4 ms·

> You'd have to be slightly insane to say that "x^2 = y^2 + z^2" is a more intuitive way to explain what a circle is than a picture of a circle. I would expect

by dls2016 5y ago

> You'd have to be slightly insane to say that "x^2 = y^2 + z^2" is a more intuitive way to explain what a circle is than a picture of a circle. I would expect any freshmen STEM student to recognize sqrt(x^2+y^2) as the distance of an (x,y) point to the origin. And the equation x^2+y^2=r^2 says this quantity must be fixed. Now I wouldn't expect a student to be able to write the previous two sentences so clearly. This gets to the heart of the paradox of learning. How to see past the clutter if you don't know what you're looking for? The human brain is lazy and prone to skipping logical steps, as reflected in the variation in natural language (and student's writing skills). Therefore learning a natural language does not demand simplicity up front.. unlike the up-front rigor required to learn mathematics notation or a programming language.

anon_tor_12345 5y ago
>How to see past the clutter if you don't know what you're looking for? the picture of the circle isn't clutter - in a very precise sense it is the definition (platonic ideal). the algebraic definition "x^2 = y^2 + z^2" clearly isn't the definition because it's missing things (which field is this polynomial over?). there are many other definitions (another is R/Z). >mathematics notation or a programming language notation isn't a formal language - there is no formal grammar written down somewhere and in fact there can be none (cf godel). whether you realize it or not it's a natural language that evolves, has slang, and is spoken to various degrees of competency by many successful mathematicians.
- dls2016 5y ago
  > 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. Writing is hard.
  - anon_tor_12345 5y ago
    >Which precise sense? in the sense that "mathematical entities are abstract, have no spatiotemporal or causal properties, and are eternal and unchanging" https://en.wikipedia.org/wiki/Philosophy_of_mathematics#Platonism https://en.wikipedia.org/wiki/Philosophy_of_mathematics#Plat... >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)
    - dls2016 5y ago
      > 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!
      - anon_tor_12345 5y ago
        >oh THAT sense, why didn't you just say so? 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?

anon_tor_12345 5y ago

>How to see past the clutter if you don't know what you're looking for? the picture of the circle isn't clutter - in a very precise sense it is the definition (platonic ideal). the algebraic definition "x^2 = y^2 + z^2" clearly isn't the definition because it's missing things (which field is this polynomial over?). there are many other definitions (another is R/Z). >mathematics notation or a programming language notation isn't a formal language - there is no formal grammar written down somewhere and in fact there can be none (cf godel). whether you realize it or not it's a natural language that evolves, has slang, and is spoken to various degrees of competency by many successful mathematicians.

dls2016 5y ago
> 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. Writing is hard.
- anon_tor_12345 5y ago
  >Which precise sense? in the sense that "mathematical entities are abstract, have no spatiotemporal or causal properties, and are eternal and unchanging" https://en.wikipedia.org/wiki/Philosophy_of_mathematics#Platonism https://en.wikipedia.org/wiki/Philosophy_of_mathematics#Plat... >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)
  - dls2016 5y ago
    > 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!
    - anon_tor_12345 5y ago
      >oh THAT sense, why didn't you just say so? 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?