4 ms·

A Note on Two Problems in Connexion with Graphs (1959) [pdf]

21 points by guiambros 11y ago

punnerud 11y ago
Is there a visualization showing this?
- jonsen 11y ago
  https://www.cs.usfca.edu/~galles/visualization/Dijkstra.html https://www.cs.usfca.edu/~galles/visualization/Dijkstra.html
- Someone 11y ago
  Heresy :-) https://www.cs.utexas.edu/users/EWD/transcriptions/EWD06xx/EWD696.html https://www.cs.utexas.edu/users/EWD/transcriptions/EWD06xx/E...: "The habit of using pictorial aids, like any habit, is very difficult to get rid of. If, however, we take any responsibility for the effectiveness of our thinking habits, we should try to get rid of the habit as quickly as possible, for it is a bad habit, confusing and misleading up to the point of being paralysing"
  - jonsen 11y ago
    I don't think Dijkstra denies the possibility of "A picture is worth a thousand words". He just says that the opposite is certainly true, and warns against the habit of relying on pictures.
    - Someone 11y ago
      Not quite. He opposed words, too. https://www.cs.utexas.edu/users/EWD/transcriptions/EWD12xx/EWD1239.html https://www.cs.utexas.edu/users/EWD/transcriptions/EWD12xx/E...: "But in order to reason effectively, the very least we should be able to do is establishing whether two statements mean the same thing or whether two definitions are equivalent, and here we immediately run into trouble with our so-called “natural languages”, which (among other shortcomings) are not very well suited to precision" So, prose isn't his optimal way op communication, either. Further on, he wrote: "A picture may be worth a thousand words, a formula is worth a thousand pictures." The problem Dijkstra had with pictures is, that there is no way to show that you have considered all their variants without the use of logic. Also, logic alone is sufficient. So, why bother drawing pictures that may give rise to false intuitions?
  - daralthus 11y ago
    "One of the bad things about pictures is that they are almost always overspecific. One cannot make a picture of "an arbitrary triangle": as soon as one has made it, it has either an obtuse angle or not, whereas for "the arbitrary triangle" the property of having an obtuse angle is explicitly undefined." Still one can make a picture of all the arbitrary triangles at the same time and in an interactive visualization explore them to get the complex behaviour and constrains the notion "arbitrary triangle" actually represent. For a good example have a look at: http://worrydream.com/LadderOfAbstraction/ http://worrydream.com/LadderOfAbstraction/
    - Someone 11y ago
      But how do you know that, while playing with such an interactive visualization, you looked at all possible cases? Also, let's say a theorem says two lines are parallel, two distances are equal, or three lines intersect in a single point, how are you going to judge that that's the case from such a display, even for one particular choice of points?
  - yodsanklai 11y ago
    I believe the opposite. Isn't most mathematical intuition and knowledge based on them? can we really understand any abstraction without some type of graphical representation?

21 points by guiambros 11y ago

punnerud 11y ago

Is there a visualization showing this?

jonsen 11y ago
https://www.cs.usfca.edu/~galles/visualization/Dijkstra.html https://www.cs.usfca.edu/~galles/visualization/Dijkstra.html
Someone 11y ago
Heresy :-) https://www.cs.utexas.edu/users/EWD/transcriptions/EWD06xx/EWD696.html https://www.cs.utexas.edu/users/EWD/transcriptions/EWD06xx/E...: "The habit of using pictorial aids, like any habit, is very difficult to get rid of. If, however, we take any responsibility for the effectiveness of our thinking habits, we should try to get rid of the habit as quickly as possible, for it is a bad habit, confusing and misleading up to the point of being paralysing"
- jonsen 11y ago
  I don't think Dijkstra denies the possibility of "A picture is worth a thousand words". He just says that the opposite is certainly true, and warns against the habit of relying on pictures.
  - Someone 11y ago
    Not quite. He opposed words, too. https://www.cs.utexas.edu/users/EWD/transcriptions/EWD12xx/EWD1239.html https://www.cs.utexas.edu/users/EWD/transcriptions/EWD12xx/E...: "But in order to reason effectively, the very least we should be able to do is establishing whether two statements mean the same thing or whether two definitions are equivalent, and here we immediately run into trouble with our so-called “natural languages”, which (among other shortcomings) are not very well suited to precision" So, prose isn't his optimal way op communication, either. Further on, he wrote: "A picture may be worth a thousand words, a formula is worth a thousand pictures." The problem Dijkstra had with pictures is, that there is no way to show that you have considered all their variants without the use of logic. Also, logic alone is sufficient. So, why bother drawing pictures that may give rise to false intuitions?
- daralthus 11y ago
  "One of the bad things about pictures is that they are almost always overspecific. One cannot make a picture of "an arbitrary triangle": as soon as one has made it, it has either an obtuse angle or not, whereas for "the arbitrary triangle" the property of having an obtuse angle is explicitly undefined." Still one can make a picture of all the arbitrary triangles at the same time and in an interactive visualization explore them to get the complex behaviour and constrains the notion "arbitrary triangle" actually represent. For a good example have a look at: http://worrydream.com/LadderOfAbstraction/ http://worrydream.com/LadderOfAbstraction/
  - Someone 11y ago
    But how do you know that, while playing with such an interactive visualization, you looked at all possible cases? Also, let's say a theorem says two lines are parallel, two distances are equal, or three lines intersect in a single point, how are you going to judge that that's the case from such a display, even for one particular choice of points?
- yodsanklai 11y ago
  I believe the opposite. Isn't most mathematical intuition and knowledge based on them? can we really understand any abstraction without some type of graphical representation?