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Looking back at arithmetization

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 history 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.

A modest caveat

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.

Challenges from non-standard analysis

In 1960, Abraham Robinson developed non-standard analysis. He did this by (rigourously) extending the field of reals to include infinitesimal numbers and infinite numbers. The new, extended field is called the field of hyperreal numbers. His goal was to create a system of analysis that was more intuitively appealing than standard analysis but without losing any of the rigor of standard analysis.

In summary and informally, the definition of the hyperreal field is as follows:

• Axiom A – $\mathbb{R}$ is a complete ordered field.
• Axiom B – $\mathbb{R}^*$ is a proper ordered field extension of $\mathbb{R}$.

• An element, $x$, of $\mathbb{R}^*$ is an infinitesimal if $|x| < r$ for all positive real $r$.
• Two elements $x, y$ of $\mathbb{R}^*$ are said to be infinitely close if $x \approx y$ and if $x - y$ is an infinitesimal. Thus, $x$ is an infinitesimal if and only if $x \approx 0$.
• An element, $x$, of $\mathbb{R}^*$ is an infinite if $|x| > r$ for all real $r$.

A fundamental property of non-standard analysis the transfer principle, which states "every proposition true in classical analysis is also true in non-standard analysis," and vice versa. Here, for example, are two, equivalent definitions of a limit:

• a standard $\varepsilon, \delta$ definition of a limit

the limit of $f (x)$ as $x \to c$ is L if,

for every real $\varepsilon > 0$, there is a real $\delta > 0$, such that

whenever $x$ is real and $0 < |x - c| < | \delta$, then $| f (x) - L | < \varepsilon$

• and an equivalent, non-standard definition:

whenever $x \approx c$ but $x \neq c$ then $f (x) \approx L$

Given a sentence $\phi$, we may take its *-transform *$\phi$ as follows:

• replacing a function $f$ with $f^*$
• replacing a relation $R$ with $R^*$, and
• replacing any bound $P$ occurring in $x \in P$ by $x \in$ $P^*$.

The Transfer Principle states that if we are working over the language of $\mathbb{R}$, then, a sentence $\phi$ is true if and only if $\phi^*$ is true.

As an interesting example, we can look at the Archimidean property of the reals:

$\forall x \in \mathbb{R^+} (\exists n \in \mathbb{N}) (nx >$ $1)$

The *-transfer of this statement is

$\forall x \in \mathbb{R^*}^+ (\exists n \in \mathbb{N}^*) (nx >$ $1^*$)

This second statement is true, yet it is not equivalent to the Archimidean property. In fact, $\mathbb{R}^*$ is not Archimidean in the standard sense. No repeated addition of $[1/n]$ will ever bring it above $1$. The Transfer Principle says that It doesn't really matter whether we work in the standard or the non-standard setting. Yet, there is a lesson here, namely, that some properties transfer very naturally between $\mathbb{R}$ and $\mathbb{R}^*$, while others may transfer in ways that may not be intuitive and/or useful.

An alternate expression of the Archimedean axiom or property with respect to the theory of ordered fields is this::

$(\forall x > 0) (\forall \varepsilon > 0) (\exists n \in \mathbb{N}) (n \varepsilon > x)$

or equivalently

(3) $(\forall \varepsilon > 0) (\exists n \in \mathbb{N}) (n \varepsilon > 1 )$.

A number system satisfying (3) above is an Archimedean continuum.

In the contrary case, there is an element $\varepsilon > 0$ called an infinitesimal such that no finite sum $\varepsilon + \varepsilon + . . . + \varepsilon$ will ever reach 1; in other words,

(4) $(\exists \varepsilon > 0) (\forall n \in \mathbb{N}) (\varepsilon \leq 1/n)$.

A number system satisfying (4) above is a Bernoullian continuum (i.e., a non-Archimedean continuum).

The development of non-standard analysis has given rise to the following questions, the answers to which challenge the standard account of arithmetization:

• What is the role of intuition in mathematical theory and thinking?
• What is the place of infinitesimals in analysis?
• What is the nature of rigour in mathematical definitions and proofs?
• Are there useful alternate methods of teaching (and doing) analysis?

The remainder of this section relates some answers that have arisen from non-standard analysis.

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 addressed these two questions:

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.

Even so, and in somewhat of an about face, Pierpont concludes with this comment about the "extreme arithmetizations" ushered in by that school of Weierstrass:

It is, however, not to be overlooked that the price paid for this clearness is appalling, it is total separation from the world of our senses.

Today, more than 100 years after Pierpont's address, intuition is present in mathematics in at least two, quite different, respects:

1. as an accompaniment to the reintroduction of infinitesimals by non-standard analysis
2. as an essential part of what is involved in mature, high-level mathematical thinking

Intuition in mathematical theory

According to the standard account, the arithmetization program excised intuition from the foundations of mathematics. With the development (or, as it has alternatively been put, the "discovery") of non-standard analysis, intuition was restored to legitimacy in the foundations of mathematics. Robinson maintained that results achieved using non-standard analysis could not have been achieved "just as well" by standard methods and that translation of non-standard results into standard terms "usually complicated matters considerably." He explained the reason for this difficulty as follows:

Our approach has a certain natural appeal, as shown by the fact that it was preceded in history by a long line of attempts to introduce infinitely small and infinitely large numbers into Analysis.

The "natural appeal" to which Robinson refered is grounded in two factors:

1. non-standard analysis is often simpler and more intuitive in a very direct, immediate way than standard approaches
2. the concept of infinitesimals had always seemed natural and intuitively preferable to more convoluted and less intuitive sorts of rigor.

Robinson's extension of the number concept has been shown to be of use in various domains, including areas of analysis, the theory of complex variables, mathematical physics, and economics. As a consequence, while the methods of non-standard analysis can be avoided in these domains, the cost of doing so may be more complicated proofs and less intuitive arguments.

The "natural advantages" of using infinitesimals could be exploited, now that non-standard analysis had shown that their use was "safe for consumption" in mathematics!

Intuition in mathematical thinking

In a very different sense, intuition is present in mathematical thinking at the highest levels, quite apart from and irrespective of the nature of foundations theory. In the remarks quoted above, Pierpont noted 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

The standard account asserts that arithmetization "banished" infinitesimals from mathematics. Certainly, the elimination of infinitesimals was considered an important reason for pursuing the arithmetization program. Here, very briefly, is some history:

For many centuries, early mathematicians and physicists would solve problems by considering infinitesimally small pieces of a shape, or movement along a path by an infinitesimal amount.... In particular, the construction of calculus was first motivated by this intuitive notion of infinitesimal change.... Although intuitively clear, infinitesimals were ultimately rejected as mathematically unsound, and were replaced with the common $\varepsilon, \delta$ method of computing limits and derivatives.

And here, equally brief, is a bit more:

When Newton and Leibnitz practiced calculus, they used infinitesimals, which were supposed to be like real numbers, yet of smaller magnitude than any other type of postive real number. In the 19th century, mathematicians realized they could not justify the use of infinitesimals according to their sense of rigor, so they began using definitions involving $\varepsilon$'s and $\delta$'s. Then, in the early 1960's, the logician Abraham Robinson figured out a way to rigorously define infinitesimals, creating a subject now known as non-standard analysis.

Several comments are relevant. First, the work on non-Archimedean systems actually continued unabated both during and after the period of arithmetization. Indeed, there was "a rich and uninterrupted chain of work" on non-Archimedean or infinitesimally-enriched systems or what some have called Bernoullian continua:

Second, as noted above, in 1960 came a very significant "correction" to arithmetization, namely, the development of non-standard analysis:

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 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 $a$. The claim that Weierstrass eliminated infinitesimals has been termed an oversimplification of the history of analysis:'"`UNIQ--ref-00000014-QINU`"' : If the epsilon-delta methods had not been discovered, then infinitesimals would have been postulated entities (just as ‘imaginary’ numbers were for a long time). Indeed, this approach to the calculus–enlarging the real number system–is just as consistent as the standard approach, as we know today from the work of Abraham Robinson [. . . ] If the calculus had not been ‘justified’ Weierstrass style, it would have been ‘justified’ anyway. There are a variety of possible conceptions of the continuum. Mathematicians have considered at least two different types of continua: * Archimedean continua, or A-continua for short * infinitesimal-enriched (Bernoulli) continua, or B-continua for short Neither an A-continuum nor a B-continuum corresponds to a unique mathematical structure. The notion that there is a single coherent conception of the continuum, and it is a complete, Archimedean ordered field has been called "an academic dogma."'"`UNIQ--ref-00000015-QINU`"' : the collection of all number systems is not a finished totality whose discovery was complete around 1600, or 1700, or 1800, but . . . has been and still is a growing and changing area, sometimes absorbing new systems and sometimes discarding old ones, or relegating them to the attic. Accordingly, there are not one but two separate tracks for the development of analysis:'"`UNIQ--ref-00000016-QINU`"' : (A) the Weierstrassian approach (in the context of an Archimedean continuum) and : (B) the approach with indivisibles and/or infinitesimals (in the context of a Bernoullian continuum). In summary, the effect (at least in some quarters) of Robinson's work in non-standard analysis was to usher in "a rehabilitation of the use of infinitesimals in mathematics" and to show, as Robinson himself proposed, that (at least some of) the infinitesimal methods used by the founders of calculus to be "correct and consistent."'"`UNIQ--ref-00000017-QINU`"' ==='"`UNIQ--h-7--QINU`"'The nature of rigour in definitions and proofs=== The ''standard account'' of the arithmetization program claims for it an increase in rigour, both in mathematical thinking and in mathematical definitions and proofs:'"`UNIQ--ref-00000018-QINU`"' : A foundational rock of the received history of mathematical analysis is the belief that mathematical rigor emerged starting in the 1870s through the efforts of Cantor, Dedekind, Weierstrass, and others, thereby replacing formerly unrigorous work of infinitesimalists from Leibniz onward. Some (Dauben, Katz et. al.) working with the new non-standard methods challenged this claim. Robinson himself, in his own historical writing, refuted the claim that increased rigour resulted from of the success of Cauchy-Weierstrassian epsilontics over infinitesimals. Certainly one of his most important achievements in non-standard analysis was the conclusive demonstration that such a claim is mere historicism.[12]'"`UNIQ--ref-00000019-QINU`"' ==='"`UNIQ--h-8--QINU`"'An alternate method of teaching calculus=== There is little dispute that arithmetization brought about a shift in emphasis from geometric to algebraic reasoning. Certainly this shift was 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 was taught.'"`UNIQ--ref-0000001A-QINU`"' The development of non-standard analysis and the consequent return of infinitesimals to mathematics have given rise to the possibility of a further shift in the way mathematics is taught. The year 1971 saw the publication of Jerome Keisler's textbook, ''Elementary Calculus: An Approach Using Infinitesimals''. This text used non-standard analysis to explain, in an introductory course, the basic ideas of calculus. In 1973-74, a study was done to determine the pragmatic value of using this textbook in teaching calculus. The study examined this pedagogical claim:'"`UNIQ--ref-0000001B-QINU`"' : from this non-standard approach, the definitions of the basic concepts [of the calculus] become simpler and the arguments more intuitive. The results of the study, both as measured by a calculus test given to the students and by the comments of the instructors, are remarkable in their support of the heuristic value of using non-standard analysis in the classroom.'"`UNIQ--ref-0000001C-QINU`"' It is worth looking at the single question in the test that brought out the greatest difference between the experimental group and the control group: ::Define $f(x)$ by this rule: \[ f(x) = \begin{cases} x^2 & \text {for } x ≠ 2 \\ 0 & \text {for } x = 2 \end{cases} \]

Prove using the definition of limit: \[ \begin{align} \displaystyle \lim_{x \to 2} f(x) = 4 \end{align} \]

The study results summarized the comments of the instructors as follows:

The group as a whole responded in a way favorable to the experimental method on every item: the students learned the basic concepts of the calculus more easily, proofs were easier to explain and closer to intuition, and most felt that the students end up with a better understanding of the basic concepts of the calculus.

References

• Bair, Jacques, et. al., "Is Mathematical History Written by the Victors?" Notices of the American Mathematical Society, (60) No 7, pp. 886-904, URL: http://www.ams.org/notices/201307/rnoti-p886.pdf

• Dauben, Joseph W., (1985). "Abraham Robinson and Nonstandard Analysis: History, Philosophy, and Foundations of Mathematics," URL: http://www.mcps.umn.edu/philosophy/11_7dauben.pdf

• Davis, Isaac, (2009). "An Introduction to Nonstandard Analysis," URL: http://www.math.uchicago.edu/~may/VIGRE/VIGRE2009/REUPapers/Davis.pdf

• Ehrlich, P. (2006). "The rise of non-Archimedean mathematics and the roots of a misconception. I. The emergence of non-Archimedean systems of magnitudes." Archive for History of Exact Sciences (60), No. 1, pp. 1–121.

• Keisler, H. J. (1976) Foundations of Infinitesimal Calculus, Boston: Prindle, Weber &Schmidt. Inc.

• Klein, F. (1932). Elementary Mathematics from an Advanced Standpoint. Vol. I, Arithmetic, Algebra, Analysis. Translation by E. R. Hedrick and C. A. Noble [Macmillan, New York, 1932] from the third German edition [Springer, Berlin, 1924]. Originally published as Elementarmathematik vom höheren Standpunkte aus (Leipzig, 1908).

• Laszczyko, Piotr B., Katz, Mikhail G., Sherry, David, (2012) "Ten Misconceptions from the History of Analysis and their Debunking," URL: http://arxiv.org/pdf/1202.4153.pdf

• Parker, Frieda (2006) "Infinitesimals: Intuition and Rigor," Prepared in fulfillment of the requirements for the course Math 531, URL: http://www.unco.edu/NHS/mathsci/facstaff/parker/math/Infinitesimal_Paper.pdf.

• Putnam, H. (1975): "What is mathematical truth?", Proceedings of the American Academy Workshop on the Evolution of Modern Mathematics (Boston, Mass., 1974). Historia Mathematica (2), No. 4, pp. 529–533, URL: http://ac.els-cdn.com/0315086075901160/1-s2.0-0315086075901160-main.pdf?_tid=4c03c38c-45e4-11e4-b1af-00000aab0f26&acdnat=1411780935_ff148c2ea34d192b4319dbd82629724c.

• Robinson, Abraham (1965). "On the Theory of Normal Families," Acta Philosophica Fennica 18:159-84.

• Robinson, Abraham (1973). "Numbers—What Are They and What Are They Good For?" Yale Scientific Magazine (47), p. 14-16.

• Sullivan, Kathleen (1976). "The Teaching of Elementary Calculus Using the Nonstandard Analysis Approach," American Mathematical Monthly (83) pp. 370-75, URL: http://academic.brcc.edu/johnson/Projects/Calculus%20with%20Infinitesimals/Files/Sullivan%20(1976).pdf.