Zero and Infinity in the History of Mathematics
“The Sultan Abdul Hamid the Second, perpetrator of the
terrible nineteenth-century Armenian massacres, had his censors, they say,
remove any reference to H20 from chemistry books entering his empire, [as
he was] convinced that the symbol stood for ‘Hamid the Second is Nothing!’”
“You flatter me – flatten, I mean,” said the Tortoise;
“for you are a heavy weight, and no mistake! Well now, would you like to
hear of a race-course, that most people fancy they can get to the end of
in two or three steps, while it really consists of an infinite number of
distances…?”
As children – and perhaps as adults – each of us presumably
wondered about infinity and nothingness: in the seeming infinity of a Möbius
Band, the thought of what it would be like had we not been born, or the endlessness
of the night sky: “twinkle, twinkle, little star, how I wonder what you are….”
Perhaps in grammar school we told our classmates, “ I know how to count higher
than infinity… Infinity plus one!”. Maybe someone read us Can you count to
a Googol? by Robert Wells, and we wondered when they told us numbers did
not stop at googol. Throughout the ages, the Infinite and the Nothing have
been a perennial motif for all thinkers, woven in and out of the fabric of
culture and time, politics, religions, and peoples; but they have been a
particular hobbyhorse of theologians, mathematicians, philosophers, cosmologists,
and mystics.
Modern studies of infinity and zero still leave us spinning.
As G.K. Chesterton penned, “ The poet only asks to get his head into the
heavens. It is the logician who seeks to get the heavens into his head. And
it is his head that splits.” Perhaps advances toward the understanding
of nothingness and infinity have simply qualified the void and not quantified
it. Nonetheless, our limited understanding of zero and infinity has proven
to be useful and significant for the development of mathematics and society.
Zero’s exact birthday into the world of written numbers
is not known. It made “…shadowy appearances only to vanish again almost as
if mathematicians were searching for it yet did not recognize its fundamental
significance even when they saw it.” Zero has two distinct functions,
namely a symbol for null (the way we use a “0” by itself) and a placeholder
in a place-value number system. In modern science, zero is also used to indicate
the degree of accuracy. The last digit of the number is the only number that
involved subjectivity, e.g. in the case of 3.000, 3.00 is absolutely accurate
and the last zero approximated. These functions of zero developed independently.
Especially for those who only calculated so-called “real” problems (e.g.
A farmer owned five horses and two died; how many are left?), zero was“…far
from an intuitive concept.”
One of those so-called “real” problems was astronomy, a developing science
whose calculations and inductions depended on the ability to keep lists.
While the ancient Greek mathematicians, such as Euclid, were working on deductive
proofs using magnitudes and not numerical quantities, astronomers were working
on inductive proofs based on empirical data. Although ancient Greek mathematicians
had little need for the number zero, the Greek astronomers used the null
to record what they saw (perhaps it was used more often on cloudy nights).
Here, in ancient Greece, was the first appearance of our modern symbol for
zero, the “O”. However, the concept of zero had not developed; the
omicron “O” simply represented nothing, of which the Greeks cared to understand
nothing.
One of the important uses of zero is as a placeholder
in a place-value number system. The ancient Babylonians were already using
sexagesimal (base-60) place-value system for complex calculations at the
time of Hammurabi (ca. 1800BC). In the first millennium BC (the “Seleucid”
period), a separator symbol was introduced to indicate an empty space within
a number. For example, the ancient reader would need to infer whether
“2 6” meant 26, 206, or 2006 based on the context. To prevent confusion,
Scribes put a small imprint between “2” and “6” when they wanted to indicate
the number 206. This little imprint was a forerunner of one function of our
modern zero.
By 665 AD, the Mayans had developed a vigesimal (base-20)
place-value system complete with a null-value placeholder similar to the
Babylonians. The primary uses were calendrical or astrological. Unfortunately,
Mayan mathematics all but died out with the European conquest and colonization.
Mayan culture has no extant heirs.
Although the Babylonians, Greeks, and Mayans began to
use zero in their own right, its true conceptual development took place in
India. Some historians have argued that the Indian mathematicians just developed
near-Mediterranean concepts: “Pygmies placed on the shoulders of giants see
more than the giants themselves.” However, there is evidence that Indian
mathematics were already highly developed at the peak of Greek astronomy:
A symbol for null was used in India as early as 200 BC, marked by the binary
system described in Pingala’s Chandra-sutra. Even before this, Panini
used a zero operator in his Astadhyayi. Kak states, “The
development of the zero sign in India was motivated by numerical calculations….
contrasted from the manner in which the zero signs arose in Babylon and Mexico
[the Mayans], where the motivation was from the areas of astronomy and calendrical
calculations.”
Three prominent Indian mathematicians attempted to discover
the functions zero: Brahmagupta, Mahavira, and Bhaskara. In ca. 600 AD, Brahmagupta
drafted several accurate laws for addition, subtraction, and multiplication
operations involving zero. In the case of his statements regarding zero and
division, he sometimes grappled with the truth, e.g. n divided by zero is
a fraction with numerator n and zero as the denominator, or missed the truth
altogether, e.g. zero divided by zero is zero. No major developments
were made in the two hundred years following Brahmagupta. Mahavira clarified
some of Brahmagupta’s ideas; but as indicated by his err in Ganita Sara Samgraha,
division remained an enigma. Three centuries later, in ca. 1100 AD, Bhaskara
outlined a number of laws concerning zero and came close to understanding
division by zero. He wrote (n/0 = 8), which is close: today we know that
division by zero is impossible.
About one century later, Arabic culture fell heir to the Indian number system,
which included zero, but the concept of zero as a number was lost or weakened
at best. Fibonacci introduced the Hindu-Arabic number system to Europe at
around the same time, but zero was not granted equal status with real numbers.
Perhaps the dominant philosophical ideas hindered it: St. Thomas Aquinas’
philosophy of Ens, that “There is an Is”, dominated the intellectual climate
of Europe at the time. Mortal Ens (“being”) is predicated of God’s
infinite, perfect, and eternal Ens. Most likely, null was not seen as an
entity but as a non-entity and was therefore deemed unnecessary. It was certainly
not in vogue. It was not until the seventeenth century that zero had filtered
down into common use.
Infinity’s history is much more juicy. Hilbert’s Infinite
Hotel (a common demonstration of 8 + 1) may be more like a haunted house,
trapping the hapless transient with mirror rooms. Unlike zero, infinity
is not a quantity we encounter in usual mathematical situations; we cannot
find it at the grocery, the office, the home, or the street corner. We rarely
encounter it, if ever, in scientific work, except in irrational numbers,
such as p or the square root of two, but we “round” the decimal or our calculator
does rounds it for us. Usually we only use the word “infinity” when speaking
about non-mathematical things. We speak seldom of actual infinity but often
of potential infinity:“I could look into her eyes forever” or “It seems I
have an infinite number of papers on my desk”. Perhaps for these reasons,
mathematicians did not see infinity as a worthy pursuit and therefore relegated
it to the philosophers and mystics.
Infinity’s salad days were spent amongst the dreamers.
The first of these dreamers, Zeno of Elea (495-435 BC), is best known for
his paradoxes, especially “Achilles and the Tortoise,” also demonstrated
by his Dichotomy of the Room: Achilles can never overtake the Tortoise if
he makes each step half as long as the preceding step. This illustrates “the
unlimited density of points in any interval on the real number line.”
The second of these dreamers, Pythagorus (569-500 BC), was the father of
a philosophical movement. His fascination with infinity, irrational numbers,
and numbers in general became what we now call number mysticism, or the worship
of numbers. The Pythagoreans did make some accurate contributions to mathematics,
namely the Pythagorean theorem, and to music, namely the understanding of
intervallic ratios.
Plato (428-347 BC) adopted the Pythagorean emphasis on
number without the accompanying mysticism. The mathematicians trained at
his school in Athens began to work with magnitudes instead of numbers. Euclid
of Alexandria (c. 330-275 BC), for example, does not use actual numbers in
his Elements. While he touched on potential infinity in his definition of
a line or a plane, it was merely that one could extend these things to infinity
if he wanted to, not that he would. Eudoxus of Cnidus (408-355 BC), also
of Plato’s academy, used the idea of potential infinity in his “computation
of the total area of a volume of a curved surface.” Eudoxus’ ideas
were extended by Archimedes of Alexandria approximately a century later.
At this point, infinity passed out of the hands of the
mathematicians into the hands of the mystics and philosophers. Kabbalah,
founded by Jewish Rabbi Joseph ben Akiva (50-132 AD), were a group of such
mystics. By the thirteenth century, the Kabbalah believed that “Since God
is Infinity and cannot be comprehended, the Sefirot [ten permutations of
the letters of God’s name, YHWH] are the finite aspects… gleaned from the
immensity of the Ein Sof [the infinite]…” which can be “studied and meditated
upon, and prayed with.” The Kabbalah believed that Yahweh had given
them a finite, bite-sized piece of His infinity. The infinite could be perceived.
St. Augustine of Hippo (354-430 AD) also contemplated infinity and time.
He demonstrates a complex understanding of infinity and the 8 + 1 principle
in City of God, Book XII, Chapter 18. However, this understanding is used
solely for the purpose of religious contemplation: “Does [God’s] knowledge
extend only to a certain height in numbers, while of the rest He is ignorant…?
All infinity is in some ineffable way made finite to God, for it is comprehensible
to His knowledge.” Nicholas of Cusa (1401-1464) used the concept of
infinity for similar meditations.
Through the ingenious mathematical and astronomical cookery
of Galileo Galilei (1564-1642), Infinity returned to the menu of mathematicians
as “Infinity Florentine.” Ecclesiastics saw Galileo’s iconoclastic tendencies
and his challenge to the earth-centered view of the universe as a recipe
for disaster and eventually condemned Galileo to house arrest. Knowing that
the authorities had cooked his proverbial goose, Galileo focused his attention
on pure mathematics, and especially on the infinite, because he was no longer
at liberty to conduct experiments. In his Dialogues on Two New Sciences (1638),
he wrote that the number of squares of integers in an infinite set is not
less than the number of integers (essentially 82 is not < 8 in an infinite
set). In common language, this means, “an infinite set is not less
than a part of itself.”
Infinity lay dormant like the sleeping beauty in academic
ivory towers for another two centuries, only to be awakened in Europe in
the nineteenth century. Bernhard Bolzano (1781-1848) discovered what Zeno
searched for in his paradoxes: aggregate infinity, or the concept that there
are unlimited infinitesimally small numbers between any two points on the
number line. He went on to prove that there are equally infinite numbers
between zero and one as there are between zero and two.
Infinity qua infinity was not articulated until the research
of Georg Cantor (1845-1918). As infinity was for the Kabbalah, the Jewish
number mystics, so it was for the Jewish mathematician, Cantor: to comprehend
infinity was to comprehend the divine Mind. Cantor’s studies of infinity
are possibly the most confusing and openly metaphysical of all mathematics.
Cantor applied set theory to infinity. (Set theory was first drafted by Guiseppe
Peano (1858-1932). “Zero [was defined as] the empty set. One was then defined
as the set containing the empty set. Two was the set that contained the empty
set and the set containing the empty set.” ) He assigned “¿”, or Aleph,
the first letter of the Hebrew alphabet, to an infinite set. The notation
for the infinity of integers and rational numbers, ¿0 , enabled the
articulation of some laws of infinity, namely the following: ¿0 +
1 = ¿0 ; ¿0 + n = ¿0 ; ¿0 + ¿0 =
¿0 ; and ¿0 x ¿0 = ¿0. Infinity could now
be expressed in equations. Cantor then proved that there are equally infinite
numbers of points in one dimension (the line) as in two (the plane), three
(cube), four dimensions, …ad infinitum. Thus, with infinity, it was possible
for the part to be equal to the whole, contrary to Euclid’s fifth Common
Notion.
Cantor also sought to classify and differentiate levels
of infinity. We must remember Peano’s set theory (vide supra): for every
set of numbers, the power set (the set of all possible subsets) is greater.
In the case of the number three, the power set would be the set of all possible
subsets one could form the set from the set of the three elements. The power
set of the set of three elements has eight elements: 23 = 8. Cantor then
applied this logic to Infinity. Thus, the cardinality of the numerical continuum
could be expressed thus: c=n¿0 , where c is the cardinality and n
is any number on the numerical continuum. In the case of infinity, the power
set was a “greater” infinity. It was later expanded by Felix Hausdorff (1868-1942)
to the general statement 2¿a = ¿a+1. The number of ¿’s
could thus go on infinitely, because ¿n always had a power set of
greater cardinality.
Cantor recognized the infinite number of infinites – that
there was no cardinal number that bound the infinite – but he could not finish
without implying finality. He named the set of all infinites z, or
taf, the last letter of the Hebrew alphabet. Later, Kurt Gödel (1906-1978)
developed the incompleteness theorem, part of which implies that the cardinality
of z must be outside of the realm of the universe.
Friedrich Nietzsche (1844-1900) once said, “If you gaze
for long into the abyss, the abyss also gazes into you.” Many of those who
attempt to fathom the infinite have lost their sanity (e.g. Cantor, Gödel,
Rabbi Ben Zoma). At the climax of The Man Who Was Thursday, Chesterton’s
captivating “nightmare,” Gogol, Syme, and Bull are all in fast pursuit of
Sunday (who represents Nature distinct from God). As he is running,
Sunday throws a parcel high in the air. Gogol stops to examine it, but finds
it “…to consist of thirty-three pieces of paper of no value wrapped one round
the other. When the last covering was torn away it reduced itself to a small
slip of paper, on which was written: ‘The word, I fancy, should be ‘pink’’…”
Is there anything at the end of infinity? Does it matter? It seems the infinite
is a bridge between material and immaterial and the physical and metaphysical.
In the end, what is the function of zero and infinity?
Some mathematicians and philosophers would say that these concepts are dead
ends in themselves, only useful by their application. Others, like Plato,
would claim that contemplation of incommensurables is worthwhile because
it sets the mind on higher things. Zero is a much more commonly used and
understood than infinity, perhaps because of how common it has become in
the past four centuries, but more likely because we can grasp it. Numerous
new “sciences” depend on it, especially Statistics. Although we cannot contain
“nothingness” in a box as with the bag of wind Aeolus gave to Odysseus in
Homer’s Odyssey, we see it quite often. As a student in college, it was:
I have zero money; as a new mother: I have no time of my own; as a city-dweller:
I have no backyard. It only seems fitting that our numerical vocabulary should
include null. Using zero as a placeholder in an empty column was not intuitive,
however, but it certainly was the largest advance toward the immediate clarity
and intelligibility of written numbers. It eliminated the possibility for
multiple interpretations of one number. Infinity, on the other hand, has
not had practical application in the world of commerce or technology. However,
infinity and zero are not much ado about nothing; they have had a tremendous
impact during their brief entrances on the world stage. Consider the confusion
over whether it was 2000 or 2001that marked the new millennium. Infinity
and zero remain as an idée fixe for many mathematicians and philosophers,
a wildcard in the theologian’s hand, and as an enigma for all who dare to
think about nothing.
Works Cited and Selected Bibliography
St. Augustine of Hippo, City of God, trans. Marcus Dods (New York: Modern
Library, 2000).
James Brackett, “Children’s conceptualizations of infinity,” Journal of Interdisciplinary
Mathematics Vol. 1, No. 1. (1998).
Lewis Carroll, “What the Tortoise Said to Achilles,” Mind 4, No. 14 (April
1895): 278- 280. Also found online at http://www.ditext.com/carroll/tortoise.html.
G.K. Chesterton, Orthodoxy (Colorado Springs, Colorado: Shaw Books, 2001).
G.K. Chesterton, The Annotated Thursday, ed. Martin Gardner (San Francisco:
Ignatius Press, 1999),
Joseph Dauben, Georg Cantor (Princeton: Princeton University Press, 1979).
Stuart Hollingdale, Makers of Mathematics (London: Penguin Books, 1989).
Robert Kaplan, The Nothing that Is (Oxford: Oxford University Press, 2000).
Eli Maor, To Infinity and Beyond (Princeton: Princeton University Press,
1987).
Richard Morris, Achilles in the Quantum Universe (New York: Henry Holt and
Company, 1997).
J.J. O’Connor and E.F. Robertson, “A history of Zero”,
http://www-history.mcs.st-andrews.ac.uk/history/HistTopics/Zero.html