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| Lester Zick |
 Posted: Thu Oct 21, 2004 1:15 pm |
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On Thu, 21 Oct 2004 11:02:03 -0400, "robert j. kolker"
<nowhere@nowhere.net> in comp.ai.philosophy wrote:
[quote:2c35804725]
Lester Zick wrote:
This post was originally written six months ago but is still relevant.
- LZ
The Origin of Mathematics Symbols
--------------------------
Let's indulge in a little historical speculation on the origin of
mathematical symbols.
The math symbols [+ - = < >] have been in use in the west since the
late 15th century [-] to be followed subsequently by [+] by the early
16th century. The equal sign [=] is noted by the mid 16th century and
the [< >] symbols around the first third of the 17th century. (The
reference for this is an article at ThinkQuest.org)
These symbols (with the exception of = > and < ) are purely conventional.
[/quote:2c35804725]
So are the "+" for addition and the "-" symbol for subtraction or
difference. In fact in chronological terms the minus symbol takes
historical precedence, and my speculation is that etymologically
"+" just represents compound subtraction, "=" identification of
commutative properties etc.
[quote:2c35804725]It just occured to me that O for zero i.e. nothing is a very natural and
suggestive symbol. The hollow figure does suggest emptiness and
nothingness.
[/quote:2c35804725]
I can agree with this. However 0 for zero is a collateral conceptual
development occuring long after basic arithmetic concepts emerged in
western science.
[quote:2c35804725]Of course we know that zero is not nothing, it is the
identity for the additive group. The same symbol or one very much like
it represents the empty set. The mathematician C.S. Peirce used some
schematic or quasi pictorgraphic symbols in his works on logic. They did
not make the cut because they were not discursive (all in a line) and
were not loved by the typesetting people. Now that we have computer
generated symbols it seems that we could add to the mathematical
repetoire suggestive or pictographic symbols. Somewhite like the circle
through the integral sign indicating a line integral about a closed path.
It would be very interesting to identify the symbols that, in some
sense, "look like" the concepts they represent. We have a few already,
to wit = > < O. To this I would add the arrow -> indicating approach as
in limit or convergence. The arrow suggests "go this way".
We take very pride in our phonetic alphabet but we use quite a few
pictograms in every day activities. What about the red slash through
some object meaning not or dismissal or the heart meaning love, as in I
heart> New York?
[/quote:2c35804725]
Didn't the circle-slash originate with traffic signs? I can remember a
couple of guys who were 86'd from a bar with little signs containing
their names with a circle-slash around them. Quite amusing.
Regards - Lester |
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| Lester Zick |
| Posted: Thu Oct 21, 2004 1:20 pm |
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On Thu, 21 Oct 2004 11:18:03 -0500, Albert <alwagner@tcac.net> in
comp.ai.philosophy wrote:
[quote:7dc85d7702]Lester Zick wrote:
snip
Personally, I suspect that math symbols originated just as did
software jargon, by borrowing from symbols commonly used by
working people; i.e. the '-' probably started life as just a tick
mark, as digits were checked off on a list. Then the '+' was a
'-' slashed through because it had been marked mistakenly,
thereby returning it to it's previously unmarked state; i.e. '2'
and '+2' were equivalent.
[/quote:7dc85d7702]
But your speculation is contradicted by the historical emergence of
the "-" for subtraction before the "+" for addition. Besides your idea
would have required a left hand ticker and southpaws were definitely
frowned upon as sinister from Roman times onward.
[quote:7dc85d7702]--
"Don't you see that the whole aim of Newspeak is to narrow the
range of thought? In the end we shall make thoughtcrime literally
impossible, because there will be no words in which to express it."
-- George Orwell as Syme in "1984"
[/quote:7dc85d7702]
Regards - Lester |
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| Lester Zick |
| Posted: Thu Oct 21, 2004 1:24 pm |
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On Thu, 21 Oct 2004 18:45:50 +0100, David Hartley <me9@privacy.net> in
comp.ai.philosophy wrote:
[quote:ad14c38df7]In message <10nfoaub5n70ib7@corp.supernews.com>, Albert
alwagner@tcac.net> writes
Lester Zick wrote:
snip
Personally, I suspect that math symbols originated just as did software
jargon, by borrowing from symbols commonly used by working people; i.e.
the '-' probably started life as just a tick mark, as digits were
checked off on a list. Then the '+' was a '-' slashed through because
it had been marked mistakenly, thereby returning it to it's previously
unmarked state; i.e. '2' and '+2' were equivalent.
I thought '+' was generally supposed to have evolved from 'et'.
[/quote:ad14c38df7]
Hard to say where it originated exactly. My own speculation seems
consistent with the history of the signs themselves.And it coordinates
very nicely with my own ideas of processes of arithmetic definition in
mechanical terms.
Regards - Lester |
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| Ken Pledger |
| Posted: Thu Oct 21, 2004 1:47 pm |
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In article <41787e59.6568672@netnews.att.net>,
lesterDELzick@worldnet.att.net (Lester Zick) wrote:
[quote:265ee64aed]On Thu, 21 Oct 2004 09:05:01 +1300, Ken Pledger
Ken.Pledger@mcs.vuw.ac.nz> in comp.ai.philosophy wrote:
In article <41775733.83469314@netnews.att.net>,
lesterDELzick@worldnet.att.net (Lester Zick) wrote:
....
Let's indulge in a little historical speculation on the origin of
mathematical symbols....
There's no need to speculate where a lot of historical research
has been done. The standard reference is Florian Cajori, "A History of
Mathematical Notations." Brief summaries of origins of symbols can also
be found on Jeff Miller's web site
http://members.aol.com/jeff570/mathsym.html
Ken Pledger.
My objective is the sequence of development of math concepts and the
reflection of that in symbol derivation rather than simple history.
Regards - Lester
[/quote:265ee64aed]
Well, to temper this fanciful thread with a sample fact, here's
why Robert Recorde invented the "=" sign (cf. Cajori Vol.1, pp.164-5).
In "The Whetstone of Witte" (1557) he wrote:
And to avoide the tediouse repetition of these woordes: is equalle to: I
will sette as I doe often in woorke use, a paire of paralleles, or
Gemowe lines of one lengthe, thus: ======, bicause noe. 2. thynges, can
be moare equalle.
Ken Pledger. |
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| Lester Zick |
| Posted: Thu Oct 21, 2004 2:14 pm |
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On Fri, 22 Oct 2004 08:47:59 +1300, Ken Pledger
<Ken.Pledger@mcs.vuw.ac.nz> in comp.ai.philosophy wrote:
[quote:9d417a0de3]In article <41787e59.6568672@netnews.att.net>,
lesterDELzick@worldnet.att.net (Lester Zick) wrote:
On Thu, 21 Oct 2004 09:05:01 +1300, Ken Pledger
Ken.Pledger@mcs.vuw.ac.nz> in comp.ai.philosophy wrote:
In article <41775733.83469314@netnews.att.net>,
lesterDELzick@worldnet.att.net (Lester Zick) wrote:
....
Let's indulge in a little historical speculation on the origin of
mathematical symbols....
There's no need to speculate where a lot of historical research
has been done. The standard reference is Florian Cajori, "A History of
Mathematical Notations." Brief summaries of origins of symbols can also
be found on Jeff Miller's web site
http://members.aol.com/jeff570/mathsym.html
Ken Pledger.
My objective is the sequence of development of math concepts and the
reflection of that in symbol derivation rather than simple history.
Regards - Lester
Well, to temper this fanciful thread with a sample fact, here's
why Robert Recorde invented the "=" sign (cf. Cajori Vol.1, pp.164-5).
In "The Whetstone of Witte" (1557) he wrote:
And to avoide the tediouse repetition of these woordes: is equalle to: I
will sette as I doe often in woorke use, a paire of paralleles, or
Gemowe lines of one lengthe, thus: ======, bicause noe. 2. thynges, can
be moare equalle.
[/quote:9d417a0de3]
But surely he was referring to cardinal things? Or not? It also
doesn't suggest where his view of equality originated or what it
amounted to. I'm prepared to be wrong on the issue because it is
certainly speculative. But fanciful?
Regards - Lester |
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| Lester Zick |
| Posted: Thu Oct 21, 2004 2:18 pm |
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On 21 Oct 2004 11:19:39 -0700, rick303@hotmail.com (rick++) in
comp.ai.philosophy wrote:
[quote:479942e192]I imagine you're trying to tie this all together in your theory of
differences, but I suggest that humans started with the idea of
addition first, and somewhat later discovered subtraction, and many
centuries later multiplication and division. Even H.erectus realized
that having 2 turkey legs to eat was better than 1. Also, my pile is
bigger than your pile. People naturally count on their fingers by
successively increasing the number held up, on and on.
Much of this is recorded in evolution of numeral systems and in linguistic
artifacts, e.g. in a set of books by Georges Ifrah.
Two is probably the first numeral to be invented and zero the last.
[/quote:479942e192]
Wouldn't one have been the first numeral invented as evidenced by
singular versus plural? Two is just one of many plural cases, but one,
like zero, is unique.
[quote:479942e192]Two requires the concept that similar things can be grouped and counted.
Zero requires an algebraic leap of abstraction that nothingness can be
a symbol that particpates in arithmetic.
Some languages have singular, dual, and plural cases, e.g. Hebrew.
The dual is almost totally gone from English except in rare exceptions like "pants".
[/quote:479942e192]
Regards - Lester |
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| Glen M. Sizemore |
| Posted: Thu Oct 21, 2004 4:09 pm |
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Alpo: And what if they never saw a blue cube before?
GS: It would still be likely that they could do things that they learned to
do when they learned to see cubes and sense blue. Of course, it goes without
saying that they have been exposed to the contingencies that allow them to
observe their own perceptual behavior. There is also something of the "do
what is described repertoire" involved (i.e., "Imagine a cat with ears like
a donkey's"). One's behavior would have to be under stimulus control of
mands such as "imagine." This is all very difficult and subtle stuff.
"AlphaOmega2004" <OmegaZero2003@yahoo.com> wrote in message
news:1098382711.UMbVw33DJVgoenjt/mc4HA@teranews...
[quote:5defec25ca]And what if they never saw a blue cube before?
"Glen M. Sizemore" <gmsizemore2@yahoo.com> wrote in message
news:20041021133131.185$7P@news.newsreader.com...
"AlphaOmega2004" <OmegaZero2003@yahoo.com> wrote in message
news:1098378750.m6VRBe9nKxVxAi5oC5R0gw@teranews...
What is your "explanation" for seeing a blue cube in the "mind's" eye?
The person is doing some of the same things they learned to do when they
were exposed to the contingencies that produce the behavior called
"seeing
a
blue cube."
"Glen M. Sizemore" <gmsizemore2@yahoo.com> wrote in message
news:20041021115218.959$ZA@news.newsreader.com...
I don't explain behavior with behavior. I explain behavior in terms of
selection contingencies that operate at the level of species,
individuals
and cultures.
"AlphaOmega2004" <OmegaZero2003@yahoo.com> wrote in message
news:1098372240.X0e9IbKrQLBIO3wMdHS5Vg@teranews...
Simply invoking behavior to explain all that exists is no
explanation
at
all.
"Glen M. Sizemore" <gmsizemore2@yahoo.com> wrote in message
news:20041021065559.155$a0@news.newsreader.com...
Someone should tell Dumb Dan that simply inventing "ideas" to
explain
behavior is really no explanation at all.
"dan michaels" <feedbackdroids@yahoo.com> wrote in message
news:8d8494cf.0410202024.46bd5076@posting.google.com...
lesterDELzick@worldnet.att.net (Lester Zick) wrote in message
news:<41775733.83469314@netnews.att.net>...
This post was originally written six months ago but is still
relevant.
- LZ
The Origin of Mathematics Symbols
--------------------------
Let's indulge in a little historical speculation on the origin
of
mathematical symbols.
The math symbols [+ - = < >] have been in use in the west since
the
late 15th century [-] to be followed subsequently by [+] by the
early
16th century. The equal sign [=] is noted by the mid 16th
century
and
the [< >] symbols around the first third of the 17th century.
(The
reference for this is an article at ThinkQuest.org)
Now what I'd like to suggest is that there is some kind of
derivation
implied in the development of these symbols. The symbol [-] is
the
first to have been defined for subtraction in a book from 1489
in
which both addition and subtraction are mentioned but the [+]
symbol
does not occur. And the article concludes that the origin for
the
[-]
is not knowable for that reason.
What is suggested here is that all four of these symbols are
derived
from the symbol for subtraction according to various properties
for
differences in general. In other words it is possible that the
nature
of arithmetic addition was predicated on the taking of
successive
differences such as A + C is taken to mean A - - C.
I imagine you're trying to tie this all together in your theory
of
differences, but I suggest that humans started with the idea of
addition first, and somewhat later discovered subtraction, and
many
centuries later multiplication and division. Even H.erectus
realized
that having 2 turkey legs to eat was better than 1. Also, my pile
is
bigger than your pile. People naturally count on their fingers by
successively increasing the number held up, on and on.
Interestingly, I've just discovered that it's *much much* easier
bio-mechanically to open fingers up successively one at a time
from
a
closed fist than to bend the fingers back into a fist one at a
time
from an open hand. Try it.
And if so it is
also plausible that the use of the [+] sign to indicate
addition
just
represents the compounding of two [-] symbols by crossing them.
Of course we know that differences in general are not
distributive.
However any difference for which the rule [+ = - -] held
would
be
distributive because the double negative difference would be
the
same
as positive original. And we would find that +/- operations
have
the
same logical definitions.
[=] illustrates commutativity for similar reasons and
[< >] symbols reflect neither distributivity nor commutativity
and
reflect composites of difference symbols joined at skewed
angles.
Regards - Lester
[/quote:5defec25ca] |
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| dan michaels |
| Posted: Thu Oct 21, 2004 6:26 pm |
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[quote:fe872b7062]I imagine you're trying to tie this all together in your theory of
differences, but I suggest that humans started with the idea of
addition first, and somewhat later discovered subtraction, and many
centuries later multiplication and division.
A fascinating coincidence, Dan. But after I'd gotten the idea of
differences down and shown that they could be used to define
arithmetic, I decided to check the fossil record to see if there might
be any evidence one way or the other. And lo and behold, there it is,
written evidence that subtraction or differences really came first.
Your idea that addition comes first is certainly the prevalent notion.
I suspect it's a result of Euler's approach to math and definition via
set theory. But the fact remains that we can't define subtraction
through addition without hidden assumptions, but we can define
addition through subtraction as the primary arithmetic operation.
You asked for experimental evidence; there it is.
[/quote:fe872b7062]
I was thinking back to when the conceptual ideas may have been formed
..... ie, somewhat before the mathematical formulations ... maybe by
50,000 years.
[quote:fe872b7062]Even H.erectus realized
that having 2 turkey legs to eat was better than 1. Also, my pile is
bigger than your pile. People naturally count on their fingers by
successively increasing the number held up, on and on.
Interestingly, I've just discovered that it's *much much* easier
bio-mechanically to open fingers up successively one at a time from a
closed fist than to bend the fingers back into a fist one at a time
from an open hand. Try it.
That's probably a function of differences too. At least the difference
of being able to grip remaining fingers while opening the fist one
finger at a time.
[/quote:fe872b7062]
One is much more natural and easier to do than the other. Might that
have any bearing on how the organism learns to conceptualize? They do
this about 5 years before learning the math. |
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| Albert |
| Posted: Thu Oct 21, 2004 6:38 pm |
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Lester Zick wrote:
[quote:69100ff227]On Thu, 21 Oct 2004 11:18:03 -0500, Albert <alwagner@tcac.net> in
comp.ai.philosophy wrote:
Lester Zick wrote:
snip
Personally, I suspect that math symbols originated just as did
software jargon, by borrowing from symbols commonly used by
working people; i.e. the '-' probably started life as just a tick
mark, as digits were checked off on a list. Then the '+' was a
'-' slashed through because it had been marked mistakenly,
thereby returning it to it's previously unmarked state; i.e. '2'
and '+2' were equivalent.
But your speculation is contradicted by the historical emergence of
the "-" for subtraction before the "+" for addition. Besides your idea
would have required a left hand ticker and southpaws were definitely
frowned upon as sinister from Roman times onward.
[/quote:69100ff227]
LOL. Maybe so.
--
"Don't you see that the whole aim of Newspeak is to narrow the
range of thought? In the end we shall make thoughtcrime literally
impossible, because there will be no words in which to express it."
-- George Orwell as Syme in "1984" |
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| Jesse F. Hughes |
| Posted: Fri Oct 22, 2004 3:05 pm |
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lesterDELzick@worldnet.att.net (Lester Zick) writes:
[quote:74d3059b7a]A fascinating coincidence, Dan. But after I'd gotten the idea of
differences down and shown that they could be used to define
arithmetic, I decided to check the fossil record to see if there might
be any evidence one way or the other. And lo and behold, there it is,
written evidence that subtraction or differences really came first.
[/quote:74d3059b7a]
I hesitate to ask, but what evidence could the fossil record provide?
And what has that to do with the very recent introduction of the
current arithmetic notations? After all, mathematics had been done
for *centuries* prior to the introduction of +, -, etc., so these
symbols can't tell us diddly about philosophical or historical
priority.
--
Jesse F. Hughes
"There's a thrill that's gone that I'll probably not have in quite the
same way again. After all, FLT was a unique animal, and we had a
great dance." -J.S. Harris on "proving" Fermat's last theorem |
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| Lester Zick |
| Posted: Fri Oct 22, 2004 5:46 pm |
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On Fri, 22 Oct 2004 23:05:58 +0200, jesse@phiwumbda.org (Jesse F.
Hughes) in comp.ai.philosophy wrote:
[quote:b7a6e6a935]lesterDELzick@worldnet.att.net (Lester Zick) writes:
A fascinating coincidence, Dan. But after I'd gotten the idea of
differences down and shown that they could be used to define
arithmetic, I decided to check the fossil record to see if there might
be any evidence one way or the other. And lo and behold, there it is,
written evidence that subtraction or differences really came first.
I hesitate to ask, but what evidence could the fossil record provide?
[/quote:b7a6e6a935]
Evidence of your extraordinary obtusity perhaps.
[quote:b7a6e6a935]And what has that to do with the very recent introduction of the
current arithmetic notations? After all, mathematics had been done
for *centuries* prior to the introduction of +, -, etc., so these
symbols can't tell us diddly about philosophical or historical
priority.
[/quote:b7a6e6a935]
Well, they can certainly tell us diddly about your philosophical and
historical priority.
Regards - Lester |
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| Jesse F. Hughes |
| Posted: Fri Oct 22, 2004 7:03 pm |
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lesterDELzick@worldnet.att.net (Lester Zick) writes:
[quote:178ebb12b6]On Fri, 22 Oct 2004 23:05:58 +0200, jesse@phiwumbda.org (Jesse F.
Hughes) in comp.ai.philosophy wrote:
lesterDELzick@worldnet.att.net (Lester Zick) writes:
A fascinating coincidence, Dan. But after I'd gotten the idea of
differences down and shown that they could be used to define
arithmetic, I decided to check the fossil record to see if there might
be any evidence one way or the other. And lo and behold, there it is,
written evidence that subtraction or differences really came first.
I hesitate to ask, but what evidence could the fossil record provide?
Evidence of your extraordinary obtusity perhaps.
[/quote:178ebb12b6]
Perhaps. But my obtusity isn't directly relevant for your claim.
[quote:178ebb12b6]And what has that to do with the very recent introduction of the
current arithmetic notations? After all, mathematics had been done
for *centuries* prior to the introduction of +, -, etc., so these
symbols can't tell us diddly about philosophical or historical
priority.
Well, they can certainly tell us diddly about your philosophical and
historical priority.
[/quote:178ebb12b6]
Very illuminating response, that. My questions remain.
--
Jesse F. Hughes
"C is for Cookie. That's good enough for me."
Cookie Monster |
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| Christopher J. Henrich |
| Posted: Fri Oct 22, 2004 9:04 pm |
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In article <876552xy2x.fsf@phiwumbda.org>, Jesse F. Hughes
<jesse@phiwumbda.org> wrote:
[quote:f9c8d45321]lesterDELzick@worldnet.att.net (Lester Zick) writes:
A fascinating coincidence, Dan. But after I'd gotten the idea of
differences down and shown that they could be used to define
arithmetic, I decided to check the fossil record to see if there might
be any evidence one way or the other. And lo and behold, there it is,
written evidence that subtraction or differences really came first.
I hesitate to ask, but what evidence could the fossil record provide?
Well, I think Mr. Zick was not being entirely serious in his choice of[/quote:f9c8d45321]
the phrase "fossil record" for "really old books."
[quote:f9c8d45321]
And what has that to do with the very recent introduction of the
current arithmetic notations? After all, mathematics had been done
for *centuries* prior to the introduction of +, -, etc., so these
symbols can't tell us diddly about philosophical or historical
priority.
Notations, and other means of expression, are certainly influenced by[/quote:f9c8d45321]
what their introducers are trying to express. Conversely, they affect
what their users /can/ express, and perhaps influence what their users
will think. (Actually, there is a lot less to this than meets the eye;
it goes back to an argument that language molds thought, introduced by
Whorf, and severely contested by later linguists.)
In the present discussion I think Mr. Hughes has a better case. Our
notations for arithmetic go back to 1400 or maybe 1200 at the earliest.
But people in Egypt and Babylonia mst have been adding and subtracting.
Some of their math "books" are still extant. Now, do these have
anything to suggest that subtraction is prior to addition?
--
Chris Henrich
God just doesn't fit inside a single religion. |
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