That was a miserable day four years ago at UCLA when our MIT secretary Jane sent out the email announcing that Gian-Carlo passed away in his Cambridge house. Ever since, I have not been able to live a single day without frequent flashes of Gian-Carlo in my mind and soul.

I could see his smile in the sparkling raindrops upon the standing glass of my living room. I could hear his voice in the running waves of the Mississippi River during my jogging.  And I often chat with him, as he sits among millions of shining stars that are far away above in the distant Heaven...

---

Gian-Carlo was neither a heavenly saint nor a holy scholar without opponents. In fact, due to his outspoken nature and undisguised writing, he had many. But few were personal. As one of the most productive mathematicians and professional philosophers,  Gian-Carlo kept on writing, not only on combinatorial mathematics and phenomenological philosophy, but also on the human and social facets of a very special species called mathematicians.

What he had probably failed to realize is that mathematicians are a very sensitive species on this planet. They often pretend to be indifferent to others' opinions, and only engage themselves in the epsilon's and delta's. Deep into their bones and genes, however, they are really not the "pencil-and-paper" type as they attempt to appear. Yes, they do care about what and how you comment on them, in fact a whole lot more than you could have imagined. They carry a great sense of pride and egoism. Gian-Carlo made numerous playful observations, such as which professor hid his money beneath bed sheets or inside spare shoes. So he had enemies. But he had more friends, especially among the young generations.

What make Gian-Carlo so irresistible, to both his friends and opponents, are his outstanding character as a professional philosopher, and his unique philosophical way of approaching mathematics.  

As mathematics has evolved into such a complex tree with seemingly unstoppable branched growing and infinitely detailed crystallization, the majority of contemporary mathematicians tend to proudly expertize in their specific branches, just as a silkworm satisfied with a sufficiently large green leaf. Gradually,  the very essence, the true meaning, and the global picture of this mathematics tree start to fade away into distant canyons, and no one bothers to call upon them, nor shows any interest in hearing their echoes.  Gian-Carlo was the exception, in his way of teaching and exploring  mathematics...

In his courses, Gian-Carlo usually started with his pleasant light Italian accent: "Let me first finish this part, then I will explain  the underlying philosophy." As a professional philosopher, Gian-Carlo had his teaching and research thoroughly blended with philosophy.

For thousands of years, a great scholar must have to be a great philosopher as well, from Plato and Aristotle of the Occidental, to Laotze and Confucius of the Oriental.  However, ever since Newton and Leibnitz started up the Calculus Inc., mathematicians have been able to feed on mathematics alone, and philosophy has gradually faded away, or only become a 5-second relaxation comment in one's lectures. Who suffer the most from such separation? The students. They sacrifice all their movie and party time, laboring on Calculus, Linear Algebra, and Ordinary Differential Equations. Their sweat only helps them draw the choking conclusion that mathematics only means endless streams of problem sets, quizzes, and exams. Seldom have they been told the integrated picture of mathematics - its soul, essence, history, and future. So is the situation for young scholars. Gian-Carlo was an exception. He was not only clearly aware of this, but  incredibly skillful in weaving mathematics and philosophy into a colorful and lively carpet, on which you would love to stay.

Gian-Carlo once wrote in Cambridge's Philosophy Society: "A mathematician starts from definitions, while where a philosopher ends up. "[not exact quotation] For example, for thousands of years, philosophers have taken pains in  the definition and meaning of "TRUTH," while mathematicians, aha, take a cheating shortcut - "let me define 1 to be true and 0 false." Mathematicians, therefore, are the least trustworthy jurors inside court rooms.

In all these years in my undergraduate courses, I have made a ground-breaking discovery, whose impact and echo will probably lure Gian-Carlo up in the Heaven to travel down again. That is, good students are most often philosophers, in Gian-Carlo's sense, of course.  Once in my lecture I defined "the distance between two points p=(x, y) and q=(u, v) is ||p-q||=sqrt((x-u)²+(y-v)²)," from which I developed the theory of angles, perpendicularity, etc. "Jackie, why does distance have to be defined using this complicated formula? Why not simply the length of the line segment connecting them?" one of my students came up forward to the blackboard at the end of the lecture and complained. Obviously she was questioning my definition. That is, she was looking for the very meaning of the definition.  I could have responded in a more artistic way. Instead, I simply exclaimed mindlessly, "Look! that is exactly what I have defined!"

Gian-Carlo's another famous statement is "To be a mathematician is to be consistent." You start from definitions and basic rules, which must serve as the base for all the results you attempt to develop later. Now you feel annoyed, "Nonsense, Gian-Carlo! Consistency is required in every corner of human life."  Really? No, that is nothing more than an illusion, one that is heavily polluted by personal or group interest, and our own unconscious psychology. Consistency is characteristic of mathematics. Few politicians could  or have to become consistent. They modify, either secretly or openly, their sets of rules and principles, to fit the demands of  time and people, and to increase their approval ratings.  How about health information? Yesterday they could have said that chemical B is good for your health since it kills certain virions, while today they could claim that B is a bad guy since it increases blood pressure quite noticeably, and tomorrow USAtoday might have a headline saying that it is not chemical B but a substrate of B that stops the growth of certain tumor cells. Thus eventually you become furious - "What for God's sake do you health scientists want to say?!"   Now how about your boyfriends or girlfriends? Love seems to be the very opposite of consistency. "I love you" could easily deteriorate to, after a few days or months, "I could not see the point of this relationship. It is a house but no longer a home to me (as the popular Spanish singer Marc Anthony once sang)!"

Philosophers often pay less attention to details. After getting deep down to the bottom of something, you often lose interest in certain surface details. This is certainly true in Gian-Carlo's lectures. Here I specifically mean graduate courses, since he prepared his undergraduate lectures in full details. In contrast, his graduate lectures often contain some minor errors. He seemed never bothered by the sign errors, + or -. (This is quite opposite in his undergraduate courses, in which if you could catch one sign  or any minor error, Gian-Carlo will reward you a Hirsch chocolate bar.) Once in his course of Topics in Combinatorics, we were all puzzled by one formula from Tensor and Clifford Algebra. Gian-Carlo spent 20 minutes trying to get it right. But the time was up. "Well," Gian-Carlo turned back from the blackboard, and as usual, cast us a childish smile, "philosophically I know it has to be correct, I mean from the bottom line of physics." After a few hours' work, one of my friends in the class and I did get it proved. Gian-Carlo was certainly right, not only philosophically.

I have ever since secretly copied this trick in my teaching. Whenever I could not get something right but the time is up, as long as I have solid "philosophy" backing up my intuition, I would say "Well, philosophically it has to be right." The history of science tells us that sometimes it could become very risky and irresponsible saying like this. As far as a lecture is concerned, however, it  is a lovely "period," if a lecture is to be compared to a sentence. A general sentence has to be closed up by a period, otherwise the incompleteness will haunt in the air for at least two days.

Recently I have been carefully reflecting on the hidden and consistent philosophical structures underlying Gian-Carlo's mathematical world. One of the founders of modern Combinatorics,  Gian-Carlo is certainly most well known for his master works of putting Combinatorics on the solid foundation of algebra. Less obvious to most others is that Gian-Carlo had maintained everlasting interest and enthusiasm in probability and statistical mechanics.

After all these years' personal interaction,  recollection, and my own scholastic growth, I am now eventually able to see and understand the probability ghosts wandering in his combinatorial world, as well as his algebraic and combinatorial spirit flowing in the flesh of his probability and statistical mechanics.

Gian-Carlo deeply believed that probability could be taught and survive without measure theory. Most mathematicians would start to sneeze,  "All right, Gian-Carlo. Don't just say it. Show us how far you could go without measure theory!" Measure theory has become so fundamental in modern probability that I personally would believe that 99.9999% probabilists on this planet would have become jobless without it, purer than 24K gold! Kolmogrov needed measure theory for his 0-1 law or backward and forward equations. Norbert Wiener (Gian-Carlo was the Wiener Professor of MIT at the time he died in 1999) needed measure theory for his Brownian motion. dot dot dot

Yes, Gian-Carlo thus was the only singularity,  one with ingenious courage and insights. Though I still disagree with the surface meaning of his assertion,  I  could completely understand his philosophy and line of thinking. Gian-Carlo did go far enough. He formulated and invented Umbral Calculus to deal with what I call algebraic random variables, i.e., umbrae. They are linear functionals on polynomials and power series, and therefore completely encode the information of exponential generating functions (EGF). EGFs certainly contain the same amount of information as the Characteristic Functions (CF) of true random variables. By only focusing on algebraic random variables and their algebraic structures, Gian-Carlo was able to beautifully hybrid probability with combinatorics. Thus he developed systematic theory on Bell numbers, partition of sets, binomial sequences, orthogonal polynomials, apolar forms, umbral approach to roots of cubic and quintic polynomials, Abel polynomials, inversion of power series, cumulants, etc.

In short, Gian-Carlo attempted to discover and develop all the essential algebraic structures hidden in probability theory. Once he wrote a paper on the key quantity in statistical mechanics, the entropy, purely from the algebraic point of view. His latest monograph was on  Geometric Probability Theory, in which this effort became more explicit. For example, he endows symmetric (polynomial) functions insightful probability meaning, which he also explained using plain English to the whole MIT audience in MIT's annual Killiam Lecture.

Meanwhile, Gian-Carlo's interest in statistical mechanics was not only an innate corollary to his probability taste, but also clearly traceable to his personal friendship with Nobel Laureate C.-N. Yang. Once I was working with Gil, my dear Ph.D. advisor, on a signal processing problem which required the investigation of the zeros of high order optimal polynomials. Gian-Carlo immediately referred me to a class of polynomials and a theorem on them by Yang and Li, part of their Nobel work (on asymmetry). Gian-Carlo told me that there were many interesting polynomials in statistical mechanics related to partition functions.

Later on during my Ph.D. work, my work with Gil on wavelets was thus naturally influenced by Gian-Carlo's algebraic and probabilistic philosophy. We found that the two-scale equations defining shape functions in the multiresolution framework could simply be written as: a = (b + a)/2 using Gian-Carlo's umbral notation. This is a perfect example where probability may ask too much while  umbrae still work well, since probability density functions must be kept non-negative, while for wavelets design, sign oscillations are crucial. By hybridizing the algebraic invariances of  both the heat equations and wavelets, we have been able to develop a theory of the so-called heatlets.

Ever since, I have maintained an everlasting interest in probability and statistical mechanics, from inhomogeneous Markov chains arising from wavelets cascading algorithms, ensembles of random matrices motivated by numerical linear algebra,  Bayesian inference in vision and image analysis, to the statistical mechanics approach to pattern  and information analysis.

It is perhaps a subconscious polarization under Gian-Carlo's influence.

I know HE is now watching me from the Heaven, my dear friend, mentor, and hero, Gian-Carlo Rota, of whom all the good memories will remain evergreen...

---