[FOM] Quotation of Brouwer on Cantor

Hans van Ditmarsch hans at cs.otago.ac.nz
Thu Sep 8 17:16:58 EDT 2005

Dear Patrick,

On Wed, 7 Sep 2005, Patrick Caldon wrote:
> "Cantorian mathematics was a sickness which is best forgotten"
> but I can't remember the precise statement, or even enough
> of the statement to get a reference.  Can someone help me here?

I also was vaguely worried about such a statement which seemed unlikely to
me. I checked "Gnomes in the Fog", by Dennis E. Hesseling, on possible
occurrences of such or similar statements. This is an (excellent)
sourcebook on (subtitle anyway) "The reception of Brouwer's intuitionism
in the 1920s", so anything sensational said on such issues can almost
certainly be found there. Of course the more obvious but also more time
consuming source would be the collected works presented by Heyting and
Freudenthal in the 1970s and the recent Brouwer two-volume biography by
van Dalen - I have an (abridged) Dutch version of that, that I also
consulted. This is the closest to sensational that I found:

(...) an incorrect theory, even if it cannot be checked by any
contradiction that would refute it, is none the less incorrect, just as a
criminal policy is none the less criminal even if it cannot be checked by
any court  that would curb it.
(see Hesseling 62 - this is an English translation of the Dutch original
LEJ Brouwer, Over de rol van het principium tertii exclusi in de wiskunde,
in het bijzonder in de functietheorie, Wis- en Natuurkundig Tijdschrift II
(1923), pp 1-7. Apparently also found in translation in vHeijenoort's

More specifically concerning Cantor, thesis 13 of Brouwer's PhD thesis
(1907) states:

"The second class of numbers of Cantor does not exist"

(i.e., transfinite ordinals omega, omega+1, etc.). (See the vDalen
biography - in Dutch - page 93.)

On other Cantor related matters, it is noted that some of Cantor's
results, such as the main theorem 2^A > A, do not hold intuitionistically.
But also - I rather liked finding that - that "Cantor had admitted that
the trichotomy law of cardinals was not proved, and he intentionally
refrained from using it."  (Hesseling 62, note 256, now this refers to
Moore 1980, p 102 (Beyond First-Order Logic, journal: History and
Philosophy of Logic 1: 95-137.) In other words, lots of nuances.

Kind regards, Hans van Ditmarsch

* Hans van Ditmarsch                  Department of Computer Science *
* hans at cs.otago.ac.nz                 University of Otago            *
* +64 3 479 8475   fax 479 8529       PO Box 56                      *
* www.cs.otago.ac.nz/staffpriv/hans/  Dunedin            New Zealand *

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