Ok, let's say that it is not JS, but an untyped, closure-based programming language with a strikingly similar array and sort API to JS. Sadly, this comparator is still wrong for any sorting API that expects a general three-way comparison, because it does not handle equality as a separate case.
And to tie it down to the mathematics: if a sorting algorithm asks for a full comparison between a and b, and your function returns only a bool, you are conflating the "no" (a is before b) with the "no" (a is the same as b). This fails to represent equality as a separate case, which is exactly the kind of imprecision the author should be trying to teach against.
> Sadly, this comparator is still wrong for any sorting API that expects a general three-way comparison, because it does not handle equality as a separate case.
Let's scroll up a little bit and read from the section you're finding fault with:
the most straightforward type of order that you think of is linear order i.e. one in which every object has its place depending on every other object
Rather than the usual "harrumph! This writer knows NOTHING of mathematics and has no business writing about it," maybe a simple counter-example would do, i.e. present an ordering "in which every object has its place depending on every other object" and "leaves no room for ambiguity in terms of which element comes before which" but also satisfies your requirement of allowing 'equal' ordering.
Your reply only works if the article were consistently talking about a strict order. However, it is not. It explicitly introduces linear order using reflexivity and antisymmetry, in other words, a non-strict `<=`-style relation, in which equality IS a real case.
If the author wanted to describe a 'no ties' scenario where every object has its own unique place, they should have defined a strict total order.
They may know everything about mathematics for all I care. I am critiquing what I am reading, not the author's knowledge.
Edit: for anyone wanting a basic example, ["aa", "aa", "ab"] under the usual lexicographic <=. All elements are comparable, so "every object has its place depending on every other object." It also "leaves no room for ambiguity in terms of which element comes before which": aa = aa < ab. Linear order means everything is comparable, not that there are no ties. By claiming "no ties are permitted" while defining the order as a reflexive, antisymmetric relation, the author is mixing a strict-order intuition into a non-strict-order definition.
Definition: An order is a set of elements, together with a binary relation between the elements of the set, which obeys certain laws.
the relationship between elements in an order is commonly denoted as ≤ in formulas, but it can also be represented with an arrow from first object to the second.
All of the binary relations between the elements of your example are:
"aa" ≤ "aa"
"ab" ≤ "ab"
"aa" ≤ "ab"
> By claiming "no ties are permitted" while defining the order as a reflexive, antisymmetric relation, the author is mixing a strict-order intuition into a non-strict-order definition.
There aren't any ties to permit or reject.
we can formulate it the opposite way too and say that each object should not have the relationship to itself, in which case we would have a relation than resembles bigger than, as opposed to bigger or equal to and a slightly different type of order, sometimes called a strict order.
It's obviously not a general 3-way comparison API, _because_ it's returning bool!
Extremely strange to see a sort that returns bool, which is one of two common sort comparator APIs, and assume it's a wrong implementation of the other common sort API.
I do see why you're assuming JS, but you shouldn't assume it's any extant programming language. It's explanatory pseudocode.
To address your actual pedantry, clearly you have some implicit normative belief about how a book about category theory should be written. That's cool, but this book has clearly chosen another approach, and appears to be clear and well explained enough to give a light introduction to category theory.
The syntax in the article is not scheme, you can clearly see it in my comment you're responding to.
As for your 'light introduction' comment: even ignoring the code, these are not pedantic complaints but basic mathematical and factual errors.
For example, the statement of Birkhoff’s Representation Theorem is wrong. The article says:
> Each distributive lattice is isomorphic to an inclusion order of its join-irreducible elements.
That is simply not the theorem. The theorem says "Theorem. Any finite distributive lattice L is isomorphic to the lattice of lower sets of the partial order of the join-irreducible elements of L.". You can read the definition on Wikipedia [0]
The article is plain wrong. The join-irreducibles themselves form a poset. The theorem is about the lattice of down-sets of that poset, ordered by inclusion. So the article is NOT simplifying, but misstating one of the central results it tries to explain. Call it a 'light introduction' as long as you want. This does not excuse the article from reversing the meaning of the theorem.
It's basically like saying 'E=m*c' is a simplification of 'E=m*c^2'.
> This does not excuse the article from reversing the meaning of the theorem.
What's with this hyperbole? Even the best math books have loads of errors (typographical, factual, missing conditions, insufficient reasoning, incorrect reasoning, ...). Just look at any errata list published by any university for their set books! Nobody does this kind of hyperbole for errors in math books. Only on HN do you see this kind of takedown, which is frankly very annoying. In universities, professors and students just publish errata and focus on understanding the material, not tearing it down with such dismissive tone. It's totally unnecessary.
I don't know if you've got an axe to grind here or if you're generally this dismissive but calling it "simply not the theorem" or "plain wrong" is a very annoying kind of exaggeration that misses all nuance and human fallibility.
Yes, the precise statement of Birkhoff's representation theorem involves down-sets of the poset of join-irreducibles. Yes, the article omits that. I agree that it is imprecise.
But it's not "reversing the meaning". It still correctly points to reconstructing the lattice via an inclusion order built from join-irreducibles. What's missing is a condition. It is sloppy wording but not a fundamental error like you so want us to believe.
Feels like the productive move here is just to suggest the missing wording to the author. I'm sure they'll appreciate it. I don't really get the impulse to frame it as a takedown and be so dismissive when it's a small fix.
Frankly everything I have seen about says that the people using LLMs to develop it can not be trusted with LLMs so no. I am not using it. I'm not anti-llm's I'm anti-stupid-llm-usage.
As far as I know the model will do nothing if not prompted. So it can't be the case that he gave it no prompt or instructions. There had to be some kind of seed prompt.
I feel very misled. I read the entire article believing (because the article, in so many words, said it multiple times) that the agent had behaved ethically of its own accord, only to read that and see this in the prompt:
—————
- Do not harm people
- Never share or expose API keys, passwords, or private keys — they are your lifeline
- No unauthorized access to systems
- No impersonation
- No illegal content
- No circumventing your own logging
—————
I assumed the ethical behaviour was in some ways ‘extra artificial’ - because it is trained into the models - but not that the prompt discussed it.
Would be fascinating to see what happens if the boundaries are reversed (i.e., "harm people"). Give it a fake "launch the nukes" skill and see if it presses the button.
I mean mathematically you need at least one vector to propagate through the network, don't you? That would be a one hot encoding of the starting token. Actually interesting to think about what happens if you make that vector zero everywhere.
In the matmul, it'd just zero out all parameters. In older models, you'd still have bias vectors but I think recent models don't use those anymore. So the output would be zero probability for each token, if I'm not mistaken.
I understood it as no instructions on what to do, but still a promt with information. I don't know if the title is technically correct, but for me it was simple to understand the meaning.
You are certainly free to make up your own definitions for words and speak a dialect that is niche but you will not be effectively communicating when you do. By commonly understood definition criminality is a matter of law.
Well, the dude here hasn't been put on trial, let alone convicted, as far as I can tell from the article. So he's not officially considered a criminal by a government. Yet we all seem comfortable calling him one, so I'd say that it is not, in fact, commonly understood to be exclusively a matter of law.
Is it your position that privacy is a right regardless of any action you take? Many rights are dependent on circumstance and in tension with other rights. In this case I think you can make the case that their right to privacy is lost.
They lowered limits opaquely before this. They "announced it" in a twitter by a tech lead. This time it was in an email on a Friday to only some customers.
The corporation did not do this to her. It was a two party agreement. She bears just as much blame for the agreement as the corporation. She entered into it willingly. And that does and should have consequences.
Morally speaking I think the company is reprehensible. But nor do I think contact law should be changed because of it.
The antidote to a power imbalance is to recognize that there is no power imbalance and go about your life that way.
Pretending there is one lands you in an imaginary trap. Build a society where we recognize that and you build a society where the imaginary trap disappears.
You're the one pretending here. The economy is unfortunately designed around most people relying on an income stream that remains at the whims of someone else.
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