Did Leibniz use infinite numbers?

Question

Arthur's recent article

Arthur, Richard T. W. Leibniz's actual infinite in relation to his analysis of matter. G. W. Leibniz, interrelations between mathematics and philosophy, 137–156, Archimedes, 41, Springer, Dordrecht, 2015

was recently reviewed by Knobloch, who wrote in part:

Despite his rejection of an infinite number and infinite collections, he admitted an actual, but syncategorematic, infinite. There is an infinity of things; that is, there are always more of them than can be specified. Secondly, Arthur refutes Rescher's two-tiered ontology and Russell's pseudo-dilemma. One has to distinguish between division and resolution, between composable, resolvable, and divisible. Arthur essentially bases his deductions on Leibniz's correspondence with Arnauld and de Volder. A body is actually divided into parts by its different motions, but resolved into unities. There are actually infinitely many monads. This infinite is understood syncategorematically. "There are actually infinite aggregates but—in contrast to Cantor—there are no infinite numbers" (p. 155).

This seems surprising to me since Leibniz on many occasions spoke precisely of infinite numbers (while rejecting infinite collections). Have Arthur and Knobloch gotten it right?

They don't fully address the distiction between infinite multitudes/wholes/collections, on the one hand, and infinite numbers, on the other. I haven't seen anything in Leibniz that would suggest that he thought that infinite numbers led to contradiction, and on the contrary Leibniz often exploits them as "useful fictions".

In a recent article in HOPOS: History of Philosophy of Science we defend the thesis that Leibniz used infinite numbers galore. We also argue against the so-called syncategorematic interpretation first developed by Ishiguro allegedly following Russell (according to which Leibniz did nothing of the sort). See here.

Answer 1

It appears that they are right in the "metaphysical" sense, here is a passage from Leibniz's letter to Bernoulli (1699):"I concede an infinite multitude, but this multitude forms neither a number nor one whole. It only means that there are more terms than can be designated by a number; just as there is for instance a multitude or complex of all numbers; but this multitude is neither a number nor one whole". The main problem for Leibniz was the violation of the part-whole axiom, an infinite whole would end up being equal to a proper part of itself, as in Galileo's example with natural numbers and their squares.

However, in On Some Relations between Leibniz’ Monadology and Transfinite Set Theory Friedman argued that Leibniz nonetheless accepted infinite wholes and numbers as useful fictions for calculation (he also saw imaginary roots and infinitesimals as useful fictions). In a letter to Des Bosses (1706) Leibniz wrote:"...properly speaking, an infinity consisting of parts is neither one nor a whole, and can only be conceived of as a quantity by a mental fiction".

See more in van Atten's Note on Leibniz’s Argument Against Infinite Wholes.

Answer 2

Richard Arthur's article can be complemented with a Leibnizian fragment edited into:

• Gottfried Wilhelm Leibniz, The Labyrinth of the Continuum: Writings on the Continuum Problem, 1672-1686, edited by Richard Arthur (2001), page 82: Infinite numbers (ca.April 1676).

Consider the conclusion, page 99:

Whenever it is said that a certain infinite series of numbers has a sum, I am of the opinion that all that is being said is that any finite series with the same rule has a sum, and that the error always diminishes as the series increases, so that it becomes as small as we would like. For numbers do not in themselves go absolutely to infinity, since then there would be a greatest number [Nam ipsi per se absolute numeri in infinitum non eunt, daretur enim numerus maximus]. [...] Therefore we conclude finally that there is no infinite multiplicity, from which it will follow that there is not an infinity of things, either. [...]

Thus if you say that in an unbound [series] there exists no last finite number that can be written in, although there can exist an infinite one: I reply, not even this can exist, if there is no last number. The only other thing I would consider replying to this reasoning is that the number of terms is not always the last number of the series. That is, it is clear that even if finite numbers are increased to infinity, they never - unless eternity is finite, i.e.never - reach infinity. This consideration is extremely subtle.