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Good article. At age 11, I went downstairs to announce I was ready for tomorrow's maths exam, where we would need to plot equations like y = x - 3. My dad sai

Good article. At age 11, I went downstairs to announce I was ready for tomorrow's maths exam, where we would need to plot equations like y = x - 3. My dad said, "well, here's my question - what is a line?" This angered me greatly as I couldn't answer.

boomboomsubban 3y ago
That doesn't really make sense. How did they explain what plotting was without defining a line? I guess they could have taught how to plot a line segment then explained they go on forever.
- gizajob 3y ago
  In this context, a line is what you draw with a ruler to get the correct answer.
  - boomboomsubban 3y ago
    My math teachers would have marked it incorrect if I didn't indicate it went on forever, unless the problem stipulated bounds.
  - lozenge 3y ago
    This is it. He also asked me what the width of the line was and how could I say a line doesn't have a width.
    - gizajob 3y ago
      “Because Euclid says so in the Elements”
- thaumasiotes 3y ago
  > How did they explain what plotting was without defining a line? Most plots aren't lines, so there's no reason this task would require defining a line. In fact, the shape formed by a graph is called a "curve" for just this reason.
  - deleted 3y ago
    [deleted]
  - masklinn 3y ago
    Interestingly in Elements that is a line: a line was defined only by having no breadth (= being 1-dimensional). What is now generally called a line would have to be qualified as straight.
    - ndsipa_pomu 3y ago
      Does that mean that a point is also a line as a point has no breadth
      - masklinn 3y ago
        No because a line does have a length. In Elements a point is dimension-less.
        ndsipa_pomu 3y ago
        Okay, so "a line was defined only by having no breadth" should be "a line was defined only by having length and no breadth"
- ndsipa_pomu 3y ago
  > How did they explain what plotting was without defining a line? Plotting is simply putting a point for a specific value. Drawing the line is inferring the un-plotted points from the plotted ones which may or may not be correct though for y = x - 3, it's easy enough to prove that a straight line covers all the solutions. If "plotting" is only defined in terms of lines, then a student could attempt to plot y = x ^ 2 using only two points and a straight line.
  - deleted 3y ago
    [deleted]
thaumasiotes 3y ago
The simple answer is "it's the set of all points in a plane that are equidistant from two particular distinct fixed points", but that just gives you a clean definition that won't capture any of the properties that make lines interesting. You want something like "it's the shortest path between two points, if you extend it to be infinitely long", which glosses over how you know how to do that correctly, or "it's a 180 degree angle" [same problem]. What did your dad say? This question reminds me that one of Euclid's postulates was "all right angles are equal". We don't consider that an axiom today, and we don't understand why, in Greek mathematics, this was considered to require noting, or what exactly it meant to them. They obviously had something in mind when they said this, but we don't know what that was. Thinking of a line as being defined by the 180 degree angle it forms at every point of itself feels like a similar sort of thing to whatever it was that the Greeks were thinking about right angles.
- masklinn 3y ago
  > This question reminds me that one of Euclid's postulates The question is one of Euclid's postulates (or the combination of two, 1 which defines a straight line segment, and 2 which extends the straight line segment to a straight line).
  - thaumasiotes 3y ago
    Postulate 1 doesn't define a straight line segment. It tells you that it's possible to connect two points with a straight line segment. A definition would tell you whether or not something qualified as a straight line segment. The definition appears to be given, appropriately enough, in the "definitions" section that precedes the postulates, which says "a straight line is one which lies evenly with the points on itself". (As translated in https://farside.ph.utexas.edu/books/Euclid/Elements.pdf https://farside.ph.utexas.edu/books/Euclid/Elements.pdf ) But although that is indeed how people recognize lines, it's not so easy to define what it means.
- ndsipa_pomu 3y ago
  > it's the set of all points in a plane that are equidistant from two particular distinct fixed points I don't like that definition as how do you determine the distance from a fixed point without using a line? > it's the shortest path between two points, if you extend it to be infinitely long That's probably a better definition

boomboomsubban 3y ago

That doesn't really make sense. How did they explain what plotting was without defining a line? I guess they could have taught how to plot a line segment then explained they go on forever.

gizajob 3y ago
In this context, a line is what you draw with a ruler to get the correct answer.
- boomboomsubban 3y ago
  My math teachers would have marked it incorrect if I didn't indicate it went on forever, unless the problem stipulated bounds.
- lozenge 3y ago
  This is it. He also asked me what the width of the line was and how could I say a line doesn't have a width.
  - gizajob 3y ago
    “Because Euclid says so in the Elements”
thaumasiotes 3y ago
> How did they explain what plotting was without defining a line? Most plots aren't lines, so there's no reason this task would require defining a line. In fact, the shape formed by a graph is called a "curve" for just this reason.
- deleted 3y ago
  [deleted]
- masklinn 3y ago
  Interestingly in Elements that is a line: a line was defined only by having no breadth (= being 1-dimensional). What is now generally called a line would have to be qualified as straight.
  - ndsipa_pomu 3y ago
    Does that mean that a point is also a line as a point has no breadth
    - masklinn 3y ago
      No because a line does have a length. In Elements a point is dimension-less.
      - ndsipa_pomu 3y ago
        Okay, so "a line was defined only by having no breadth" should be "a line was defined only by having length and no breadth"
ndsipa_pomu 3y ago
> How did they explain what plotting was without defining a line? Plotting is simply putting a point for a specific value. Drawing the line is inferring the un-plotted points from the plotted ones which may or may not be correct though for y = x - 3, it's easy enough to prove that a straight line covers all the solutions. If "plotting" is only defined in terms of lines, then a student could attempt to plot y = x ^ 2 using only two points and a straight line.
- deleted 3y ago
  [deleted]

thaumasiotes 3y ago

The simple answer is "it's the set of all points in a plane that are equidistant from two particular distinct fixed points", but that just gives you a clean definition that won't capture any of the properties that make lines interesting. You want something like "it's the shortest path between two points, if you extend it to be infinitely long", which glosses over how you know how to do that correctly, or "it's a 180 degree angle" [same problem]. What did your dad say? This question reminds me that one of Euclid's postulates was "all right angles are equal". We don't consider that an axiom today, and we don't understand why, in Greek mathematics, this was considered to require noting, or what exactly it meant to them. They obviously had something in mind when they said this, but we don't know what that was. Thinking of a line as being defined by the 180 degree angle it forms at every point of itself feels like a similar sort of thing to whatever it was that the Greeks were thinking about right angles.

masklinn 3y ago
> This question reminds me that one of Euclid's postulates The question is one of Euclid's postulates (or the combination of two, 1 which defines a straight line segment, and 2 which extends the straight line segment to a straight line).
- thaumasiotes 3y ago
  Postulate 1 doesn't define a straight line segment. It tells you that it's possible to connect two points with a straight line segment. A definition would tell you whether or not something qualified as a straight line segment. The definition appears to be given, appropriately enough, in the "definitions" section that precedes the postulates, which says "a straight line is one which lies evenly with the points on itself". (As translated in https://farside.ph.utexas.edu/books/Euclid/Elements.pdf https://farside.ph.utexas.edu/books/Euclid/Elements.pdf ) But although that is indeed how people recognize lines, it's not so easy to define what it means.
ndsipa_pomu 3y ago
> it's the set of all points in a plane that are equidistant from two particular distinct fixed points I don't like that definition as how do you determine the distance from a fixed point without using a line? > it's the shortest path between two points, if you extend it to be infinitely long That's probably a better definition