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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Received on Wed Feb 11 2009 - 13:01:45 PST