What was Cantor's philosophical reason for accepting the infinite but rejecting the infinitesimal?

Question

I have begun inquiring recently into mathematical aspects of Georg Cantor's theory of transfinite numbers and sets, which he developed between the years of 1874 and 1897. Throughout his theory, Cantor captured the so called actual infinity and thus revived the controversy as to the role and place of infinity in mathematics. Cantor fought for the acceptance of the actual infinite, but nevertheless rejected the infinitesimal. Why is however not very clear to me, and hence my question:

Why did Cantor reject the infinitesimals? What was his argument? And on what grounds can anyone who accepts the infinite reject the infinitesimal? Is it not true that the infinity and the infinitesimal are reciprocal (or rather - two equivalent sides of the same thing)?

Answer 1

Here is Cantor in his own words (from his influential 1887 letter to Weierstrass):

"I begin from the supposition of a linear magnitude ζ which is so small that its product by n , ζ · n, for every finite whole number n however great is smaller than unity, and then prove, from the concept of a linear magnitude and with the help of certain propositions from transfinite number theory, that then ζ · ν is less than every finite magnitude however small, where ν is an arbitrarily great transfinite ordinal number (i.e., cardinal or type of a well-ordered set) from any arbitrarily high number class. But this means that ζ cannot be made finite by any actually infinite multiplication of any power, and hence surely cannot be an element of finite magnitudes. But then the supposition made contradicts the concept of a linear magnitude, which is such that every linear magnitude must be thought of as an integrated part of other ones, and in particular of finite ones... Hence, the so-called Archimedean Axiom is not an axiom at all but a proposition which follows with logical necessity from the concept of linear magnitude".

See a modern reconstruction of the argument, and how it fails for infinitesimals of the non-standard analysis, in Moore's Cantorian Argument Against Infinitesimals.

Peano in 1892, and Russell in 1903 gave their variations on the theme, but, according to Moore,

"in none of its incarnations is the argument particularly easy to follow, and though there are resemblances among the three versions it is not even clear that they are in fact versions of a single argument".

In the passage, Cantor's issue seems to be that the infinitesimals, traditionally intuited as the "inverses" of infinities, can not be the "inverses" of his transfinite numbers, which, in the time honored tradition, he saw as the only "true" infinities. In other words, one can not produce a finite magnitude out of infinitesimals, even concatenated transfinitely many times. As Moore shows, this is essentially because Cantor's ordinals only allow well-ordered concatenations.

It is hard to argue with Cantor's "concept of a linear magnitude", which excludes infenitesimals, just as it is with Kant's "pure intuition of space", which excludes multiple parallels. With concepts like that, to each their own. From the modern point of view, Cantor is conflating cardinality with measure, but then the modern concept of magnitude isn't Cantor's, just as the modern concept of geometry isn't Kant's. At the time, the axiomatic method and the measure theory were still in the womb, du Bois-Reymond's non-Archimedean speculations were not up to Weierstrass's standards, and Weierstrass's analysis had no use for infinitesimals.

In addition, infinitesimals had a bad reputation among philosophers since (even before) Berkeley's Analyst declared them "ghosts of departed quantities". Cantor did not want them mixed up with his transfinite numbers, acceptance of which as respectable mathematical entities he was advocating at the time. This in itself was contrary to the long standing tradition: the scholastic concept of numbers derived from Aristotle and supported by arguments like "annihilation of a number", see How does actual infinity (of numbers or space) work? The irony is that, as Dauben writes,

"Cantor condemned this kind of argument... on the grounds that it was fallacious to assume that infinite numbers must exhibit the same arithmetic characteristics as did finite numbers".

Yet he indulged in the same sort of reasoning when it came to the infinitesimal magnitudes.

Answer 2

The concept of infinitesimal small and infinitely large numbers has been been formalized by the mathematical domain of non-standard analysis.

The field of rationals (QQ,+,*) embedds into the ring (Omega_QQ,+,*). Elements of the latter are the equivalence classes of sequences of rational numbers; two sequences are considered equivalent when their difference is zero for all but finitely many elements of the sequence.

As a consequence, Omega_QQ contains the classes of the particular sequences small-omega with elements a_n = 1/n and big-omega with elements b_n = n. small-omega is infinitesimal because it is non-zero and smaller than every fixed positive number. While big_omega is infinite because it is greater than any fixed positive number.

Cantor rejected infinitesimals because the product of an infinitesimal with a fixed, finite non-zero remains infinitesimal. The Archimedean axiom does not hold in Omega_QQ. Hence - according to Cantor - infinitesimals do not represent the length of any line; they are no "linear numbers". In addition, Cantor considered the multiplication of infinitesimals with transfinite ordinal; but it remains unclear how Cantor thought this multiplication to operate.

Note that the product small_omega * big_omega = 1. Hence multiplying an infinitesimal by an infinite may result in a finite number.

Answer 3

Infinities in the sense of Cantor are cardinalities. There is no cardinal equivalent to infinitesimals. If you are counting, the smallest thing other than zero is one.

I do not know Cantor's own argument on this account, so I have to skip over your first two questions. But modern mathematicians do not see the two as in any way related, with a few exceptions, although there have been attempts to wrap both up in a single overarching approach.

You need a system with a notion of division for infinitesimals to mean something, and generally, also a notion of continuity. If you are not focussed on continuity, the idea that division just blows up at zero does not faze you: you just consider it a function with a hole in its domain of definition. The Differential Calculus is the first place where we worry about smoothly handling functions viewed infinitely closely.

There are lots of ways to embed notions of infinity into the real numbers (or any other number system) to make differentiation work more easily (or to inject its equivalent and simplify reasoning in the domain), but they are basically linguistic tricks to encode the outcome of a process (global limit-taking) into a word with specific grammar (an infinitesimal variable) that keeps it from getting you in trouble by contradicting the assumptions that hold up the process. The magic that proves that that grammar is clear and tells you how to derive different versions of it for different applications, is called L̸os's theorem, and it is the cornerstone of one of the approaches to Nonstandard Analysis.

So these two approaches to infinity cannot be explained in terms of one another, do not overlap in domains of application, and arise from very different points of view. So the answer to the last question is 'definitely not'. There is no single 'infinity' in any system of infinitesimals equivalent to, for instance 'countably many', there are many. And there is no reciprocal of 'continuously many', only of different scales of infinity unrelated to countability or projection.

An approach that tries to combines them into a more cohesive and global approach to infinities in general is proposed by John Conway in the "Surreal Numbers", which introduces interpolation as a basic operation, rather than deriving it from division and the rest of arithmetic, and approaches continuity via infinite division, rather than topologically through pre-images.

Answer 4

Cantor's reason was the holy bible and the writings of St Augustin. Both talk about the infinitely large but not about the infinitely small. Further Cantor denied already the existence of atoms.

Every single finite cardinal number (1 or 2 or 3 etc.) is contained in the divine intellect (from St Augustin: De civitate Dei, lib. XII). (Cantor, letter to I. Jeiler)

The Lord rules in eternity and beyond, from Exodus 15,18. I think this "and beyond" hints to the fact that omega is not the end but that something is existing beyond."(Cantor, letter to R. Lipschitz)

It is obvious that the belief in the whole set of all numbers requires the belief in someone who has created this wholeness. "Cantor is probably the last exponent of the Newtonian attitude with respect to religion." (Meschkowski and Nilson (eds): Georg Cantor Briefe, Springer, Berlin (1991) p. 15)

EDIT: Here are more statements of Cantor:

Unter einem A.-U. ist dagegen ein Quantum zu verstehen, das einerseits nicht veränderlich, sondern vielmehr in allen seinen Teilen fest und bestimmt, eine richtige Konstante ist, zugleich aber andrerseits jede endliche Größe derselben Art an Größe übertrifft. Als Beispiel führe ich die Gesamtheit, den Inbegriff aller endlichen ganzen positiven Zahlen an; diese Menge ist ein Ding für sich und bildet, ganz abgesehen von der natürlichen Folge der dazu gehörigen Zahlen, ein in allen Teilen festes, bestimmtes Quantum, ..., das offenbar größer zu nennen ist als jede endliche Anzahl 3. Footnote 3: Man vgl. die hiermit übereinstimmende Auffassung der ganzen Zahlenreihe als aktual-unendliches Quantum bei S. Augustin (De civitate Dei. lib. XII, cap. 19): Contra eos, qui dicunt ea, quae infinita sunt, nec Dei posse scientia comprehendi. (E. Zermelo (ed.): Georg Cantor Gesammelte Abhandlungen, Springer, Berlin (1932) p. 401)

Indem nun der h. Augustin die totale, intuitive Perzeption der Menge (nu), "quodam ineffabili modo", a parte Dei behauptet, erkennt er zugleich diese Menge formaliter als ein aktual-unendliches Ganzes, als ein Transfinitum an, und wir sind gezwungen, ihm darin zu folgen. loc cit p. 402.

Im Gegensatz zu Augustin findet sich bei Origines eine entschiedene Stellungnahme gegen das Aktual-Unendliche, daß es fast scheinen möchte, er wolle selbst die Unendlichkeit Gottes nicht behauptet wissen. loc cit p. 403.

Dies stimmt völlig mit demjenigen überein, was S. Augustin in dem pag. 32 abgedruckten Kapitel seiner Hauptschrift De Civitate Dei, lib. XII, cap. 19, sagt: "Ita vero suis quisque numerus proprietatibus terminatur, ut nullus eorum par esse cuicumque alteri possit. Ergo et dispares inter se atque diversi sunt, et singuli quique finiti sunt, et omnes infiniti sunt." loc cit p. 419.

Answer 5

Is it not true that the infinity and the infinitesimal are reciprocal (or rather - two equivalent sides of the same thing?

This is one way of intuitively understanding the infinitesimal and infinite. It can be made rigourous in non-standard analysis or in Conways surreal numbers. This is very different from the mainstream way of understanding the infinitesimal which is via limits.

A fourth way of understanding the infinitesimal is synthetically as the square root of zero. This of course is surprising as normally we would say standardly that the square root of zero is just zero. However the theory uses intuitionistic rather than classical logic and hence opens up a space for the introduction of infinitesimals. Its arguably closest to the notion of infinitesimals of Newton. Philosophically speaking it is a way of rigourously formulating Hegels insight (to Zenos paradox) that an infinitesimal is something that is both here and not here.