John Stuart Mill, Britain’s major philosopher of the nineteenth century, gave formulations of his country’s empiricist and liberal traditions of comparable importance to those of John Locke. He united enlightenment reason with the historical and psychological insights of romanticism. He held that all knowledge is based on experience, believed that our desires, purposes and beliefs are products of psychological laws of association, and accepted Bentham’s standard of the greatest total happiness of all beings capable of happiness – the principle of ‘utility’. This was Mill’s enlightenment legacy; he infused it with high Romantic notions of culture and character.
In epistemology Mill’s empiricism was very radical. He drew a distinction between ‘verbal’ and ‘real’ propositions similar to that which Kant made between analytic and synthetic judgments. However, unlike Kant, Mill held that not only pure mathematics but logic itself contains real propositions and inferences, and unlike Kant, he denied that any synthetic, or real, proposition is a priori. The sciences of logic and mathematics, according to Mill, propound the most general laws of nature and, like all other sciences, are in the last resort grounded inductively on experience.
We take principles of logic and mathematics to be a priori because we find it inconceivable that they should not be true. Mill acknowledged the facts which underlie our conviction, facts about unthinkability or imaginative unrepresentability, and he sought to explain these facts in associationist terms. He thought that we are justified in basing logical and mathematical claims on such facts about what is thinkable – but the justification is itself a posteriori.
What then is the nature and standing of induction? Mill held that the primitive form of induction is enumerative induction, simple generalization from experience. He did not address Hume’s sceptical problem about enumerative induction. Generalization from experience is our primitive inferential practice and remains our practice when we become reflectively conscious of it – in Mill’s view nothing more needs to be said or can be said. Instead he traced how enumerative induction is internally strengthened by its actual success in establishing regularities, and how it eventually gives rise to more searching methods of inductive inquiry, capable of detecting regularities where enumerative induction alone would not suffice. Thus whereas Hume raised sceptical questions about induction, Mill pushed through an empiricist analysis of deduction. He recognized as primitively legitimate only the disposition to rely on memory and the disposition to generalize from experience. The whole of science, he thought, is built from these.
In particular, he did not accept that the mere fact that a hypothesis accounts for data can ever provide a reason for thinking it true (as opposed to thinking it useful). It is always possible that a body of data may be explained equally well by more than one hypothesis. This view, that enumerative induction is the only authoritative source of general truths, was also important in his metaphysics. Accepting as he did that our knowledge of supposed objects external to consciousness consists only in the conscious states they excite in us, he concluded that external objects amount only to ‘permanent possibilities of sensation’. The possibilities are ‘permanent’ in the sense that they can be relied on to obtain if an antecedent condition is realized. Mill was the founder of modern phenomenalism.
In ethics, Mill’s governing conviction was that happiness is the sole ultimate human end. As in the case of induction, he appealed to reflective agreement, in this case of desires rather than reasoning dispositions. If happiness was not ‘in theory and in practice, acknowledged to be an end, nothing could ever convince any person that it was so’ (1861a: 234). But he acknowledged that we can will to do what we do not desire to do; we can act from duty, not desire. And he distinguished between desiring a thing as ‘part’ of our happiness and desiring it as a means to our happiness. The virtues can become a part of our happiness, and for Mill they ideally should be so. They have a natural base in our psychology on which moral education can be built. More generally, people can reach a deeper understanding of happiness through education and experience: some forms of happiness are inherently preferred as finer by those able to experience them fully.
Thus Mill enlarged but retained Bentham’s view that the happiness of all, considered impartially, is the standard of conduct. His account of how this standard relates to the fabric of everyday norms was charged with the nineteenth century’s historical sense, but also maintained links with Bentham. Justice is a class of exceptionally stringent obligations on society – it is the ‘claim we have on our fellow-creatures to join in making safe for us the very groundwork of our existence’ (1865b: 251). Because rights of justice protect this groundwork they take priority over the direct pursuit of general utility as well as over the private pursuit of personal ends.
Mill’s doctrine of liberty dovetails with this account of justice. Here he appealed to rights founded on ‘utility in the largest sense, grounded on the permanent interests of man as a progressive being’ (1859: 224). The principle enunciated in his essay On Liberty (1859) safeguards people’s freedom to pursue their own goals, so long as they do not infringe on the legitimate interests of others: power should not be exercised over people for their own good. Mill defended the principle on two grounds. It enables individuals to realize their potential in their own distinctive way, and, by liberating talents, creativity and energy, it institutes the social conditions for the moral development of culture and character.
Theological Underpinnings of the Modern Philosophy of Mathematics.
