The files from July 14 and July 15 have
been again uploaded, albeit with few changes, because I was so
wasted yesterday I managed to upload the last but one version of
July 15 instead of the last.
I am still not well physically at all and it still feels hot and humid
where I am, so I restrict myself today to two brief remarks having
to do with mathematical and psychological matters - and in fact the
first is mostly linguistic and logical and the second is mostly of
interest to psychologists.
Tomorrow there may be another translation, namely from another
Amsterdam mafiosi in the garb of civil servant, but in case you had
hoped for this today, my message is to look again the coming days,
about which I cannot say much, because I am quite PEM'ed
1. What does "reading Feferman" mean?
I have repeatedly said the last week that I was "reading Feferman"
or intended to do so, which let to questions what I possibly might
Here is the answer: It has nothing whatsoever to do with ME but a
lot with me, for Solomon Feferman is a well-known (among those who
know mathematical logic, to be sure) mathematical logician whose
papers I always liked, but often could not afford, apart from
photocopies, because they were published in books or journals I
could (and can) not afford and it so happens that I discovered about
two weeks ago that he has had the kindness to put a large amount of
his papers, mostly in excellent pdf editions, on line.
Feferman is at Stanford, and his papers are here:
Since I mentioned logic and psychology, here are links to two fine
papers of his that may be of interest to psychologists (with some
this riddle has been solved.
2. Professor Freudenthal also read
I studied psychology for three reasons: (1) I'd read
William James, and hoped for more of that quality (no such luck); (2) my
wife wanted to study psychology, because she had been a psychological
assistant in a lunatic asylum, giving tests of various kinds to inmates,
and wanted to understand what these tests were based on, and I thought
that this study could not be a difficult one (and indeed); and (3) I was
and am most interested in the very broad field of human reasoning of all
kinds since I was 15 and found out for myself that was a really
interesting subject, so I was curious what the study of psychology could
teach me about it (nothing that isn't in James, that wasn't in the
courses I took, because that would have been far too difficult for 95% of
the students, who moreover didn't want any of that, but clear, simple,
easily summarizable coursebook summaries of things).
But what has this to do with
Well... to start with, I have linked their names with the Wikipedia
entries for them:
Hans Freudenthal was a German-Dutch mathematician, who
started as an admirer and assistant of
Brouwer, one of the founders of
mathematics, but got into a quarrel with Brouwer (many did: real geniuses
are often not the friendliest of persons), and then more or less went his
own way as professor of mathematics in Utrecht (Holland).
I read several of
books, notably one about Lincos, which is - mathematical subject,
readers! - a language designed to communicate as easily as is possible
with extra-terrestrial intelligence (which means that it also is about
the foundations of mathematics and logic, whatever extra-terrestrial
intelligence there is or fails to be (**)), and a
very fine one about probability. He also wrote about mathematical
education, where I will arrive in a moment, after briefly introducing the
other person I mentioned.
Jean Piaget was a Swiss developmental psychologist, who
was quite famous both inside psychology, because of his radical new ideas
about learning, especially in children, and who wrote a large number of
books, that have several characteristics that are deplorable, in spite of
his deserved fame: For one thing, he preferred a truly awful grandiose
terminology, that makes his theories a lot more difficult to understand
than they would be otherwise.
His ideas are interesting though, and gave rise to
rather a lot of experimental psychology, which is a subject that is not
close to my heart, for which reason I only mention it in passing.
(***) He also wrote a number of mostly theoretical
books, notably about mathematics and learning, and I'd read some of these
before studying psychology, and they had lodged the strong conviction in
my mind that Piaget when writing about mathematics (i) did not really
umderstand his subject (Piaget on "transformation groups" is certainly
not what Sophus Lie
had in mind, when he wrote about it!) and (ii) had a very personal
understanding of many expressions written in precisely the same letters
in the same sequence in mathematics.
It so happened that the few psychologists I met who were
truly intelligent were also much taken by Piaget, and I could never
really clarify why I personally was not very much impressed by Piaget at
all, while also liking some of his results.
Well... it so happens that today I opened a thick volume
that Hans Freudenthal wrote late in life, "Mathematics as an
Educational Task", that I read about half of in 1980, and found today
that it has an Appendix I "Piaget and the Piaget school's
investigations on the development of mathematical notions". (p.
I was very pleased to find that professor Freudenthal
had just the same impressions as I had, for which reason I have listed it
here, in case it might enlighten some. In fact, I could quote rather a
lot, but will restrict myself to one single subject, that of
the empty set
(link to my
In brief, the empty set is a name for nothing at all:
The set of square circles, the set of even prime numbers greater than 2,
and the set of legally married elephants are all precisely the same set
the empty set, there being in either case no such things as mentioned
Not so for professor Piaget: He did experiments with
children with cards on which there were various pictures, about which
Piaget wrote (in Freudenthal's translation)