Henry Sidgwick, answered from the texts and cited to the page.
I set out to show that ethics could be a rational science, its first principles as evident as those of geometry. I found, at the end of the most careful inquiry I could conduct, that two of those principles are both self-evident and mutually contradictory. The principle of rational benevolence directs me to seek the good of all — that, from the point of view of the universe, my own good is of no more importance than the good of any other, and that a rational being is bound to aim at good generally.
The principle of rational egoism directs me to seek my own good, and it too, when I examine it, presents itself to my reason with the character of self-evidence: that my own good on the whole is what I have most reason to promote.1 In the ordinary course these principles coincide, for the general happiness and my own are usually served together.
But where they part — where the good of all demands the sacrifice of my own good on the whole — I can find no principle, self-evident or demonstrable, that adjudicates between them. Reason speaks with two voices and does not tell me which to obey. This is the dualism of practical reason, and I will not paper it over. If it cannot be resolved, then the Cosmos of Duty is really reduced to a Chaos, and the effort to frame a wholly rational ideal of conduct has failed.
I reported this because it is what I found, not because I wished it. Only a demonstration that the universe is so ordered that duty and self-interest finally coincide could close the gap, and that demonstration I could not supply.23
The commonest view of the principle would no doubt be that the present pleasure or happiness is reasonably to be foregone with the view of obtaining greater pleasure or happiness hereafter: but the principle need not be restricted to a hedonistic application; it is equally applicable to any other interpretation of 'one's own good,' in which good is conceived as a mathematical whole, of which the integrant parts…The Methods of Ethics
I think that a 'plain man,' in a modern civilised society, if his conscience were fairly brought to consider the hypothetical question, whether it would be morally right for him to seek his own happiness on any occasion if it involved a certain sacrifice of the greater happiness of some other human being,--without any counterbalancing gain to any one else,--would answer unhesitatingly in the negative.The Methods of Ethics
On reflection it did not seem to me really to meet the difficulty which had led me from Mill to Kant: it did not settle finally the subordination of Self-interest to Duty.The Methods of Ethics
The most famous unresolved problem in the history of ethics and the conclusion that has made Sidgwick central to contemporary moral philosophy. Derek Parfit (On What Matters, 2011) called Sidgwick the greatest moral philosopher and made the dualism the pivot of his own project. Singer and de Lazari-Radek (The Point of View of the Universe, 2014) accept the framing but argue the egoistic axiom can be debunked as an evolutionary artifact, leaving benevolence in sole possession — thus dissolving the dualism in utilitarianism's favor. Others (following Broad, and more recently some Kantians) argue the dualism is an artifact of Sidgwick's hedonism and his particular conception of the axioms, not a permanent feature of practical reason. That the greatest systematizer of utilitarianism ended in an honest impasse rather than a proof is itself the most-discussed fact about the book.