6 ms·

I really didn't get the article. I mean, I've got that we have some mapping f: R -> R, or equivalently f: (x, y) \in RxR, so we could represent it by drawing on

I really didn't get the article. I mean, I've got that we have some mapping f: R -> R, or equivalently f: (x, y) \in RxR, so we could represent it by drawing on 2d plane, but no further... Author claimed, that he had been missing something essential regarding that stuff, but I wonder what exactly does he talk about?

ColinWright 11y ago
People think of "drawing a graph of a function" as drawing a line, so if y=sin(x) they end up with a wriggly line going infinitely far left and right. What they don't think of is that the line is actually the collection of points (x0,y0) such that the equation y0=sin(x0) is true. Kids in high school don't think of the graph of a function as being a subset of points of the plane, being: { (x,y) : y=sin(x) is true } Realising that opens the doors to equivalences between different ways of thinking. We can think of a permutation of objects as both the act of permuting them, and as the result of applying that permutation to the default initial position. We can think of a vector (4,6,9) as a location in space, and as the movement to get from (x,y,z) to (x+4,y+6,z+9). We can think of "3" as a location on the number line, or as the action of adding 3 to something, or as the action of multiplying 3 my something, and so on. We can think of the graph you draw as a line, or as a subset of the plane, and we shift effortlessly between them, deliberately blurring the distinction, and from that blurring can come power. Does that help?
- xpinguin 11y ago
  Thanks! Seems like I'd been getting it right since the beginning, so I've missed the point.
- kbenson 11y ago
  In my first calculus class, they introduced integrals to us through Riemann sums, and ever increasing numbers of regions for those sums. It makes thinking in this manner somewhat easier to come by, as it was touched on in the introduction.
- sago 11y ago
  This hurts when they deal with equations like x^2 + y^2 = k and can't even begin to understand how you could graph something like that. It took me a while to grok it, so I second his experience of this not being well stressed (30 years ago, admittedly). I did further maths A-level (and got an A), but didn't grok it until uni.
  - bitwize 11y ago
    Having long been fascinated with computer graphics this bit was easy for me to get. I imagined a computer raster-scanning the plane and plotting a point when the condition in the equation was true. I just had to scale it up in my head from screen resolutions to infinite resolution and infinite extent and... oh boy, this is already starting to make my head hurt... But the gist of it I got. And I admit, I'm unusual.
- thaumasiotes 11y ago
  > Kids in high school don't think of the graph of a function as being a subset of points of the plane This is literally the entire point of graphing inequalities.
gniv 11y ago
It's another way of thinking of the graph. Basically, the points on the graph are the points that satisfy y = f(x), so in that sense "the graph is a picture of all the points that make the function true". From there he generalizes the idea to drawing the set of points that makes a statement true, without necessarily having a formal function defined.
contravariant 11y ago
He didn't seem to have realized fully what a 'graph' was. He probably went with the 'standard procedure' for drawing a graph (pick a x, and draw a point f(x) places above the x-axis, repeat). He later realized that functions are a type of relation, (which is made explicit with the notation "y = f(x)") and that the graph is the set of points that satisfy that relation. Of course if you realize that you can also generalize to g(x) = f(y), f(x,y) = 0 etc. Which he seems to have done. It also seems he hadn't completely realized that a graph is merely a collection of points, not a nice line-like object. This last is more explicit in non-continuous functions, but you don't usually see those in high school.
- xpinguin 11y ago
  > It also seems he hadn't completely realized that a graph is merely a collection of points, not a nice line-like object. This last is more explicit in non-continuous functions, but you don't usually see those in high school. Well, I find whole thing rather controversial, so to say. Depends on what do one mean by "collection" and by "point", eg. if we take point literally: (x, y) \in RxR, then we have to assume that "collection" is uncountable (which seems counterintuitive, esp. from the school student's pov). OTOH, one could treat "point" as representation of that point, eg. pencil mark, so that point is no longer single element, but rather an uncountable subset containing the point of interest, along with its neighbourhood => our "collection" becomes countable, so we could think about it as bunch of indexed points (much more intuitive, isn't it?), shifting our attention towards mapping's representation from the mapping itself. tl;dr Treating graph as a collection of points is no more correct/better, compared to treating it as a continous line. Different problems require different approaches and different "angles" of view. Barely an insight from the author - the most of the mathematics is about abstractions and pattern matching.
  - rsy96 11y ago
    > Well, I find whole thing rather controversial, so to say. Depends on what do one mean by "collection" and by "point", eg. if we take point literally: (x, y) \in RxR, then we have to assume that "collection" is uncountable (which seems counterintuitive, esp. from the school student's pov). OTOH, one could treat "point" as representation of that point, eg. pencil mark, so that point is no longer single element, but rather an uncountable subset containing the point of interest, along with its neighbourhood => our "collection" becomes countable, so we could think about it as bunch of indexed points (much more intuitive, isn't it?), shifting our attention towards mapping's representation from the mapping itself. The whole paragraph does not make sense to me. For example, how is a collection uncountable somehow unintuitive? Why should a point not be a single element, but a subset or a pencil mark? Are you confusing the theoretical graph with the graph actually drawn (which has finite width)?
Mz 11y ago
Some people cannot see the trees for the forest. Others cannot see the forest for the trees. If you do not understand that they are both true and real and right there in front of you, you have missed something important somewhere along the way.
- xpinguin 11y ago
  Actually, only tree is real. "Forest" is a mental construction - just region of space with the density of trees being high enough. Play with that "high enough" variable, and a meaning of "forest" will drift.
  - mc808 11y ago
    "Tree" is also a mental construction. It is just a bunch of particles that form a recognizable pattern in the features and scale we've evolved to discern in the world. Examine two trees in enough detail, and the vast differences may make the tree-scale similarity seem utterly insignificant. But alas, "particle" is also just a mental construction...

ColinWright 11y ago

People think of "drawing a graph of a function" as drawing a line, so if y=sin(x) they end up with a wriggly line going infinitely far left and right. What they don't think of is that the line is actually the collection of points (x0,y0) such that the equation y0=sin(x0) is true. Kids in high school don't think of the graph of a function as being a subset of points of the plane, being: { (x,y) : y=sin(x) is true } Realising that opens the doors to equivalences between different ways of thinking. We can think of a permutation of objects as both the act of permuting them, and as the result of applying that permutation to the default initial position. We can think of a vector (4,6,9) as a location in space, and as the movement to get from (x,y,z) to (x+4,y+6,z+9). We can think of "3" as a location on the number line, or as the action of adding 3 to something, or as the action of multiplying 3 my something, and so on. We can think of the graph you draw as a line, or as a subset of the plane, and we shift effortlessly between them, deliberately blurring the distinction, and from that blurring can come power. Does that help?

xpinguin 11y ago
Thanks! Seems like I'd been getting it right since the beginning, so I've missed the point.
kbenson 11y ago
In my first calculus class, they introduced integrals to us through Riemann sums, and ever increasing numbers of regions for those sums. It makes thinking in this manner somewhat easier to come by, as it was touched on in the introduction.
sago 11y ago
This hurts when they deal with equations like x^2 + y^2 = k and can't even begin to understand how you could graph something like that. It took me a while to grok it, so I second his experience of this not being well stressed (30 years ago, admittedly). I did further maths A-level (and got an A), but didn't grok it until uni.
- bitwize 11y ago
  Having long been fascinated with computer graphics this bit was easy for me to get. I imagined a computer raster-scanning the plane and plotting a point when the condition in the equation was true. I just had to scale it up in my head from screen resolutions to infinite resolution and infinite extent and... oh boy, this is already starting to make my head hurt... But the gist of it I got. And I admit, I'm unusual.
thaumasiotes 11y ago
> Kids in high school don't think of the graph of a function as being a subset of points of the plane This is literally the entire point of graphing inequalities.

gniv 11y ago

It's another way of thinking of the graph. Basically, the points on the graph are the points that satisfy y = f(x), so in that sense "the graph is a picture of all the points that make the function true". From there he generalizes the idea to drawing the set of points that makes a statement true, without necessarily having a formal function defined.

contravariant 11y ago

He didn't seem to have realized fully what a 'graph' was. He probably went with the 'standard procedure' for drawing a graph (pick a x, and draw a point f(x) places above the x-axis, repeat). He later realized that functions are a type of relation, (which is made explicit with the notation "y = f(x)") and that the graph is the set of points that satisfy that relation. Of course if you realize that you can also generalize to g(x) = f(y), f(x,y) = 0 etc. Which he seems to have done. It also seems he hadn't completely realized that a graph is merely a collection of points, not a nice line-like object. This last is more explicit in non-continuous functions, but you don't usually see those in high school.

xpinguin 11y ago
> It also seems he hadn't completely realized that a graph is merely a collection of points, not a nice line-like object. This last is more explicit in non-continuous functions, but you don't usually see those in high school. Well, I find whole thing rather controversial, so to say. Depends on what do one mean by "collection" and by "point", eg. if we take point literally: (x, y) \in RxR, then we have to assume that "collection" is uncountable (which seems counterintuitive, esp. from the school student's pov). OTOH, one could treat "point" as representation of that point, eg. pencil mark, so that point is no longer single element, but rather an uncountable subset containing the point of interest, along with its neighbourhood => our "collection" becomes countable, so we could think about it as bunch of indexed points (much more intuitive, isn't it?), shifting our attention towards mapping's representation from the mapping itself. tl;dr Treating graph as a collection of points is no more correct/better, compared to treating it as a continous line. Different problems require different approaches and different "angles" of view. Barely an insight from the author - the most of the mathematics is about abstractions and pattern matching.
- rsy96 11y ago
  > Well, I find whole thing rather controversial, so to say. Depends on what do one mean by "collection" and by "point", eg. if we take point literally: (x, y) \in RxR, then we have to assume that "collection" is uncountable (which seems counterintuitive, esp. from the school student's pov). OTOH, one could treat "point" as representation of that point, eg. pencil mark, so that point is no longer single element, but rather an uncountable subset containing the point of interest, along with its neighbourhood => our "collection" becomes countable, so we could think about it as bunch of indexed points (much more intuitive, isn't it?), shifting our attention towards mapping's representation from the mapping itself. The whole paragraph does not make sense to me. For example, how is a collection uncountable somehow unintuitive? Why should a point not be a single element, but a subset or a pencil mark? Are you confusing the theoretical graph with the graph actually drawn (which has finite width)?

Mz 11y ago

Some people cannot see the trees for the forest. Others cannot see the forest for the trees. If you do not understand that they are both true and real and right there in front of you, you have missed something important somewhere along the way.

xpinguin 11y ago
Actually, only tree is real. "Forest" is a mental construction - just region of space with the density of trees being high enough. Play with that "high enough" variable, and a meaning of "forest" will drift.
- mc808 11y ago
  "Tree" is also a mental construction. It is just a bunch of particles that form a recognizable pattern in the features and scale we've evolved to discern in the world. Examine two trees in enough detail, and the vast differences may make the tree-scale similarity seem utterly insignificant. But alas, "particle" is also just a mental construction...