Entropy

This is high temperature and chaotic, which is called Ising model.

Jij>0

This is like Big Bang.

You can also write this.

Expansion is high temperature.

Therefore, you can divide it.

The division is cooling.

Tr is trace.

∴

Exterior Product

Is God a Mathematician?

Descartes ignored God, and he is still influential in this century.

There is Technological Singularity which nobody knows, although we can perceive it through mathematics.
I don't know whether the decision is based on scientific truth or not. It may be democratic or your preference.
It is better to keep good impression like politicians or movie stars. I don't know who is perfect.

Hermann Grassmann knows the infinity.

AB+BC=AC

BA=-AB and CB=-BC

AB=AC+CB＝AC-BC

∴AB+BC=AC

Then you can expand dimensions.

(1)

(2)

(3)


All vectors in any spaces are reduced to the same line.

Soddy's hexlet

Soddy's hexlet is the mirror image.

Two spheres which are green and red are connected.

There are 6 spheres which are also connected each other around the two of them.

OP*OP'=R^2

OP and OP' are spheres, and R is the radius of OP'.
Therefore, OP=OP'=R
This is absurd, but P' is point at infinity.

The same spheres are connected, and they are mirror image.
Moreover, 6 spheres around OP and OP' are also mirror image.

∴

1/r1+1/r4=1/r2+1/r5=1/r3+1/r6

P^2 + Σ2P

There is no dream code.

You need infinite files to prove it.
You have to wait for compiling like C programming language.

You can find prime numbers in 100 by spreadsheet.


mod(B7,3)=0=true


3^2+Σ2*3 is blue.




5^2+Σ2*5 is blue.



7^2+Σ2*7 is blue.



Prime numbers are no color.

All colored cells are True, except for 3,5,7 because of the only numbers.
Therefore, prime numbers are False.

Observable sets

It is important to know the normal world. Arts and sciences are complicated, but we need to perceive the common sense. I will describe it mathematically.

1=objects, 2=devices or machines (eyes), 3=observers (brain)

This case is very complicated, and it is almost artificial intelligence, according to John von Neumann.
At first, our eyes are not devices like Google Glass.
When you see the objects, the information is sent to your brain. If your brain were broken like me, this must be disorder.

Math isn't chaos.

However, your eyes and brain are connected, so your perception may be sometimes different from others. You are the subject, not objects like devices. You may say that reality is abstract. You are also egocentric. This is our existential problem. You may be the transcendence like Buddha, and you are myself and there is no boundary in this world. I see and you also see. This is integration, no duality.

Math isn't religion.

We share common knowledge. This is commutative.

	A	B	C	D	E	F	G
A	○	×	○	○	○	○	○
B	×	○	×	○	○	○	○
C	○	×	○	○	○	○	○
D	○	○	○	○	○	○	○
E	○	○	○	○	○	×	○
F	○	○	○	○	×	○	○
G	○	○	○	○	○	○	○

[X,Y]=XY-YX≠0→×=noncommutative

[X,Y]=XY-YX=0→○=commutative

A and B are noncommutative, and B and C are also noncommutative. They are linked.

S={A,B,C,D}

S and S' are commutative. A-B-C is noncommutative, so it isn't S'. D and G are commutative, so they are S'. E-F is noncommutative, but D and G are commutative. Therefore, it is S'.

S'={D,E,F,G}

S'' is double commutative, so it includes D and G.

D is the center, but G connect S and S'.

D and G are commutative, so S and S'' are commutative.

S=S''

Quaternion

Prime numbers include the imaginary space in two dimensions. William Rowan Hamilton has found the quaternions. It extends complex numbers.

H is the sum of quaternions.
R is space of numerical vectors.

H={(a,b,c,d)|a,b,c,d∈R}
H=a+bi+cj+dk

i^2=j^2=k^2=ijk=-1

-1=ijk
-i=i^2jk
-i=-jk
∴i=jk

×	1	i	j	k
1	1	i	j	k
i	i	-1	k	-j
j	j	-k	-1	i
k	k	j	-i	-1

∴
ij=k, ji=-k, jk=i, kj=-i, ki=j, ik=-j

ij≠ji

H=a+bi+cj+dk (i=jk,j=ki,k=ij)

You are dazing.

Hausdorff space

It is hard to define emptiness. Hausdorff space is known as empty set Φ.



X is a topological space.

X=(x,y)

x and y are dots.

U∩V=Φ

U is vicinities of x.
V is vicinities of y.

X×X→Δ={(x,x)|x∈X} is closed.