Some philosophers think maths exists in a mysterious other realm. They’re wrong. Look around: you can see it
What is mathematics about? We know what biology is about; it’s about living things. Or more exactly, the living aspects of living things – the motion of a cat thrown out of a window is a matter for physics, but its physiology is a topic for biology. Oceanography is about oceans; sociology is about human behaviour in the mass long-term; and so on. When all the sciences and their subject matters are laid out, is there any aspect of reality left over for mathematics to be about? That is the basic question in the philosophy of mathematics.
People care about the philosophy of mathematics in a way they do not care about, say, the philosophy of accountancy. Perhaps the reason is that the certainty and objectivity of mathematics, its once-and-for-all establishment of rock-solid truths, stands as a challenge to many common philosophical positions. It is not just extreme sceptical views such as postmodernism that have a problem with it. So do all empiricist and naturalist views that hope for a fully ‘scientific’ explanation of reality and our knowledge of it. The problem is not so much that mathematics is true, but that its truths are absolutely necessary, and that the human mind can establish those necessities and understand why they must be so. It is very difficult to explain how a physical brain could do that.
One famous philosopher who finds mathematical necessity an inconvenience is Peter Singer. In one of his best-selling books on ethics, he argues that we cannot rely on intuiting ethical truths, since the most convincing case of intuition, in mathematics, is not correct. ‘The self-evidence of the basic truths of mathematics,’ he says, ‘could be explained… by seeing mathematics as a system of tautologies… true by virtue of the meanings of the terms used.’ Singer is wrong to claim that this philosophy of mathematics, called logicism, is ‘widely, if not universally accepted’. It has not been accepted by any serious philosopher of mathematics for 100 years. But it is clear why anyone who, like Singer, wishes to explain away the strange power of human intuition might want a deflationary philosophy of mathematics to be true. [ … ]