Philosophy has always been concerned with the largest questions of what exists and what is the case. Not specific questions like “Is there extraterrestrial life?” or “What is the speed of light in a vacuum?” but maximally general questions about what kinds of things exist and what kinds of facts there are—basic kinds that cannot be defined in terms of anything still more basic, and from which everything else is constituted. This is the subject matter of metaphysics. A related question, how we can know about such things, is the subject matter of epistemology.
There is disagreement about how far a philosophical account of existence, truth, and knowledge can depart from the assumptions that underlie ordinary, commonsense thought and discourse about what there is, what is the case, and what we know. Radical doubt is not unusual: the basic reality of many things—matter, mind, space, and time—has been called into question at one time or another; but among the domains or areas of thought that are perennially believed to pose problems, and that continue to be the focus of much discussion today, are morality and mathematics.
We frequently make moral judgments—that it was wrong of X to accuse Y of a theft of which X was in fact guilty, or that helping someone to commit suicide is not always wrong—but what, if anything, could make these judgments true or false? They seem not to describe what exists or happens in the world but to say something further. What kind of fact could this be? We also have mathematical beliefs, for example, that 7+8=15. Of course we use these beliefs in practical calculations about the material world, but we also think they are true in themselves—timelessly true. But what are they about, and what makes them true? The number 7 itself doesn’t seem to be part of the natural world we see around us.
It is characteristic of our age that moral truth and mathematical truth are found philosophically puzzling because they seem so different from scientific truth about the natural world. Existence, truth, and knowledge in the natural sciences are regarded as relatively straightforward, though they depend on taking for granted the validity of the observational evidence and logical or mathematical reasoning used in the confirmation of scientific theories. Natural science is widely thought to provide a standard for robust reality that is difficult for the objects of moral judgment and pure mathematics to meet. That is why it is so easy to wonder whether numbers or wrongness really exist, outside of our thoughts.
Such doubts depend, however, on a crucial assumption. They assume that there is a universally applicable standard of existence and truth that determines the reality or unreality of any type of thing or fact. This means that the reality of what we are talking about when we talk about morality or mathematics must…
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