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We are a school of philosophers of mathematics in Sydney, Australia. Our line is realist (about structure or pattern), but Aristotelian rather than Platonist: we hold that mathematics studies real properties of things such as symmetry and continuity . . . Since a main obstacle to the understanding of realism in mathematics is ignorance about Aristotelian realism in general, we provide a tutorial introduction to that topic
home.

"Orange is closer to red than to blue." That is a statement about colours, not about the things that have the colours - or if it is about the things, it is only about them in respect of their colour. There is no way to avoid reference to the colours themselves.

Colours, shapes, sizes, masses are the repeatables or "universals" or "types" that particulars or "tokens" share. A certain shade of blue, for example, is something that can be found in many particulars - it is a "one over many" in the classic phrase of the ancient Greek philosophers. On the other hand, a particular electron is a non-repeatable. It is an individual; another electron can resemble it, but cannot literally be it.

Science is about universals. There is perception of universals - indeed, it is universals that have causal power. We see an individual stone, but only as a certain shape and colour, because it is those properties of it that have the power to affect our senses. Science gives us classification and understanding of the universals we perceive - physics deals with such properties as mass, length and electrical charge, biology deals with the properties special to living things, psychology with mental properties and their effects, mathematics with ... well, we'll get to that; see intro.

Not everyone agrees with the foregoing. Nominalism holds that universals are not real, but only words or concepts; not very plausible in view of the ability of all things with the same shade of blue to affect us in the same way - "causality is the mark of being". Platonism holds that there are universals, but they are pure Forms in an abstract world, the objects of this world being related to them by a mysterious relation of "participation". That too makes it hard to make sense of the direct perception we have of shades of blue. Aristotelian realism about universals takes the straightforward view that the world has both particulars and universals, and the basic structure of the world is "states of affairs", such as this table's being approximately square.

• Q. Are universals the meaning of words? Are particulars like nouns and universals like adjectives?

  A. What universals there are is a matter for science, not for linguistics or logic. Whether "sacred" is a real property of things is for inquiry, not for fiat.
  Nevertheless, it is hardly surprising that language, which is for usefully describing the world, should loosely reflect its basic ontological structure. The subject-predicate form of many simple sentences is useful because of the state-of-affairs structure of the reality it describes. Likewise many prepositions describe relations. One should not be fundamentalist about language, however, as it intended for many human concerns other than description, for example, entertainment through fictions.

• Q. What about relations, like "being shorter than"? Do they exist too?

  A. Sure. Since lengths exist, the relations between them do too.
  A certain blindness to the reality of relations in Western thought (book) has bedevilled the philosophy of science and mathematics. For example, it is hard to appreciate mathematics and science wihout a solid grasp of the reality of ratios.

• Q. What about uninstantiated universals?

  A. An uninstantiated shade of blue (if there is one) seems an unproblematic universal - it belongs in the blue continuum and the science of colour can deal with it on an equal footing with the instantiated shades. Very large numbers are in a similar position. Truly alien universals that are not properties of anything in existence and that are beyond our imagination will be hard to know about, but there seems little reason to deny their possibility.

• Q. What about epistemology? How are universals known?

  A. A simple instantiated universal can affect the sense organs directly - we perceive a particular only as having universals: we perceive a ball as yellow and round, and can only perceive it because it is yellow and round. Different sense organs are sensitive to different properties. Some more intellectual operation, called "abstraction", is needed to explicitly isolate universals, and provide a basis for understanding the similarities and other relations between universals and the truths about uninstantiated ones.

• Q. Is there anything to particulars over and above the universals they have? Is a particular just a "bundle of universals"?

  A. This theory was defended by Bertrand Russell but it seems hard to get the particularity of particulars out of pure universals without some particular "substance" for them to inhere in.

• Q. Are sets universals?

  A. No. The set {Sydney, Hong Kong} is no more repeatable than the cities themselves are. Blue is a universal but the set of all blue things is a particular. One Aristotelian account of what sets are is D.M. Armstrong, `Classes are states of affairs', Mind 100 (1991), 189-200.

• Q. Are truths about universals necessary?

  A. Sometimes, at least. Surely there is no possible world in which orange is between blue and green instead of between red and orange, or in which orange is not a colour.
  It is harder to say about the relations between universals that constitute laws of nature. It seems possible that the attraction between masses described by Newton's law of gravity should be other than it is.

• Q. Are there dispositional properties (like brittleness, which only comes into play when something is struck) as well as categorical ones like shape?

  A. Yes, dispositions are needed to support the truth of counterfactuals like "if the glass were struck hard, it would break" (which are true even if the glass, or any glass, is never struck). It is debated whether or not dispositions can in some way be reduced to categorical properties (discussion, book). Mathematics is mercifully free of dispositions.

• Q. How many properties does a thing have?

  A. This question is probably too hard. Aristotle attempted a theory of categories, classifying the kinds of properties that a things could have, but there is no agreed list. Even with a property that is well understood, like shape, it is hard to say whether it is one property or many.

W.v.O. Quine, `On what there is' (1948). (this classic article attacking realism about universals illustrates the problems of the assumption that Platonism and nominalism are exhaustive alternatives. Its faults are less visible for it being unavailable online; but see here for a discussion that is aware of the alternatives)
• J Franklin, Corrupting the Youth: a history of philosophy in Australia
  (contains a chapter on Australian philosophy of science, including the theory of universals and laws of nature)