Of course, suggestions have been made as to how to get around these problems, but none is adequate. One suggestion is that large categories, such as the collection of all groups, should be regarded as proper classes, that is, subcollections of the hierarchy which do not themselves occur in the hierarchy, and which cannot be members of any other collection. But this will not work. First, proper classes, if we are to take them seriously (and not just as façons de parler), are a masquerade. The cumulative hierarchy was proposed as an analysis of the notion of set. It is supposed to contain all sets. If we are forced to admit that there are sets outside the hierarchy, this just shows that the analysis is wrong. And calling them by a different name is just a trivial evasion. Moreover, the insistence that proper classes cannot be members of other collections can have no satisfactory rationale. If they are determinate collections with determinate members, there is no reason why we should not consider them to be members of other collections, for example their singletons. [G. Priest: "In contradiction", 2nd ed., Oxford Univ. Press, Oxford (2006) p. 34]

All that Cantor has proved is that X is not on the list above n, no matter how large n may be. But he has not proved that X is not in the (not-yet-listed – and indeed unlistable, because infinitely large) part of the table after n ... and he cannot prove that! After all, no matter how large n gets, the part of the table after n still remains infinitely larger than the part of the table before n. And note that no matter how large n gets, the list after n can never be "brought up" to be included in the part of the list before n. This is because the part of the list before n is necessarily finite, while the part of the list after n must be infinitely long! [A. Mehta: "A simple argument against Cantor's diagonal procedure" (2001)]

Often it has been said that mathematics should start with definitions. The mathematical theorems should be deduced from the definitions and the postulated principles. But definitions themselves are an impossibility, as Kirchhoff used to say, because every definition needs notions which have to be defined themselves, and so on. We cannot, like Hegel's philosophy does, develop the being from the nothing. [...] The whole mathematics is there to be applied. [...] Mathematics is a natural science – not better, not more complete, and not simpler the phenomena can be described than mathematically. [...] If nothing else but the digits up to a certain position are known of decimal fractions, then not even two of them can be added. [...] Again and again it is confused whether a decimal fraction is known up to a certain digit or whether its formula n=1 f(n)/10n is known, i.e., f(n) for every n. [L. Kronecker: "Über den Begriff der Zahl in der Mathematik", Public lecture in summer semester 1891 at Berlin – Kronecker's last lecture. Retranscrit et commenté par Jacqueline Boniface et Norbert Schappacher: Revue d'histoire des mathématiques 7 (2001) pp. 225f & 251 & 252f & 268f]

I believe that we will succeed one day to "arithmetize" the whole contents of all these mathematical theories, that is to base it solely on the concept of number in its stricter sense, i.e., to get rid of the added modifications and extensions (namely the irrational and continuous magnitudes) which mainly have been caused by the applications on geometry and mechanics. [K. Hensel (ed.): "Leopold Kroneckers Werke" III, Teubner, Leipzig (1895-1931) p. 253]

I don't know what predominates in Cantor's theory – philosophy or theology, but I am sure that there is no mathematics there. {{I could not verify this quote, but Cantor authenticates it in a letter to G. Mittag-Leffler of 9 Sep 1883:}} "Kronecker, who visited me at the beginning of July, declared with the friendliest smile that he had much correspondence about my last paper with Hermite in order to demonstrate to him that all that was only 'humbug'."

WMs mantra : philosophy or theology, but no mathematics

On 12/27/2021 5:27 AM, WM wrote:
> Of course, suggestions have been made as to how to get around these problems, but none is adequate. One suggestion is that large categories, such as the collection of all groups, should be regarded as proper classes, that is, subcollections of the hierarchy which do not themselves occur in the hierarchy, and which cannot be members of any other collection. But this will not work. First, proper classes, if we are to take them seriously (and not just as façons de parler), are a masquerade. The cumulative hierarchy was proposed as an analysis of the notion of set. It is supposed to contain all sets. If we are forced to admit that there are sets outside the hierarchy, this just shows that the analysis is wrong. And calling them by a different name is just a trivial evasion. Moreover, the insistence that proper classes cannot be members of other collections can have no satisfactory rationale. If they are determinate collections with determinate members, there is no reason why we should not consider them to be members of other collections, for example their singletons. [G. Priest: "In contradiction", 2nd ed., Oxford Univ. Press, Oxford (2006) p. 34]
>
> All that Cantor has proved is that X is not on the list above n, no matter how large n may be. But he has not proved that X is not in the (not-yet-listed – and indeed unlistable, because infinitely large) part of the table after n ... and he cannot prove that! After all, no matter how large n gets, the part of the table after n still remains infinitely larger than the part of the table before n. And note that no matter how large n gets, the list after n can never be "brought up" to be included in the part of the list before n. This is because the part of the list before n is necessarily finite, while the part of the list after n must be infinitely long! [A. Mehta: "A simple argument against Cantor's diagonal procedure" (2001)]
>
> Often it has been said that mathematics should start with definitions. The mathematical theorems should be deduced from the definitions and the postulated principles. But definitions themselves are an impossibility, as Kirchhoff used to say, because every definition needs notions which have to be defined themselves, and so on. We cannot, like Hegel's philosophy does, develop the being from the nothing. [...] The whole mathematics is there to be applied. [...] Mathematics is a natural science – not better, not more complete, and not simpler the phenomena can be described than mathematically. [...] If nothing else but the digits up to a certain position are known of decimal fractions, then not even two of them can be added. [...] Again and again it is confused whether a decimal fraction is known up to a certain digit or whether its formula n=1 f(n)/10n is known, i.e., f(n) for every n. [L. Kronecker: "Über den Begriff der Zahl in der Mathematik", Public lecture in summer semester 1891 at Berlin – Kronecker's last lecture. Retranscrit et commenté par Jacqueline Boniface et Norbert Schappacher: Revue d'histoire des mathématiques 7 (2001) pp. 225f & 251 & 252f & 268f]
>
> I believe that we will succeed one day to "arithmetize" the whole contents of all these mathematical theories, that is to base it solely on the concept of number in its stricter sense, i.e., to get rid of the added modifications and extensions (namely the irrational and continuous magnitudes) which mainly have been caused by the applications on geometry and mechanics. [K. Hensel (ed.): "Leopold Kroneckers Werke" III, Teubner, Leipzig (1895-1931) p. 253]
>
> I don't know what predominates in Cantor's theory – philosophy or theology, but I am sure that there is no mathematics there. {{I could not verify this quote, but Cantor authenticates it in a letter to G. Mittag-Leffler of 9 Sep 1883:}} "Kronecker, who visited me at the beginning of July, declared with the friendliest smile that he had much correspondence about my last paper with Hermite in order to demonstrate to him that all that was only 'humbug'."