This is a short resume about what I understood about the masterpiece GEB, from Douglas Hofstadter
"How do we get I's from None I's?"
This book is about 'I' and how a bunch of many things construct another thing.
Looking for the construction: particles -> atoms -> molecules -> proteins -> tissues -> organs -> Human Been
Where 'Human been' is 'I'. What makes the amount of meaningless primitives (atoms, proteins ...) be 'I'? What is the relations about those thing and 'I'?
Meaningless things can construct things which can refers to itself. Hofstadter got it making an equivalence between this problem and the Gödel way of represent a mathematical system.
According to Gödel, the formal system:
2+2=4
is equivalent to say:
--p--q----
In another words, they are just [different meaningless] symbols arranged to express the same idea.
The question is: how do we go from meaningless symbols to something that refers to itself and have a meaning?
To deep into this question, We will use five important tools for thinking:
- Isomorphism
- Recursion
- Paradox
- Infinity
- Formal Systems
###Isomorphism:
What is the Isomorphism between a car and a skateboard?
- carries a person
- have four wheels (...)
What we do is construct a map that can also convert to the inverse (car <--> skateboard). You can go either way and preserve informations, preserve structure.
According to Hofstadter definition:
"The word ‘isomorphism’ applies when two complex structures can be mapped onto each other, in such a way that to each part of one structure there is a corresponding part in the other structure, where corresponding’ means that the two parts play similar roles in the respective structures."
###Recursion:
The concept of recursion here used is not different of the habitual concept that we use in our days. One simple example of recursion just to illustrate the Idea is the Fibonacci Series:
F(0) = 0
F(1) = 1
F(n) = F(n-1)+F(n-2)
The point about recursion is that we define the thing from itself. On the case of Fibonacci sequence, we use two previous values of the sequence to achieve the next one. This is the use of itself in a lower level to construct a more complex structure.
Another example of recursion making primitives structure create a more complex one is the case of fractals. Fractals clearly uses the concept of 'using small structures to create a more complex one":
This image show how several less complex identical structures (triangles) can arrange themselves to create a more complex structure.
Trivia: there is a simple way to prove that fractals lives in a dimension between 1 and two (this one that we have just showed lives in d=lg3). In a near future this prove can be appended into this post.
###Paradox
The definition of paradox is a seeming true statement (or group of statements) that lead to a contradiction or a situation which seems to defy logic or intuition. [from Wikipedia]
One good way to show a paradox is the Zeno's paradox. The idea behind the Zeno's paradox is:
If you wish to move from some point A to another point B (B different from A), first you must move half the way. After that, you must move from half the way to the half of the remainder path, an than move half, and half ... infinitely you will have to move half the remainder path, so, the conclusion is, the movement will never ends.
####Veridical Seems valid. I.e. Zeno's paradox ####Fasidical Seems absurd. I.e. Birthday Paradox ####Antinomy Real paradoxical sentences. Lies in sentences like: "This sentence is not true". Here we can put also the Russel's paradox (or Barber's paradox). The Russel's paradox says: For a set of name Omega which contains all sets that doesn't contains themselves, does Omega contains itself? (At this moment I recommend you to stop and conjecture a little about this paradox. It is very important for us, and changed a lot the history of the Math. For further information, I also recommend you to read the book Logicomix).
###Infinity Nothing to much to say about it except what I presume you already know.
###Formal Systems A formal system is nothing more than a set of symbols and rules which are used to derive an expression from one or more premises.
A simple example showed in the book is the MU problem, where we have three symbols: M, I and U, and the following rules to proceed our derivation:
- Add a U to the end of any string ending in I. For example: MI to MIU.
- Double any string after the M (that is, change Mx, to Mxx). For example: MIU to MIUIU.
- Replace any III with a U. For example: MUIIIU to MUUU.
- Remove any UU. For example: MUUU to MU.
With this rules and symbols, take the axiom:
MI
and derive it into MU using the rules above (now is when you stop reading and try it by yourself).