“Young man, in mathematics you don't understand things. You just get used to them.”
John Von Neumann (Genius Hungarian-American Mathematician).
We all know numbers. Two plus two is four, five times two is ten, six divided by two is three, etc. We learn multiplication tables and long division in Primary School and think we know what numbers are all about.
In pure mathematics “numbers” are much more interesting here is a glimpse into that universe.
Consider these statements.
People who are used to adding with their fingers, apples or sacks of sand will be puzzled by these statements.
Let’s reconsider these statements involving marbles you used to collect as a child.
Then the statements can be recast in terms of marbles.
In marble-speak this is complete nonsense. In math-speak it makes total sense.
Math is psychedelic baby.
Depending on your mathematical level of sophistication this is how you can explain this non-nonsense.
Let us go over these statements one by one.
Anyone who has filed a tax form is anxious about the last number on their form. If it is negative you owe “them” money and if it is positive “they” owe you money. So money can be both negative and positive.
Negative numbers “exist” in people’s minds.
What is an irrational number? It is a number that cannot be written as a fraction of two whole numbers. Examples of rational numbers are 1 = 1/1, 0.9 = 9/10 and so forth.
Well what about the square root of two? It is defined as the length of the diagonal of a unit square as depicted above.
The question is: can the square root of two be written as a fraction? Answer is nope.
[One liner: assume sqrt(2) = p/q. p and q have no common factors so either p is even or q is even but not both because then you can simplify the fraction. Assume p is even so p = 2k. Now square everything: sqrt(2)^2 = 4k^2/q^2. So 2 = 4k^2/q^2. So q^2 = 2k^2. So q is even! Contradiction!]
Ok so some numbers like the square root of two cannot be written as a fraction.
Rational numbers? Seriously? Who came up with this Patois?
Silly name: actually irrational numbers overwhelm rational numbers. Just like irrational people outnumber rational people.
Between every two rational numbers there is an irrational number and a rational number. Thanks for the latter.
A German mathematician named Georg Cantor proved that there are more irrational numbers than rational numbers with his famous diagonal argument. This implies a hierarchy of infinites. Some infinities are inferior to other infinities. But sometimes less is more like in the case of rational people versus irrational people.
Cantor’s argument goes as follows. Ok so you can list all real numbers just like the integers. But hey wait I can show a number that is not in the list. And voilà there is a contradiction. Hence there are more real numbers than fractions. I think I almost passed Hilbert’s test.
“A mathematical theory is not to be considered complete until you have made it so clear that you can explain it to the first man whom you meet on the street.”
David Hilbert (German Mathematician and one of the last Universalists.)
“That is trippy dude thanks for your time but can you spare a Loonie?
Sure but only if you can promise me to say that aleph zero is smaller than aleph one.
Are you one of those fundamentalists’ dudes that give me money to try to convert me to whatever they preach?
Some mathematicians called constructivists are against these kinds of arguments.
(There is a bit hipster irony at work because a number was actually constructed. It is the meta argument of proof by contradiction that is the problem. Look it up if you are into math and philosophy.)
But why take the fun out of fun math? Leopold Kronecker, another German math dude and a fierce enemy of Georg, claimed that: “God made the integers, all else is the work of man.” Chill dude: yeah whatever math is invented by man or revealed by some God. In the end we can have fun with math. No wonder that people refer to Cantor’s paradise. Cantor went insane and died in an asylum. How sad. Here is someone who opened up a can of a new genre of cool mathematics, a new paradise, only to be shot down in flames by an old grumpy geezer.
Irrational numbers “exist” in people’s minds like the diagonal of a unit square.
Let’s go from line-Land to flat-Land. We are travelling to Kansas. Usually we think of “numbers” being nicely aligned on a straight line. The number one is below the number two, and our dear irrational square root of two lies somewhere between one and two even though it is irrational.
Not so with complex numbers, also known as imaginary numbers. I call them awesome numbers.
They are awesome because these numbers are free to roam all over Kansas. Freedom at last: polynomials of degree n always have exactly n roots! How cool is that? A lot of mathematical problems become easier with awesome numbers.
That is why they are awesome.
In Kansas a number multiplied by it-self can be equal to minus one. Multiplication in Kansas gets you out of the line into the plane. So multiplying a vector also known as “i” is a rotation by ninety degrees counterclockwise. So if you multiply “i” by itself it will result in a vector pointing in the negative direction. So i^2 = -1. In vector speak (0,1)^2 = (-1,0).
It also explains Euler’s famous formula:
By the way awesome numbers are extremely useful in physics and engineering.
Awesome numbers are awesome in Kansas and in your backyard. With some training they will “exist” in your mind. Think flat not linear.
Here is the pièce de résistance.
A = 1+2+3+4+5+6+7+8+9+10+11+12+13+14+… = -1/12
What does this even mean? You add up numbers that obviously eventually have to be equal to infinity. The thing is most people think of mathematics as being just being about marbles. The Devil is in the three dots.
What about this sum?
S = 1-1+1-1+1-1+1-1+1-1+1-1+1-1+1-1+1-1+1+... = ?
Can you guess what the sum is? Let us try two approaches.
S = 1+(-1+1)+(-1+1)+(-1+1)+(-1+1)+(-1+1)+(-1+1)+(-1+1)+(-1+1)+(-1+1)+... = 1+0+0+0+0+0+0+0+0+0+… = 1
S = (1-1)+(1-1)+(1-1)+(1-1)+(1-1)+(1-1)+(1-1)+(1-1)+(1-1)+... = 0+0+0+0+0+0+0+0+0+... = 0.
You can mess with infinite sums like this. They are bad ass. You can tame them by taking averages. It goes by the fancy name of Césaro sums. Then you can prove that this sum converges to one half. Yeah just the average between the two alternatives I showed you earlier. Kind of makes sense in a Diplomatic manner. Why not? What's not to love?
Now we can prove our crazy sum of all integers assuming that S = ½.
Here it goes.
Introduce a third infinite sum.
T = 1 - 2 + 3 – 4 + 5 – 6 + 7 – 8 + 9 + … = ?
Here is the crucial trick:
2T = 1+1 -2-2 +3+3 -4-4 +5+5 -6-6 +7+7 -8-8 +9+9 + ... =
1 + (-2+1) + (3-2) + (-4+3) + (5-4) + (-6+5) + (7-6) + (-8+7) + (9-8) + … =
1 – 1 + 1 - 1 + 1 - 1 + 1 – 1 + 1 + … = S = ½.
So 2T = 1/2 and hence T = ¼.
Similarly: A - T = 4A. So A – ¼ = 4A. And hence:
A = -1/12.
This is all high school math. I think even a 4 year old can understand this argument except for the "..." part. A good starting point is: "why are there an infinite amount of numbers?" Anyone who has a kid knows they will ask it at age three once you open the infinity can of worms.
Anyway, what is the relationship to Bosonic String Theory? Here is a passage from one of the main textbooks written by Polchinski.
Weird psychedelic math can be useful in speculative physical theories.
I think the high school level explanation is more convincing than the Bosonic explanation. By the way the latter theory explains why we live in 26 dimensions. Because according to some constraints in Bosonic Theory the A mentioned in the textbook above must be -1. So: -1 = (D-2)/2 (-1/12) and thus D = 24+2 = 26. Now you know where those crazy dimensions come from in String Theory.
Some crazy results in math can be explained simply with the use of "..." A simple notational device that means "and so on and you figured out the pattern." So I do not have to go on and type an infinite amount of numbers. It is a device of the Devil. Maybe Kronecker was right after all. But who cares, the Devil doesn't exist anyway but beautiful and fun math does.
I hope you went through the geeky details and enjoyed this blog as much as I enjoyed writing it.
Numbers are more than marbles where they originated from. Mathematical numbers have evolved into many exotica just for the heck of it. That is from pure intellectual curiosity. But it turns out that these numbers can have profound implications and applications in physics and engineering. That is a big mystery to me. Yeah and it can also help you to write better code.