Saturday, Nov. 25, 2017

Maths Is Not What You Think: These Proofs Will Make You See Everything Differently

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November 3, 2017

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Maths Is Not What You Think: These Proofs Will Make You See Everything Differently


This article is going to be a far cry from the kind of maths you did in high school. High school maths is the real bare essentials. Most of it is the stuff that was discovered by the Ancient Greeks at the beginnings of when maths was created (operations, geometry, units). There’s a lot to say about the way that maths is taught in schools. It is very utilitarian (it teaches you what is useful), but it rarely inspires kids to try hard or develop an interest in, let alone see the beauty and profound philosophical significance of what is really the intricate rules that make up the universe. Maths seems just about the opposite to ‘beauty’ and ‘philosophy’ in the minds of most people into adulthood, even though it is probably the most beautiful, most philosophical thing.

We are going to stray far away from repetitive, functional mathematics where everything fits together like Lego and is set in stone. We are going towards the cutting-edge fringes of pure theoretical mathematics to see the most shocking, incomprehensible and controversial mathematical proofs, theories and results.

Numbers So Large That If You Were To Comprehend Them Fully, They Would Collapse Your Brain Into A Black Hole (No Joke, This Would Literally, Physically Happen)

Graham’s Number

This is the most iconic incomprehensibly super-stupendously enormous number in mathematics because it actually has practical uses (namely in modelling high dimensions and combinations of groups).

But you don’t believe that it would really collapse your head into a black hole do you? Maybe you don’t realise just how large it is. Graham’s Number is so large that if you were to fully comprehend it, it would exceed the maximum entropy that could be contained within the space occupied by your brain without collapsing into a black hole. There is no way to comprehend this number (not by psychological, but by physical limitations), it’s just nuts.

What is Graham’s Number? To describe it we need to use an up-arrow notation (that even itself needs to use ellipses in all directions to capture Graham’s Number)

Up-arrow notation:

a↑3 = aaa

Graham’s Number:

g1 = g↑↑↑↑3 (the number g to the power of itself three times equals the number of times that g is to the power of itself equals the number of times that it is to the power of itself equals the number of times that it is actually to the power of itself. This is already just beyond astronomical)

g2 = g↑···{g1}···↑3 (g with g1 number of up-arrows. This is insane)

g64 = g↑···{g63}···↑3 (This pattern is taken to the 64th position, and that is Graham’s Number)

So what is Graham’s Number exactly? It is too large for us to even know, but we do know that it is still 0% of infinity. There are infinity numbers larger than Graham’s number.

How complicated is this universe…………

TREE(3)

We knew you had a thirst for more, so now for something even larger than Graham’s number. TREE(3) is the third number in another in incomprehensibly rapidly increasing series. In this series, the third term is already larger than Graham’s Number (which was the 64th item in its series). The TREE sequence emerges out of graph theory.

The rule defining the sequence is that you build ‘trees’ out of three colours of dots connected by lines. You start with a dot and you keep expanding it outwards from there, but as long as a tree doesn’t contain any previous tree in the sequence anywhere within it. From this the series is:

TREE(1) = 2

TREE(2) = 5

TREE(3) = a number vastly vastly larger than Graham’s Number

It just takes off so incredibly fast that it astounds mathematicians. How do these simple rules produce such a dramatic result. What property causes this? Once again it is a case of exponentials to exponential amounts of exponentials. TREE(3) achieves a certain critical amount of complexity that it opens the possibility of branches to other branches and etc., which grow to vast numbers and which do end surprisingly, but only after a very long time. Meaning….it is still 0% as large as infinity.

For a longer but more visual demonstration, check out the Numberphile episode:

1 + 2 + 3 + 4 + 5 + … = -1/12

Wait what!? This is one of the most counter-intuitive results in all of mathematics and thus creates a lot of controversy whenever it is brought up. However, the proof is so simple and logical that it convinces most people.

The sum of the natural numbers is infinity right?

Well, in string theory, the result -1/12 is sometimes relied upon.

This is concerning and the proof is going to drive you insane

S       = 1 + 2 + 3 + 4 ....
S₁      = 1 - 1 + 1 - 1 .... = 1/2
(if you stop it after a ‘1’ you get 1, if you stop it after a ‘-1’ you get -1, so the answer is the average of those two possibilities)
2S₂     = 1 - 2 + 3 - 4 ....
                   + 1 - 3 + 3 - 4 ....
        = 1 - 1 + 1 - 1 + 1 ...
(We multiplied the two previous series and doubled it, but adding the second one on ‘shifted’ one number along. Then we arrived at S₁ again surprisingly)
        = 1/2
S₂      = 1/4
S - S₂  = 1 + 2 + 3 + 4 ....
        -(1 - 2 + 3 - 4 ....)
        = 4   +   8   +   12 ....
        = 4(1 + 2 + 3 + 4 ....)
        = 4S
∴
S - 1/4 = 4S
3S      = -1/4
S       = -1/12

This was also a proof from a Numberphile video (a great YouTube channel). You can read the comments to see some of the contention surrounding this result. One view is that equals should not be used, but instead ‘is represented by’.

There Are Truths That Will Always Be Beyond The Limits Of Mathematics

Gödel’s incompleteness theorems confirmed the fear at the heart of mathematics—that there are some truths that cannot be proven by a system of rules.

How did Gödel manage to prove something so absolute about all that which is unprovable? His proof is very unique and interesting.

Gödel investigated paradoxes. A verbal example of a paradox is the Liar Paradox which is most simply expressed as “I am lying”. If the person who says this is lying, they are telling the truth and therefore not lying. If they are telling the truth, they are lying and therefore telling the truth. This statement doesn’t yield True or False, it is a contradiction.

Voicing verbal paradoxes doesn’t create black holes, but it does express paradoxes that may actually exist in the logical/mathematical basis of the universe. Gödel proved that there are ‘inconsistencies’ in mathematics, i.e. that paradoxes exist that simply cannot be proved

The classic example of a mathematical paradox is ‘The set of all sets that don’t contain themselves’. If this set doesn’t contain itself, then it satisfies the criteria for the set, so it must contain itself and therefore is disqualified from the criteria from the set. This is very much like a mathematical translation of the Liar’s paradox. So Gödel asks, can this ever be proved?

Gödel assigned every mathematical statement a unique code number and proved that any true mathematical statement would have a code number that could be deduced from the code numbers of the basic truth axioms in mathematics. For instance ‘a + b = b + a’ is an axiom of mathematics that is used along with other basic axioms to solve complex problems. Incorrect statements like ‘2 + 2 = 5’ have code numbers that can’t be derived from the axioms. However, what about the statement ‘This statement cannot be proved from the axioms’? This statement has a truth that cannot be proven within rules of any axiomatic system.

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