If God is dead, then everything is permitted, worried Dostoevsky. Without a realm of moral facts independent of our minds, objective moral truth seemed impossible. Yet, argues Justin Clarke-Doane, moral anti-realism is perfectly compatible with objective moral truth. Conversely, if we insist on realism about mathematics – the idea that our mathematics tracks a reality outside of our minds – then we must give up on the idea that mathematical claims are objectively true. Realism and objectivity, far from always going together, seem here to be in tension.

Realism and objectivity

To what extent are the subjects of our thought and talk real? This is the question of realism. Realism about a subject says that there are facts about it, and these facts hold independently of our beliefs. For example, many of us are realists about mathematics. We believe that there are facts about numbers, sets, and tensors, and these facts do not depend on us. Realism contrasts with constructivism, according to which the facts about a given subject do depend on the minds of enquirers. Many of us are constructivists about aesthetics, for instance.

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Philosophers have widely associated, and even identified, realism and objectivity. But this is a mistake. As we’ll see, realism and objectivity are in tension in domains like mathematics and morality.

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What is the relationship between realism and objectivity? Objectivity about a subject opposes relativism, the idea that there are only relative facts about the subject. Philosophers have widely associated, and even identified, realism and objectivity. But this is a mistake. As we’ll see, realism and objectivity are in tension in domains like mathematics and morality.

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Consider geometry. Suppose that you pass two straight lines, A and B, through a third line, C, and on one side of line, C, the sum of the angles that A and B make with C equals less than 180° – that is, their sum is less than the sum of two right angles. Must A and B intersect? This is the famous Parallel Postulate question. We could understand it as a question about lines (“geodesics”) in physical spacetime. But let us ask the Parallel Postulate question as a question of pure mathematics. Is it true then?

The question is misconceived. In a sense that is distinct from mind-independence, the Parallel Postulate question lacks a non-relative, i.e., objective, answer. There are simply different geometries, and these give different answers to the Parallel Postulate question. Relative to Euclidean geometry, the answer is “yes.” But relative to, say, hyperbolic geometry the answer is “no.” All we would learn in answering the Parallel Postulate question is something about how we use words. We would only learn which geometrical structures we were talking about, rather than learning which there were.

To be clear: it is not that geometrical reality depends on us. If mathematical realism is true, then how things are with Euclidean, hyperbolic, or variably curved spaces depends entirely on the mind-independent mathematical facts. The point is that there is no once-and-for-all answer to the Parallel Postulate question. Geometrical reality is so rich as to furnish any answer to it that we might have proffered. Even if geometry is real, it is not objective.

Anti-Objective Realism about Mathematics