Who gets credit for the real numbers?

Question

If Simon Stevin already pioneered the unending decimal representation for every number (rational, surd, etc.) at the end of the 16th century, why do Cantor and Dedekind (who certainly gave a more detailed account) routinely get credit for the real numbers?

Stevin did some detailed work (rather than vague general ideas) with unending decimals, including a proof of the intermediate value theorem for polynomials. Newton was in fact inspired by infinite decimals to introduce his general theory of power series.

An interesting point was raised in an answer by Peter Diehr. The so-called Archimedean property (which is one of the defining characteristics of the real number field; though of course it does not suffice to characterize them as the rationals also satisfy it) was considered by authors like Euclid (Elements V.4) considerably earlier. However as far as giving an actual construction (rather than axiomatic definition) Stevin seems to have been the first.

Note 1. To clarify, Stevin developed specific notation for decimals (more complicated than the one we use today) and did actual technical work with them rather than merely envisioning their possibility, unlike some of his predecessors.

Note 2. One useful source for this is Malet, Antoni. Renaissance notions of number and magnitude. Historia Math. 33 (2006), no. 1, 63–81.

Note 3. As Malet notes, "Stevin does not justify his definition" which identifies number and "quantity of anything" because to him the identification is obvious, and the implementation of number is his unending decimals. This was an appropriate move indeed since we know today that the Cantor-Dedekind postulate identifying the number line and the line in physical space is untenable based on what modern physics teaches us; similar remarks apply to magnitude/quantity. Stevin of course was not aware of "transcendental" numbers but no such knowledge is required in order to define the real numbers by means of unending decimals; namely this could have been done even if Liouville did not prove the existence of transcendental numbers.

Note 4. I should clarify that Stevin dealt with unending decimals in his book l"Arithmetique rather than the more practically-oriented De Thiende meant to teach students to work with decimals (of course, finite ones).

Note 5. As far as using the term real to describe the numbers Stevin was concerned with, it should be clarified that the first one to describe the common numbers as real may have been Descartes and at any rate this usage is later than Stevin. On the other hand, if we talk about representing common numbers (including both rational and not so), Stevin not only speculated about the possibility of a representation scheme using decimals, but (unlike some of his predecessors) developed a specific notation (though different from what we use today) and moreover did work with this notation.

Note 6. Cantor thought that Cauchy Completeness (CC) was sufficient to characterize the real numbers axiomatically. Today we know this is not the case, as one also needs the Archimedean property. I found out recently that Dedekind was convinced he had a proof of the existence of an infinite set; see here. Do these misconceptions by Cantor and Dedekind indicate a shortcoming of the constructions of the real numbers they proposed? Hardly so. Stevin's approach to representing all common numbers by unending decimals similarly could not be held at fault because Stevin was not aware of certain future developments.

Answer 1

Many people get credit, because this was a long story beginning in the ancient Greece. Euclid has a theory of proportions (based on earlier research) which is equivalent to modern theory of real numbers. Infinite decimal expansions were gradually introduced since 17th century (Napier, Stevin), and the modern theories are due to Cantor and Dedekind. So the development took 2000 years, and it is impossible to credit one person.

Answer 2

The Archimedean property as it is called, was used as an axiom by Archimedes, and he credited Eudoxus of Cnidus, who predates Euclid; also see this.

In Section 7: Stevin, Malet says:

In fact Stevin does not justify his first definition (“Number is that by which one can tell the quantity of anything”)

So it appears that, like Archimedes, Simon Stevin assumes that every point of a line corresponds to a distance from its origin; that is, magnitudes correspond to points of the line. The nice mathematical distinctions that appear in the 19th century which sort out the details of the Real numbers, are not important to Stevin; what is important is that the decimal notation provides a convenient method for recording these magnitudes.

His work was intended to teach students how to work with decimal numbers. Since even the concept of transcendental numbers does not appear until the 19th century, I don't see how any earlier work could be cited as referring to the Real numbers, except as an axiom.

For reference: Archimedes' Axiom and Archimedean axiom

Answer 3

I've only recently begun reading on the subject of the history of mathematics, and my readings are currently limited to a single text; Boyer's A History of Mathematics. However, according to Boyer, and supported by the wikipedia entry for Simon Stevin, I believe your claim that Stevin dealt with "all" real numbers, "(rational, surd, etc.)", is an overreach.

Quoting Boyer :

Viète, ... , in 1579 had urged the replacement of sexagesimal fractions by decimal fractions. In 1585 an even stronger plea for the use of ten-scale fractions, as well as integers, was made by the leading mathematician in the Low Countries, Simon Stevin of Bruges.

This appears to stop short of claim that Stevin's work was conceptually embracing all real numbers. The linked paper by Malet makes the clear claim that Stevin also considered (some) irrational numbers :

That “any root whatsoever is number” [Stevin, 1585, 8] is also a consequence of identifying numbers and measures

Thus, according to Malet, Stevin does consider algebraic numbers, but again this stops short of claiming that Stevin was in possession of the correct notion of "all real numbers". In other words, although we know now that all real numbers can be represented in this way, it is not clear that Stevin was aware of the true and correct nature of the real numbers and their different types. Perhaps this provides some explanation for why Stevin does not get full credit for the real numbers.

As final point, it may also be worth mentioning that Boyer notes :

It is clear that Stevin was in no sense the inventor of decimal fractions, nor was he the first systematic user of them. More than incidental use of decimal fractions is found in ancient China, in medieval Arabia, and in Renaissance Europe; by the time of Viète's forthright advocacy of decimal fractions in 1579 they were generally accepted by mathematicians on the frontiers of research. Among the common people, however, and even among mathematical practitioners, decimal fractions became widely known only when Stevin undertook to explain the system in full elementary detail.

Answer 4

Man, as a collective, is credited in a famous quote: "God made the integers; all else is the work of man" (or "Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk"). The quote is often attributed to Leopold Kronecker, see for instance "Philosophies of Mathematics", p. 13, Alexander George, Daniel J. Velleman, 2001. Apparently, the authenticity of the quote is disputed.

This book also kind of credits Dedekind:

Of particular note in this connection is the accomplishment, due primarily to the German mathematician Richard Dedekind (1831-1916), of defining the integers, rationals and reals, taking only the system of natural numbers for granted.

From my education, with a western twist, the Dedekind cut was a construction of real numbers that allowed me to get a instantaneous (perhaps faulty) inner picture of reals, based on (natural) rationals, which I did not grasp before (with unending decimal representations). I learned about it while studying rational (Diophantine) approximations of polynomial roots in the real field and in finite fields (continuing fractions).

In the history of science, the person who gets credits is not always the first. In the West, C. Columbus often gets credits for discovering America, which is probably unfair. Should the first one proving that there were at least one irrational number be credited too?

I begin (after your comments) to think that answers depend of "what kind of real numbers?", in other words, which which structure? As points on a line, as a succession of figures, as a ring or field structure, as a vector space or an an algebra, as a "sense" of continuity?

From my second hand knowledge, some say that arabic/muslim (in a wide sense) mathematicians were the first to treat irrational numbers as algebraic objects (possibly only surds), and indian ones developed trigonometric series (Ideas of Calculus in Islam and India, Katz, 1995). And the first time I heard about (one instance of) the Hamel basis,

a basis for the real numbers \mathbb{R} as a vector space over the field \mathbb{Q} of rational numbers

I understood that my level in mathematics was too narrow to understand what real numbers really were. Since Heron of Alexandria is sometimes credited with the first (western) notion of imaginary numbers, can we expect the real was discovered after the complex?

Answer 5

The history of decimal point is much older than Simon Stevin.

According to Joseph Needham and Lam Lay Yong, decimal fractions were first developed and used by the Chinese in the 1st century BC. But the oldest Chinese book introducing a decimal mark equivalent, is from the thirteenth century.

Around the middle of the tenth century, Al-Uqlidisi wrote "Kitab al-fusul fi al-hisab al-Hindi" (Chapter's book of Indian Arithmetic), which is the earliest surviving book that presents the Indian system.(Survived in a copy of the original which was made in 1157).

In the fourth part of this book Al-Uqlidisi showed how to modify the methods of calculating with Indian symbols, which had required a dust board, to methods which could be carried out with pen and paper. This requirement of a dust board had been an obstacle to the Indian system's acceptance.

Al-Uqlidisi's book is also historically important as it is the earliest known text offering a direct treatment of decimal fractions. Al-Uqlidisi used a decimal dash, above the first digit of the fractional part of the decimal number.( Very simple if compared to Stevin's notation. But similar to the decimal point used by Bartholomaeus Pitiscus).

While the Persian mathematician Jamshīd Al-Kāshī claimed to have discovered decimal fractions himself in the 15th century, J. Lennart Berggrenn notes that he was mistaken, as decimal fractions were first used five centuries before him by Al-Uqlidisi as early as the 10th century.

Rashed puts Al-Kashi's important contribution into perspective. He shows that the main advances brought in by Al-Kashi are:-

(1) The analogy between both systems of fractions; the sexagesimal and the decimal systems.

(2) The usage of decimal fractions no longer for approaching algebraic real numbers, but for real numbers such as π.

(Sorry, I am not permitted to post more than two links.)

http://vedicsciences.net/articles/history-of-numbers-part-2.html

http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Al-Uqlidisi.html