The concept of infinity refers to something without any limit or any end. The symbol “∞” usually denotes infinity. This symbol was invented by the English mathematician John Wallis in 1655. Infinity can be understood in three different discourses: mathematics, physics, and metaphysics or philosophy. Georg Cantor has distinguished between infinities in these three categories. Infinity in mathematics involves the endless number sequence 1,2,3,4…. and so on, or the infinite number of points possible on a continuous line. In physics, the concepts of temporal and spatial infinity are studied, and questions about the universe’s infinite nature are discussed and debated. In metaphysics, the infinite nature of God or the ultimate entity of the Absolute are some of the concepts associated with infinity.
From the classical Greek period onwards, philosophers have debated the philosophical nature of infinity. With the introduction of the infinity symbol (∞) in the 17th century and the development of infinitesimal calculus, the study of infinity took a more mathematical turn. Mathematicians such as Guillaume de l’Hopital and Johann Bernoulli started working on infinite series and the concept of infinitely small quantities. The idea of infinity in mathematics, in a way, elaborates and improves upon the old philosophical concept.
Infinity in Mathematics
The concept of infinity has become one of the core issues in mathematics. In 1930, the German mathematician, physicist and philosopher began one of his mathematical and philosophical addresses by saying, “Mathematics is the science of the infinite”. Philosophers and mathematicians have postulated the concept of infinity since classical Greek times. The ancient Greeks used the word ‘Apeiron’ to express infinity. ‘Apeiron’ meant the undefined, the indefinite, or something without form. One of the earliest occurrences of infinity in the study of mathematics is in the ratio between the diagonal and the side of a square.
Pythagoras was one of the first mathematicians to discover this possibility of infinity. Plato and Aristotle were against the conceptualization of infinity. Aristotle rejected the notion of actual infinity (temporal, spatial or numerical), which he differentiated from the idea of potential infinity, that is, being able to count without an end. In the philosophy of mathematics, actual infinity refers to the assumption of infinite entities as a given; they are considered to be authentic and complete in themselves, for example, natural numbers. It is said that the Hellenistic Greeks (323 BCE – 30 BCE) had a supposed ‘horror of the infinite’, which explains Plato and Aristotle’s rejection. To evade the use of actual infinity, Archimedes and Eudoxus of Cnidus developed what later came to be known as the method of exhaustion.
Further development occurred in the study of infinity in mathematics when Isaac Newton and Gottfried Wilhelm Leibniz discovered calculus while working on infinitely small numbers in the late 1600s. Subsequently, the concept of infinity has developed in modern mathematics in the study of real and complex analyses, nonstandard analysis, set theory, the cardinality of the continuum, geometry, fractals and the infinite dimension. There is also a skeptic branch within the philosophy called finitism which only accepts the existence of finite mathematical objects. Leopold Kronecker was one such mathematician.
Infinity in Physics
Infinity has not been studied in physics in as much depth as mathematics. The primary reason is the uncertain status of physical infinities. In physics, infinity can be understood in time, space, dimensionality or divisibility. There are theories of the three-dimensional space being infinite. Some cosmologists believe that the universe is set or rooted in a higher-dimensional superspace, which could be endless. Thomas Digges was the first person to propose that the universe was infinite in 1576. In 1584, Giordano Bruno, the Italian philosopher and astronomer, gave a similar proposal. From the big bang model of the universe’s origin, cosmologists believe that although the universe has a finitely long past, it may or may not have an infinite future. Under the infinite future theory, there is one view that space is expanding as the galaxies are drifting apart, albeit slowly.
In contrast, physicists who believe in a finite future postulate that the universe will collapse in a cosmic catastrophe. But there is also speculation that this end of the universe will be followed by the birth of a new universe, indicating the possibility of an infinite universe. Infinity, in physics, is also studied in the idea of infinitely divisible matter, which states that each object is potentially made up of an infinite number of particles. There is also the possibility of infinite universes, that is, the multiverse hypothesis which the astrophysicist Michio Kaku explains as the existence of an infinite number and variety of universes.
Infinity in Philosophy
In philosophy, theology and spirituality, the idea of infinity is theorized or understood in relation to God, the Absolute and Zeno’s paradoxes. The concept of infinity has existed since ancient civilization. For example, the ancient Greeks or the Indians did not define infinity in the way done in modern mathematics. They took a more philosophical approach. In Brihadaranyaka Upanishad, one of the principal Upanishads of Hinduism, the Shanti mantra delineates infinity in terms of the Brahman, which refers to the ultimate reality of the universe. It is the infinite, eternal truth and forms one of the core concepts of Hindu philosophy. The first Greek philosopher to propose the idea of an endless universe was Anaximander.
A pre-Socratic Greek philosopher, he gave the theory that infinity was the foundational and primitive basis of reality. Most theologians and metaphysicians who came after Plotinus have believed the Absolute to be infinite. By Absolute, they could mean God, the universal mind or simply the infinite possibilities of thought. In theology, infinity is often understood in terms of the perfection of God or the Divine. While philosophers debate about the case of the infinity of matter, theologians generally attribute the infinite only to God. God is infinite, and His qualities are present in him in a proper manner for an infinite Being. For example, in Judeo-Christian theology, the infinite nature of God is understood as God being without any limitation or restriction, someone who is the All-Being. Modern philosophical views about infinity are mainly studied under the philosophy of mathematics, especially set theory.