9 ms·

The Power of Mathematics - John Conway [pdf]

138 points by niyazpk 15y ago

abecedarius 15y ago
I've only read as far as the first example, and wow, it's beautiful -- you don't even need to read the proof, the diagram says it all.
- deleted 15y ago
  [deleted]
- Someone 15y ago
  If you read it really carefully, you will notice that it does not say it all. The picture can only be drawn if m < 2n. You would have to proof that separately. I think you will be able to draw the picture that illustrates such a proof.
kevinalexbrown 15y ago
This is great. The only nitpicky thing I have is this line: This new proof was created by a friend of mine called Stanley Tennenbaum, who has since dropped out of mathematics. There's kind of a snobbery in mathematics that goes along the lines of Pure Mathematics > Applicable Math > Applied Math > Programming the Math. Theoretical Comp Sci falls somewhere around Applicable. Why is it that he "dropped out of mathematics" and not "decided he liked [whatever he does now]"? Otherwise, a great read. Math is so much cooler than high school calculus drills make it seem.
- deleted 15y ago
  [deleted]
- dvanduzer 15y ago
  Do a little more research on Stanley Tennenbaum. This discovery happened around the 1950s. Tennenbaum didn't drop out of mathematics so much as he dropped out of institutional mathematics. More here, including a post from one of his children in the comments: http://divisbyzero.com/2009/10/06/tennenbaums-proof-of-the-irrationality-of-the-square-root-of-2/ http://divisbyzero.com/2009/10/06/tennenbaums-proof-of-the-i... Stanley's up in heaven now.
- hyperbovine 15y ago
  That's ironic because John Conway does a lot of applied math. I think he might have just been explaining why the audience did not know the name. Mathematics is still a relatively small field.
  - kevinalexbrown 15y ago
    Oh yeah, Game of Life and all that. Oops, thanks for pointing that out.
- ionfish 15y ago
  I should probably point out that snobbery is very far from the actual attitude towards Tennenbaum, at least as far as logicians are concerned. For example, in 2006 there was a conference organised to celebrate his contribution to mathematical logic. http://mamls.org/Tennenbaum http://mamls.org/Tennenbaum I recently attended a workshop devoted entirely to Tennenbaum's Theorem, his famous result that the only recursive model of PA is the standard model. http://www.bbk.ac.uk/philosophy/our-research/ppp/tennenbaumposterfinal.pdf http://www.bbk.ac.uk/philosophy/our-research/ppp/tennenbaump...
jerfelix 15y ago
The first example is pretty cool... Not to nitpick, but as he states the problem: Could there be two squares with side [sic] equal to a whole number, n, whose total area is identical to that of a single square with side equal to another whole number, m? Given that he's speaking of whole numbers, the number zero comes to mind, which satisfies this. So his whole proof is shot. -- John Conway's greatest contribution to my life (as opposed to the game of life), and one I use about five times a week is The Doomsday Rule: http://en.wikipedia.org/wiki/Doomsday_rule http://en.wikipedia.org/wiki/Doomsday_rule
- dvanduzer 15y ago
  It is not non-controversial to include zero in the set of whole numbers. A square with side n, where n=0 would be a point. Except it wouldn't, because we have to define these things differently, or geometry would be incoherent. Incidentally, there is no need for a [sic] there because you only need one side to define a square.
- kevinalexbrown 15y ago
  > So his whole proof is shot. Not really. Just add the special case of zero. If you want to sound cool say "There are no two nontrivial squares with side ..."
- Arjuna 15y ago
  "Given that he's speaking of whole numbers, the number zero comes to mind, which satisfies this. So his whole proof is shot." Both Wikipedia [1] and Wolfram [2] indicate that there are varying interpretations regarding which integers are included in the definition of the term whole number. Since Conway is discussing two-dimensional distance in order to determine area, his definition of the term would not include integers that are less than or equal to zero. [1] http://en.wikipedia.org/wiki/Whole_number http://en.wikipedia.org/wiki/Whole_number "Whole number is a term with inconsistent definitions by different authors. All distinguish whole numbers from fractions and numbers with fractional parts. Whole numbers may refer to: natural numbers in sense (1, 2, 3, ...) - the positive integers natural numbers in sense (0, 1, 2, 3, ...) - the non-negative integers all integers (..., -3, -2, -1, 0, 1, 2, 3, ...)" [2] http://mathworld.wolfram.com/WholeNumber.html http://mathworld.wolfram.com/WholeNumber.html "0 is sometimes included in the list of "whole" numbers (Bourbaki 1968, Halmos 1974), but there seems to be no general agreement."
38leinad 15y ago
always a good idea to already insult your reader in the second sentence...
- ejenkinsiii 15y ago
  Ha...I agree but I do appreciate this thread I currently had to review geometry while studying C++ and most of what's explained on the document is what I discovered while reviewing so it's nice to have it documented, now it's time to get the logic down math is everything I'm discovering when it comes to C.S./Programming, it defines how you build Data Structures/Algorithms well apologies for the rant it's just good to see a thread which helps in an area of my current study, and it's true using geometry makes programming and math easier to comprehend
palish 15y ago
This is great! And the best part is, this sort of "visual math" is applicable to every single field of mathematics; by visualizing every equation you come across, you'll find that you eventually gain a crisp (and often intuitive) understanding of the math. For example: when I think of "x times y", I picture a rectangle whose sides are lengths x and y respectively; so naturally, the area of the rectangle is x times y. Next time you come across an equation similar to "(x0 + x1)(y0 + y1)", you might try picturing it like this: http://content.screencast.com/users/shawnpresser/folders/Jing/media/43e220e1-2f69-49c4-a2a3-1a4801f0950d/2011-11-19_1213.png http://content.screencast.com/users/shawnpresser/folders/Jin... ... you'll discover all kinds of interesting things. E.g., Karatsuba noticed that (x0y1 + x1y0) can be computed as follows: "Find the area of the entire rectangle (x0 + x1)(y0 + y1); then subtract the area of the purple rect (x1y1); then subtract the area of the gray rect (x0y0); thus giving the answer." This was a major breakthrough in mathematics at the time, because it meant you could calculate the product of two arbitrarily large integers with N digits in less than N-squared time: http://en.wikipedia.org/wiki/Karatsuba_algorithm http://en.wikipedia.org/wiki/Karatsuba_algorithm (and http://dl.dropbox.com/u/315/books/Karatsuba%20Algorithm%20%28Fast%20Multiplication%3B%20History%29.pdf http://dl.dropbox.com/u/315/books/Karatsuba%20Algorithm%20%2... was his original paper). Another random example: Have you ever played the game "pipe dream", where you have to connect the pipes together before the liquid fills them up and spills out? Well... the way I visualize "integrating a function" is: imagine the graph of the function. Now start "filling up the graph" from left to right --- just like Pipe Dream. The answer is: the total volume of the "liquid" above the zero line, minus the volume below the zero line. http://upload.wikimedia.org/wikipedia/commons/9/9f/Integral_example.svg http://upload.wikimedia.org/wikipedia/commons/9/9f/Integral_...) Visualization tricks are a great way to get a "gut feeling" about "what does this equation actually mean? e.g., how might I relate the equation to some real-world (or imaginary-world) phenomenon?" ... you just have to be careful that your visualization is exactly equivalent to the mathematics, since an inaccurate visualization would throw off your intuition.
- jberryman 15y ago
  Reminds me of how the ancient Greeks thought of mathematics in a purely geometric way. The pythagoreans wouldn't buy a proof that you couldn't demonstrate geometrically, with whole numbers. They couldn't conceive of infinite or infinitesimal entities, so they explained them away, making calculus one of those ideas that "the Greeks could have had but didn't".
tzs 15y ago
There are two well known mathematicians named John Conway. A bit of Googling confirms that this paper is from John H Conway (the one well known in programming circles for the cellular automata game of "Life)), not John B Conway (the well known functional analyst and author of one of the leading textbooks on complex variables).
anrope 15y ago
A bit off topic, but I was always a bit puzzled about how knots are described mathematically. This is a fantastic intro to knots in the mathematical sense.
alf 15y ago
Visual proofs are uniquely beautiful. See the 2 & 3 proofs of Pythagorean Theorem: http://www.cut-the-knot.org/pythagoras/index.shtml http://www.cut-the-knot.org/pythagoras/index.shtml

138 points by niyazpk 15y ago

I've only read as far as the first example, and wow, it's beautiful -- you don't even need to read the proof, the diagram says it all.

deleted 15y ago
[deleted]
Someone 15y ago
If you read it really carefully, you will notice that it does not say it all. The picture can only be drawn if m < 2n. You would have to proof that separately. I think you will be able to draw the picture that illustrates such a proof.

kevinalexbrown 15y ago

This is great. The only nitpicky thing I have is this line: This new proof was created by a friend of mine called Stanley Tennenbaum, who has since dropped out of mathematics. There's kind of a snobbery in mathematics that goes along the lines of Pure Mathematics > Applicable Math > Applied Math > Programming the Math. Theoretical Comp Sci falls somewhere around Applicable. Why is it that he "dropped out of mathematics" and not "decided he liked [whatever he does now]"? Otherwise, a great read. Math is so much cooler than high school calculus drills make it seem.

deleted 15y ago
[deleted]
dvanduzer 15y ago
Do a little more research on Stanley Tennenbaum. This discovery happened around the 1950s. Tennenbaum didn't drop out of mathematics so much as he dropped out of institutional mathematics. More here, including a post from one of his children in the comments: http://divisbyzero.com/2009/10/06/tennenbaums-proof-of-the-irrationality-of-the-square-root-of-2/ http://divisbyzero.com/2009/10/06/tennenbaums-proof-of-the-i... Stanley's up in heaven now.
hyperbovine 15y ago
That's ironic because John Conway does a lot of applied math. I think he might have just been explaining why the audience did not know the name. Mathematics is still a relatively small field.
- kevinalexbrown 15y ago
  Oh yeah, Game of Life and all that. Oops, thanks for pointing that out.
ionfish 15y ago
I should probably point out that snobbery is very far from the actual attitude towards Tennenbaum, at least as far as logicians are concerned. For example, in 2006 there was a conference organised to celebrate his contribution to mathematical logic. http://mamls.org/Tennenbaum http://mamls.org/Tennenbaum I recently attended a workshop devoted entirely to Tennenbaum's Theorem, his famous result that the only recursive model of PA is the standard model. http://www.bbk.ac.uk/philosophy/our-research/ppp/tennenbaumposterfinal.pdf http://www.bbk.ac.uk/philosophy/our-research/ppp/tennenbaump...

jerfelix 15y ago

The first example is pretty cool... Not to nitpick, but as he states the problem: Could there be two squares with side [sic] equal to a whole number, n, whose total area is identical to that of a single square with side equal to another whole number, m? Given that he's speaking of whole numbers, the number zero comes to mind, which satisfies this. So his whole proof is shot. -- John Conway's greatest contribution to my life (as opposed to the game of life), and one I use about five times a week is The Doomsday Rule: http://en.wikipedia.org/wiki/Doomsday_rule http://en.wikipedia.org/wiki/Doomsday_rule

dvanduzer 15y ago
It is not non-controversial to include zero in the set of whole numbers. A square with side n, where n=0 would be a point. Except it wouldn't, because we have to define these things differently, or geometry would be incoherent. Incidentally, there is no need for a [sic] there because you only need one side to define a square.
kevinalexbrown 15y ago
> So his whole proof is shot. Not really. Just add the special case of zero. If you want to sound cool say "There are no two nontrivial squares with side ..."
Arjuna 15y ago
"Given that he's speaking of whole numbers, the number zero comes to mind, which satisfies this. So his whole proof is shot." Both Wikipedia [1] and Wolfram [2] indicate that there are varying interpretations regarding which integers are included in the definition of the term whole number. Since Conway is discussing two-dimensional distance in order to determine area, his definition of the term would not include integers that are less than or equal to zero. [1] http://en.wikipedia.org/wiki/Whole_number http://en.wikipedia.org/wiki/Whole_number "Whole number is a term with inconsistent definitions by different authors. All distinguish whole numbers from fractions and numbers with fractional parts. Whole numbers may refer to: natural numbers in sense (1, 2, 3, ...) - the positive integers natural numbers in sense (0, 1, 2, 3, ...) - the non-negative integers all integers (..., -3, -2, -1, 0, 1, 2, 3, ...)" [2] http://mathworld.wolfram.com/WholeNumber.html http://mathworld.wolfram.com/WholeNumber.html "0 is sometimes included in the list of "whole" numbers (Bourbaki 1968, Halmos 1974), but there seems to be no general agreement."

38leinad 15y ago

always a good idea to already insult your reader in the second sentence...

ejenkinsiii 15y ago
Ha...I agree but I do appreciate this thread I currently had to review geometry while studying C++ and most of what's explained on the document is what I discovered while reviewing so it's nice to have it documented, now it's time to get the logic down math is everything I'm discovering when it comes to C.S./Programming, it defines how you build Data Structures/Algorithms well apologies for the rant it's just good to see a thread which helps in an area of my current study, and it's true using geometry makes programming and math easier to comprehend

palish 15y ago

This is great! And the best part is, this sort of "visual math" is applicable to every single field of mathematics; by visualizing every equation you come across, you'll find that you eventually gain a crisp (and often intuitive) understanding of the math. For example: when I think of "x times y", I picture a rectangle whose sides are lengths x and y respectively; so naturally, the area of the rectangle is x times y. Next time you come across an equation similar to "(x0 + x1)(y0 + y1)", you might try picturing it like this: http://content.screencast.com/users/shawnpresser/folders/Jing/media/43e220e1-2f69-49c4-a2a3-1a4801f0950d/2011-11-19_1213.png http://content.screencast.com/users/shawnpresser/folders/Jin... ... you'll discover all kinds of interesting things. E.g., Karatsuba noticed that (x0y1 + x1y0) can be computed as follows: "Find the area of the entire rectangle (x0 + x1)(y0 + y1); then subtract the area of the purple rect (x1y1); then subtract the area of the gray rect (x0y0); thus giving the answer." This was a major breakthrough in mathematics at the time, because it meant you could calculate the product of two arbitrarily large integers with N digits in less than N-squared time: http://en.wikipedia.org/wiki/Karatsuba_algorithm http://en.wikipedia.org/wiki/Karatsuba_algorithm (and http://dl.dropbox.com/u/315/books/Karatsuba%20Algorithm%20%28Fast%20Multiplication%3B%20History%29.pdf http://dl.dropbox.com/u/315/books/Karatsuba%20Algorithm%20%2... was his original paper). Another random example: Have you ever played the game "pipe dream", where you have to connect the pipes together before the liquid fills them up and spills out? Well... the way I visualize "integrating a function" is: imagine the graph of the function. Now start "filling up the graph" from left to right --- just like Pipe Dream. The answer is: the total volume of the "liquid" above the zero line, minus the volume below the zero line. http://upload.wikimedia.org/wikipedia/commons/9/9f/Integral_example.svg http://upload.wikimedia.org/wikipedia/commons/9/9f/Integral_...) Visualization tricks are a great way to get a "gut feeling" about "what does this equation actually mean? e.g., how might I relate the equation to some real-world (or imaginary-world) phenomenon?" ... you just have to be careful that your visualization is exactly equivalent to the mathematics, since an inaccurate visualization would throw off your intuition.

jberryman 15y ago
Reminds me of how the ancient Greeks thought of mathematics in a purely geometric way. The pythagoreans wouldn't buy a proof that you couldn't demonstrate geometrically, with whole numbers. They couldn't conceive of infinite or infinitesimal entities, so they explained them away, making calculus one of those ideas that "the Greeks could have had but didn't".

tzs 15y ago

There are two well known mathematicians named John Conway. A bit of Googling confirms that this paper is from John H Conway (the one well known in programming circles for the cellular automata game of "Life)), not John B Conway (the well known functional analyst and author of one of the leading textbooks on complex variables).

anrope 15y ago

A bit off topic, but I was always a bit puzzled about how knots are described mathematically. This is a fantastic intro to knots in the mathematical sense.

alf 15y ago

Visual proofs are uniquely beautiful. See the 2 & 3 proofs of Pythagorean Theorem: http://www.cut-the-knot.org/pythagoras/index.shtml http://www.cut-the-knot.org/pythagoras/index.shtml