sciences.

Note that a question like this about the classification of intellectual disciplines appears to be a philosophical question itself.

We support this claim with three kinds of evidence:

- the history of mathematics is different from the history of science.
- the objects of mathematics are different from those of the sciences.
- the methods of mathematics are different from those of science.

Mathematics is of course far older. There is no essential difference between mathematics as practiced by the ancient Greeks and mathematics as we practice it except the vast extension of the subject matter which has occurred recently. (This is itself a bold claim to which one could take exception: there are differences, such as the extension of rigorous methods to arithmetic as well as geometry and the modern attitude toward the actual infinite).

An amusing historical question: given the role of the Pythagorean
school
in the foundation of Greek mathematics, is mathematics a contemplative
religion?

It took longer for it to be recognized that the geometry studied in mathematics is not to be identified with the geometry of physical space. Right into the 19th century, Euclidean geometry could be regarded (as by Kant) as a discipline which revealed logically necessary truths about the actual space of the world. The discovery of non-Euclidean geometry cast doubt on this view of the relationship between the space of geometry and the space of the physical world, and the observational successes of the theory of general relativity led most scientists to conclude that the geometry of Euclid was not the geometry of the physical world.

The further objects of mathematics at the time of the discovery of non-Euclidean geometry were numbers (of many varieties, some of which caused philosophical disquiet: rational and irrational, positive and negative, real and complex) and the functions which were the object of study in calculus. These had possible physical interpretations (though there were doubts about the physical meaning of, for example, imaginary numbers). At about the same time that non-Euclidean geometry was discovered, mathematicians began to investigate further abstractions, such as higher-dimensional geometries and hypercomplex numbers, whose connection to the physical world was clearly more remote.

The further enterprise of the late 19th and early 20th century in developing the foundations of mathematics in terms of set theory made it clear that the ontology of mathematics was not based on physical reality in any obvious way.

Even my discussion of the objects of mathematics makes it clear that the objects (especially geometrical) of mathematics were clearly confused with aspects of physical reality until recently, and there are those who maintain that the abstract turn of mathematics in the late 19th century was a mistake and the real mathematics is that which models physical phenomena. Unfortunately for this argument, one of the most active areas of applied mathematics nowadays is exactly the pure mathematics of the foundations (logic and aspects of set theory) which turns out to be very useful in computer science.

The strongest point in my argument is the clear difference in method between mathematics and science. Historically, arithmetic and its offspring algebra and calculus were not pursued with the same logical rigor as geometry had been since the time of the Greeks. Mathematicians put this right in the late 19th and early 20th centuries: not only was arithmetic put on a firm logical foundation, but geometry was explained in terms of arithmetic (an outcome whose possibility was foreshadowed by the algebraic geometry of Descartes). This would have surprised the Greeks. The fact that arithmetic was then further explained in terms of set theory would have been even more surprising to them.

Mathematics is a rationalist discipline apparently studying a realm of "abstract" objects and engaged in determining necessary truths about these objects. Natural science is an empiricist discipline engaged in studying objects with which we are only acquainted by observation and about which we do not claim (in our more careful moments) to be determining necessary truths. This seems like a mismatch. In practice, it is enormously successful. Why? (This is more a question about philosophy of science)

The objects of mathematics have only an abstract nodding acquaintance with the objects of physics (at best we may identify some mathematical objects as idealizations of physical objects: think about elementary school level discussions of the difference between the circle we draw on paper and a real circle). The official ontology of mathematics explains all mathematical objects in terms of sets, which certainly do not seem to be physical objects. Geometric objects (and perhaps the functions of calculus) are actually fairly readily confused with (and often successfully understood in terms of) physical models: but this is not true of numbers. The number 17 seems to be something we learn about successfully in school, and so is the square root of negative one. But what are they? If they are not physical objects in the everyday world, what are they and how do we know about them? If there are no such objects, how do we succeed in talking about them (and, even more embarrassingly, in talking about them usefully?) This is more a question about philosophy of mathematics.

For Pythagorean atomism to work successfully, all lengths needed to be commensurable (I'll explain what this means). They discovered (and I will prove) that the length of the diagonal of a unit square is not commensurable with the lengths of its sides. They concluded on the basis of this logical proof that their view of reality was wrong (in this lies their greatness!) and thereafter the Greeks, who did not understand real number as we do, preferred geometry to arithmetic.

I may further briefly outline the definition of real numbers due to Dedekind (restricting myself to positive real numbers to minimize technical complexities).

The point of this bit is to illustrate by example what it means to reduce mathematics (in this case arithmetic) to set theory. As soon as the real numbers are explained in terms of sets, notice that points in the usual Euclidean plane can be identified with pairs of real numbers using the older (but philosophically analogous) reduction of geometry to the arithmetic of the real numbers by Descartes (problematic when it was first proposed because geometry was at that time on a firmer logical footing than the real numbers!)