[FOM] A last lesson by Ken Lopez-Escobar

walter.carnielli at cle.unicamp.br walter.carnielli at cle.unicamp.br
Fri Sep 16 20:03:35 EDT 2011

Dear colleagues:

I deeply regret to inform, to the FOMers that perhaps do not know yet, the
passing of our friend Professor E. G. K. (Ken) Lopez-Escobar on September
1, 2011 in a hospital in Annapolis, MD.

Ken worked on model theory and proof theory, with special interests in
intuitionistic proof systems and also in the history of mathematics. He
received his Ph.D. in 1965 at the University of California, Berkeley,
under the supervision of Dana Scott.

Two months before his death He had sent the following open e-mail (with
copy to several colleagues) to the administrators of the UMD Math Dept.,
which as he himself puts it, "explains my current physical status and
throws in some mathematics".

His message is very touching, but at the same time reveals his passion for
the history and for the foundations of mathematics and his brave

Ken has been many times to Brazil, and was a member of my PhD committee
back to 1982.
I have learned many lessons from him, and I take the liberty to reproduce
one of Ken's las lessons here as a homage to him.

Sad regards,

Walter Carnielli


Dear Jim, Denny and Brian:

The cancellation of SS I Math 310 has turned out to be very helpful to me.
First of all it allowed me to spend some weeks with my elder brother whom
I had not seen for many years. Secondly it allowed me to be interned for 8
days in the hospital to take care of a cancer flare up without having to
worry about the disruption it would have caused to the Math 310 students'.
Now although all my doctors (cardiologists, urologists, nephrologists,
oncologists, surgeons, ....) have put me back together again, in a
condition far better than I was last Spring, they also discovered that my
cancer had changed from a superficial one to a more invasive type and thus
I will shortly start a more aggressive treatment to eliminate it.

There is a good chance that the treatment itself will be over by
September, but since there will be a recuperation period I believe it
would not be fair to the students to attempt to teach all my Fall classes.
Consequently I have decided to retire and devote more of my energy on
treating the cancer.

Please do not construe this email as an official request to retire as of
today; the principal reason for this note is to facilitate the class
assignments for the Fall. Also since I have had very bad experiences when
the medical insurance is perturbed in the middle of treatment I would like
to be able to complete the treatment before I change the medical insurance
from that of a full time employee to that of a retired employee.

The classes that I was scheduled to teach in the Fall were Math 406
(Introduction to Number Theory) and Math 274 (History of Mathematics).

The only problem that may occur with Math 406 is that I had chosen as
textbook LeVeque's "Fundamentals of Number Theory", Dover Publications
(about $10, used), and it requires, specially towards the beginning of the
course, that the instructor fleshes out the proofs given in the book (and
also explain what is an acceptable mathematical proof). Of course this
becomes a moot point if it is not too late to change the text book.

I have taught Math 274 quite a few times and I have become convinced that
the course is misclassified; if you want it to be a 200 series course then
it should be changed to "A History of Mathematicians", while the "History
of Mathematics" should be a 400 level course and reserved for students who
have had a fair amount of what might be called "Pure Mathematics"; Math
406 would be an ideal prerequisite.

Let me explain my reasoning. By having it as a 200 level course and with
"History" as part of the title, many students, specially those whose
arithmetical skills are well below that of the ancient egyptian scribes,
see this as a course as a simple course in which they can easily get an
"A" or "B" by simply memorizing names, dates and events. Now although
knowing that Girolamo Cardamo (Cardan) was found guilty of heresy and was
going to be burnt at the stake (he had cast a horoscope of Jesus
Christ---the heresy being that it implied that God was controlled by the
Stars and thus God was not omnipotent---) is in itself an interesting
event in the life of Cardamo and also gives us an insight into the social
mores of his times; it gives us very little information about the
development of Mathematics. In other words it belongs much more to a
"History of Mathematicians" than to a "History of Mathematics"! What would
pertain to the "History of Mathematics" would be an analysis of the method
by which Cardamo solved the cubic equation and that completely overwhelms
the marginal student.

But the problem is deeper than simply of not understanding the method. To
the student who has never tried to "manually" (i.e. without using a
graphing calculator or computer) solve a cubic equation the whole
operation appears to be a waste of time and the student then complains
that the course is not a "History of Mathematics" but rather an advanced
course on "Number Theory".

So what should be the principal content of a "History of Mathematics"
course? A clue can be found in alternate names that have been used for
Mathematics, namely "Science of Number" and "Science of Measurement". I
doubt that anyone would disagree that the beginning of Mathematics is
intimately connected with the beginning of Number and thus that an
"(Early) History of Mathematics" is in fact a "History of Number".

In fact I venture to suggest that Mathematics begins when Homo Sapiens
replaces "number of ....." by simply "number", even though at the time
there was not a clearly defined concept of "number" and there was a great
deal of confusion between "number", "numeral" and "number word" (it was
historically much later that it was realized (postulated) that the number
two was what was common to (i) a pair of doves, (ii) a brace of hounds,
(iii) a yoke of oxen ...) . In any case, starting from the concept of
"number of .....", the Greeks were able to justify the Rational Numbers.
But what are the grounds for calling the (positive) square root of two a
"number"? Or for that matter, when can a mathematical object be called a
"number"? In my view the best answer to "What is a number" was given by R.
Dedekind [1888]:

 " The numbers are free creations of man's mind, they serve as a means of
apprehending the difference of things more easily and more sharply. "

(I would go as far as to replace "number" by "mathematics".)

It is clear that a student who had not experienced, or at the very least
is aware of, some of the wonderful achievements of Mathematics, would be
completely bored in such a course.

Thus my recommendations for Math 274 (History of Mathematics) is that it
be returned to the History Department--where it would become a "(Social)
History of Mathematics" and that the Mathematical "History of Mathematics"
become either a 400 level course, with say Math 406 as a prerequisite, or
else an undergraduate seminar--in which the students get appropriate
credits towards graduation--.




Professor E. G. K. Lopez-Escobar
Mathematics Department
University of Maryland
College Park, MD 20742
egkle at math.umd.edu
egkleye at gmail.com

Walter Carnielli
Centre for Logic, Epistemology and the History of Science – CLE
State University of Campinas –UNICAMP
Rua Sérgio Buarque de Holanda, 251
CEP 13083-859 - Campinas-SP - Brazil
Phone: (+55) (19) 3788-6519
Fax: (+55) (19) 3289-3269
e-mail: carniell at cle.unicamp.br
Website: http://www.cle.unicamp.br/prof/carnielli

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