractions either. Simple abstractions like generic containers are also not about ignoring errors.

You're just ignoring that these things when implemented in computers actually do have error, because you find it convenient. You're also apparently not self-aware enough to realize that this agrees with my central premise. Think about your thinking and you'll notice that this failure is indicative of your own minds belief in wrong abstractions being right - otherwise you wouldn't have been able to make this error.

> Honestly, the random segues to diagonalization arguments, game theory, continuous functions, RL and Bellman equations(WTF!) sound like stream of conciousness random ramblings and an attempt at "out jargoning", much less a "formal proof". Reminds me of this story by Tadelis.

We're talking about learning as it relates to abstraction. If you can't see why a learning problem formulation is relevant to discussion about whether learning an abstraction is appropriate that says a lot more about your reasoning than it does my articulation.

As a reminder the OP said:

> Teaching people to think about abstractions wrong is not in any way helpful. Everyone I know including myself had to spend years writing bad code before realizing that this way of thinking is counterproductive.

I've disagreed with the claim that teaching a bad abstraction that has to be unlearned is bad. I've shown that in learning problems, teaching a bad abstraction can actually be good. I've given proofs to the effect that there are times in which teaching a bad abstraction produces algorithms which terminate in situations where teaching a perfect abstraction doesn't terminate. I've given you links to mathematics showing that given a bad abstraction as a starting point you can produce agents which outcompete something which doesn't use an abstraction. The paper in question takes a bad abstraction and improves upon it after getting it during the challenge of a more specific problem.

> "We use Lagrange multipliers," one of them said.

Hacker News guidelines call for more thoughtful points, not less thoughtful points, as threads get deeper. It also calls for assumption of good faith. It calls for increasing nuance [3]. What you are doing here - it isn't that. At the risk of starting something painfully obvious, talking about random shit people that aren't me said that you found dumb isn't actually relevant to the discussion. This is a subthread that is on the topic of learning and abstraction. I'm talking about both topics. You aren't.

> random ramblings

I'm sharing something deeply counterintuitive but true because I think people might find it interesting. Right algorithm + correct data can be worse than right algorithm + wrong data. This is deeply counterintuitive and I find it very fascinating, but it falls out of the formal definitions of utility under multiple different learning frameworks. Something isn't bad because it has abstractions with error in it. Teaching an abstraction with error in it isn't bad either. It isn't even bad when you can find the error - because you're in a more specific situation the learning problem changes, in the general case it was too hard to compute the unabstracted best thing to do, but you can reduce the error in your abstraction when you get more specific because there are less states and so the learning problem becomes more tractable.

[1]: https://www.cs.virginia.edu/~robins/Turing_Paper_1936.pdf

[2]: https://floating-point-gui.de/

A person might try to respond to this point by claiming arbitrary precision numbers exist. That person is lying to themselves. They don't exist. The abstraction error is just in a different place. Go back to the definition of Turing Machines and observe again that not all numbers are computable - we are in an abstraction subject to error and you can't escape this while staying in the Turing framework [1].

[3]: https://news.ycombinator.com/newsguidelines.html

[random314]

> Bullshit. Your claim that there is no abstraction error in linear algebra (when run on computers - we're in subdiscussion related to programming) is false.

The fact that you had to change my statement to add computers to it already makes it clear that you know my statement is correct. Don't put words into my mouth to prove statements that I have never made false. You also failed to realize that the tweaked statement you put in to my mouth is still correct!

You fail to realise that we can prove theorems about linear algebra using computers without actually using floating point numbers. Linear algebra can be infinite dimensional and not just be defined on the complex field, and this doesn't introduce any errors in to the abstraction or theorems being proved, with or without computers. Your assumption that linear algebra on computers is exclusively about floating point number crunching tells me that you don't understand what abstraction linear algebra represents. Linear algebra is the study of linear maps. This is a good book for you to get started [1]

You also failed to address the gazillions of abstractions that don't have errors, some of which I have listed along with linear algebra. For those familiar with proofs, Just one counter example can prove your sweeping statement "all abstractions have errors" false. We are discussing math, not physics. And if you want to restrict yourself to computable functions- modulo arithmetic with groups, rings and fields suffices as a counter example.

It isn't even clear what you mean by abstraction at all.

> We're talking about learning as it relates to abstraction.

The OP was not discussing learning at all. Your segue into machine learning concepts is completely unrelated to the topic being discussed. So is game theory. I am familiar with virtually all the topics you are discussing, so you can skip the citations. I find no coherence to any of your segues.

> You're just ignoring that these things when implemented in computers actually do have error, because you find it convenient.

You are completely out of touch [2][3]

[1] https://linear.axler.net/LinearAbridged.pdf

[2] https://en.wikipedia.org/wiki/Univalent_foundations

[3] Mathematician Kevin Buzzard https://www.microsoft.com/en-us/research/video/the-future-of...

[JoshCole]

> I am familiar with virtually all the topics you are discussing, so you can skip the citations.

I really can't, because other people might believe you if I don't; I don't hate them. I want them to know the truth. So I'll oppose you strongly, for their sake, so they can discriminate between my counterintuitive truth and your rejection of the truth on that basis of confusion.

Go read page 173 of Artificial Intelligence: A Modern Approach. I'll quote Norvig here. "Because calculating optimal decisions in complex games is intractable, all algorithms must make some assumptions and approximations." Now go to page 172. I'll quote Norvig again. "One way to deal with this huge number is with abstraction: i.e. by treating similar hands as identical. For example, it is very important which aces and kings are in a hand, but whether hand has a 4 or a 5 is not as important, and can be abstracted away."

But, the discerning might ask, what of the talk of the infinite? Why does Josh speak of such an absurd thing? Isn't it irrelevant? It is not. Go to page 611, "Non-Cooperative Game Theory". I'll quote him again for you, "With this observation in mind, the minimax trees can be thought of as having infinitely many mixed strategies the first player can choose." The thing to notice in this quote is that we have a simple game - very simple. Yet Norvig just explained that in this simple game we have the quality of a tree of infinite size. This growth to infinite is actually very normal - mixed strategies are continuous and we have proofs that mixed strategies are the solution for a variety of different games involving imperfect information.

[JoshCole]

> It isn't even clear what you mean by abstraction at all.

I'm using abstraction in the sense of 'blueprint abstraction' from game theory. This is basically compression of the input to your learning algorithm. There exists lossless compression - perfect one to one abstraction. There is also lossy compression - given one compressed form it could be any number of uncompressed forms. Abstraction with error is then compression with error. What I was trying to prove and what I still believe to have proved is that some algorithms when given an input of unbounded size have the property of not terminating. What I then tried to show was that abstraction breaks the proof of non-termination, because it breaks the core assumption of diagnolization - that there is a one to one mapping. So the proof of non-termination doesn't hold.

> I am familiar with virtually all the topics you are discussing, so you can skip the citations.

I literally linked to a paper that used abstractions in the way I meant it. So maybe you should not skip citations? Clearly you don't know the fields as well as you think you do.

> I find no coherence to any of your segues.

It isn't a segue; it is what you asked for. I gave you a proof that unabstracted learning problems can terminate where abstracted problems terminate.

> You are completely out of touch [2][3]

This is such an ironic statement; here we were discussing what abstraction techniques we should teach when teaching computer programming and you're trying to complain that I'm the one who is out of touch when I say computers use abstractions. They definitely do. In fact, they use abstractions with error.

[JoshCole]

> You fail to realise that we can prove theorems about linear algebra

Obviously I realize we can prove theorems. If I didn't realize it was possible to prove things I wouldn't claim things were provable. This is false by contradiction with my previous statements. Your worldview of me isn't consistent.

> You also failed to address the gazillions of abstractions that don't have errors, some of which I have listed along with linear algebra.

Why should I have to prove that abstractions without error don't exist? They do. I never claimed they didn't. My point was that abstractions with error reduce computational complexity. This gives them room to outcompete perfect abstraction for sufficiently complex problems. It honestly seems insane to me to not believe what I'm saying is true, because it is true. I can't fathom how the concept would be impossible to grasp. Showing that we regularly use error filled abstractions is more important than demonstrating something I don't intend to show.

> The OP was not discussing learning at all.

I literally quoted him saying there was never a reason to teach bad abstractions. Teaching is related to learning. Abstractions are related to abstractions. So abstraction and learning were a topic of discussion. Even the original post we're under is about teaching programming technique. Which is about learning, because of the relationship between teaching and learning. It is also about abstraction, because problem modeling is very related to abstraction.

> "all abstractions have errors"

Ctrl + F shows no instance of this except for you saying it. What do you think I'm claiming? I'm really confused? You seem to think I'm saying something I'm definitely not saying.

Notice how when I said it I put parantheses and explained my reasoning as to why I felt your claim was your claim? You meanwhile misquote me. Strict quotes imply actual attribution, but I never said what you claimed I said.

I don't want to talk to you anymore. I very much don't appreciate your comparison to someone who just makes stuff up. I found that very rude and insulting. A kind person would help correct me if my reasoning was wrong and I would appreciate it, but you haven't done that. When you quoted me, it wasn't even something I said or tried to argue. If you were trying to make someone have a worse day, congratulations, you did.

[random314]

>>>> One abstraction that is very useful is linear algebra which is an abstraction without errors.

>>> Bullshit. Your claim that there is no abstraction error in linear algebra (when run on computers - we're in subdiscussion related to programming) is false

>> You fail to realise that we can prove theorems about linear algebra USING COMPUTERS

> Obviously I realize we can prove theorems. If I didn't realize it was possible to prove things

You have been arguing in bad faith by either putting words in my mouth (when run on computers) or deleting key phrases from my reply (USING COMPUTERS - reinserted by me)

These 2 statements by you contradict each other

"Bullshit, Your claim that there is no abstraction error in linear algebra (when run on computers - we're in subdiscussion related to programming) is false "

" Obviously I realize we can prove theorems" -----> USING COMPUTERS

It is clear that you thought of computers as IEEE floating point number crunching machines, with implicit floating point errors. All of your arguments rested on this irrelevant point. You even condescended to teach me about floating points using citations not needed by anyone who has attended CS101.

You failed to realize that finite sized representations of computable real numbers exist, by definition[1]. One such representation could be a finite sum on surd basis, instead of using a binary basis eg sqrt(2) instead of 1.414.... and the most general form is a theorem prover like Lean.

All of these misunderstandings in your head because you failed to realize the gist of the Church Turing thesis ' - Everything that you do with your brain and paper can be duplicated on a computer.

In any case, I am glad you learned something today. Computers don't relate to math via IEEE floating pointing numbers, the connection is a lot deeper[1]. You won't acknowledge this, but frankly speaking, I can't really stand out- jargoning pretending to be an honest discussion. I did give an opportunity to you to correct yourself with my first comment. But you only doubled down - more jargon, bad paraphrasing of diagonalization, putting words in my mouth, followed by removing key phrases from my replies amongst other bad faith arguments.

[1]

https://en.wikipedia.org/wiki/Church%E2%80%93Turing_thesis

To establish that a function is computable by Turing machine, it is usually considered sufficient to give an informal English description of how the function can be effectively computed, and then conclude "by the Church–Turing thesis" that the function is Turing computable

[JoshCole]

Serious question. Putting aside all the stuff that basically make me feel like you are just calling me a moron and wasting my time: what do you disagree about? You keep treating the things I say like they are irrelevant jargon and not bothering to engage. Can you please try for a moment to explain why you think my argument that error in abstraction is useful is wrong by addressing the actual premises that are within that argument. Literally, address one of these claims:

1. Do you disagree with my claim that the runtime of learning algorithms depends depends on the graph size in both game theory and reinforcement learning problem formulations?

2. Do you disagree with my claim that abstraction reduces the number of states in the graph?

3. Do you disagree with my claim that since abstraction reduces the number of states in the graph the learning algorithms which run against them can complete more quickly because there are less states?

4. Do you disagree with my claim that algorithms which can compute a solution can have a better solution than algorithms which don't compute the solution?

5. Or if you don't disagree, can you admit that we agree on these things? Because when you just act like I'm not making any points it makes me feel like you are trolling me and being a jerk, not actually trying to talk to me.

I'm still just as convinced of the truth of the idea that it can be very wise to accept a bad abstraction, one that has error, rather than a perfect abstraction. I can't even fathom how to go about the opposite. How would a child go from knowing nothing to knowing everything perfectly without moving through areas of bad abstraction along the way?

Which numbered point do you feel is incoherent?

[JoshCole]

Waiting for your rebuttal. Worth nothing the problem isn't theoretical. We actually run into this 'we can't compute it fast enough' problem in practice.

- When we tried to solve chess we couldn't, the branching factor was too much.

- Go, it was horrendous there too.

- Poker, terrible there too.

But you want to dismiss me on the basis of jargon right? So here you go. Bellman coined the term curse of dimensionality. Combinatorial explosions happen because of branching factors in game graphs. Computational complexity for algorithms are defined with respect to this graph in both time and space for many learning algorithms. Because the games get so big the curse of dimensionality forces problem relaxation. I used ~words~. I must be an idiot. Feel free to dismiss me, I guess. I heard you heard someone else use words once and they were ~wrong~.

Hey wait a second. You're using words too. Does that mean everything you say is wrong?

[JoshCole]

> Everything that you do with your brain and paper can be duplicated on a computer.

Can you stop pretending I'm talking about things that are computable when talking about things that don't terminate? When someone says that computation is only defined for the computable numbers responding with the claim that they don't understand "that finite sized representations of computable real numbers exist" is honestly either stupid or malicious.

[JoshCole]

I made the claim that problem simplification through an abstraction that has error can reduce computational complexity leading to improved solution quality. You responded by talking about people who are talking about Lagrange points. I found that extremely insulting. I still find you to be extremely insulting. I don't think my claims are so hard to understand. I find your intentional misinterpretation of my points annoying.

I despise that you lied about whether we were talking about learning and abstraction. I don't like talking with people who blatantly lie. I consider lying bad.

I also dislike that you misquoted me. You put quotes around words I didn't say. I didn't do that to you. You say that I did. You lie when you say that. I gave you my interpretation of what I felt you were claiming. I even explained why I felt you claimed that. This wasn't under the quote symbol. Yours was inside quotes. Yours was a lie. Mine had your original quote, unaltered, with my interpretation below it. I was in error. I admit that. I was trying to get at the heart of my point - that erroneous abstractions aren't inherently bad. Outcome error is much more important than input error.

I don't agree that you've taught me anything - you just try to call me incoherent because what I'm saying is true but you employ motivated reasoning to avoid having to refute it. If you actually understood what I'm saying - which obviously you don't, which is a big part of the problem here, you would agree with me. Or at least, I think you would.

That simple problems are easier to solve and sometimes an actual solution is better than no solution really isn't that complicated a thing. Or controversial. I'm sure plenty of people understand it.

There are so many times in life where my point holds. The use of floating point is one. Perhaps you didn't notice that ML engineers frequently choose to move from float64 to float32 to float16 to float8? Perhaps you didn't notice that services all throughout the computing industry choose to meet an SLA, minimizing latency sometimes at the cost of optimal solutions whose computation isn't realistic given their computing budget. I don't know. But you're definitely not actually teaching me anything. Your just not understanding me. So this conversation is pointless.

I'm still just as convinced of the truth of the idea that it can be very wise to accept a bad abstraction, one that has error, rather than a perfect abstraction. I can't even fathom how to go about the opposite. How would a child go from knowing nothing to knowing everything perfectly without moving through areas of bad abstraction along the way?

I feel you are mean. I'd rather we stop talking about this together if we're not going to actually engage with each other on the topic under discussion.

[JoshCole]

> The OP was not discussing learning at all

Yes they were. They were discussing teaching, which is related to learning.

Moreover, the person they were responding to - they also were talking about learning. They talked about how it was good to learn an abstraction that wasn't perfect.

Even the article is about whether we should teach one abstraction or another.

> You are completely out of touch

Gaslighting is abusive. Stop abusing me.

> I am familiar with virtually all the topics you are discussing, so you can skip the citations. I find no coherence to any of your segues.

This is an argument from authority. It is a fallacious argument. If my segue is incorrect, you need to show it from the structure of the argument, not by appealing to your authority, which is irrelevant; you might be great - I'm not saying your not.

You are great; I'm not saying you're not. I'm sure you're intelligent and smart and witty and cool. But who we are - it doesn't matter. We're irrelevant. The ideas are all that matters.

If I'm wrong - why isn't chess solved? Why do we approximate solutions? Meanwhile, why is checkers solved? Why do games with simpler graphs get solved perfectly but more complex games with more complex graphs not get solved perfectly? Please back up the ideas you would be advancing were it the case you actually disagreed with what I claim is provable. If you actually understood my point, than you should know that the opposite of my point isn't that perfect abstractions exist - it is that the run time of specific learning algorithms aren't correlated with their input size. Your trying to get me to defend conclusions I didn't make, treating me like I'm stupid and comparing me to people who are rambling. You are being abusive. Lying. Gaslighting. Attacking me as a person rather than my ideas. Appealing to authority.

I'm sorry that I thought you were referring to linear algebra when used as abstraction in the way I meant it - I'm talking about computational abstractions; stuff with many dimensions, compressed to be in fewer dimensions. That happens when we do linear algebra on computers in practice. I thought it was reasonable to point this out, because my claim is closer to "abstractions with errors in them can be useful" than it is to "abstractions without error don't exist". But I feel like you do understand me in this - because you seem desperate not to admit this. To pretend we can represent all things in finite space, when we can't, because the infinite things can't all be represented in finite space. And it seems to me the only reason you would be so desperate not to admit this - to try and throw up so much confusion about this idea - is that you do understand me. And you understand that if you surrender on that point, you admit I'm right. So I think you already know I'm right. And you're just being mean intentionally. I think I probably offended you. It best explains the inconsistency in your reasoning. So I'm sorry about that - and I assume it was sometime in the past, because my first replies didn't deserve your malice.