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Clearly The Author Does Not Know What The Natural Numbers Are

What we’ve got here is failure to communicate

Suppose I were to say: “any natural number can be written uniquely, up to order, as a, possibly empty, finite product of prime number(s).”

This seems possibly correct, and possibly even careful. Though, one may have to look up the terms (such as “prime number“) to know what it means. And we even get the number one correct, as the convention for empty products is they equal to one.

However, a reader could respond with “Clearly the author does not know what the natural numbers are. Zero is a natural number and is not the product of any finite list of prime numbers.” In my opinion such a response is just wrong, as it relies on the disingenuous position that there is one and only one definition for the phrase “natural numbers.”

There are in fact two definitions of “natural numbers” that are common enough in practice that a reader should consider the possibility that the author may not be using the one the reader guessed. Obviously authors being unambiguous is better, but often that is at odds with brevity and clarity. The reader doesn’t have to clean up for the author, but it would be unfair to try both definitions with the sole intent of finding one that does not work.

How the pros do it

The currently dominant definition of the natural numbers is: the non-negative integers {0, 1, 2, …}. Sources that use this convention include:

  • ISO 80000-2 (includes the international system of quantities) (source Wikipedia).
  • Bourbaki’s germinal Theory of Sets (III., 4., 1., Definitions of Integers. Definition 1 “… A finite cardinal is also called a natural integer …”; when combined with the common convention that the empty set has cardinality zero).
  • The Presburger arithmetic axiomatizes structures like the non-negative integers (source).
  • Mac Lane, Birkhoff, Algebra, 3rd Edition: ch 1, sec 4 The Natural Numbers: “Intuitively the set N = {0, 1, 2, …} … .”
  • Ebbinghaus et al. Numbers, Springer, 1991, 1. para2, pg. 14: “The natural numbers for a set N, containing a distinguished element 0 … .”

Note, we are not trying to make an argument by authority that the natural numbers must be the non negative integers in the above; we are merely trying to prove by example some common sources use the non-negative integers as the “natural numbers.”

And there are sources that use the positive integers {1, 2, 3, …} as the “natural numbers.”

  • H. L. Royden Real Analysis, 2nd Edition, MacMillan, 1968, Sec 1, Introduction, p. 6: “The natural numbers (positive integers) play such an important rile in this book that we introduce the special symbol N for the set of natural numbers.”
  • Peano’s 1889 axioms of arithmetic starts the numbers at 1 (“COMMENTARY: Peano starts his natural numbers at 1; most modern versions start at 0.” from here).
  • The Merriam-Webster dictionary admits to both possibilities.

    Definition of natural number

    1. the number 1 or any number (such as 3, 12, 432) obtained by adding 1 to it one or more times : a positive integer
    2. any of the positive integers together with 0 : a nonnegative integer

    (Merriam-Webster dictionary)

Whose side is Peano really on anyway?

Notice we put Peano’s axioms in the “starts from one, not zero” camp. The Wikipedia discusses this issue as follows:

The first axiom states that the constant 0 is a natural number:

  1. 0 is a natural number.

Peano’s original formulation of the axioms used 1 instead of 0 as the “first” natural number.[6] This choice is arbitrary, as axiom 1 does not endow the constant 0 with any additional properties. However, because 0 is the additive identity in arithmetic, most modern formulations of the Peano axioms start from 0.

Definitions depend on history and background

How could something so foundational as the natural numbers have more than one definition? How could the definition of the Peano axioms change over time?

The answer is: we don’t live in a single fully finished textbook. Mathematics is a living field that is only approaching consensus.

Definitions get adjusted and tuned. Mathematics today is more algebraic than earlier versions of the discipline, so introducing a semi-group unit (such as 0) is very desirable. Whereas, more geometric treatments of number theory often do not treat 0 as a run-of-the-mill number (for example see “The Greek Natural Numbers” here, zero and one are both considered to be exceptional cases) [This definition may seem very odd, but it is somewhat compatible with current number theory where “composite” and “prime” are not applied to the numbers 0 and 1 as they are zero and “a unit.”]

So I think we are seeing a move from geometry to algebra in treatments of basic arithmetic.

Rota has a sparkling explanation of why we don’t see this sort of change discussed more often.

Philosophers and psychiatrists should explain why it is that we mathematicians are in the habit of systematically erasing our footsteps. Scientists have always looked askance at this strange habit of mathematicians, which has changed little from Pythagoras to our day.

Rota, G. “Two turning points in invariant theory.” Math Intelligencer 21, 20–27 (1999).
(as reported by Doron Zeilberger).

And, as to be expected, Edsger W. Dijkstra has an even more pointed view.

… in an emotional outburst, one of my mathematical colleagues at the University —not a computing scientist— accused a number of younger computing scientists of “pedantry” because —as they do by habit— they started numbering at zero. He took consciously adopting the most sensible convention as a provocation. … I think Antony Jay is right when he states: “In corporate religions as in others, the heretic must be cast out not because of the probability that he is wrong but because of the possibility that he is right.”

Edsger W. Dijkstra “Why numbering should start at zero”, EWD831.

(Note the Dijkstra quote very much reminds me of Chesterton’s Policeman in The Man Who was Thursday: “The moderns say we must not punish heretics. My only doubt is whether we have a right to punish anybody else.”)


Uncharitable reads can be considered uncharitable acts. (Note: I did not in fact write the initial example!)

Appendix: Note on Euclid’s Elements

A note on the very odd issue of what are numbers in Euclid’s Elements. In Heath, The Thirteen Books of The Elements, Dover, 1956, 2nd Edition Unabridged, Vol. 2, Book VII we have Euclid’s writing on “numbers”- which we take to mean some set of integers as Book V already dealt with “magnitudes” which we take to be lengths. In this writing there is a unit (standing in for 1) and definition 2 says “a number is a magnitude composed of units.” In the discussion of this definition on page 180 we have that this is likely summarizing earlier writing that is translated as “multiple of individuals”, “several ones”, and similar phrases. This hints that we may need the “Let A be a Pedant and Let B be a Pedant” style reading of “a number is a magnitude composed of units” as “a number is a magnitude composed of two or more units.”

This may seem like an extremely strained reading of the text. But it is compatible with other sources. For example the Wikipedia History/Ancient_Roots section of the Natural Number entry says:

The first systematic study of numbers as abstractions is usually credited to the Greek philosophers Pythagoras and Archimedes. Some Greek mathematicians treated the number 1 differently than larger numbers, sometimes even not as a number at all.[f] Euclid, for example, defined a unit first and then a number as a multitude of units, thus by his definition, a unit is not a number and there are no unique numbers (e.g., any two units from indefinitely many units is a 2).

Wikipedia citing Mueller, Ian Philosophy of mathematics and deductive structure in Euclid’s Elements. Mineola, New York: Dover Publications, (2006). p. 58. ISBN 978-0-486-45300-2. OCLC 69792712.

The issue is, the ancient and even classical Greek mathematicians seemed to be incredibly late to adopt the use of zero as a mundane integer. Euclid is thought to have been active in the mid 4th century BC. However we don’t see formal use of zero possibly until AD 150 and Ptolemy:

The ancient Greeks had no symbol for zero (μηδέν), and did not use a digit placeholder for it.[19] They seemed unsure about the status of zero as a number. They asked themselves, “How can nothing be something?”, leading to philosophical and, by the medieval period, religious arguments about the nature and existence of zero and the vacuum. The paradoxes of Zeno of Elea depend in large part on the uncertain interpretation of zero.[20]

By AD 150, Ptolemy, influenced by Hipparchus and the Babylonians, was using a symbol for zero[21][22] in his work on mathematical astronomy called the Syntaxis Mathematica, also known as the Almagest.[23] This Hellenistic zero was perhaps the earliest documented use of a numeral representing zero in the Old World.[24] Ptolemy used it many times in his Almagest (VI.8) for the magnitude of solar and lunar eclipses.

Wikipedia 0 citing:

  • 20. Huggett, Nick (2019), Zalta, Edward N. (ed.), “Zeno’s Paradoxes”, The Stanford Encyclopedia of Philosophy (Winter 2019 ed.), Metaphysics Research Lab, Stanford University, retrieved 9 August 2020.
  • 21. Neugebauer, Otto (1969) [1957]. The Exact Sciences in Antiquity (2 ed.). Dover Publications. pp. 13–14, plate 2. ISBN 978-0-486-22332-2.
  • 22. Mercier, Raymond, “Consideration of the Greek symbol ‘zero'” (PDF), Home of Kairos.
  • 23. Ptolemy (1998) [1984, c.150], Ptolemy’s Almagest, translated by Toomer, G. J., Princeton University Press, pp. 306–307, ISBN 0-691-00260-6
  • 24. O’Connor, J J; Robertson, E F, A history of Zero, MacTutor History of Mathematics

This is indeed strange. Actually a lot of we “remember” as Euclidian geometry is actually a descendent of Hilbert’s masterful re-axiomatization of the topic.

The above sort of crazy pedantry is actually still current, consider program fragments such as “sprintf("%d error(s)", errors)“. If people weren’t worried about the “s” they wouldn’t sometimes put it in parenthesis.

In my opinion this is a great example of why truly interactive classes are better than books and videos. Instead of worrying about the above until one’s ears bleed, one could raise one’s hand and ask “are 0 and 1 numbers in your formulation?”

Categories: Expository Writing Mathematics Opinion Tutorials

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Data Scientist and trainer at Win Vector LLC. One of the authors of Practical Data Science with R.

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