Why Europeans long time did not love zero?
“About you, my zero and zeroes,
Ya you loved, I love you! are Rather treated by
the Touch to zero!
... When I will die, do not put,
do not buy to me a wreath,
A better a zero put
On my sad hillock“.
(N. Oleynikov) I already wrote
In the previous article how the concept of “Zero“ was hard acquired by the European mathematics. As it seemed to me to that the - world outlook - the reasons.
Space in doctrines of Ancient Greek philosophers was material, and the universe - in detail. For antique reason emergence of the world from nothing was not representable. The universe could be born from the disorder Chaos, water, fire, apeyron and other primary elements - from anything, but only not from emptiness. Not without reason Pythagoras considered as a numerical basis of the universe unit, and all other numbers arose already from it. The non-existence, bessushchnost, as well as concept of number, nothing not estimating, for Greeks were equally absurd. Also two well-known antique principles - Natura abhorret vacuum (“The nature does not suffer emptiness“) and Ex nihilo nihil eloquently testify to it (“From nothing nothing will appear“). Even for Democritus operating with concept of emptiness, this emptiness was only a condition for spatial existence and the movement of atoms - the same initial, eternal, indivisible and qualitatively invariable primary elements.
Well and if the same Claudius Ptolemaeus had to use position system and to designate lack of the category a sign, then this sign was not perceived by number at all, it was just replacement to the word “anything“. And the similar relation of the European scientists to zero held on some more centuries. Fibonacci obviously refuses to zero an equivalence to other figures. And Rene Descartes who much made for final “legalization“ of zero himself considered it as number “false“, “artificial“ (there it carried also negative numbers). Like, it is necessary for mathematical operations, but seriously you should not treat it. In many zero dictionaries as the number is still characterized only through arithmetic operations.
“The Soviet encyclopedic dictionary“, 1980:
“Number 0 from addition (or subtraction) which to any number the last does not change“.
For us this sign and remained insignificant “zero“ which becomes “mighty zero“ only in combination with other “real“ numbers. Not without reason such expressions as “Full zero“, “Zero without stick“, characterize the insignificant person, and the verb “cancel“ demonstrates elimination something.
Other business - east consciousness. Emptiness did not frighten Hindus and Buddhists. Besides - only bessushchny, bezveshchestvenny, not representable was also considered real, true as opposed to “Maya`s cover“ - that to the illusory fragile changing peace which surrounds us. Antique attitude had nothing similar to a Buddhist Nirvana or Chinese Dao - concepts by definition not giving in to the description.
Therefore idea of zero as about NUMBER again - arose in India. In the same place for the first time tried to describe mathematical operations with zero. Addition and subtraction were given to Indians simply. They coped and with multiplication, having defined that multiplying number by zero, we receive the same zero. And it is valid - to increase number by zero, it to take this number zero times, that is, not to take in general and consequently also the result will be zero.
The real problems arose with division into zero. Since the childhood we learned that it is impossible to divide into zero, but very few people remember why it is impossible. The Indian mathematicians tried to divide into zero desperately. So Brakhmagupta (7th century) wrote that “zero on zero is zero“, and about the number divided into zero, writes very carefully - “The positive or negative number divided into zero is fraction with zero in a denominator“. More courageous attempt to realize division into zero was made by Bkhaskara.
Bkhaskara, “Siddkhanta - siroman“ (“A science wreath“) apprx. 1150:
“The size divided into zero becomes fraction with zero in a denominator. This fraction is called infinite size. This size consists of the size having zero as a divider, it is constant in spite of the fact that it is possible to add a lot of things to it and a lot of things from it to take, just as God is infinite and invariable even then when are created or the whole worlds stop existing and the set of beings is absorbed or “thrown up“. to
of the Logician of the Indian mathematician it is clear - at reduction of a denominator the fraction automatically increases, so if the denominator becomes nothing, then the result necessarily has to turn back infinity. Not representable absence addresses the same not representable never-ending presence. Zero here as if acts as the antagonist Beskonechnosti and Eternity.
Already one it is enough that the European mathematicians began to hate division into zero and renounced it. Of course, and they operate with concept of infinity, for example, recognize infinity of a numerical row and aspiration to infinity (we will remember schedules which constantly come to axes of coordinates, but never with them are crossed). However and in these cases mathematicians deal with CERTAIN numbers. The pure infinity is inexpressible in number, and arithmetic operations with it are simply deprived of sense. “Go you … to philosophers!“ - as if tell mathematics.
To. F. Gauss, 1831: “I object
to use of infinite sizes as something complete, it is not admissible in mathematics. The infinity is only a turn of speech which real value - a limit which certain relations while it is allowed to others to increase infinitely beyond all bounds approach“. the Ban of division into zero mathematicians explain
quite logically. Let m:0 = n. Then also the return operation - n · 0 = m. Has to be performed but we know that multiplication by zero is always yielded by zero, therefore, the previous result was wrong. Well, you will tell, but that it is possible to divide zero into zero-. It is valid, apparently, that expression 0:0=0 is true, also the return action is true - 0 · 0 = 0. However we will not hurry with conclusions. Let`s take not less true expression 4 · 0 = 0 also we will make the return action. It turns out that 0:0 = 4! And why not five, not thirty, not hundred twenty three? Any number increased by zero will give zero. Therefore, zero divided into zero, have to give ANY number too that nullifies an essence of numbers. A set of cunning mathematical pseudo-proofs, like the fact that “twice two - five“, contain the camouflaged division into zero in the actions. All this disgrace also led to the fact that mathematicians forbade division into zero. They could not refuse the zero any more … the Real way to equality zero began
after emergence in mathematics of negative numbers. I think, now you not strongly are surprised to the fact that this discovery was made by the same Hindus. This time about a lshy role would be played by everyday practice, but not abstract thinking. Not without reason Indians called negative numbers “debts“, anticipating accounting and widespread expression “to go to minus“, that is, to suffer losses instead of profit. “the zero balance“ where expenses and profit mutually repay each other became border between profit and losses.
So zero, earlier only filling in empty categories, began “career“ of a dividing barrier between affine numbers. In the 17th century Descartes enters into use of mathematics the system of coordinates (probably, not without influence of the geographical grid invented still by ancient Greeks). Together with it zero found a graphic vision, having become a point of intersection of abscissa axes and ordinates - a point in which both quantitative, and qualitative characteristics of numbers disappeared.
A. Stepanov in the work “The number and culture“ considers, as the European rationalism in relation to zero was shown here. He writes: “Zero - same “point“, as well as other numbers. The geometry assumes a touch zrimost, presentation and about what “true“ absence then can there be a speech? Europeans from the very beginning “materialized“ zero. Besides, arithmetics and geometry in general are essentially various“.
the Most known dividing zero is of course zero scale of the thermometer. At first the provision of zero was defined by the most minimum temperature which scientific Fahrenheit could receive in the laboratory (it was temperature of mix of salt and ice). The temperature scale of Celsius in which zero degrees is the temperature of melting of ice is more familiar to us. For the live organism consisting generally of water, and living in a water environment, this reference system was the most convenient. It is “below zero“ expression and give rather real idea of weather behind a window “above zero“. However Calvin developed an absolute temperature scale where he any is not present “below zero“ for scientists. Here zero is called absolute and makes minus 273 degrees Celsius. At such temperature all movement of atoms and molecules has to stop completely. “Has to“ - it is told not incidentally because (if laws of thermodynamics are right) it is considered that in reality the absolute zero is unattainable. It is explained by the fact that the closer the system approaches absolute zero, the more work needs to be spent for its further cooling.
After introduction of coordinates zero was approved in mass consciousness as a starting point. Today nobody is surprised by expression “to start from scratch“ or time 00:00. And here ancient Greeks would never say “zero hours“. Midnight for them would be twenty fourth hour then it would be reckoned hour of the first - and any zero there. The similar account is in many respects not deprived of sense, quite often a habit to treat zero as to a reference point, results in absurdities. To take at least memorable celebration of so-called “millennium“. People of the world amicably celebrated the beginning of the new millennium on January 1, 2000, having tempted beautiful round date. Though it is not necessary to be the scientist to understand that two-thousand year - not the first, but the last year of the millennium.
Nevertheless, “time number zero“ in modern science looks not such absurdity. As zero reference point of time it is considered to be the moment of the Big Bang - the moment of formation of the Universe. Scientists prefer not to argue on what was before - to deal with it!
So zero, at all convenience of its use, still remains the most mysterious number, moreover - the sign and a symbol which is beyond mathematics to the field of pure philosophy where the same mysterious concepts as Eternity and Infinity dominate. Thanks to zero, people could operate with what they cannot present. And unless it not a miracle?