Yep, mega-impact is a classic example of an ad-hoc hypothesis. For example, Moon formation is much better explained by multi-impact hypothesis, which also requires less assumptions.
Dang, I can't post anything in this thread without someone thinking I agree with them that science is BS. Giant impacts aren't actually surprising in an early solar system that hasn't hit steady state yet. If multiple impacts better explain all the evidence for Theia, including the weird patterns of isotopes and possible fragments deep in the mantle, that's news to me.
Yep, asking questions directly in the notebook, then editing the answer and executing it would be very powerful. I think `jupyter-ai` is close to this, judging by description, but I didn't try it.
Category theory is actually a ‘simplified’ graph theory, i.e. you can see categories as a restricted class of graphs. E.G. ‘Category Theory for Computing Science’ introduces categories this way (a category is a directed graph with associative composition and identity; the free category on a graph is the graph with all identities and compositions filled in). But the restrictions (associative composition and identity) are harmless and natural for programming applications where there's always a notion of ‘do nothing’ or ‘do one thing after another’, and unlock a lot of higher structure.
It helps you build an intuition for categories, if you're used to graphs :)
If you have a working intuition for categories then in most cases the specific formulation you choose as a foundation doesn't matter, just as most mathematicians work nominally in set theory without worrying about the subtleties of ZFC.
IMO the right intuition about a tool comes from applying it in the context where it provides a real leverage. In case of Category Theory that would be advanced algebraic topology (not re-phrasing basic things which are easier to understand without CT).
Rephrasing things is useful when it allows you to draw equivalences with other things that might have results or insights that translate across, which is something a lot of people use CT for. As with most maths the payoffs aren't always immediate, though, which leads to a lot of frustration with from programmers who expect immediate results from their insights.
From a programming perspective, though, a fair few important things have (by now) turned out to be useful that have come as a result of rephrasing programming in mathematical terms. Some top contenders are monads (directly from CT — think Haskell, but also LINQ, JavaScript Thenables, async/await, Rust Option/Result combinators, parser combinators), proof assistants and modern approaches to type systems, linear/substructural logic (think Rust or C++ move semantics), functional programming (which is now embraced at least to some extent by all mainstream programming languages), functional reactive programming (the core ideas behind things like React), et cetera.
If you allowed infinite graphs maybe. How would you define a functor or natural transformation in graph theory? Seems like you would need to construct a conceptual system that is just equivalent to category theory
No, but if you want to talk about composing those arrows (and a sensible notion of composition should probably be associative, and perhaps have a unit) you eventually end up reinventing category theory.
I think your reply overstates the importance of category theory in mathematics and doesn't give any hint on what it is about.
IMO a better reply would be: category theory appeared to unify the concepts around using discrete objects to prove the properties of continous objects in topology, like fundamental groups, homology groups and homothopy groups. It is only practically useful for very advanced proofs like 2nd Weil Conjecture. Any usage of it in programming is only an analogy and is not mathematically rigorous (see https://math.andrej.com/2016/08/06/hask-is-not-a-category/)
Insofar as ‘computation’ is about mapping one state or value to another state or value, it has a lot to do with CT!
The question of whether CT is _useful_ for studying computation is different, and there are certainly other lenses you can see computation through that some people would argue are better. But it's hard to deny that they're _related_.
I mean, technically almost all of math can be related to other math one way or another. To say the CT has a lot to do with computation is definitely a stretch. CT is not a recognized Computer Science subject. It's mostly used in the functional programming community to name certain concepts and theorems, but then applied to a specific type system (so it's not actually doing CT, since your restricting yourself to a single category, whereas CT is really about connecting different categories by generalizing over them).
Good point, algebraic topology is mostly concerned with Top -> Grp functors to prove the properties of continuous transformation, while Haskel community focuses on Hask -> Hask endofunctors to use fancy names for mundane things.
And yet the categorical concepts in Hask are undoubtedly practically useful, more so than an arbitrary sample of concepts, and compose extraordinarily well. Does that have nothing to do with those concepts deriving from (even more general concepts of) category theory?
> To say the CT has a lot to do with computation is definitely a stretch.
I think the argument I presented above adequately justifies CT as fundamentally connected to computation, at least as we study it today, though I think there are other formalisms as well that are just as worthy.
> CT is not a recognized Computer Science subject
I'm not sure who decides this :) Certainly when I studied computer science we had modules on category theory, along with other applicable ‘discrete math’ subjects.
CS is a bit of a grab-bag of a discipline, so it's very hard to say what does or doesn't belong on a CS curriculum. But I think programming language semantics, which is the field where you see CT pop up the most, is probably uncontroversially CS, or at least I don't know what other discipline would claim it.
> It's mostly used in the functional programming community to name certain concepts and theorems
It's mostly used in the PL community; its use in functional programming is an attempt to connect the language more directly to that lineage (or because functional programming languages traditionally came from that community, where the language is already understood and available to describe things).
> but then applied to a specific type system (so it's not actually doing CT, since your restricting yourself to a single category, whereas CT is really about connecting different categories by generalizing over them)
I agree that there's not a lot of interesting category theory to do when restricted to a single category, but this isn't quite correct. Some type systems, e.g. simply typed lambda calculus, are adequately _described_ by a single category, though in more advanced cases you usually need more than one (e.g. dependently typed languages are often described by ‘categories with families’, which are a slightly more sophisticated category-theoretical construction). But even within a single type system you usually find many categories; for example a type system can often be described by a category of terms that map between its typing contexts, but within that category you also have a category of types and functions between them, a category of (functorial) type constructors and natural transformations, et cetera. An important family of examples are the Kleisli categories over monads: every monad actually defines a category of its own, whose categorical laws give the good behaviour of monads that make them useful for composing effectful computations. Interactions between these different categories are where inspiration comes from for CT-flavoured API design, e.g. the ‘interpreter’ pattern that's popular in Haskell (https://softwareengineering.stackexchange.com/questions/2427...). Even in pure CT the study of ‘connecting different categories’ can be seen mostly as taking a sufficiently close look at the category `Cat` of categories and functors!
I'm using Ergodox for 9 years and still can touch type fluently on laptop keyboard. However, I never do heavy programming tasks on the laptop, only messaging, notes and small fixes.
I think the more customized your Ergodox layout is the easier it is for the brain to treat 2 different keyboards as 2 different input devices.
