Animal, Vegetable, or Minister? By Charles Seife.
Note: x^2 means x to the second power or "squared".
Let a and b each be equal to 1. Since a and b are equal,
b^2 = ab (eq. 1)
Since a equals itself, it is obvious that
a^2 = a^2 (eq. 2)
Subtract equation 1 from equation 2. This yields
a^2 – b^2 = a^2 – ab (eq. 3)
We can factor both sides of the equation; a^2 – ab equals a(a – b). Likewise, a^2 – b^2 equals (a + b)(a – b). Substituting into equation 3 we get
(a + b)(a – b) = a(a – b) (eq. 4)
So far, so good. Now divide both sides of the equation by (a – b) and we get
a + b = a (eq. 5)
Subtract a from both sides and we get
b = 0 (eq. 6)
But we set b to 1 at the very beginning of this proof, so this means that
1 = 0 (eq. 7)
Going further, we know that Winston Churchill has one head. But one equals zero by equation 7, so that means that Winston has no head. Likewise, Churchill has zero leafy tops, therefore he has one leafy top. Multiplying both sides of equation 7 by 2, we see that
2 = 0 (eq. 8)
Churchill has two legs, therefore he has no legs. Churchill has two arms, therefore he has no arms. Now multiply equation 7 by Churchill's waist size in inches. This mans that
(Winston's waist size) = 0 (eq. 9)
This means that Winston Churchill tapers to a point. Now what color is Winston Churchill? Take any beam of light that comes from him and select a photon. Multiply equation 7 by the wavelength, and we see that
(Winston's photon's wavelength) = 0 (eq. 10)
But multiplying equation 7 by 640 nanometers, we see that
640 = 0 (eq. 11)
Combining equations 10 and 11, we see that
(Winston's photon's wavelength) = 640 nanometers
This means that this photon – or any other photon that comes from Mr. Churchill – is orange. Therefore Winston Churchill is a bright shade of orange.
To sum up, we have proved, mathematically, that Winston Churchill has no arms, no legs; instead of a head, he has a leafy top; he tapers to a point; and he is bright orange. Clearly, Winston Churchill is a carrot.
Algebraically speaking there is nothing wrong with the above. In fact, an almost identical proof is made in one section of Einstein's theory of relativity. (Don't blame him too much, he did after all hire a mathematician to help him with that one, and was undoubtedly unaware of that man's questionable use of algebra, specifically substitution.)
If you haven't figured it out yet, that was a tricky way to essentially do a divide by zero. You know, that pesky number that was outlawed by the Greeks and off and on by most of the West. Even calculus, modern man's way of dealing with zero, is often accused of simply ignoring zero by constantly rounding up or down away from it (what does 1/3 equal?).
But what do you think? Is it necessarily wrong? Scientists toy with the idea of infinite dimensions with infinite possibilities. In one of those dimensions, Winston Churchill would indeed, be a carrot. So perhaps the proof is accurate.. if not for this place.
Calculus' handy little rounding tool only works if a number approaches something. That's why a divide by zero is so dangerous. It doesn't approach anything, it is everything. But geometry gives us the same problem. Between any two points in the plane, there are an infinite number of points between. For any distance except zero and infinitely small, this is true. There is no approaching a number, they are all infinite. Zeno took this paradox to prove that movement is impossible. And he believed it. He believed what we consider movement is simply a misconception of our perceptions – something difficult to disprove, but which we simply don't accept.
Others, like the Greeks, answered the paradox by saying neither zero nor infinite exists (and even executed those who had different ideas – the Greeks were really into math). This is because their math of choice was geometry, and as we have seen, these ideas can be a tad dangerous to geometry in particular.
Taking the Greek ideas further, some say that there is a minute point at which there is no smaller, but it is not infinitely small. This solves movement, but is a bit disturbing itself. After all, that means that we do skip space, jump as it were. You know, exactly like animation. The smallest unit on a monitor is a pixel, thus we move our pictures by pixel increments. So basically, we're all part of one big flip book.
OK, enough with these theories, I want to hear what you all think of this. What is the way of things? Proofs? Theories?
-Justin
