On 12 Feb 2009, at 18:17, John Mikes wrote:
> 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 <marchal.domain.name.hidden>
> 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?
The religion of the mystic, perhaps. But like consciousness (true but
you can't prove it), which can be seen has an elementary mystical
state already.
Surely John, you can sole the problem of extending the sequence I II
III ...
That is the mystery. Later we will see that the "..." is even
necessarily ambiguous, but then we (the humans) can progress.
>
> If you teach: III and IIIIIII "mean" 3 and 7, then you said
> nothing, just named them.
I said nothing indeed, but I did not even name them. If you look
carefully. Obviously by "7" I was referring to the meaning I am
supposing you already know.
>
> 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.
I did not switch. Cardinal numbers (and natural number are both
cardinal and ordinal numbers) are the elementary quanta of quantities.
>
> 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.
I am hesitating but I think I will do it. The order of the
presentation is not easy to decide. Thanks for your "cannot wait". Asap!
>
>
>
>> 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)
yes but a position notation emphasize the role of position. The Romans
like the Greeks were a bit crazy about the Decade.
Best regards,
Bruno
>
>
>
>
>
>
>> (...)
>> 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.
>>
>> (...)
>>
>> Your teachings made an enjoyable reading, thank you. I confess: I
>> 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.
>>
>> (...)
>
> 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 <marchal.domain.name.hidden>
>> 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/
>
>
>
>
>
> >
http://iridia.ulb.ac.be/~marchal/
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Received on Fri Feb 13 2009 - 13:07:55 PST