Re: The Seventh Step 1 (Numbers and Notations)

```My present inserts in Italics - some parts of the posts erased for brevity
John```
```

On Thu, Feb 12, 2009 at 10:32 AM, Bruno Marchal <marc...@ulb.ac.be> wrote:

>
>  On 11 Feb 2009, at 23:46, John Mikes wrote:
>

>             (...)
>
Not that if I see  'I'  that means 1, but if I see 'III' that does not
> mean 3 to me, it means 111. You have to teach first what those funny
> 'figures'  (3,7,etc.) mean.
>
> I don't have to do that. If you follow the thread, you will even understand
> why I cannot do that. *The existence and nature of numbers as well as our
> understanding of it will remain a mystery.* But assuming comp (and thus
> the numbers), we can understand why this is a necessary mystery. It is part
> of the unbridgeable gap which has to remain if we want to remain bot
> scientist and consistent.
>  *JM: *
>
*like a religion?*

>   If you teach: III and IIIIIII "mean" 3 and 7,  then you said nothing,
> just named them.
>
> Br:
> That was my point. To talk on notation. I just hope people understand
> enough the number so that if I ask them to give me 3 euros, they will not
> give me two or four.
>

>  *JM:* *now you swithch to quantity.*
>

>  BR: Later we will axiomatize the theory of numbers. But I prefer to wait
> to be sure people understand the notion of number before axiomatizing. If I
> do the axiomatization too early, some people will believe I am rigorously
> defining the numbers, but this is a grave error. I will axiomatize the
> number to reason about them and to interview machines about the numbers.
>
**
*JM: "interviewing machines" is no evasion of the topic. Axiomatizing in my
vocabulary means to invent some unreal statement that justifies the
otherwise not justified theory. I don't fight it in this case: with your
numbers it may be (excusably) needed.*

>
> Numbers are as mysterius as consciousness and time. That is why
> mathematicians does not even try. But wait for the next thread, *I will
> give a definition of numbers* (which sometimes makes some mathematician
> believed we have a definition). But it will not be a definition, just a
> representation in term of another notion, av-ctually the notion of set. of
> course the notion of set is richer and even less definable than numbers.
>
*JM: can't wait for your definition. Set is introduced? a "many" looking
like a "one"? with lots of characteristics hidden? A table of 9 loose
letters is no 'set' **by itself. *
**

>
>
>  No content meant. Quantity???(vs. number?)
> (...) [to: Romans...]
>
> Br: "decimal"? Without zero there is no position based notation for the
> number.
>

>  *JM: I consider a decimal system as more than just positioned numbers*
>  *The Romans emphsised the exceptional role of 10 (X) 100 (C) 1000(M)
> (even if I play down V,L,D as auxilieries)*
>

>
>
>
>
>
>  (...)
> I think your teaching is fine, but one has to know it before learning it.
> And: as a nun said to a friend when she had questions 'upon thinking':
> "you  should not "think", you should believe.
>
> (...)
>
> count the 'I'-s just believed that there are 2009 of them. It is not
> magical, in other calendar-countings the year has quite different number of
> 'I'-s.
>
> (...)
>
> Br: Thanks for those kind and funny remarks and questions,
>
> Best,
>
> Bruno
>
>  *JM:I take it lightly*
> *John M*
>
>
>
> On Wed, Feb 11, 2009 at 1:01 PM, Bruno Marchal <marc...@ulb.ac.be> wrote:
>
>>
>> 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/
>>
>>
>>
>>
>>
>>
>>
>>
>  http://iridia.ulb.ac.be/~marchal/
>
>
>
>
> >
>

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