Is There a Multi-dimensional Mathematical World Hidden in the Brain’s Computation?

Algebraic Topology” can detect the Multi-dimensional neural network in our brain – by studying the Homology (同调) and co-Homology (上同调) with the help of Linear Algebra (multi-dim Matrix) &  Computers.

Homology = compute the number of “holes” in multi-dim space. 

Neurons formed in the brain can be modeled in Math (Topology) by Simplex 单纯 (plural : Simplices), billions of them interconnected into a complex – “Simplicial Complex” (单纯复体)。

What is math? 

What is Math ? Interesting article below:

  • Mathematics = “that which is learned“ –(Pythagoras)

Math is not about calculation, it is understanding the nature, the universe, the philosophy (logic, intelligence – both “human” and “artificial”)…

What is Axiom, Lemma, Proposition ? Why rigorous Calculus was needed hundred years after Newton & Leibniz had invented it – “Epsilon-Delta” Analysis.

Difference between Riemann Integral & Lebesgue Integral ?

Parallel and Concurrent Haskell 


1-1: Introduction

Everything in Haskell is PURE (function), including side effects (print, I/O such as open files, update data, …)

“Pure”: f (x) = a, regardless of ‘x’ value may change, always returns the result ‘a’

2.1 Function

f a b = function f, arg a & b

sqDist (x, y ) = x^2 + y^2
main = print $ sqDist (3, 4)
dist pt  = sqrt  $ sqDist pt

dist = sqrt . sqDist

Note: pronounce “.” as “AFTER

id  :: a -> a [Signature] # Polymorphic
id  x = x [Implementation]

flop :: (a, b) -> (b, a)
flop p = = (snd p,fst p)

flop’ (x,y) = (y,x)

print $ flop (5, “Hello!”)

sqDist :: Double ->Double -> Double
Equivalent to: sqDist 1 arg but return a function (Double -> Double)
sqDist :: Double -> (Double -> Double)

sqDist x y = x^2 + y^2

Equiv: sqDist (x,y) # but difficult to partial application.

Partial Application (pass fewer args)
sqDist 0 = Double -> Double

:: Double -> Double

= sqDist 0

Parallel and Concurrent Haskell: