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.
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-i...
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.
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.
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
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.
"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.
"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
Where I study, it's a general consensus that in Math, N does not include 0 unless explicitly included as N index 0. In CompSci on the other hand, 0 is assumed to be included for practical and technical reasons.
No, that is a nitpick, I'm afraid. If you think otherwise may I invite you to draw a square with sides of length zero.
Hint: it doesn't exist.
Second hint: it is trivial to demonstrate that it doesn't exist. Consider the triangle. Can you define it as a quadrilateral with one side of length zero? What is the angle at the null vertex? Proceed from there.
Finally, the problem states "another whole number". Zero is not different from zero. As with any exam, you should read the question carefully first.
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
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/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 (and 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_...)
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.
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".
The red circle is a "wheel" which happily rolls back and forth along our f(x) graph. The green line is the derivative of f(x). It's intuitively obvious what that green line is -- if you had to fit a flat piece of paper between the wheel and where it touches the graph, then you'd get that line. That's the derivative, also called the "tangent line" or "slope of the graph" at that point.
(The wheel will always touch the graph only once, because we can visualize it as small as we need it to be --- infinitely small, even.)
It's common knowledge that a curve's derivative is zero at its maximum / minimum. (In other words, "when the wheel reaches the top or bottom of a 'hill', then its derivative is zero, aka the tangent line is perfectly horizontal".) So right away, we immediately understand where the zeroes of the f'(x) graph must be located.
But the real power is, we also have a rough idea of how the f'(x) graph must look, by "watching the wheel's tangent line as we roll it along the graph", in our mind's eye.
So that's a neat trick, but why does it matter?
Well, here's one example: It's the most natural and obvious thing in the world that the derivative of cos(x) is -sin(x).
We start at x=0; we see that the wheel is "resting on the very top of the slope", meaning the tangent line is flat, so we know right off the bat that the derivative of cos(0) is 0.
From there, the wheel "rolls downhill"; therefore the derivative "goes negative"; and it goes back to zero when the wheel reaches the bottom of the hill (which we see is at x=pi).
The wheel then begins climbing uphill, so its derivative "goes positive"; and it goes back to zero when the wheel reaches the top of the slope (at x=2pi).
So we see that the graph of the derivative of cos(x) has zeroes at 0, pi, and 2pi; and we also intuitively understand that it's negative from 0 to pi; and we see that it's positive from pi to 2pi.
Now all that's left is to think about the graph of -sin(x). It's exactly what we just described: zero at x=0, x=pi, and x=2pi; negative from 0 to pi; and positive from pi to 2pi. So that's why "it's obvious to us that the derivative of cos(x) is -sin(x)", QED.
So if you've had to worry about rote memorizing those sorts of formulas, I'd urge you to give these visual techniques a shot. You'll never again need to memorize seemingly-arbitrary equations, which is pretty sweet IMO.
Sorry for rambling on; I just get so excited about presenting math "in a visual way". It's a lot of fun, and likely wouldn't hinder the ambitious mathematician at all --- just the opposite, in fact: Feynman, for example, had an arsenal of similar techniques, which probably contributed to his intuitive understanding of the physics behind the formulas and his ability to "see past" the raw equations.
Awesome example about the derivative! I think the best part of geometric thinking is running ideas on your "native hardware".
Our brains are massively parallel, visual-processing monsters... and we try to think about ideas linearly, like we're a turing machine. No -- use that GPU! :).
When I think visually, I can immediately grok the idea (it's running in hardware) and my software layer (my conscious thought process) can work on stitching together deeper ideas.
I would say it was a matter of interest led by cultural biases, not lack of imagination. It is impossible to look at the ideas of Archimedes and not be dumbfounded by his supreme creativity. Archimedes was able to think of infinitesimals and come up with some of the basics of integral calculus but seemed to think of his methods as an unwholesome but useful tool so he never published them. Who knows where things would have gone if he had. http://en.wikipedia.org/wiki/The_Method_of_Mechanical_Theore...
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.
Not every math field uses equations, and most of those that do, use equation, where terms are very far from being ordinary (integer, real, complex) numbers, and picturing a rectangle when you have xy will only mislead you. For instance, sometimes xy is not equal to yx, sometimes x and y are not 0, but xy is 0.
Of course, there are other ways to employ visual intuition to gain better understanding of the problem -- for instance, if our multiplication comes from group structure, we can imagine an object the symmetries of which are group elements. Nevertheless, the history of mathematics teaches us that the greatest thing started to happen when we ditched our geometric and intuitive understanding of math and turned to more formal ways. I'm certainly happy to see a geometric interpretation of some definition, theorem, or proof, but I'm more interested in logical argumentation than appealing to intuition.
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).
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.
27 comments
[ 5.1 ms ] story [ 64.9 ms ] threadYou would have to proof that separately. I think you will be able to draw the picture that illustrates such a proof.
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.
Stanley's up in heaven now.
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/tennenbaump...
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
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.
Not really. Just add the special case of zero. If you want to sound cool say "There are no two nontrivial squares with side ..."
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
"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
"0 is sometimes included in the list of "whole" numbers (Bourbaki 1968, Halmos 1974), but there seems to be no general agreement."
Hint: it doesn't exist.
Second hint: it is trivial to demonstrate that it doesn't exist. Consider the triangle. Can you define it as a quadrilateral with one side of length zero? What is the angle at the null vertex? Proceed from there.
Finally, the problem states "another whole number". Zero is not different from zero. As with any exam, you should read the question carefully first.
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/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 (and 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_...)
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.
For example we can visualize "taking the derivative of the function f(x)" by "rolling a wheel over the top of its graph": http://dl.dropbox.com/u/315/random_pics/derivative_wheel.png
The red circle is a "wheel" which happily rolls back and forth along our f(x) graph. The green line is the derivative of f(x). It's intuitively obvious what that green line is -- if you had to fit a flat piece of paper between the wheel and where it touches the graph, then you'd get that line. That's the derivative, also called the "tangent line" or "slope of the graph" at that point.
(The wheel will always touch the graph only once, because we can visualize it as small as we need it to be --- infinitely small, even.)
It's common knowledge that a curve's derivative is zero at its maximum / minimum. (In other words, "when the wheel reaches the top or bottom of a 'hill', then its derivative is zero, aka the tangent line is perfectly horizontal".) So right away, we immediately understand where the zeroes of the f'(x) graph must be located.
But the real power is, we also have a rough idea of how the f'(x) graph must look, by "watching the wheel's tangent line as we roll it along the graph", in our mind's eye.
So that's a neat trick, but why does it matter?
Well, here's one example: It's the most natural and obvious thing in the world that the derivative of cos(x) is -sin(x).
http://dl.dropbox.com/u/315/random_pics/derivative_wheel_cos...
We start at x=0; we see that the wheel is "resting on the very top of the slope", meaning the tangent line is flat, so we know right off the bat that the derivative of cos(0) is 0.
From there, the wheel "rolls downhill"; therefore the derivative "goes negative"; and it goes back to zero when the wheel reaches the bottom of the hill (which we see is at x=pi).
The wheel then begins climbing uphill, so its derivative "goes positive"; and it goes back to zero when the wheel reaches the top of the slope (at x=2pi).
So we see that the graph of the derivative of cos(x) has zeroes at 0, pi, and 2pi; and we also intuitively understand that it's negative from 0 to pi; and we see that it's positive from pi to 2pi.
Now all that's left is to think about the graph of -sin(x). It's exactly what we just described: zero at x=0, x=pi, and x=2pi; negative from 0 to pi; and positive from pi to 2pi. So that's why "it's obvious to us that the derivative of cos(x) is -sin(x)", QED.
So if you've had to worry about rote memorizing those sorts of formulas, I'd urge you to give these visual techniques a shot. You'll never again need to memorize seemingly-arbitrary equations, which is pretty sweet IMO.
Sorry for rambling on; I just get so excited about presenting math "in a visual way". It's a lot of fun, and likely wouldn't hinder the ambitious mathematician at all --- just the opposite, in fact: Feynman, for example, had an arsenal of similar techniques, which probably contributed to his intuitive understanding of the physics behind the formulas and his ability to "see past" the raw equations.
Our brains are massively parallel, visual-processing monsters... and we try to think about ideas linearly, like we're a turing machine. No -- use that GPU! :).
When I think visually, I can immediately grok the idea (it's running in hardware) and my software layer (my conscious thought process) can work on stitching together deeper ideas.
Shameless plug, but I've written about visualizing Euler's Identity (http://betterexplained.com/articles/intuitive-understanding-...):
* Radians are distance from the mover's point of view
* e is continuous growth
* i is a rotation (so e^i*anything is imaginary growth, i.e. growth in a perpendicular direction)
Combining ideas like this is just so enjoyable! It brings out the natural beauty in learning.
Not every math field uses equations, and most of those that do, use equation, where terms are very far from being ordinary (integer, real, complex) numbers, and picturing a rectangle when you have xy will only mislead you. For instance, sometimes xy is not equal to yx, sometimes x and y are not 0, but xy is 0.
Of course, there are other ways to employ visual intuition to gain better understanding of the problem -- for instance, if our multiplication comes from group structure, we can imagine an object the symmetries of which are group elements. Nevertheless, the history of mathematics teaches us that the greatest thing started to happen when we ditched our geometric and intuitive understanding of math and turned to more formal ways. I'm certainly happy to see a geometric interpretation of some definition, theorem, or proof, but I'm more interested in logical argumentation than appealing to intuition.
See the 2 & 3 proofs of Pythagorean Theorem: http://www.cut-the-knot.org/pythagoras/index.shtml