The nature of infinity, 2

On Thursday, the first in a series of public discussions on scientific topics was put on by an organisation called Mass Interaction (the name comes from a statement by Richard Feynman that “all mass is interaction”). John Barrow and I discussed the nature of infinity, moderated by the driving force behind the event, Max Sanderson.

The event was held in a new venue called AirSpace, in Oxford Street near Tottenham Court Road station. This area is a huge building site at the moment because of Crossrail works, so I had a bit of trouble finding my way to the venue; but I did make it, only a little late.

I will write here something which is not a summary of the discussion but was certainly inspired by it.

We agreed, more or less, that there are three kinds of infinity:

  • The mathematicians’ infinity. This included the paradise opened to us by the work of Georg Cantor, with its dizzying vision of infinitely many different sizes of infinity, but is not just that: it also includes the infinitesimals of Newton’s calculus, over which he was criticized by Berkeley (who called them “the ghosts of departed quantities”), and more besides. Do these infinities exist? In mathematics, existence does not necessarily mean correspondence with a physical object in the universe; self-consistency is enough for mathematical existence.
  • The physicists’ infinity. This has several subdivisions, which I will indicate by questions: is the Universe infinitely large? Has it existed for infinitely long? Will it continue for infinitely long? Is space infinitely divisible? Is time infinitely divisible? Is it possible to have a point within our finite part of the universe where a physical quantity such as density or temperature is infinite? These infinities may potentially exist; it is reasonable to believe that the question of whether the universe has existed for an infinite time already has a definite answer, even if we cannot be certain what the answer is.
  • The theologians’ or mystics’ infinity. This is much harder to discuss. Many people have had a mystical experience of union with the infinite, but it is notoriously difficult to put the experience into words afterwards. If we cannot agree on what infinity is, it is unlikely that we can decide whether it really exists or not.

Of course, the different types are not sharply separated. The Buddha’s disciple Mālunkyaputta went to his master with a number of questions, to which he desperately wanted answers:

  • whether the world is eternal or not eternal,
  • whether the world is finite or not,
  • whether the soul (life) is the same as the body, or whether the soul is one thing and the body another,
  • whether a Buddha (Tathagata) exists after death or does not exist after death, whether a Buddha both exists and does not exist after death, and whether a Buddha is non-existent and not non-existent after death.

Are these questions about the physical universe, or are they transcendental? It doesn’t really matter, they are surely the great questions that have troubled humans for as long as there have been humans.

(The Buddha refused to answer these questions. He said, “Whether the view is held that the world is eternal, or that the world is not eternal, there is still re-birth, there is old age, there is death, and grief, lamentation, suffering, sorrow, and despair.”)

Indeed, I would add here that our first exposure to infinity comes at the point in our childhood where it suddenly comes home to us that, at a certain point, the world will carry on but we will cease to exist. We naturally wonder whether the world is infinite in time, and whether in some other sense we are also infinite. I think it takes great courage to face these questions squarely. This is perhaps why most of us put them aside and don’t think about them any more.

The great civilisations of the first millennium BCE thought about infinity, and left us with records of their thought. The Greeks had the greatest influence on subsequent European thought. Partly as a result of problems such as Zeno’s paradox, Aristotle forbade consideration of actual infinity and permitted only “potential infinity” (such as the progression from a natural number to the next). This damped down European speculation for a long time, abetted by the Catholic Church, which taught that humans were uniquely creatures of God and an infinite universe would have no distinguished centre for us to inhabit. (Giordano Bruno was burnt at the stake for maintaining an infinite universe.) Albert of Saxony and Galileo both considered the problem of infinity but decided that it was too difficult, and it was only faced in a constructive way by Cantor in the late nineteenth century. The Indians were much more bold, aided perhaps by their invention of zero (the counterpart and “inverse” of infinity).

However, as I outlined above, I think that our ancestors thought about infinity long before this. One member of the audience objected that, prior to the present, people were too preoccupied with the problem of getting enough to eat to engage in speculation. I disagree, on several grounds.

  • Even if young children are put to work, they can still think. I grew up on a farm, and I vividly remember summing geometric series while chasing the cows in to be milked. If anything, the present age of screens and light pollution discourages children from looking up at the stars and wondering if they are infinite.
  • There is evidence that there was actually much more leisure in hunter-gatherer societies than in farming societies. Also, signs in the sky are important to both. Moreover, the Ishango bone (an artefact from the Congo basin now in a museum in Brussels) shows that our ancestors were mathematicians long before they were farmers.

Could we do without infinity? Our present civilization rests on mathematics to such a great extent that it is clear we could not do without mathematics. I believe that we cannot have mathematics without infinity. (A large majority of mathematicians would agree, though not all: there are constructivists who insist on restricting to the finite.) A vast amount of engineering design depends on differential equations, which depends on real numbers. Electronics depends on quantum theory, which depends on complex numbers. Real and complex numbers are essentially infinite objects: although we can only ever use real numbers which have a finite description, we need the entire set in order for properties of functions such as continuity and differentiability to work properly. My students struggle with the construction of the real numbers from the natural numbers; they would struggle much harder if I tried to construct them from finite fragments of the natural numbers. So infinity in mathematics is here to stay!

On the other hand, as John mentioned, physics has the Cosmic Censorship hypothesis, according to which a singularity within the finite part of the universe we inhabit can only occur behind an event horizon, and so can have no effect on us. This has been shown (except in a few cases) to be a consequence of general relativity. Physicists still abhor infinity, and where it appears (e.g. in studying point charges) they try to develop a theory (such as string theory) which will smooth it out.

If you enjoyed this, you can find more in my short course on The Infinite Quest at the Institute of Art and Ideas. (You have to work: the lecture is interrupted by questions, so you have to turn on your brain from time to time!) Also, we are promised that there will be a podcast and more on-line material about the Mass Interaction event at some point: I will publicise it when this happens.

About Peter Cameron

I count all the things that need to be counted.
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3 Responses to The nature of infinity, 2

  1. Love it! Infinities can engage us all irrespective of academic, or otherwise, backgrounds.

  2. Pingback: The nature of infinity, 2 | Guzman's Mathematics Weblog

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