Hi Kim, Hi Marty and others,
So it is perhaps time to do some math. Obviously, once we are open to
the idea that the fundamental reality could be mathematical, it is
normal to take some time to do some mathematics. Many people seems
also to agree here that the computationalist hypothesis could be
interesting, and this should motivate for some amount of theoretical
computer science, or recursion theory. This is a branch of mathematics
which study computability, and mainly uncomputability, as opposed to
computation theory which study all aspect of computation.
You can easily show that something is computable, by giving an
explicit procedure to compute it. But to show that something is NOT
computable, you need a very solid notion of computability. Now,
computationalism suggest that the interesting and fundamental things,
like life, consciousness, even matter, "lives" somehow on the border
of the computable and the non computable, so that it is perhaps time
to dig a little bit deeper in those direction.
I have already explain this on this list, but never from scratch,
having in mind those who are, for whatever reason, the mathematical
basis.
So I guess that many of view will find those preliminaries a bit too
much simple. yet, by experience, I know that difficulties will appear,
and it is frequent that I met people with very big baggages in
mathematics who have some difficulties to understand the final point.
So I would encourage everyone to be sure everything is clear. For
those who have already a thorough understanding of UDA1-7, and in
particular have grasped the difference between a computation and a
description of a computation, I ask them to just be patient with the
list. There will be nothing new here, nothing really original, and
nothing controversial. All what I will explain has been anticipate by
Emil Post in the 1920, and found or rediscovered by many
mathematicians independently in USA, and in the ex USSR.
The present post just give a preview, and the beginning . I intent to
send only short posts, or the shorter as possible. So here is the plan.
1) Set (probably a dozen of posts)
2) Function (I don't know how many posts, could depend on the replies.
It is the key notion of math)
3) Language, machine and computable function
4) Universal machine, universal language, universal function,
universal dovetailer, universal number ...
How will I proceed?
By exercise only. I will ask question, and I will wait for either the
answer. I expect that those "who know" wait for Kim's answer, or for
answer by those who are not supposed to "know".
Kim seems to courageously accept the role of the candid, but if
someone else want to answer it is OK for me.
I will give only VERY SIMPLE exercise. The goal is to be short and
simple, and as informal as possible. I will explain by examples, and I
will avoid as much as possible tedious definitions. And when we will
meet a more difficult question, I will just solve it myself, and even
explain why it is difficult. So, those preliminaries will not be very
funny. I will of course adapt myself to the possible replies, and fell
free to comment or even metacomment.
Obviously this is a not a course in math, but it is an explanation
from scratch of the seven step of the universal dovetailer argument.
It is a shortcut, and most probably we will make some digression from
time to time, but let us try not to digress too much.
Kim, you are OK with this? I have to take into account the problem you
did have with math, and which makes this lesson a bit challenging for
me, and I guess for you too.
I begin with the very useful and elementary notion of set, as
explained in what is called "naive set theory", and which is the base
of almost all part of math.
============================================= begin
===============================
1) SET
Informal definition: a set is a collection of object, called elements,
with the idea that it, the collection or set, can be considered itself
as an object. It is a many seen as a one, if you want. If the set is
not to big, we can describe it exhaustively by listing the elements,
if the set is bigger, we can describe it by some other way. Usually we
use accolades "{", followed by the elements, separated by commas, and
then "}", in the exhaustive description of a set.
Example/exercise:
1) The set of odd natural numbers which are little than 10. This is a
well defined, and not to big set, so we can describe it exhaustively by
{1, 3, 5, 7, 9}. In this case we say that 7 belongs to {1, 3, 5, 7, 9}.
Exercise 1: does the number 24 belongs to the set {1, 3, 5, 7, 9}?
2) the set of even natural number which are little than 13. It is {0,
2, 4, 6, 8, 10, 12}. OK? Some people can have a difficulty which is
not related to the notion of set, for example they can ask themselves
if zero (0) is really an even number. We will come back to this.
3) The set of odd natural numbers which are little than 100. This set
is already too big to describe exhaustively. We will freely describe
such a set by a quasi exhaustion like {1, 3, 5, 7, 9, 11, ... 95, 97,
99}.
Exercise 2: does the number 93 belongs to the set of odd natural
numbers which are little than 100, that is: does 93 belongs to {1, 3,
5, 7, 9, 11, ... 95, 97, 99}?
4) The set of all natural numbers. This set is hard to define, yet I
hope you agree we can describe it by the infinite quasi exhaustion by
{0, 1, 2, 3, ...}.
Exercise 3: does the number 666 belongs to the set of natural numbers,
that is does 666 belongs to {0, 1, 2, 3, ...}.
Exercice 4: does the real number square-root(2) belongs to {0, 1, 2,
3, ...}?
5) When a set is too big or cumbersome, mathematician like to give
them a name. They will usually say: let S be the set {14, 345, 78}.
Then we can say that 14 belongs to S, for example.
Exercise 5: does 345 belongs to S?
A set is entirely defined by its elements. Put in another way, we will
say that two sets are equal if they have the same elements.
Exercise 6. Let S be the set {0, 1, 45} and let M be the set described
by {45, 0, 1}. Is it true or false that S is equal to M?
Exercise 7. Let S be the set {666} and M be the set {6, 6, 6}. Is is
true or false that S is equal to M?
Seven exercises are enough. Are you ready to answer them. I hope you
don't find them too much easy, because I intend to proceed in a way
such that all exercise will be as easy, despite we will climb toward
very much deeper notion. Feel free to ask question, comments, etc. I
will try to adapt myself.
Next: we will see some operation on sets (union, intersection), and
the notion of subset. If all this work, I will build a latex document,
and make it the standard reference for the seventh step for the non
mathematician, or for the beginners in mathematics.
Bruno
http://iridia.ulb.ac.be/~marchal/
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