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Looking back at these efforts

As stated at the outset, this article is intended to be a brief history of the arithmetization program. It seeks to set forth the achievements of mathematicians working in that program that have led to what is important in mathematics today. The view presented has been dubbed "the standard account" by some, for the simple reason that it is not the only account. In such history articles, it is usual to include alternate views of events within the main narrative. That practice is not adhered to in this article. Instead, summaries of those alternate views are gathered here, at the end of the article, with the intention of providing a platform for inserting additional entries and, if appropriate, for linking to additional articles.

Concurrent "corrections" to arithmetization

One modern commentator, forthright in his support of the arithmetization program, generally, and of the contribution of his countryman Bolzano, in particular, nevertheless advances somewhat of a caveat in the form of the following contrast:

• it was essential to move analysis off of its intuitive/geometric base onto a rigourous/arithmetic base
• mathematics continued to develop during the arithmetization period, somewhat as a "correction" to it

In his own words:

[By] the first half of the nineteenth century the building of mathematical analysis was raised to such a height that continuing its construction without fortifying its foundations was unthinkable. This brought a period of great revision of the foundations of analysis...; the development of the other branches of mathematics continued, of course, simultaneously and in mutual interaction.

It seems evident that ... the revision could not follow other direction than that of consequential arithmetization of analysis.... [Yet,] this arithmetization ... was later corrected by the modern development of mathematics; after all, even in the period mentioned the dialectics of this process can be observed: so, for example, B. Riemann who on the one hand contributed considerably to the arithmetization of analysis by his theory of integral, was on the other hand the ingenious builder of the geometric theory of analytic functions.

A role for intuition in mathematics

What today are commonplace notions in undergraduate mathematics were anything but commonplace among practicing mathematicians even a quarter century after the 1872 achievements of Cantor, Dedekind, and Weierstrass. In 1899, addressing the American Mathematical Society, James Pierpont spoke to show these two things:

1. why arithmetical methods form the only sure foundation in analysis at present known
2. why arguments based on intuition cannot be considered final in analysis

In a later, printed version of his address, Pierpont prefaced his words with the following:

We are all of us aware of a movement among us which Klein has so felicitously styled the arithmetization of mathematics. Few of us have much real sympathy with it, if indeed we understand it. It seems a useless waste of time to prove by laborious $\varepsilon$ and $\delta$ methods what the old methods prove so satisfactorily in a few words. Indeed many of the things which exercise the mind of one whose eyes have been opened in the school of Weierstrass seem mere fads to the outsider. As well try to prove that two and two make four!

The term "arithmetization of mathematics," which Pierpont here ascribed to Klein, has also been credited to Kronecker -- perhaps to others as well? In any case, Pierpont ended his 1899 address with this paean to the labours of Weierstrass and others:

The mathematician of to-day, trained in the school of Weierstrass, is fond of speaking of his science as die absolut klare Wissenschaft. Any attempts to drag in metaphysical speculations are resented with indignant energy. With almost painful emotions he looks back at the sorry mixture of metaphysics and mathematics which was so common in the last century and at the beginning of this. The analysis of to-day is indeed a transparent science. Built up on the simple notion of number, its truths are the most solidly established in the whole range of human knowledge.

Pierpont here addresses what may seem to be a very uncontroversial result among several that are said to have proceeded from the arithmetization program, namely, the replacement of "arguments based on intuition" with "arithmetic methods" as the foundation of analysis. Even so, an objection has been raised in the form of a suggestion that an untoward fixation on "rigourous formalism" can lead to a loss of the mathematical intuition that is essential to working at a mature level:

The point of rigour is not to destroy all intuition; instead, it should be used to destroy bad intuition while clarifying and elevating good intuition. It is only with a combination of both rigorous formalism and good intuition that one can tackle complex mathematical problems; one needs the former to correctly deal with the fine details, and the latter to correctly deal with the big picture. . . . So once you are fully comfortable with rigorous mathematical thinking, you should revisit your intuitions on the subject and use your new thinking skills to test and refine these intuitions rather than discard them. . . .

The ideal state to reach is when every heuristic argument naturally suggests its rigorous counterpart, and vice versa.

A place for infinitesimals in mathematics

Certainly a very significant "correction" of arithmetization, the development of non-standard analysis, was described generally by another modern commentator as follows:

Weierstrass' definition of limit appeared to finally nail the coffin of the departed quantities and led to a complete abandonment of the original idea of infinitesimals. However, in the 1960s the ghosts have been resurrected by Abraham Robinson and placed on the sound foundation of the non-standard analysis thus vindicating the intuition of [Newton and Leibniz,] the founding fathers.

Here, from the same source, is Robinson's definition of limit:

$\displaystyle \lim_{x \to a} f(x) = L$, if and only if $f(x)$ is infinitely close to $L$ whenever $x ≠ a$ is infinitely close to $a$.

It was widely believed that, until the development of non-standard analysis by Abraham Robinson in the 1960s, arithmetization had banished infinitesimals from mathematics. Certainly, the banishment of infinitesimals was considered an important reason for pursuing the arithmetization program In reality, however, the work on non-Archimedean systems continued unabated during and after the period of arithmetization, as documented by P. Ehrlich.

One modern commentator has characterized the effect (in some quarters) of Robinson's work in non-standard analysis as "a rehabilitation of the use of infinitesimals in mathematics."

• Robinson himself proposed that the techniques of non-standard analysis would show (at least some) of the infinitesimal methods used by the founders of calculus to be "correct and consistent."

Arithmetization and rigour

Some (Dauben, Katz et. al.) working with the new non-standard methods challenged what is reasonably termed "the standard account" among both 19th and 20th century mathematicians, namely, that the arithmetization program resulted in an increase of rigor.

Set theoretic foundations

The following contrast highlights a modern issue in our understanding of the arithmetization program:

• to the mathematicians who developed it, the arithmetization program signified efforts to develop a foundation for analysis, i.e. the calculus, in terms of the natural numbers
• after the development of naive set theory by Cantor, the arithmatization program came to signify the set-theoretic definition of function and the set-theoretic construction of the real line

There were several important consequences of this shift in meaning and intent of the arithmetization program, some of which were and remain contentious.

Kronecker and other intuitionists rejected such efforts to extend the arithmetization program by defining the natural numbers using set theoretic concepts, arguing instead as follows:

• all the definitions appearing in the field of analysis can be reduced to the whole numbers and their properties
• the whole domain of this branch of mathematics can be explained basically from arithmetic

Kronecker's view was that mathematics "should be constructed rigidly on the basis of intuition of natural numbers."

Consequences for teaching

As has been noted by many, arithmetization brought about a shift in emphasis from geometric to algebraic reasoning. Again, this shift was certainly considered an important reason for pursuing arithmetization. What is not widely appreciated is that an important consequence of this shift was a change in the way mathematics is taught today.

Finally, the shift in the meaning of arithmetization led to logicism, a currently prevalent philosophical position that all of mathematics should be derivable from logic and set theory. Logicism, in turn, led to Hilbert's program, to the theorems of Gödel, Turing, and Chaitin on undecidability and incompleteness, and to non-standard analysis.