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Infinity: the story on HearLore | HearLore
Infinity
The ancient Greeks lived in fear of the unbounded, a psychological barrier that shaped their mathematics for centuries. Anaximander, a pre-Socratic philosopher active around 546 BC, introduced the word apeiron to describe the unbounded or indefinite, yet this early philosophical concept remained distinct from the rigorous definitions that would emerge millennia later. Aristotle, writing in 350 BC, drew a sharp line between potential infinity and actual infinity, arguing that actual infinity was impossible because it produced logical paradoxes. This deep-seated horror of the infinite influenced the great Euclid, who wrote around 300 BC. When proving that prime numbers are infinite, Euclid did not state that there are an infinity of primes. Instead, he carefully phrased it to say that prime numbers are more than any assigned multitude of prime numbers, a linguistic maneuver designed to avoid the implication that infinity was a completed, tangible entity. Even the parallel postulate, which Euclid formulated, was translated by some as lines produced indefinitely, a phrasing that deliberately avoided suggesting the lines actually extended to infinity. For over two millennia, the prevailing view was that infinity was a process that could never be finished, rather than a number or a set that could be manipulated.
Achilles and the Tortoise
Zeno of Elea, who lived between 495 BC and 430 BC, did not propose a theory of infinity, yet his paradoxes forced mathematicians to confront the inadequacy of their understanding of motion and limits. His famous paradox of Achilles and the Tortoise describes a race where Achilles gives a tortoise a head start. In the first step, Achilles runs to the tortoise's starting point while the tortoise moves forward. In the second step, Achilles advances to where the tortoise was at the end of the first step, while the tortoise moves yet farther. This process repeats indefinitely, creating the illusion that Achilles never overtakes the tortoise. Zeno, a member of the Eleatics school, believed motion itself was an illusion, so he saw the paradox as a mistake in assuming Achilles could run at all. However, subsequent thinkers found this solution unacceptable and struggled for over two thousand years to find weaknesses in the argument. It was not until 1821 that Augustin-Louis Cauchy provided a satisfactory definition of a limit and a proof that resolved the paradox. If Achilles runs at 10 meters per second and the tortoise walks at 0.1 meters per second with a 100-meter head start, the duration of the chase fits Cauchy's pattern, and Achilles does overtake the tortoise. The paradox, described by Bertrand Russell as immeasurably subtle and profound, highlighted the need for a rigorous mathematical framework to handle infinite processes.
Common questions
When did Anaximander introduce the word apeiron to describe the unbounded?
Anaximander introduced the word apeiron to describe the unbounded around 546 BC. This early philosophical concept remained distinct from the rigorous definitions that would emerge millennia later.
What did Aristotle argue about actual infinity in 350 BC?
Aristotle argued that actual infinity was impossible because it produced logical paradoxes. He drew a sharp line between potential infinity and actual infinity in his writing from 350 BC.
Who introduced the infinity symbol known as the lemniscate in 1655?
John Wallis introduced the infinity symbol known as the lemniscate in 1655 within his work De sectionibus conicis. This symbol has since been used outside mathematics in modern mysticism and literary symbology.
When did Georg Cantor revolutionize the study of infinity by introducing infinite sets?
Georg Cantor revolutionized the study of infinity in the late 19th century by introducing the concept of infinite sets. He demonstrated that there are different levels of infinity and defined two kinds of infinite numbers: ordinal numbers and cardinal numbers.
Who proposed an unbounded universe in 1584 and what did they state?
Giordano Bruno proposed an unbounded universe in 1584 in On the Infinite Universe and Worlds. He stated that innumerable suns exist and innumerable earths revolve around these suns in a manner similar to the way the seven planets revolve around our sun.
What is the Koch snowflake and what are its properties?
The Koch snowflake is a fractal curve with an infinite perimeter and finite area. It is an object whose structure is reiterated in its magnifications, meaning it can be magnified indefinitely without losing its structure and becoming smooth.
John Wallis, an English mathematician, introduced the infinity symbol, known as the lemniscate, in 1655 within his work De sectionibus conicis. Before this, European mathematicians had used infinite numbers and expressions in a systematic fashion, but Wallis gave them a visual identity that persists today. In his 1656 work Arithmetica infinitorum, Wallis indicated infinite series, infinite products, and infinite continued fractions by writing down a few terms or factors and then appending &c., as in 1, 6, 12, 18, 24, &c. The symbol, encoded in Unicode and LaTeX as \infty, has since been used outside mathematics in modern mysticism and literary symbology. Isaac Newton, writing in 1699 in his work De analysi per aequationes numero terminorum infinitas, explored equations with an infinite number of terms, building on the foundation Wallis had laid. Gottfried Leibniz, one of the co-inventors of infinitesimal calculus, speculated widely about infinite numbers and their use in mathematics. To Leibniz, both infinitesimals and infinite quantities were ideal entities, not of the same nature as appreciable quantities, but enjoying the same properties in accordance with the Law of continuity. The symbol became a cornerstone of calculus, allowing mathematicians to denote unbounded limits and manipulate infinite series with precision.
Cantor's Transfinite Revolution
Georg Cantor, working in the late 19th century, revolutionized the study of infinity by introducing the concept of infinite sets and showing that they could be of various sizes. Before Cantor, infinity was often viewed as a single, undifferentiated concept, but Cantor demonstrated that there are different levels of infinity. He defined two kinds of infinite numbers: ordinal numbers and cardinal numbers. Ordinal numbers characterize well-ordered sets, or counting carried on to any stopping point, including points after an infinite number have already been counted. Cardinal numbers define the size of sets, meaning how many members they contain. The smallest ordinal infinity is that of the positive integers, and any set which has the cardinality of the integers is countably infinite. If a set is too large to be put in one-to-one correspondence with the positive integers, it is called uncountable. Cantor's theorem shows that there is no largest cardinal number, meaning that for any infinite set, there exists a larger set. One of Cantor's most important results was that the cardinality of the continuum, the set of real numbers, is greater than that of the natural numbers. This meant there are more real numbers than natural numbers, a result that was initially controversial but eventually prevailed. The theory of ordinal and cardinal numbers has been further developed since Cantor, with large countable ordinals and large cardinals currently being studied in mathematical logic.
The Continuum and the Curve
Giuseppe Peano, in 1890, introduced space-filling curves, which are curved lines that twist and turn enough to fill the whole of any square, or cube, or hypercube, or finite-dimensional space. These curves can be used to define a one-to-one correspondence between the points on one side of a square and the points in the square, proving that the number of points in a real number line is equal to the number of points in any segment of that line, and also equal to the number of points on a plane and, indeed, in any finite-dimensional space. This result, proved by Cantor in 1878, only became intuitively apparent in 1890 with Peano's work. The continuum hypothesis states that there is no cardinal number between the cardinality of the reals and the cardinality of the natural numbers, a hypothesis that cannot be proved or disproved within the widely accepted Zermelo, Fraenkel set theory, even assuming the Axiom of Choice. Cardinal arithmetic can be used to show not only that the number of points in a real number line is equal to the number of points in any segment of that line, but also that this is equal to the number of points on a plane and, indeed, in any finite-dimensional space. The first of these results is apparent by considering, for instance, the tangent function, which provides a one-to-one correspondence between the interval and the real line.
The Universe and the Multiverse
The first published proposal that the universe is infinite came from Thomas Digges in 1576. Eight years later, in 1584, the Italian philosopher and astronomer Giordano Bruno proposed an unbounded universe in On the Infinite Universe and Worlds, stating that innumerable suns exist and innumerable earths revolve around these suns in a manner similar to the way the seven planets revolve around our sun. Cosmologists have long sought to discover whether infinity exists in our physical universe, asking if there are an infinite number of stars or if the universe has infinite volume. The curvature of the universe can be measured through multipole moments in the spectrum of the cosmic background radiation. To date, analysis of the radiation patterns recorded by the WMAP spacecraft hints that the universe has a flat topology, which would be consistent with an infinite physical universe. However, the universe could be finite, even if its curvature is flat, as seen in video games where items that leave one edge of the screen reappear on the other. The concept of infinity also extends to the multiverse hypothesis, which, when explained by astrophysicists such as Michio Kaku, posits that there are an infinite number and variety of universes. Cyclic models posit an infinite amount of Big Bangs, resulting in an infinite variety of universes after each Big Bang event in an infinite cycle.
Fractals and the Infinite Loop
Fractals are objects whose structure is reiterated in their magnifications, meaning they can be magnified indefinitely without losing their structure and becoming smooth. One such fractal curve with an infinite perimeter and finite area is the Koch snowflake. In computing, the IEEE floating-point standard specifies a positive and a negative infinity value, defined as the result of arithmetic overflow, division by zero, and other exceptional operations. Some programming languages, such as Java and J, allow the programmer an explicit access to the positive and negative infinity values as language constants, which can be used as greatest and least elements, as they compare greater than or less than all other values. In languages that do not have greatest and least elements but do allow overloading of relational operators, it is possible for a programmer to create the greatest and least elements. In programming, an infinite loop is a loop whose exit condition is never satisfied, thus executing indefinitely. In the arts, perspective artwork uses the concept of vanishing points, roughly corresponding to mathematical points at infinity, located at an infinite distance from the observer. Artist M.C. Escher is specifically known for employing the concept of infinity in his work, and variations of chess played on an unbounded board are called infinite chess.