Re: Set of prime numbers

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• The operation of the law of small numbers in http://www.naturalsciences.be/expo/old_ishango/en/ishango/riddle.html reminds me of some messages on this list.
Message 1 of 10 , Dec 2, 2009
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The operation of the law of small numbers in
http://www.naturalsciences.be/expo/old_ishango/en/ishango/riddle.html
reminds me of some messages on this list.

David
• ... Sorry for the late response to this thread but I ve been rather tied up with Real Life(tm) recently. There is a persuasive suggestion that the divisibility
Message 2 of 10 , Dec 23, 2009
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On Tue, 2009-12-01 at 23:54 +0000, Phil Carmody wrote:

> > > and was certainly known some 1000
> > > (maybe 25 000) years earlier.
> >
> > I'm intrigued by the "certainly";
> > I would have said "probably" for 1k BCE.
>
> I'd have said "definitely" for >3k BCE. Base 60 just screams
> knowledge of divisibility properties.

Sorry for the late response to this thread but I've been rather tied up
with Real Life(tm) recently.

There is a persuasive suggestion that the divisibility properties of
radix-60 arithmetic is a consequence of its choice, not a reason for its
choice. The argument goes as follows.

A number of cultures have independently invented quinary arithmetic, for
reasons which should be obvious. There are still relics of this in
modern culture --- the five-bar-gate tallying method, for instance.
Bi-quinary has also been widely used throughout history. This uses four
different symbols for the digits 1-4 (the symbols are frequently 1 to 4
identical lines or dots) and another symbol for 5. Digits 6 through 9
are then represented by the juxtaposition of the 5-symbol and the
appropriate symbol for 1 through 4.

A number of cultures have independently invented duodecimal arithmetic.
Many relics of this exist: 12 ounces to the Troy pound; 12 inches to the
foot; 12 pennies to the shilling and so on. The most convincing
survivors to my mind are the survival of the English words "dozen" and
"gross".

Some time around 4000 to 3500 BCE the Sumerians moved into Mesopotamia
and merged with a pre-existing culture. One culture used quinary or
bi-quinary and the other duodecimal. Neither culture supplanted the
other, rather their notations merged. Indeed, the symbols of early
Mesopotamian arithmetic and accounting documents show strong evidence
for a bi-quinary (later decimal) sub-structure in the sexagesimal
notation.

If need be, I'll try and dig up the references from my catastrophically
disorganized library.

Paul
• ... I learnt to count like that in Chi-Nyanja: modzi : one wiri : two tatu : three nai : four sanu :
Message 3 of 10 , Dec 23, 2009
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Paul Leyland <paul@...> wrote:

> Digits 6 through 9 are then represented by the juxtaposition
> of the 5-symbol and the appropriate symbol for 1 through 4.

I learnt to count like that in Chi-Nyanja:

modzi : one
wiri : two
tatu : three
nai : four
sanu : hand
sanu ndi modzi : hand-and-one
sanu ndi wiri : hand-and-two
sanu ndi tatu : hand-and-three
sanu ndi nai : hand and-four
khumi : all-together

It seemed much more sensible than counting in French :-)

David
• ... That looks like the choice of 60 precisely because of its divisibility properties. They didn t take the LCM and later make a shock discovery that it had
Message 4 of 10 , Jan 3, 2010
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--- On Wed, 12/23/09, Paul Leyland <paul@...> wrote:
> On Tue, 2009-12-01 at 23:54 +0000, Phil Carmody wrote:
> > > > and was certainly known some 1000
> > > > (maybe 25 000) years earlier.
> > >
> > > I'm intrigued by the "certainly";
> > > I would have said "probably" for 1k BCE.
> >
> > I'd have said "definitely" for >3k BCE. Base 60 just screams
> > knowledge of divisibility properties.
>
> Sorry for the late response to this thread but I've been
> rather tied up
> with Real Life(tm) recently.
>
> There is a persuasive suggestion that the divisibility
> properties of
> radix-60 arithmetic is a consequence of its choice, not a
> reason for its
> choice.  The argument goes as follows.
>
> A number of cultures have independently invented quinary
> arithmetic, for
> reasons which should be obvious.  There are still
> relics of this in
> modern culture --- the five-bar-gate tallying method, for
> instance.
> Bi-quinary has also been widely used throughout
> history.  This uses four
> different symbols for the digits 1-4 (the symbols are
> frequently 1 to 4
> identical lines or dots) and another symbol for 5.
> Digits 6 through 9
> are then represented by the juxtaposition of the 5-symbol
> and the
> appropriate symbol for 1 through 4.
>
> A number of cultures have independently invented duodecimal
> arithmetic.
> Many relics of this exist: 12 ounces to the Troy pound; 12
> inches to the
> foot; 12 pennies to the shilling and so on.  The most
> convincing
> survivors to my mind are the survival of the English words
> "dozen" and
> "gross".
>
> Some time around 4000 to 3500 BCE the Sumerians moved into
> Mesopotamia
> and merged with a pre-existing culture.  One culture
> used quinary or
> bi-quinary and the other
> duodecimal.   Neither culture supplanted the
> other, rather their notations
> merged.   Indeed, the symbols of early
> Mesopotamian arithmetic and accounting documents show
> strong evidence
> for a bi-quinary (later decimal) sub-structure in the
> sexagesimal
> notation.

That looks like the choice of 60 precisely because of its divisibility properties. They didn't take the LCM and later make a shock discovery that it had all the factors of the two original numbers, shall we say.

Phil
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