Mathematics is concerned with a lot more than arithmetic and computation. Beyond the most basic levels, a mathematician will profit greatly from being aware of this type of epistemological vocabulary and a strong sense of their underlying meaning. Whether reading or writing mathematics, we're constantly dealing with propositions, and correctly taxonomising those propositions can really help keep your mental workspace clean.
I do question the effectiveness (and accuracy) of this exercise, but its learning objectives I think are quite apt.
To be honest I do not think that these word games are helpful at all. Throughout all of my mathematical education what has always helped me to keep my "mental workspace clean", was to never abandon the model.
> and correctly taxonomising those propositions
The correct taxonomy for a proposition is true/false and proven/unproven.
I can not even fathom a mathematical model where distinguishing a "law" from a "fact" is meaningful.
And the idea of defining a "fact" as something empirically demonstrated is just ridiculous, I totally reject it.
I dislike the linked site. A lot. But counterpoint: Zermelo-Frankel with or without Axiom of Choice is a fair mathematical analogue to distinguishing laws and facts, in my opinion.
Put another way, decidability is a large area of mathematical research.
>Put another way, decidability is a large area of mathematical research.
What does ZF(C) have to do with decidability? Decidability is a question in any sufficiently complex system (Gödel's first theorem). And exactly this distinction is what I made for the taxonomy of propositions, you can group them into true and false and also into provable and unprovable. What would be a fact and what would be a law?
Regardless of that, in neither case the empiricism the site uses to define a fact would play any role.
I like the argument that every number in the row below is formed by summing two numbers from above. So each number above appears twice below. Hence the sum doubles.
Thicker walked ceramic mugs are definitely common in the UK, and probably more common now than anything else, but me and my gran both agree that a proper cup of tea needs serving in a thin-walled, fine-china cup, like this: https://fegghayespottery.co.uk/product/plain-270ml-bone-chin...
Fantastic. I used to set a competition in my school maths dept to compute as many digits of pi as possible in under one minute. Was always a highlight of the year, a kind of mathematical drag race.
Happy render customer right here.
We’ve got multiple rails apps running on render.
Renders tech support team have been very helpful wherever needed also.
Should also point out the recently released Rails 8, has as key features focussed on making rails much easier to deploy to anywhere that supports Docker.
I get where you're coming from with that last sentence - people do sometimes go against the grain because it's not cool to follow the crowd.
But I am one of those people who have felt a definite shift in their relationship to music because of streaming. I don't feel special and I don't want to hate Spotify - I still have an account and use it daily.
But I am less mindful. I don't build relationships with albums any more. I sometimes don't even know the name and artists of songs I listen to and enjoy dozens of times. I definitely feel like I have lost parts of my relationship to music (I've gained things, too, but I'm still not sure if I'm happy with the trade). It's a quiet loss that I've only started to appreciate in the previous year or two, and not yet salient enough to take the radical step of cancelling, but I really do get where the cancellers are coming from (even if there are some edgelords among them!).
Films exist without motive, plot, climax, or story. Some people love these films. If that’s the kind of story that you seek, then the teacher doesn’t need to lift a finger. Good for you.
For the rest of us, we need to want a hole before we learn how to use a drill. And that’s okay too.
OC's point is that the chain rule for partial derivatives shouldn't be assumed because the ordinary chain rule holds, there's more depth to it than that, and the proof is harder than you might instinctively expect based on the ordinary chain rule.
It's epistemically acceptable to understand these both as "the chain rule" once we're satisfied they've both been proved, and apply liberal amounts of synecdoche from there (and I don't think OC disagrees with you on that).
Actually by 'ordinary chain rule' I am referring to what you're referring to as 'the chain rule for partial derivatives'. It seems like backprop follows very quickly even from that, but it does not.
The reply you replied to was a parody of how Chat GPT responds when you correct it. They were alluding that its grandparent could be AI generated, explaining its slight vapidness.
I do question the effectiveness (and accuracy) of this exercise, but its learning objectives I think are quite apt.