MATHEMATICAL REALISM

Many mathematicians think that the things studied in mathematics are human creations. They think that integers, real numbers, imaginary numbers, groups, topological spaces, and the rest have been invented by human beings. Such things would not exist if there were no people. They came into existence as a result of human activity. They even have birthdays.

We agree with the positive claims of such antirealists: we agree that there are indeed human creations which would not exist if there were no people and which have birthdays. These human creations are words, ideas, diagrams, images, concepts, theories, textbooks, academic departments, and so forth. Yet as realists, we insist that these human creations are not the only things worth studying; there are more things than these. In addition to human creations, there are things which are not human creations, things which would still exist even if people did not, things whose birthdays, if they have any, probably coincide with the birth of the universe and certainly precede the origin of human life.

Realism about mathematics rests comfortably and naturally with scientific modal realism. It is not compulsory for all scientific realists to take a realist stand about mathematics along with all the rest of science. No one can sensibly maintain realism about absolutely all the bits and pieces bandied about in science. It must be admitted that some parts of science are merely useful fictions and do not accurately represent anything beyond themselves.