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Indian mathematicians (e.g., Brahmagupta in the 6th century) developed the idea of 0 as more than a placeholder.

In 1202, Fibonacci wrote "These are the nine figures of the Indians: 9 8 7 6 5 4 3 2 1. With these nine figures, and with this sign 0, which in Arabic is called zephyr, any number is written, as shown below."

Fibonacci seems to be saying that zero is not quite a number like the rest. But if he basing this off the Indians work from centuries before, why is this? Wasn't it well-developed by this point?

Thank you in advance!

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    $\begingroup$ See Euclid's Elements Book VII, Def.2: "A number is a multitude composed of units." $\endgroup$ Oct 26, 2023 at 7:11
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    $\begingroup$ (1) Perhaps Fibonacci had an incomplete or inaccurate view of the history of numerals and thought the figure for zero came from a different source. For instance, according to Wiki., Gerbert of Aurillac promulgated the western Arabic numerals, derived from Hindu system, for 1–9 after he became Pope (late 10th century). Fibonacci added zero to this. $\endgroup$
    – Michael E2
    Oct 26, 2023 at 15:05
  • $\begingroup$ (2) Even today, some construct the natural numbers without 0. A significant part of number theory was developed as a theory of the positive integers; what @Mauro says seems significant here. Whether to include 0 is choice, at least from some point in history; however, I think my comment (1) is a stronger argument. $\endgroup$
    – Michael E2
    Oct 26, 2023 at 15:05

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The nonzero digits are also numbers that were considered by the Greeks as existing entities (the case for 1 was seen as somewhat special as it is not composed of other entities; Simon Stevin notably fought against this attitude concerning the singularity of 1). Since 0 is not a number of anything, it was naturally viewed as a fictional entity not on par with $1,\ldots,9$. Leibniz viewed entities familiar to the Greeks as the only possible ones; other entities such as surds, imaginary roots, infinitesimals, negatives (surely including 0) were fictions. A detailed analysis appears in this publication. There is nothing surprising about Fibonacci's attitude here.

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  • $\begingroup$ I’d suggest that the wording “Since 0 is not a number of anything” be changed. At least to us Moderns, zero very much is the number of plenty of things. It is the number of females who have been elected president of the United States. It’s the number of bipartite graphs that contain odd cycles. It’s the number of positive roots of $y=x^2+8x+15$. $\endgroup$ Nov 15, 2023 at 2:33
  • $\begingroup$ @PaulTanenbaum, "at least to us moderns" : exactly :-) $\endgroup$ Nov 15, 2023 at 10:54
  • $\begingroup$ But you write “is not a number,” which—as I point out—isn’t true in the unconditional sense that your wording suggests. Perhaps “was not seen as a number.” $\endgroup$ Nov 15, 2023 at 12:12
  • $\begingroup$ @Paul, thanks for your suggestion, but I think such an edit would only obscure the meaning. When I write "0 is not a number of anything", the term anything is equivalent to something :-) $\endgroup$ Nov 15, 2023 at 12:14
  • $\begingroup$ Obviously your call, it’s your answer. But I, for one, don’t understand your reasoning. A statement like There are zero cats in the room has a unique and well understood meaning that, semantically is completely parallel to There are three cats in the room. Now, pragmatically, the statements may not be considered to be parallel, but semantically they are. $\endgroup$ Nov 15, 2023 at 12:49

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