Infinity – the idea that something is without limit or bound – challenges the reach of all but a few minds. It is nonetheless of enormous value within the field of mathematics in solving both practical and theoretical problems.
Infinity has applications across many fields, though in none of them is it easy to grasp. Within the metaphysical field, it may, for example, relate to our understanding of the nature of an infinite god – one without end or beginning, whose love is limitless etc. In physics, meanwhile, we may conjure with the concept as we attempt to understand the nature of the cosmos. How big is the universe? If it has an end, where is it? And are there infinite stars in infinite galaxies? Will the cosmos go on for ever? What was there before the universe existed?
In maths, infinity is usually related to an unending sequence of numbers. In its most simple form, we may think of infinity in terms of an unimaginably large number to which we may always add one more. However, infinity is not always big – just endless. The ancient Greek philosopher Anaximander (c. 610–c. 546 BCE) is often credited with first coming up with the concept of infinity, although Zeno of Elea (c. 490 BCE–c. 430 BCE) was the first to use it in a mathematical context. Pythagoras, in his study of geometric shapes, famously discovered ratios that went on infinitely, but probably the most well-known example of an infinite number is pi (see here). Several of the great thinkers were highly sceptical of the idea of infinity. Aristotle accepted the notional concept (in terms of always being able to ‘add one’ when you count) but rejected the idea of a ‘real’ infinity, whether spatial, temporal or numerical.
Mind-boggling infinity
In the late 19th and early 20th centuries, the German mathematician Georg Cantor did much to formalize ideas of infinity. His work in set theory (based on the simple notion that numbers may be put into sets) revealed that there are multiple types of infinity and that some of them, mind-bogglingly, are larger than others. ‘I see it, but I do not believe it,’ he is said to have commented on his discovery. In the hundred and more years since his startling revelations, mathematicians continue to struggle to get to grips with the implications. Meanwhile, innovators in fields as disparate as mechanical engineering, software development and cosmology reap the practical benefits of the mind-bending idea of numbers without end.
The idea of infinity doggedly held on. In the 17th century, the notion of infinitely small numbers that are nonetheless greater than zero were pivotal to Isaac Newton and Gottfried Leibniz independently developing their systems of calculus (see here). Meanwhile, English mathematician John Wallis may claim the glory for introducing the common symbol for infinity – ∞ – back in 1657.