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 <[email protected]> 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. > > (...) > > 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 <[email protected]> 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/ > > > > > > > --~--~---------~--~----~------------~-------~--~----~ You received this message because you are subscribed to the Google Groups "Everything List" group. 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