Georg Cantor, freedom, and infinitesimals

To defend his set theory against critics, Cantor insisted that "Mathematics is in its development entirely free, etc." (see full quotation below). Given Cantor's advocacy of freedom in mathematics, it may come as a bit of a surprise to find him attacking infinitesimals as "abominations", "bacilli", "chimeras", "fantasies", and "ghosts". What is behind Cantor's hostility toward infinitesimals? The answer is laid out clearly in Cantor's 7 september 1890 letter to Veronese. The problem with infinitesimals was that they did not fit easily into Cantor's framework, suggesting that his framework may be somehow incomplete.

(1) The idea of 'Consistency as existence' is usually attributed to David Hilbert (who was motivated in part by a battle against Brouwer over the meaning of mathematical 'existence' and was therefore interested in softening up/lowering expectations with regard to the latter), but in fact rather detailed investigations in this direction are already found in Leibniz. Leibniz was confronted with the uncomfortable situation with regard to his new mathematics which involved entities beyond the traditional material found in Euclid and Greek geometry: negative numbers, surds, imaginary roots, infinitesimals, some of which seemed not only impossible but quite heretical to some of his contemporaries. Some of them still seem heretical today though with regard to a different faith; more on this below.
Leibniz's strategy was to soften up the concept of 'impossibility' by introducing a clever distinction between 'absolute impossibility' and 'accidental impossibility'. Leibniz defines 'absolute impossibility' as one involving a contradiction, but 'accidental impossibility' involves merely a feature of the particular world we are familiar with. This ties up with the Leibnizian doctrine of 'possible worlds' but comparing this to modern concepts of 'distinct models' would be far-fetched. At any rate the idea of a mathematical entity being possible without having a counterpart in the physical world ('natura rerum') is definitely there in Leibniz. How successful Leibniz was in convincing his contemporaries can be judged from George Berkeley's reaction a few decades later :-) In fact, only about 130 years ago, Georg Cantor was acting up toward (more precisely, against) infinitesimals way before he was committed to a physical asylum. For details see item (4) below.

(2) [start of FoM comment by Marcel Ertel on 2 jul '21]
This quote from Cantor's 1883 Grundlagen einer allgemeinen Mannigfaltigkeitslehre (Foundations of a general theory of manifolds), paragr. 8, may be relevant:

Mathematics is in its development entirely free and is only bound in the self-evident respect that its concepts must both be consistent with each other and also stand in exact relationships, established by definition, to those concepts which have previously been introduced and are already at hand and established. In particular, in the introduction of new numbers it is only obligated to give definitions of them which will bestow such determinacy and, in certain circumstances, such a relationship to the older numbers that they can in any given instance be precisely distinguished. As soon as a number satisfies all these conditions it can and must be regarded in mathematics as existent and real. (Translation by W. W. Tait)

German original is in: G. Cantor, Gesammelte Abhandlungen, edited by E. Zermelo, Springer, 1932, p. 182. Note that in the same paragraph, Cantor distinguishes two types of existence: the "immanent" and the "transient" reality of concepts. According to Tait's interpretation (with which I tend to agree), the former corresponds to purely mathematical existence, the latter to a concept being instantiated in physical or psychological reality. The latter is explicitly said not to be necessary for mathematical existence, although Cantor had faith that the two de facto always occur together (for what seem to be mystic religious motives, the "Unity of the All", ibid.). See Tait's article "Cantor's Grundlagen and the Paradoxes of Set Theory" (reprinted in The Provenance of Pure Reason), where he also discusses a disagreement with Michael Hallett on this matter.
[end of comment by Marcel Ertel]

(3) [start of FoM comment by William Tait on 2 jul '21]
Cantor, in his FOUNDATIONS 1 (OF A GENERAL THEORY OF MANIFOLDS) (1883), [an expanded version of 'On infinite linear sets of points', #5], wrote:

'Mathematics in its development is entirely free and is only bound in the self-evident respect that its concepts must both be consistent with each other and also stand in exact relationships, established by definitions, to those concepts which have been previously introduced and are already at hand and established'.

So this preceded Hilbert's statement that, having stated the axioms of Euclidean geometry or the real number, for example, all that was needed for a foundation was a consistency proof. But Hilbert's call for consistency proofs didn't begin in the 1920s: it was at the turn of the century. The relation to the finitism of the twenties was this: there were two problems with the call for consistency proofs in the beginning of yer 20th century. One was that there was no precise definition of consistency, because there was no precise and adequate definition of proof. The only way to 'prove consistency' was to present a model - which won't work for the theory of a Dedekind infinite set. The second problem, raised at the time by Poincare, was that any consistency proof for number theory, for example, would require some form of complete induction, and so would be circular. It took Hilbert's at least 17 years to solve the first problem: Building on the logical work of Frege and Russell-Whitehead, he - or maybe I should say he and Bernays - developed the concept of nth order predicate logic as the possible logical frameworks for mathematical theories. He took longer to respond, in the early 1920s (unsuccessfully, as it turned out) to the second problem: that the consistency. Proof would be finitist, and finitist complete induction has an intuitive justification not present in the general use of complete induction.
[end of comment by William Tait]

(4) In connection with the issue of consistency being sufficient for mathematical existence, William Tait mentioned Cantor's well-known comment regarding mathematics and its essential freedom. There are two reasons why attempts to paint Cantor's ideas as an antecedent of Hilbert's formalism need to be put in context as much as Ebrahim Raisi's comment about being a defender of human rights.

1. In addition to publishing an alleged proof that infinitesimals are inconsistent, Cantor described infinitesimals not merely as 'paper numbers' but as (a) an abomination; (b) the cholera bacillus of mathematics (this was at the time of the cholera plague in Europe). So apparently some forms of mathematics were freer than others, in Cantor's view. To give an example, somebody who describes, say, Scholze's perfectoids in those terms (the terms used by Cantor) today would have difficulty maintaining a reputation as a defender of mathematical freedom.

2. As is well known, Cantor's set-theoretic ideas encountered resistance among his contemporary mathematicians. What is perhaps less known is the fact that Cantor's attempt to defend his theory was, paradoxically, far less Hilbertian than Leibniz's. Instead of disassociating mathematics from metaphysics and working to relax the requirement that mathematical entities must have physical counterparts, as Leibniz did, Cantor on the contrary developed a peculiar form of metaphysics to justify his theory, and adhered as much as possible to the straitjacket of a correspondence to physical reality. There is a perceptive analysis of Cantor's rather old-fashioned metaphysics in Hill's piece, which I summarized in my MathSciNet review freely accessible here: hill17MSN.pdf
In sum, if one is looking for historical antecedents of Hilbert's idea that 'mathematical consistency is enough', then Leibniz seems a better bet than Cantor.

(5) In Mark van Atten's book Essays on Gödel's reception of Leibniz, Husserl, and Brouwer from '15 on page 47 he speaks of "Leibniz' picture according to which mathematical existence is equivalent to mathematical possibility, and the latter is wholly determined by a (global) principle of non-contradiction."

On page 54 he quotes Leibniz as follows: "Possible things are those which do not imply a contradiction" with source in footnote 72: Leibniz to Joh. Bernoulli, February 21, 1699, Leibniz (1849–1863, 3:574): 'Possibilia sunt quae non implicant contradictionem.'

On page 57, note 81 he writes: "Note that for Leibniz, what makes mathematical truths true has nothing to do with possible worlds, only with the principle of contradiction."