Hi Kim,

I told you that to grasp the seventh step we have to do some "little"  
amount of math.
Now math is a bit like consciousness or time, we know very well what  
it is, but we cannot really define it, and such an encompassing  
definition can depend on the philosophical view you can have on "the  
mathematical reality".

So, if I try to be precise enough so that the math will be applicable,  
not just on the seventh step, but also on the 8th step and eventually  
for the sketch of the AUDA, that is the arithmetical translation of  
the universal dovetailer argument, I am tempted by providing the  
philosophical clues, deducible from the comp hypothesis, for the  
introduction to math.

But I realize that this would entail philosophical discussion right at  
the beginning, and that would give to you the feeling that, well,  
elementary math is something very difficult, which is NOT the case.  
The truth is that philosophy of elementary math is difficult.

So I have change my mind, and we will do a bit of math. Simply. It is  
far best to have a practice of math before getting involved in more  
subtle discussion, even if we will not been able to hide those  
subtleties when applying the math to the foundation of physics and  
cognition.

I propose to you a shortcut to the seventh step. It is not a thorough  
introduction to math. Yet it starts from the very basic things.

Let us begin. What I explain here is standard, except for the  
notations, and this for mailing technical reason.

I guess you have heard about the Natural Numbers, also called Positive  
Integers. By default, when I use the word number, it will mean I am  
meaning the natural number.

I guess you agree with the statement that 0 is equal to the number of  
occurrence of the letter "y" in the word "spelling". OK?

Then you have the number 1, 2, 3, 4, etc. OK? They are respectively  
equal to the number of stroke in I, II, III, IIII, etc. OK?

Of course the number four is not equal to "IIII". But the string, or  
sequence of symbols "IIII" is a good notation for the number four. The  
notation is good in the sense that it is quasi self-explaining. To see  
what number is denoted by a string like "IIIIIIIIIII": just count the  
strokes. OK?

If that stroke sequences are conceptually good for describing the  
numbers, it happens that it is horrible for using them, and you are  
probably used to the much more modern positional notation for the  
number. If I ask you which year we are. You will not answer me that we  
are in the year  
IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII 
IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII 
IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII 
IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII 
IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII 
IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII 
IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII 
IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII 
IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII 
IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII 
IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII 
IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII 
IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII 
IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII 
IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII 
IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII
You will most probably tell me that we are in the year 2009.

Is that not a bit magical? The explanation of that "miracle" relies in  
the very ingenuous way we can use our hands to count on our fingers or  
digits. We put 0 on a little finger, and then 1 on the next up to 4,  
and then we use the other hand to continue with 5 on the thumb, 6,  
then 7, then 8, then 9 on the last right fingers. Unfortunately we  
lack fingers to continue, so we will describe the next number by 1  
times the number of finger + 0 unities. We have 10 fingers, meaning 1  
times the number of fingers + 0.

Humans have ten fingers, that is why they use ten symbols 0, 1, 2, 3,  
4, 5, 6, 7, 8, 9.

it is very useful. Later I will perhaps explain that the benefit of  
such a notation is exponential, and in our story the exponential will  
recurre again and again and again .... Indeed a Universal Machine can  
be considered as a generalized exponential, but let me try to not  
anticipate.

For example let us take the number 378. This is an abbreviation of

(8 times 1) + (7 times 10) + (3 times (10 times 10)), or if you  
prefer:  3 times (10 times 10) + 7 times 10 + 8 times 1, like the  
number 2009 is an abbreviation of 9 + (0 times 1) + (0 times 10) +  (0  
times (10 times 10)) + (2 times (10 times 10 times 10).

Are OK with this? As you have learned in school, this notation  
provides method for adding and multiplying the numbers, and I will not  
elaborate on this now.

And now an important question about math (and not philosophy of math).  
What if God created us with 12 fingers? Would the math be different?

Well, there is a planet near Alpha Centaury where God was a bit lazy  
and decide to create creature with only one finger to each hands (and  
yes, they have two hands thanks God).

How could they notate the numbers? Well let us count on their fingers.  
They can use only two symbols, like in the Yi King and in Leibnitz: 0  
and 1. But for the number two, they already have to use the positional  
trick: 2 is really (1 times the number of fingers) + 0 unity: that is  
they wrote two as 10. And three? easy: it is 10 + 1, and this gives  
11. That is three is equal to 1 times (number of fingers) + 1. And four?

Well, let us try the addition trick you have learned:

11
+1


I start at the right, and I compute 1+1, well this gives two, that is  
10, so I write 0, and I report 1:

1
11
+1
---
    0

and 1 + 1 gives 10 again, so we get 100. Let us verify 100, is an  
abbreviation, for those extra-terrestrials for 0 + (0 times two) + (1  
times (two times two), where "two" is the number of fingers they have,  
and this gives indeed four. OK

So we get the number in their "two-fingers" positional system:

0
1
10
11
100
101
110
111
1000
1001
etc.

1001 is the number nine, it is the number of strokes in IIIIIIIII. You  
could feel like if 1001 is already long, but the gain can be shown to  
be still exponential. Indeed you can see that:

0 = 0
2 = 10
4 = 100
8 = 1000
16 = 10000
32 = 100000
...
18446744073709551616 =  
10000000000000000000000000000000000000000000000000000000000000000.
etc.


The last one is (2 times 2 times 2 times ... times 2) with 64 "two".  
64 and the numbers on the left are described in our notation system,  
and on the right their are described in the two fingers system. It is  
a number which can no more be printed on paper on this planet. Indeed  
if you want print it in the stroke notation: indeed it is  
18446744073709551616 strokes long! It is big, but this is relative,  
and is very little compared to the monstrous numbers that universal  
machine can met.


You see that "4", "IIII", and "100" are just different notations for  
the same positive integers. Tell me if you are OK with this.

Mathematical truth will have to be invariant for change of notation.  
Yet when I say that positional notation gives an exponential benefit,  
I am using math (the exponential) to talk about math notations. Well,  
even those truth about notations will have to be invariant for the  
change of notations. This "subtlety" will grow in importance au fur et  
à mesure.

Facultative exercises: 1) try to find a rule for going from the two- 
fingers notation to our ten fingers notation, and vice versa, and 2)  
what about the planet near Vega, where God, very generous that day,  
give 8 fingers to each hands for the creatures there (and yes, they  
have two hands). Hint: those are using the 16 ciphers 0, 1, 2, 3, 4,  
5, 6, 7, 8, 9, A, B, C, D, E, F.

Obligatory home work: 1) keep this post or a copy in a place you can  
find it for later reference. 2) Make sure you are OK everywhere I ask  
you if you are OK?, and if not please ask a question or make a  
comment. There will be errors, for sure.

Next lesson: numbers and other numbers. It should be more interesting,  
but the lesson of today has some role.

Best,

Bruno



http://iridia.ulb.ac.be/~marchal/




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