Socially we proceed by example. Wim has exhibited
such an example to me, granting a hope of future during the first
years of our interest in application of the modern technique of
model theory to positivity.
It is worth recalling that in the early 1960s, Wim, then an established analyst of high reputation in Banach lattices,
stood second to Abraham Robinson
in promoting the ideas of the brand-new nonstandard analysis.
The second publication on nonstandard analysis was Wim's famous article
“What Is Nonstandard Analysis?” In 2006 I was asked to give a lecture on the basics of Boolean valued analysis
to geometers and of course the title of
this lecture was a la Wim “What Is Boolean Valued Analysis?”
It was Wim who edited the famous green book
“Applications of Model Theory to Algebra, Analysis and Probability”
A few weeks ago I exchanged letters with Dana Scott who will visit the Maltcev Centennial at
Novosibirsk later this summer. He reminded me that in this book he
published the article “Boolean
Models and Nonstandard Analysis” where he forecasted the future of what we know now as Boolean valued analysis.
It so happened that Wim reviewed the majority of books on
positivity and nonstandard methods stemming from Novosibirsk and
Vladikavkaz, my textbook on functional analysis inclusively.
Of course we were and still are very proud of his evaluations
and comments. However, I first met Wim in Dresden at Positivity IV
and since then a photo of him with Tolya Kusraev and me is always
right above my left hand in my study at Novosibirsk.
Wim is a man of an exceptional understanding and marvelous sense of humor.
So I finish with a short slightly jocular
toast I prepared to welcome Wim here at Positivity VI
and sent to him yesterday.
It is an honor and pleasure for me to felicitate
Professor Luxemburg who is unfortunately absent from this Positivity meeting.
We are just a bunch of positivity adepts.
Thinking positively, we may view the realm of mathematics as
the free quite real vector space over all mathematicians.
The conic hull of all positivity humans
makes mathematics into an ordered vector space.
Most likely this yields
a vector lattice despite the ludicrous claims of some fellow mathematicians
to be the top elements in this ordered vector space.
In any case, we belong to the positive cone of mathematics and so
we are positive. Since positivity is a general mode of living, it is better
to say that a member of the positive cone has positive sense.
As usual in many dimensions, most vectors are senseless,
i.e., the relevant persons have no sense at all.
Professor Luxemburg has taught us that we live not only in the
vector lattice of mathematics but also in the
unusually nonstandard universe where we must distinguish between
the negligible and the eternal as well as between the infinite
and the infinitesimal.
Positivity is a relation and some of us are more positive than
the others. In the nonstandard world we live in, one
is definitely greater than the other, if the difference between the two
is infinitely large.
Professor Luxemburg occupies an exceptional place in positivity,
for he is definitely greater than any standard mathematician of positive sense.
Professor Luxemburg's service to mathematics is so impressive that
I am sure that no one but he has any right to repeat
the ancient words of
Het Wilhelmus: “Life and my all for others
I sacrificed, for you!”
The noble tradition calls us to raise our glasses and wish Professor
Luxemburg, the true Wilhelmus of Positivity, many happy returns of the day.
Inspiration is always genuine. You cannot counterfeit inspiration.
We all share the feeling of inspiration that stems from Wim.
A speech at Positivity VI in El Escorial (Spain) on July 23, 2009