Mathematicians describe points on the plane with coordinates (x,y). The first coordinate says how far 𝑎𝑐𝑟𝑜𝑠𝑠 you go and the second says how far 𝑢𝑝 you go. Then they describe entries of a matrix Mᵢⱼ with indices where the first says how far 𝑑𝑜𝑤𝑛 you go and the second says how far 𝑎𝑐𝑟𝑜𝑠𝑠 you go.
In each case I've had teachers who insinuate that this is the only reasonable thing to do and you'd have to be nuts to dream of doing anything else.
I think some teachers don't distinguish between facts and conventions. Actual facts are always worth thinking about - understanding them more and more deeply is a never-ending quest. But for arbitrary conventions, you should just memorize them and move on.
@johncarlosbaez When I took an undergraduate class in knot theory I had to remember most of the conventions (e.g. which type of crossing is +ve) as "the opposite of the way I think they should be".
This works fine until you get used to the convention and have to double-guess yourself.
@johncarlosbaez Reminds me of a favourite quote: "All science is either physics or stamp collecting".
I have always found mathematics so interesting that it's fairly easy to absorb conventions.
But in computing, there's much less "physics" involved and tons of "stamp collecting". Consequently, I've always gravitated towards libraries and systems software rather than application programming. (Maybe application programming is more principled than I give it credit for?)
@BartoszMilewski - it's true! There's a fact lurking there, but it's hard to express the fact without some conventions.
I like to tell students: "In category theory, we reduce all of mathematics to the study of arrows. As a result, you'll spend 90% of your time wondering which way the arrows should be pointing."
@BartoszMilewski@johncarlosbaez I think the 'left' comes from the fact that a left adjoint applies to the left varaible in Hom(l,U(r)) = Hom(F(l),r). Of course the order of arguments in Hom is itself based on our convention of writing direction. It makes sense for the source to come before the target.
The 'computer' convention for graphing also follows our writing convention, with x going across and y going down. It would be neat if we could start plotting graphs the same way.
@OscarCunningham wrote: "I think the 'left' comes from the fact that a left adjoint applies to the left variable in Hom(l,U(r)) = Hom(F(l),r)."
I think so too. Which is why I never have trouble remembering this convention.
"It makes sense for the source to come before the target."
Which is why we write (x)f: x is an element of the source, and (x)f is an element of the target.
"The 'computer' convention for graphing also follows our writing convention, with x going across and y going down. It would be neat if we could start plotting graphs the same way."
@johncarlosbaez@OscarCunningham To preserve my sanity, I taught myself to always read (f \circ g ) as "f after g". This doesn't help with "f is pre-composed with g" or "g is post-composed with f." Especially in (C(-, a)) is a functor that transforms by pre-composition.
@BartoszMilewski@johncarlosbaez@OscarCunningham I've been trying develop a stronger feeling for this (C(r,d)) notation as I work thru Mac Lane proof of #YonedaLemma, and am experimenting with writing such hom-sets as (r{\rightarrow}d). Given that we draw arrows left-to-right (as in (f:a\rightarrow b)), this makes immediately evident that — and 'why' — the hom-functor (-{\rightarrow}d) is contravariant but (r{\rightarrow}-) is covariant. There is also the advantage that seeing the braces reminds me the diagram is in (\mathbf{Set}).
“Both Descartes and Fermat used a single axis in their treatments and have a variable length measured in reference to this axis.”
I have no idea what that means.
@philippsteinkrueger and @OscarCunningham explain that Descartes and Fermat used the distance along a family of lines that meet the single axis at some fixed angle, not necessarily a right angle, as the second variable.
So the wording in Wikipedia, “measured in reference to this axis”, really means “measured from this axis, in a fixed direction that need not be orthogonal to it.”]
The passage continues:
“The concept of using a pair of axes was introduced later, after Descartes' La Géométrie was translated into Latin in 1649 by Frans van Schooten and his students. These commentators introduced several concepts while trying to clarify the ideas contained in Descartes's work.”
@gregeganSF@johncarlosbaez I don’t remember the source, but I remember things having to do with sun rays going down in parallel to the floor. This single axis may represent the ground.
@gregeganSF@johncarlosbaez It means that the distance the y axis was "given" in the diagram depended on how the x axis was set up. They only measure in one dimension, that you can move in, so things are relative to a fixed point. Think of it like drawing a map where your house is the origin and everything is described relative to your house. So things like "six blocks due east" are valid directions. Hope this helps!
@gregeganSF - I've tried, very slighlty, to understand what Descartes actually did. All I know for sure is that it was not much like what we call Cartesian coordinates! I heard he did not use a pair of axes. Separately I've heard he considered only positive coordinates. This makes some sense, since Morris Kline wrote
"Descartes partially accepted negative numbers. He rejected negative roots of equations as 'false', since they represented numbers less than nothing. However, he showed that an equation with negative roots could be transformed into one having positive roots, which led him to accept negative numbers."
You might think he'd look at the plane and notice the regions not described by positive coordinates!
@johncarlosbaez@gregeganSF You can always slide your axis so that everyting is positive. Well, no, you can't, but he might have thought you would (have to) slide it far enough so that everything you are interested in was positive.
@johncarlosbaez I've been curious about the related issue of which orientation to use for 'naturality squares'. Mac Lane draws the natural transformation as a shift left-to-right (see triangular prism, p.16), whereas Leinster and Riehl draw them as downward shifts in keeping with the 'globular' notation. Has the latter practice become more common?
--- edit ---
Oops, misreported Riehl's convention here, which matches Mac Lane's
@dcnorris - Since the diagram means the same thing either way, I''ve never thought about this. But if I draw a naturality square I tend to draw the natural morphisms pointing down, like you say Leinster and Riehl do. That's probably a spinoff of working with 2-categories and drawing globes with the 2-morphism pointing down.
@johncarlosbaez Sorry, but looking more closely at Riehl I see now she follows the Mac Lane convention. Working that way, terms 'horizontal|vertical composition' (which Riehl introduces in §1.7 on 2-categories) I think appeal only to the globular notation.
A further survey shows Fong & Spivak drawing natural transformations both ways [pp.96–7], introducing the rightgoing naturality square (3.4) but then drawing a downgoing Example 3.34, and asking, "Does this help you to see and appreciate the notation [globular diagram]?"
Like Leinster, @PeterSmith employs downgoing naturality squares, and his diagram here https://www.logicmatters.net/resources/pdfs/SmithCat-II.pdf#page=30 shows nicely how the squares themselves stack vertically under the downgoing convention.
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