"Mathematics is the initial object in the category of learning" I've had this phrase in my head for a few days, so it's time to get it out and look at it to see if it still makes sense outside my head. #mathsky #ITeachMaths #UKMathsChat - ThreadSky

mathforge.org • 4 days ago

"Mathematics is the initial object in the category of learning"

I've had this phrase in my head for a few days, so it's time to get it out and look at it to see if it still makes sense outside my head.

#mathsky #ITeachMaths #UKMathsChat

Comments

infiniterest.bsky.social•4 days ago

Depends how you define it

alexjw.bsky.social•2 days ago

Surely the self is the initial object of learning?

mathforge.org•2 days ago

The full phrase is important here: its initial in the *category* of learning.

alexjw.bsky.social•2 days ago

Ahh yes I missed that!

segarrogers.bsky.social•2 days ago

What might some of the subsequent objects be?

Also wondering what someone from another discipline would say to this. e.g. music, social studies, PE.

mathforge.org•2 days ago

The thing about being an /initial/ object in a category is that you very definitely still belong to that category, and you exist because of everything else there, not despite. So this isn't meant as saying that Mathematics is primary and everything else is secondary.

mathforge.org•2 days ago

Rather, I mean by it that Mathematics is _distilled_ learning. It is what you get when you strip away all else, and only the _learning_ is left.

So it doesn't take away from the experience of learning in another discipline, rather it illuminates and illustrates it.

segarrogers.bsky.social•2 days ago

I love the sound of this Andrew … but I don’t actually understand it! I assume I need to go away and learn the basics of category theory – correct?!

mathforge.org•2 days ago

It is a statement in the language of category theory, yes. But that's just like anything in maths - a concise way of saying something that should be understandable if unpacked. So I'll have a go at unpacking it. Let's see if this makes sense!

mathforge.org•2 days ago

The important thing about category theory is that it puts emphasis on the relationships between things. Relationships can have a direction, so in a category we have _objects_ (things) and for two things, say A & B, we have a set of _arrows_ (relationships) from A to B, write this as Hom(A,B).

jjnevis.bsky.social•4 days ago

I love thoughtful sentences that begin with "mathematics is..." and yes, of course it makes sense.

Got any more?

glittercatstudios.bsky.social•4 days ago

That's true if you look at mathematics for what it really is: the teaching of critical thinking and LOGIC (prove the argument X and do it with these numerous tools). The numbers just pay the bills, but what we're really showing them is how to SOLVE any problem. #ITeachMaths #mathsky

swillstroganoff.bsky.social•2 days ago

I wonder if language might come before mathematics.

mathforge.org•2 days ago

Curious as to your reasoning.

swillstroganoff.bsky.social•2 days ago

Just that in order to even contemplate mathematics, or really anything, it seems you need language first.

mathforge.org•2 days ago

Ah, I wondered at that.

"Initial" here has its meaning in the sense of category theory. It is not intended to imply anything temporal.

swillstroganoff.bsky.social•2 days ago

I suppose I don’t know what your morphisms are. I took them to be some kind of (logical?) dependency. In that vein I have trouble even thinking about what mathematics is without a superstructure of language(s) in which to house it.

suburbanlion.com•4 days ago

interesting from a platonic perspective, but for me mathematics seems like more of a "map" than an "object". it's a social process by which we try to establish an isomorphism between the objects in your head and the ones in mine.

mathforge.org•4 days ago

Hmm, I like that perspective!

So then I would say that the "mathematics" in my sentence is the "category" in yours (of people and "understandings" between them).

So maybe "learning" is a 2-category ...

suburbanlion.com•4 days ago

i'm still a cat theory noob, but i can't say you're wrong? swapping two ideas would define an isomorphism. from what i've learned so far, cat theory is metamathematical in nature. could we have similarly said that "categories are the terminal object of learning mathematics"?

suburbanlion.com•4 days ago

i think this conversation is a great example. you put your idea into words, i read it and translate into my own model and send it back again. "math" happens when the composition of those actions is the identity map on the abstract ideas themselves.

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