About 300 years ago, Newton and Leibniz invented calculus. Well, that's an oversimplification. Some of the ideas of calculus were already around, but they cleaned it up and knitted it together with what we now call the Fundamental Theorem of Calculus. Newton also showed some of the ways calculus can be used -- he worked out many of the basic laws of physics, and showed how to compute the orbits of the planets much more simply and accurately than anyone had ever done before. In doing so, he contributed greatly to the beginning of the Age of Enlightenment -- an age in which people realized that they can accomplish quite a lot through reasoning, and that they don't have to just live in fear, superstition, and confusion. This may have indirectly contributed to things like the industrial revolution and the birth of democracy.

Anyway, Newton and Leibniz knew how to do many of the computations that we now teach in calculus, but they didn't know how to do satisfactory proofs of the theory behind calculus. They tried to do proofs, but their explanations were a bit lacking. Many of their explanations were based on infinitesimals -- i.e., numbers that are infinitely small but not zero. For instance, in their explanations, dy/dx did not represent a limit of changing numbers. It represented a quotient of unchanging numbers, but those numbers were infinitesimals.

The computations of Newton and Leibniz were accepted by other mathematicians, but the proofs were not. The explanation of infinitesimals didn't entirely make sense, and mathematicians were uncomfortable with it. In the following centuries, Cauchy and Weierstrass produced the epsilon-delta proofs that we now find in calculus textbooks. Those proofs involve numbers that are of "ordinary" size (not infinitesimal), but the numbers would vary through many different ordinary sizes; thus we take the limit as epsilon changes toward zero. In our textbooks, dy/dx represents the limit of a changing quotient of two ordinary numbers. In the late 19th century, Dedekind finally gave a clear explanation of the real numbers (which we'll sketch at the end of this web page), and we can prove that in Dedekind's number system there are no infinitesimals. Arguments with infinitesimals were no longer needed and fell out of favor. Ultimately, infinitesimals were discredited and discarded by mathematicians (though they continued to be mentioned in some physics books many decades later).

In the 1960's, mathematician Abraham Robinson finally figured out how to make sense out of infinitesimals. Thus nonstandard analysis was born. It involved some nonstandard real numbers, among which we can find some infinitesimals. In the paragraphs below, I will give an example of an ordered field that has some infinitesimals. The discussion below is based on 20th-century ideas, not just on those of Newton and Leibniz. I should mention, however, that the example that I will present is not the approach preferred by the nonstandard analysts. They prefer an approach that is more complicated but also more powerful. (It involves making careful logical analysis of a formal first-order language, but we don't need to discuss that here.)

Some of the nonstandard analysts now actually feel that infinitesimals yield a better understanding of calculus. After all, it gave Newton and Leibniz the intuition that they needed. We can actually make rigorous mathematics, with only slight adjustments in the ideas of Newton and Leibniz. (For instance, the derivative should be the standard part of that quotient of infinitesimals; this term is explained in a later paragraph below.) But most mathematicians still prefer the epsilon-delta approach, which they feel is simpler. (Both methods are correct, and both yield the same results.) At any rate, some discussion of infinitesimals may be helpful in our explanation of ordered fields.

Some ordered fields have infinitesimals, and some don't. The ordered fields that have no infinitesimals are called Archimedean fields; we'll see later that the real number system (i.e., Dedekind's number system, also known as the standard real numbers) is Archimedean. The ordered fields that do have infinitesimals are called non-Archimedean fields; we'll give an example of such a field in the next few paragraphs.

The example will be based partly on rational functions. By a rational function in the variable t, we will mean a function of the form p(t)/q(t), where p(t) and q(t) are polynomials with standard real coefficients, and q is not the constant polynomial 0. Note that each real number can be viewed as a rational function -- for instance, the number 7 can be viewed as 7/1, where 7 and 1 are both polynomials of degree 0. Thus the set of real numbers is a subset of the set of rational functions. (Of course, to make sense of this, we have to assume that we already have some understanding of the real numbers. But we won't need a very deep understanding; the "points on a line" conception will suffice for now.)

We define addition and multiplication of rational functions in the usual fashion, as in high school algebra. However, we make this one alteration in the usual treatment of rational functions: We will consider two rational functions to be "the same" if they agree except at finitely many values of t. For instance, these two functions

are not really the same, because the first one is defined at t = -2 and the second one is not. But the two functions are identical for all other values of t, so we will view them as "the same" for purposes of the present discussion. With that convention, it can be shown that the set of all rational functions is a field.

Also, the real numbers are a subset of the rational functions. For instance, the constant 1 and the constant 7 are polynomials of degree 0, so the constant 7/1 is a rational function. In this fashion we can view every real number as a rational function.

We can make the rational functions into an ordered field, if we just define the right ordering. To do so, we will make use of the following theorem. (We will omit the proof of the theorem, which is a bit harder, but it just involves some advanced calculus and some college algebra.)

We now define an ordering on the rational functions, by saying that

if cases 1, 2, or 3 hold, respectively. In other words, one rational function is less than another if it is eventually less -- i.e., if it is less when we go far enough to the right on the graphs of the two functions. How far to the right we have to go may depend on which two functions we're looking at; but the theorem says that for each choice of two rational functions, there is some point after which one function stays below the other (unless they're the "same").

With this definition of ordering, it turns out that the set of rational functions is an ordered field. But it also turns out that the functions

are infinitesimals. Thus, the field of rational functions is non-Archimedean, when ordered as we have described.

How does this relate to Newton's view of numbers? I'm sure that Newton wasn't thinking of his infinitesimals as rational functions. But we can get some idea of his viewpoint, as follows:

There are no infinitesimals among the standard real numbers. But we could imagine that, with a sufficiently powerful microscope, we might discover some additional "nonstandard" numbers that we had not noticed before. Nestled around each standard real number r, infinitely close to it, are infinitely many new nonstandard numbers. (Then r is the standard part of any of those new numbers.) In particular, nestled around 0 are the infinitesimals. We can also get some other nonstandard numbers by taking the reciprocals of the infinitesimals; those numbers are infinitely large. The collection of all the numbers -- both "standard" and "new", together -- is an ordered field. Its ordering is the same as the ordering of the set of rational functions.

Information provided by Eric Schechter