## Δευτέρα, 27 Φεβρουαρίου 2017

## Πέμπτη, 23 Φεβρουαρίου 2017

## Κυριακή, 19 Φεβρουαρίου 2017

### Θεώρημα Viviani

0_ET_VivianisTheoremG.gif (GIF εικόνα, 470 × 491 εικονοστοιχεία)

https://lh3.googleusercontent.com/-bKTKNwZamt0/WKh_Ib6_RfI/AAAAAAAAJjE/lbhpGdDofmcCCczxJ0w7ITpzB8LkMv3ogCJoC/w470-h491-rw/0_ET_VivianisTheoremG.gif

https://lh3.googleusercontent.com/-bKTKNwZamt0/WKh_Ib6_RfI/AAAAAAAAJjE/lbhpGdDofmcCCczxJ0w7ITpzB8LkMv3ogCJoC/w470-h491-rw/0_ET_VivianisTheoremG.gif

## Παρασκευή, 3 Φεβρουαρίου 2017

### Η ιστορία του μηδενός.

The History of Zero: How Ancient Mesopotamia Invented the Mathematical Concept of Nought and Ancient India Gave It Symbolic Form – Brain Pickings

If

the ancient Arab world had closed its gates to foreign travelers, we

would have no medicine, no astronomy, and no mathematics — at least not

as we know them today.

Central to humanity’s quest to grasp the nature of the universe and

make sense of our own existence is zero, which began in Mesopotamia and

spurred one of the most significant paradigm shifts in human

consciousness — a concept first invented (or perhaps discovered) in

pre-Arab Sumer, modern-day Iraq, and later given symbolic form in

ancient India. This twining of meaning and symbol not only shaped

mathematics, which underlies our best models of reality, but became

woven into the very fabric of human life, from the works of Shakespeare,

who famously winked at zero in

Mathematician

It is, in a sense, an archetypal story of scientific discovery, wherein

an abstract concept derived from the observed laws of nature is named

and given symbolic form. But it is also a kind of cross-cultural fairy

tale that romances reason across time and space

Kaplan writes:

Like all transformative inventions, zero began with necessity — the

necessity for counting without getting bemired in the inelegance of

increasingly large numbers. Kaplan writes:

source of growing confusion for the Sumerians, who wrote by pressing

the tip of a hollow reed to create circles and semi-circles onto wet

clay tablets solidified by baking. The reed eventually became a

three-sided stylus, which made triangular cuneiform marks at varying

angles to designate different numbers, amounts, and concepts. Kaplan

demonstrates what the Sumerian numerical system looked like by 2000 BCE:

This cumbersome system lasted for thousands of years, until someone

at some point between the sixth and third centuries BCE came up with a

way to wedge accounting columns apart, effectively symbolizing “nothing

in this column” — and so the concept of, if not the symbol for, zero was

born. Kaplan writes:

it. The story follows history’s arrow from Mesopotamia to ancient

Greece, where the necessity of zero awakens anew. Kaplan turns to

Archimedes and his system for naming large numbers, “myriad” being the

largest of the Greek names for numbers, connoting 10,000. With his

notion of

came within inches of inventing the concept of powers, but he gave us

something even more important — as Kaplan puts it, he showed us “how to

think as concretely as we can about the very large, giving us a way of

building up to it in stages rather than letting our thoughts diffuse in

the face of immensity, so that we will be able to distinguish even such

magnitudes as these from the infinite.”

This concept of the infinite in a sense contoured the need for naming

its mirror-image counterpart: nothingness. (Negative numbers were still

a long way away.) And yet the Greeks had no word for zero, though they

clearly recognized its spectral presence. Kaplan writes:

continued its odyssey around the ancient world before it was given a

name. After Babylon and Greece, it landed in India. The first surviving

written appearance of zero as a symbol appeared there on a stone tablet

dated 876 AD, inscribed with the measurements of a garden: 270 by 50,

written as “27°” and “5°.” Kaplan notes that the same tiny zero appears

on copper plates dating back to three centuries earlier, but because

forgeries ran rampant in the eleventh century, their authenticity can’t

be ascertained. He writes:

undoubtedly be the mathematician and astronomer Āryabhata, whose

identity is shrouded in as much mystery as Shakespeare’s. Nonetheless,

his legacy — whether he was indeed one person or many — is an indelible

part of zero’s story.

Kaplan writes:

Kaplan goes on to explore how various other cultures, from the Mayans

to the Romans, contributed to the trans-civilizational mosaic that is

zero as it made its way to modern mathematics, and examines its profound

impact on everything from philosophy to literature to his own domain of

mathematics. Complement it with this Victorian love letter to mathematics and the illustrated story of how the Persian polymath Ibn Sina revolutionized modern science.

# The History of Zero: How Ancient Mesopotamia Invented the Mathematical Concept of Nought and Ancient India Gave It Symbolic Form

## “If you look at zero you see nothing; but look through it and you will see the world.”

### By Maria Popova

If

the ancient Arab world had closed its gates to foreign travelers, we

would have no medicine, no astronomy, and no mathematics — at least not

as we know them today.

Central to humanity’s quest to grasp the nature of the universe and

make sense of our own existence is zero, which began in Mesopotamia and

spurred one of the most significant paradigm shifts in human

consciousness — a concept first invented (or perhaps discovered) in

pre-Arab Sumer, modern-day Iraq, and later given symbolic form in

ancient India. This twining of meaning and symbol not only shaped

mathematics, which underlies our best models of reality, but became

woven into the very fabric of human life, from the works of Shakespeare,

who famously winked at zero in

*King Lear*by calling it “an O without a figure,” to the invention of the bit that gave us the 1s and 0s underpinning my ability to type these words and your ability to read them on this screen.

Mathematician

**Robert Kaplan**chronicles nought’s revolutionary journey in

**(**

*The Nothing That Is: A Natural History of Zero**public library*).

It is, in a sense, an archetypal story of scientific discovery, wherein

an abstract concept derived from the observed laws of nature is named

and given symbolic form. But it is also a kind of cross-cultural fairy

tale that romances reason across time and space

Kaplan writes:

If you look at zero you see nothing; but look through itWith an eye to the eternal question of whether mathematics is discovered or invented — a question famously debated by Kurt Gödel and the Vienna Circle — Kaplan observes:

and you will see the world. For zero brings into focus the great,

organic sprawl of mathematics, and mathematics in turn the complex

nature of things. From counting to calculating, from estimating the odds

to knowing exactly when the tides in our affairs will crest, the

shining tools of mathematics let us follow the tacking course everything

takes through everything else – and all of their parts swing on the

smallest of pivots, zero

With these mental devices we make visible the hidden laws controlling

the objects around us in their cycles and swerves. Even the mind itself

is mirrored in mathematics, its endless reflections now confusing, now

clarifying insight.

[…]

As we follow the meanderings of zero’s symbols and meanings we’ll see

along with it the making and doing of mathematics — by humans, for

humans. No god gave it to us. Its muse speaks only to those who ardently

pursue her.

The disquieting question of whether zero is out there or a

fiction will call up the perennial puzzle of whether we invent or

discover the way of things, hence the yet deeper issue of where we are

in the hierarchy. Are we creatures or creators, less than – or only a

little less than — the angels in our power to appraise?

Like all transformative inventions, zero began with necessity — the

necessity for counting without getting bemired in the inelegance of

increasingly large numbers. Kaplan writes:

Zero began its career as two wedges pressed into a wetHaving to reconcile the decimal and sexagesimal counting systems was a

lump of clay, in the days when a superb piece of mental engineering gave

us the art of counting.

[…]

The story begins some 5,000 years ago with the Sumerians, those

lively people who settled in Mesopotamia (part of what is now Iraq).

When you read, on one of their clay tablets, this exchange between

father and son: “Where did you go?” “Nowhere.” “Then why are you late?”,

you realize that 5,000 years are like an evening gone.

The Sumerians counted by 1s and 10s but also by 60s. This may seem

bizarre until you recall that we do too, using 60 for minutes in an hour

(and 6 × 60 = 360 for degrees in a circle). Worse, we also count by 12

when it comes to months in a year, 7 for days in a week, 24 for hours in

a day and 16 for ounces in a pound or a pint. Up until 1971 the British

counted their pennies in heaps of 12 to a shilling but heaps of 20

shillings to a pound.

Tug on each of these different systems and you’ll unravel a history

of customs and compromises, showing what you thought was quirky to be

the most natural thing in the world. In the case of the Sumerians, a

60-base (sexagesimal) system most likely sprang from their dealings with

another culture whose system of weights — and hence of monetary value —

differed from their own.

source of growing confusion for the Sumerians, who wrote by pressing

the tip of a hollow reed to create circles and semi-circles onto wet

clay tablets solidified by baking. The reed eventually became a

three-sided stylus, which made triangular cuneiform marks at varying

angles to designate different numbers, amounts, and concepts. Kaplan

demonstrates what the Sumerian numerical system looked like by 2000 BCE:

This cumbersome system lasted for thousands of years, until someone

at some point between the sixth and third centuries BCE came up with a

way to wedge accounting columns apart, effectively symbolizing “nothing

in this column” — and so the concept of, if not the symbol for, zero was

born. Kaplan writes:

In a tablet unearthed at Kish (dating from perhaps as farBut zero almost perished with the civilization that first imagined

back as 700 BC), the scribe wrote his zeroes with three hooks, rather

than two slanted wedges, as if they were thirties; and another scribe at

about the same time made his with only one, so that they are

indistinguishable from his tens. Carelessness? Or does this variety tell

us that we are very near the earliest uses of the separation sign as

zero, its meaning and form having yet to settle in?

it. The story follows history’s arrow from Mesopotamia to ancient

Greece, where the necessity of zero awakens anew. Kaplan turns to

Archimedes and his system for naming large numbers, “myriad” being the

largest of the Greek names for numbers, connoting 10,000. With his

notion of

*orders*of large numbers, the great Greek polymath

came within inches of inventing the concept of powers, but he gave us

something even more important — as Kaplan puts it, he showed us “how to

think as concretely as we can about the very large, giving us a way of

building up to it in stages rather than letting our thoughts diffuse in

the face of immensity, so that we will be able to distinguish even such

magnitudes as these from the infinite.”

This concept of the infinite in a sense contoured the need for naming

its mirror-image counterpart: nothingness. (Negative numbers were still

a long way away.) And yet the Greeks had no word for zero, though they

clearly recognized its spectral presence. Kaplan writes:

Haven’t we all an ancient sense that for something toOrdinarily, we know that naming is what gives meaning to existence. But names are given to things, and zero is not a thing — it is, in fact, a no-thing. Kaplan contemplates the paradox:

exist it must have a name? Many a child refuses to accept the argument

that the numbers go on forever (just add one to any candidate for the

last) because names run out. For them a googol — 1 with 100 zeroes after

it — is a large and living friend, as is a googolplex (10 to the googol

power, in an Archimedean spirit).

[…]

By not using zero, but naming instead his myriad myriads, orders and

periods, Archimedes has given a constructive vitality to this vastness —

putting it just that much nearer our reach, if not our grasp.

Names belong to things, but zero belongs to nothing. ItZero, still an unnamed figment of the mathematical imagination,

counts the totality of what isn’t there. By this reasoning it must be

everywhere with regard to this and that: with regard, for instance, to

the number of humming-birds in that bowl with seven — or now six —

apples. Then what does zero name? It looks like a smaller version of

Gertrude Stein’s Oakland, having no there there.

continued its odyssey around the ancient world before it was given a

name. After Babylon and Greece, it landed in India. The first surviving

written appearance of zero as a symbol appeared there on a stone tablet

dated 876 AD, inscribed with the measurements of a garden: 270 by 50,

written as “27°” and “5°.” Kaplan notes that the same tiny zero appears

on copper plates dating back to three centuries earlier, but because

forgeries ran rampant in the eleventh century, their authenticity can’t

be ascertained. He writes:

We can try pushing back the beginnings of zero in IndiaBut if zero were to have a high priest in ancient India, it would

before 876, if you are willing to strain your eyes to make out dim

figures in a bright haze. Why trouble to do this? Because every story,

like every dream, has a deep point, where all that is said sounds

oracular, all that is seen, an omen. Interpretations seethe around these

images like froth in a cauldron. This deep point for us is the cleft

between the ancient world around the Mediterranean and the ancient world

of India.

undoubtedly be the mathematician and astronomer Āryabhata, whose

identity is shrouded in as much mystery as Shakespeare’s. Nonetheless,

his legacy — whether he was indeed one person or many — is an indelible

part of zero’s story.

Kaplan writes:

Āryabhata wanted a concise way to store (not calculateKaplan reflects on the multicultural intellectual heritage encircling the concept of zero:

with) large numbers, and hit on a strange scheme. If we hadn’t yet our

positional notation, where the 8 in 9,871 means 800 because it stands in

the hundreds place, we might have come up with writing it this way:

9T8H7Te1, where T stands for ‘thousand’, H for “hundred” and Te for

“ten” (in fact, this is how we usually pronounce our numbers, and how

monetary amounts have been expressed: £3.4s.2d). Āryabhata did something

of this sort, only one degree more abstract.

He made up nonsense words whose syllables stood for digits in places,

the digits being given by consonants, the places by the nine vowels in

Sanskrit. Since the first three vowels are a, i and u, if you wanted to

write 386 in his system (he wrote this as 6, then 8, then 3) you would

want the sixth consonant, c, followed by a (showing that c was in the

units place), the eighth consonant, j, followed by i, then the third

consonant, g, followed by u: CAJIGU. The problem is that this system

gives only 9 possible places, and being an astronomer, he had need of

many more. His baroque solution was to double his system to 18 places by

using the same nine vowels twice each: a, a, i, i, u, u and so on; and

breaking the consonants up into two groups, using those from the first

for the odd numbered places, those from the second for the even. So he

would actually have written 386 this way: CASAGI (c being the sixth

consonant of the first group, s in effect the eighth of the second

group, g the third of the first group)…

There is clearly no zero in this system — but interestingly enough,

in explaining it Āryabhata says: “The nine vowels are to be used in two

nines of places” — and his word for “place” is “kha”. Thiskha

later becomes one of the commonest Indian words for zero. It is as if we

had here a slow-motion picture of an idea evolving: the shift from a

“named” to a purely positional notation, from an empty place where a

digit can lodge to “the empty number”: a number in its own right, that

nudged other numbers along into their places.

While having a symbol for zero matters, having the notionIn the remainder of the fascinating and lyrical

matters more, and whether this came from the Babylonians directly or

through the Greeks, what is hanging in the balance here in India is the

character this notion will take: will it be the idea of the absence of

any number — or the idea of a number for such absence? Is it to be the

mark of the empty, or the empty mark? The first keeps it estranged from

numbers, merely part of the landscape through which they move; the

second puts it on a par with them.

**,**

*The Nothing That Is*Kaplan goes on to explore how various other cultures, from the Mayans

to the Romans, contributed to the trans-civilizational mosaic that is

zero as it made its way to modern mathematics, and examines its profound

impact on everything from philosophy to literature to his own domain of

mathematics. Complement it with this Victorian love letter to mathematics and the illustrated story of how the Persian polymath Ibn Sina revolutionized modern science.

## Πέμπτη, 2 Φεβρουαρίου 2017

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