Today is the birthday (1854) of Jules Henri Poincaré, a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as “The Last Universalist,” since he excelled in all fields of the discipline as it existed during his lifetime. Outside of mathematics and science Poincaré is scarcely a household name, yet a reasonable argument can be made that both Einstein and Picasso were profoundly influenced by his work – yes, BOTH. I’ll try not to make your eyes glaze over with technicalities too much, and, instead, focus on generalities that just about anyone can grasp, including Poincaré’s ideas concerning original thinking, as well as his work habits.
As a mathematician and physicist, Poincaré, made a number of original and fundamental contributions to pure and applied mathematics, mathematical physics, and celestial mechanics. He was responsible for formulating the Poincaré conjecture, which was one of the most famous unsolved problems in mathematics until it was solved in 2002–2003. In presenting the conjecture he helped found the field of topology. In his research on the three-body problem (calculating the motion of three interacting bodies – such as planets – using the laws of motion and gravity posited by Newton), Poincaré became the first person to discover a chaotic deterministic system which laid the foundations of modern chaos theory. Poincaré was also one of the first to propose the existence of gravitational waves emanating from a body and propagating at the speed of light as a solution to problems in celestial mechanics. His proposal has proven correct experimentally.
Poincaré’s work itself was of great importance in numerous fields, but for the moment I would like to focus on that way in which he worked and how he construed the intellectual process. To begin, let me remind you that scientific discovery is almost never a step-by-step process. It requires imaginative leaps and “what-ifs” that are anything but logical. Poincaré tried to probe the mechanics of genius, and, though he explained his own process quite well superficially, he did not exactly provide much useful penetrating detail.
Poincaré’s work habits have been compared to a bee flying from flower to flower. One observer said:
Accustomed to neglecting details and to looking only at mountain tops, he went from one peak to another with surprising rapidity, and the facts he discovered, clustering around their center, were instantly and automatically pigeonholed in his memory.
The mathematician Darboux claimed he was un intuitif (intuitive), saying that this is demonstrated by the fact that he usually worked by visual representation. Poincaré himself wrote that he believed that logic was not a way to discover ideas, but a way to structure and manage ideas once imagination and intuition have uncovered them. Poincaré’s mental processes were not only interesting to Poincaré himself but also to Édouard Toulouse, a psychologist at the Psychology Laboratory of the School of Higher Studies in Paris. Toulouse wrote Henri Poincaré (published in 1910). In it, he discussed Poincaré’s personality and work habits:
He worked during the same hours each day for short periods of time. He did mathematical research for four hours a day, between 10 a.m. and noon then again from 5 p.m. to 7 p.m. He read articles in journals later in the evening.
His normal work habit was to solve a problem completely in his head, then commit the completed problem to paper.
His ability to visualize what he heard proved particularly useful when he attended lectures, since he was severely nearsighted and could not see what the lecturer wrote on the blackboard.
He was always in a rush and disliked going back for changes or corrections.
He never spent a long time on a problem since he believed that his unconscious would continue working on the problem while he consciously worked on another problem.
While most mathematicians worked from principles already established, Poincaré started from basic principles each time.
Not exactly helpful. What you get from this list is that Poincaré’s mind was a cauldron of stuff that he poked around in until he came up with something useful. I am, fortunately or unfortunately, familiar with the process, although my cauldron as not filled with as much technical stuff as his. Furthermore, filling your cauldron with stuff is not enough. You have to have ways of wading around in the stuff productively. Here Poincaré is not much help. He talks about intuition and imagination, but what are they and how do you get them? Poincaré studied his habits and gave a talk about his observations in 1908 at the Institute of General Psychology in Paris. He linked his way of thinking to how he made several discoveries. Some quotes are pertinent.
The chief aim of mathematics teaching is to develop certain faculties of the mind, and among these intuition is by no means the least valuable.
It is by logic that we prove, but by intuition that we discover.
Logic teaches us that on such and such a road we are sure of not meeting an obstacle; it does not tell us which is the road that leads to the desired end. For this, it is necessary to see the end from afar, and the faculty which teaches us to see is intuition. Without it, the geometrician would be like a writer well up in grammar but destitute of ideas.
All good up to a point. A good vocabulary and excellent command of grammar will not make you a poet; skilled brushwork and an array of paints will not make you an artist. No argument. What does make a Keats or a Picasso? Poincaré has no answer, and neither do I.
An array of pots and pans and a larder full of good ingredients will not make you a good cook either. Nor will training by the best chefs. I can give you recipes, however. They are the building blocks. Poincaré was born in Nancy, former capital of the duchy of Lorraine. Lorraine is, of course, the birthplace of quiche Lorraine, which you can find in qualities from wretched to divine the world over. It has become a rather mundane staple in many places. This recipe is serviceable, but you ought to go to Lorraine for a proper quiche. Even there you may be disappointed. You are best served by seeking the advice of a knowledgeable local. Eggs and cream in a pastry shell is not especially French, but the word “quiche” comes from Lorraine dialect (maybe from German, “kuchen”). The bacon was at one time lardons, and the cheese is a late addition also. Therefore, finding “authentic” quiche Lorraine is a lost cause.
For the crust
1 ¼ cups all-purpose flour
½ tsp salt
½ cup cold butter, cubed small
3 tbsp ice water
For the quiche
8 slices bacon
1½ cups shredded gruyere
1 shallot, minced
6 large eggs
1½ cups heavy cream
salt and black pepper
For the crust, sieve the flour and salt into a mixing bowl or food processor. Work the flour and butter together with your hands, or by pulsing in the food processor, until it resembles coarse sand.
Add the ice water one tablespoon at a time and work the mixture into a dough. Form into a disc, wrap in plastic wrap, and refrigerate until firm, (at least 30 minutes).
On a lightly floured surface, roll out crust until ¼” thick. Loosely drape it over a 9” pie plate (or quiche pan) and crimp the edges. Refrigerate until ready to use.
Preheat the oven to 350˚F/175˚C.
In a large, dry skillet over medium heat, cook the bacon until crispy. Drain and cool on wire racks and then break into bite-sized pieces.
Scatter the bacon pieces evenly on the pie crust and then spread over 1 cup of grated gruyere and the shallot.
In a large bowl, whisk together the eggs, cream, a pinch of cayenne and nutmeg, and season with salt and pepper to taste. Pour mixture over bacon and cheese. Sprinkle with the remaining cheese.
Bake for around 45 minutes until the crust is golden and the eggs are cooked through. Test by inserting a knife into the eggs near the center. It should come out clean when the eggs are cooked. Cool the quiche on a wire rack in the tin for 10 minutes before slicing into wedges and serving.