> the picture of the circle isn't clutter - in a very precise sense it is the definition (platonic ideal)
Which precise sense? I get that language is slippery, but I can't follow you if you get philosophical, bring up fields (maybe the polynomial is defined over a ring!) while simultaneously stating that your favorite definition is The One True Way (TM).
How does one use the platonic ideal to distinguish a circle from an ellipse?
> whether you realize it or not it's a natural language that evolves
Yes, I realize.
My point was that the day-to-day usage of natural language tolerates a lot of ambiguity. Whereas the classroom usage of mathematical/programming notation does not. It's hard to be informal and "break the rules" as a beginner while retaining precision. Because the web of potential associations hidden in a statement (the "clutter") hasn't yet been pruned down by experience.
>How does one use the platonic ideal to distinguish a circle from an ellipse?
entailed in the picture "definition" is constant curvature.
>while simultaneously stating that your favorite definition is The One True Way (TM).
that's not what i wrote.
>Whereas the classroom usage of mathematical/programming notation does not.
i won't speak to programming notation but every upper level or grad math class i've had has been at times extremely informal in its use of notation. take an integral on any chalkboard in any class and you're almost certain to have these questions without listening to the lecture itself (i.e. it's omitted from the notation):
1 what is the domain? R or C or something really exotic
2 is it over a closed contour/boundary or open?
3 what's the metric? i.e. are we actually integrating forms?
4 is it even an integral at all not just shorthand for solving a differential equation (e.g. stochastic integral)
> in the sense that "mathematical entities are abstract, have no spatiotemporal or causal properties, and are eternal and unchanging"
oh THAT sense, why didn't you just say so?
but seriously what does that sense have to do with a picture of a circle? how would one use the picture of a circle to distinguish from an ellipse?
edit: nobody ever checked if i knew diddly squat about "real actual academic platonism v formalism" before peforming mathematical exposition in front of numerous classrooms in at least two countries, and surely i'm not going to hold back while arguing on the internet!
i mean if you're not familiar with the real actual academic platonism v formalism debate maybe you shouldn't pontificate on what's better or worse re mathematical exposition?
Which precise sense? I get that language is slippery, but I can't follow you if you get philosophical, bring up fields (maybe the polynomial is defined over a ring!) while simultaneously stating that your favorite definition is The One True Way (TM).
How does one use the platonic ideal to distinguish a circle from an ellipse?
> whether you realize it or not it's a natural language that evolves
Yes, I realize.
My point was that the day-to-day usage of natural language tolerates a lot of ambiguity. Whereas the classroom usage of mathematical/programming notation does not. It's hard to be informal and "break the rules" as a beginner while retaining precision. Because the web of potential associations hidden in a statement (the "clutter") hasn't yet been pruned down by experience.
Writing is hard.