# Re: The Seventh Step 1 (Numbers and Notations)

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

> Dear Bruno, just lightening up a bit...you know that I graduated
> already from 2nd yr grade school so I have an open mind criticizing
> high science.
>
> 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.

> If you teach: III and IIIIIII "mean" 3 and 7,  then you said
> nothing, just named them.

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.

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

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.

> No content meant. Quantity???(vs. number?)
> Having 10 digits on 2 hands is the 2nd mental evolutionary step
> after recognizing 5 digits on 1 hand, which was the earlier stage
> (among others old Hungarians  had that and a folks music in
> pentatonic scale). The 'ancient' computer-folks have ony 2 digits on
> their mind, Yin and Yang (0 and 1) and voila they made lots of
> marvels from this simplified system already. (You have that).  And
> the French? with quatrevingtdix for nonante? XC is not XX-XX-XX-XX-
> X  - Romans still recognizing the '5' as a basic tenet (V, L, D,) as
> cornerstones in their number system.
>
> Also your digital 0,9,8,7,6 and then 5,4,3,2,1 was trouble in
> ancient Rome..
> The Romans had no zero, yet used a (quasi) decimal system.

"decimal"? Without zero there is no position based notation for the
number.

> However they did not write IIII rather IV and then for 9: IX
> anticipating V and X as the next one. They also subtracted 4 from 7
> as counting backwards: like 7,6,5,4, which made 7-4=4 in all
> calendar countings which was based on the subtraction of day-numbers
> from the next 'fix' day in the month. Can you figure the
> consequences of this in paying interest (or taxes?)
> (That may be the reason why Muslims are banned from counting
> interest).
> 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.
>
> About the 12 digital creation: In J.Cohen - J.Stewart ('Chaos' and
> 'Reality') the Zarathustran 'aliens' had an 8 based thinking
> (octimal) as best and perfect. Well, 10 gives a prime after one
> halfing, 12 after two, 8 after 3. I think there were 12 digit
> creatures but failed. 10 proved practical - maybe not because of the
> decimal as best mathematical system. It just survived...
>
> did not 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.
>
> If I should ask a question: how would one note 1 billion on the
> planet of centipeds with 8 fingers on all 100 feet? (Don't answer,
> please). (Q2: which billion? the 1000M or the MM?)

Thanks for those kind and funny remarks and questions,

Best,

Bruno

>
> 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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