Arithmetization of analysis

The phrase "arithmetization of analysis" refers to 19th century efforts to create a "theory of real numbers ... using set-theoretic constructions, starting from the natural numbers." These efforts took place over a period of about 50 years, with the following results:

1. the establishment of fundamental concepts related to limits
2. the derivation of the main theorems concerning those concepts
3. the creation of the theory of real numbers.

It is interesting, and has been noted elsewhere, that although the theory of real numbers is today the logical starting point (foundation) of analysis in the real domain, the creation of the theory was not achieved historically until the end of the period (program or movement) of arithmetization.

History

Non-mathematical issues

The history of the arithmetization of analysis was complicated by non-mathematical issues. Some authors were very slow to publish and some important results were not published at all during their authors' lifetimes. The work of other authors was, for unknown reasons, completely ignored. As a consequence, some results were achieved multiple times, albeit in slightly different forms or using somewhat different methods, by different authors.

As a first example, consider the work of Bozano. Only two of his papers dealing with the foundations of analysis were published during his lifetime. Both of these papers remained virtually unknown until after his death. A third work of his, based on a manuscript that dates from 1831-34, but that remained undiscovered until after WWI, was finally published in 1930. This work contains some results fundamental to the foundations of analysis that were re-discovered in the 19th century by others decades after Bolzano completed his manuscript. "We may ask how much Bolzano's work could have changed the way analysis followed, had it been published at the time."

As a second example, consider the 1837 essay of W.R. Hamilton on the foundations of mathematics, in which he attempted to show that analysis (which for Hamilton included algebra) alike with geometry, can be "a Science properly so called; strict, pure, and independent; deduced by valid reasonings from its own intuitive principles." Hamilton's essay contained the following:

• the notion that analysis "can be constructively inferred from a few intuitively based axioms"
• ideas used much later by others (Peano, Dedekind, others) "including a notion related to the concept of a cut in the rationals"

Even so, Hamilton's essay was ignored by other English mathematicians and had no apparent influence on the work of German mathematicians who completed the process of arithmetization later in the century.

Two pillars of mathematics

Prior to 19th century efforts at arithmetization, analysis rested more or less comfortably on two pilars: the discrete side on arithmetic, the continuous side on geometry. "The analytic work of L. Euler, K. Gauss, A. Cauchy, B. Riemann, and others led to a shift towards the predominance of algebraic and arithmetic ideas. In the late nineteenth century, this tendency culminated in the so-called arithmetization of analysis, due principally to K. Weierstrass, G. Cantor, and R. Dedekind."

The fundamental theorem of algebra

Proofs of "The fundamental theorem of algebra" have a long history, with dates (currently) ranging from 1608 (Peter Rothe) to 1998 (Fred Richman).

Gauss offered two proofs of the theorem that were significant for the arithmetization of analysis:

• the theorem itself seemed to involve a discrete result
• his proofs required the use of continuous methods

In other words, Gauss' proofs called into question the comfortable two-pillar foundation of mathematics.

All proofs offered before Gauss assumed the existence of roots. Gauss' proofs were the first that did not make this assumption:

• In 1799, he offered a proof of the theorem that was largely geometric. This first proof assumed as obvious a geometric result that was actually harder to prove than the theorem itself!
• In 1816, he offered a second proof that was not geometric. This proof assumed as obvious a result known today as the intermediate value theorem.

The intermediate value theorem

As noted above, Gauss' 1816 proof of the fundamental theorem of algebra relied on the intermediate value theorem: if f(x) is a continuous function of a real variable x and if f(a) < 0 and f(b) > 0, then there is a c between a and b such that f(c) = 0. It was Bolzano's insight that this theorem, though very plausible, needed to be proved. In 1817, he offered a proof of the theorem, which he stated in a "rather (unnecessarily) complicated" form: If f,g are two functions, both continuous in a closed interval [a, b], and if f(a) < g(a), f(b) > g(b), then there is at least one number x inside this interval, such that f(x) = g(x).

Balzano's proof relied, in turn, on an assumption, namely, the existence of a greatest lower bound: if a certain property M holds only for values greater than some quantity l, then there is a greatest quantity u such that M holds only for values greater than or equal to u." A proof of the greatest lower bound theorem needed to await the building of the theory of real numbers. However, Bolzano demonstrated the plausibility of the theorem, introducing the condition for the convergence of a sequence known today as the Bozano-Cauchy condition and attempting actually to prove the sufficiency of this condition.

The condition for continuity of a function

Bolzano saw that the intermediate value theorem needed to be proved "as a consequence of the definition of continuity." In his proof, he introduced essentially the modern condition for continuity of a function $f$ at a point $x$: $f(x + h) − f(x)$ can be made smaller than any given quantity, provided $h$ can be made arbitrarily close to zero. The caveat essentially is needed because of his complicated statement of the theorem. In effect, the condition for continuity as stated by Bolzano actually applies not at a point $x$, but within an interval.

Bolzano and Cauchy were contemporaries, "both chronologically and mathematically." They gave similar defnitions of limits, derivatives, continuity, and convergence. In 1821, Cauchy added "the final touch of precision" to the definition of continuity of a function at a point: "for each $\epsilon > 0$ there is a $\delta > 0$ such that $|f(x + h) − f(x)| < \epsilon$ for all $|h| < \delta$." Here, again, it's important to note that Cauchy's condition for continuity as he stated it, alike with Bolzano's, actually applies not at a point x, but within an interval.

It was Weierstrass who, very much later than both Bozano and Cauchy, formulated "the precise $(\epsilon,\delta)$ definition of continuity at a point."

Continuous nowhere differentiable functions

"The discovery of continuous nowhere differentiable functions shocked the mathematical community. It also accentuated the need for analytic rigour in mathematics."

References

• Arithmetization, Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Arithmetization&oldid=31486
• Fundamental Theorem of Algebra, Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php/Algebra,_fundamental_theorem_of
• "Fundamental Theorem of Algebra," Wikipedia, URL: http://en.wikipedia.org/wiki/Fundamental_theorem_of_algebra
• Grabiner, J. V. (1981), The Origins of Cauchy's Rigorous Calculus, MIT Press.
• Hamilton, W. R. (1837), "Algebra as the Science of Pure Time."
• Hatcher, William S. (2000), Foundations of Mathematics: An Overview at the Close of the Second Millenium 3.2 Aritmetization of Analysis, Switzerland: Landegg Academy, URL: http://bahai-library.com/hatcher_foundations_mathematics
• Jarník, Vojtěch; Novák, Josef; Folta, Jaroslav; Jarník, Jiří (1981). Bolzano and the Foundations of Mathematical Analysis. (English). Praha: Society of Czechoslovak Mathematicians and Physicists, pp. 33--42, URL: http://dml.cz/dmlcz/400082.
• Mathews, Jerold. "William Rowan Hamilton's 1837 Paper on the Arithmetization of Analysis."
• Pinkus, Allan Weierstrass and Approximation Theory
• Stillwell, John Colin (2013). "Arithmetization of Analysis," Encyclopaedia Britannica, URL: http://www.britannica.com/EBchecked/topic/22486/analysis/247690/Complex-exponentials.