Source: Slate Star Codex, May 2017
Source: Simons Foundation, Dec 2014
Gromov’s first bombshell was the homotopy principle, or “h-principle,” a general way of solving partial differential equations. “The geometric intuition behind the h-principle is something like this,” explained Larry Guth of the Massachusetts Institute of Technology. “If you had a sweater and you wanted to put it into a box, then because the sweater is soft, it is easy to put it into the box, and there are lots of ways to do it. But if you had to write a list of totally precise instructions about exactly how to put the sweater into the box, it would actually be hard and kind of complicated.”
In mathematics, the question was whether some high-dimensional object could be embedded into a given space. “And the only way to deal with high-dimensional objects, at least traditionally,” said Guth, “was to write down equations that say precisely where everything goes, and it’s hard to do that. Like the situation with the sweater, the only way that we could describe how to put the sweater into the box was to write a completely precise list of instructions about exactly how to do it, and it makes it look as if it is complicated. But Gromov found a good way of capturing the idea that the sweater is very soft, hence you can do almost anything and it will fit into the box.”
… a list of questions: “What is mathematics and how has it originated? Where does the stream of mathematical ideas flow from? What is the ultimate source of mathematics in the brain? Gromov rebuffed as absurd his own question about what mathematics is. But, he continued, “you can say, ‘How is it being created, how is it being studied, how is it being learned?’”
“The way I think of mathematics,” he said, “is as a physical and psychological process, not some abstraction.” And herein arises his notion of a “bug on a leaf,” which gives a nice sampling of his inquisitive excursions away from research mathematics proper and into biology, evolution, the structure of the brain, and the question of how scientific ideas evolve.
“A bug on a leaf displays two simple phenomena,” he said. “It always does the same thing, one leg, another leg, one leg, another leg. Just movement. Many, many things are done this way, including Euclidean geometry. That’s one point. The other point is information theoretic. And that is that the bug spends more time on the edge of the leaf compared to the interior, and more time on the tip of a leaf compared to the interior. And your eye does the same thing when looking at an image, your eye will spend more time around the image.”
Both of these processes, with the bug and the eye, in Gromov’s opinion, are run by universal mechanisms. “The logic of the world forces the way we think,” he said. “This is what I have been thinking about for the last 10 years: the basic principle underlying thinking, and specifically underlying thinking about mathematics. It’s very different from logic, it’s not logic.” He calls it “ergologic” — a reconsideration of traditional logic, encompassing the “ergo system” and the “ergo brain” and “ergo thinking.”
“Life is similar to how mathematics is organized in our brain. If you don’t accept it, life is impossible,” he said. And then one final Gromovian snippet I managed to catch was this: “If it is impossible, you try to do it anyway.”
Source: Scott Young’s website, Mar 2017
In college, Einstein often struggled in math, getting 5s and 6s (out of a possible 6) in physics, but getting only 4s in most of his math courses (barely a passing grade). His mathematics professor, and future collaborator, Hermann Minkowski called him a “lazy dog” and physics professor, Jean Pernet, even flunked Einstein with a score of 1 in an experimental physics course.
At the end of college, Einstein had the dubious distinction of graduating as the second-to-worst student in the class.
The difficulty Einstein had was undoubtedly due in part to his non-conformist streak and rebellious attitude, which didn’t sit well in an academic environment. This would follow him in his future academic career, when he was struggling to find teaching jobs at universities, even after he had already done the work which would later win him the Nobel prize.
Einstein learned physics, not by dutifully attending classes, but by obsessively playing with the ideas and equations on his own.
… a strong curiosity both to know how things actually work, and a belief that, “nature could be understood as a relatively simple mathematical structure.”
Einstein’s curiosity wasn’t merely to perform adequately, but to develop a deep understanding and intuition about physical concepts.
Einstein’s later education in Aarau, Switzerland, was heavily influenced by the philosophy of Swiss educational reformer, Johann Heinrich Pestalozzi. Pestalozzi claimed, “Visual understanding is the essential and only true means of teaching how to judge things correctly,” adding, “the learning of numbers and language must definitely be subordinated.”
Einstein’s ability to focus, combined with a reverence for solitude, allowed him to do some of his best work in physics.
Einstein’s most famous method for learning and discovering physics has to be the thought experiment.
One of his most famous was imagining riding on a beam of light. What would happen to the light beam as he rode alongside it at the same speed?
These thought experiments were built on his intuitive understanding of physics, which in turn was built on his experience with working through theories and problems. Their strength, however, was to draw attention to contradictions or confusions that may have been missed by a less intuitive physicist.
Discussing ideas aloud, sharing them with others, can often put together insights that were previously unconnected. Einstein made great use of this technique of discussing tricky problems with friends and colleagues, even if they were merely a sounding board rather than an active participant in the discussion.
Einstein was never much of a conformist. While his rebellious streak probably hurt his earlier academic career when he was struggling to find work in physics, it is also probably what enabled his greatest discoveries and accentuated his later celebrity.
This rebelliousness likely helped him in learning physics as he pushed against the traditions and orthodoxy he didn’t agree with. He hated the German educational system, finding in Isaacson’s words, “the style of teaching—rote drills, impatience with questioning—to be repugnant.” This rejection of the common educational method encouraged him to learn physics on his own through textbooks and practice.
Later, the same rebelliousness would be essential in revolutionizing physics. His research on the quantization of light, for instance, had been first discovered by Max Planck. However, unlike the older Planck, Einstein saw the quantization as being a physical reality—photons—rather than a mathematical contrivance. He was less attached to the predominant theory of the time that light was a wave in the ether.
Where many students would have been happy to conform to predominant educational and theoretical orthodoxies, Einstein wasn’t satisfied unless something made sense to him personally.
“Curiosity has its own reason for existing,” Einstein explains. “One cannot help but be in awe when one contemplates the mysteries of eternity, of life, of the marvelous structure of reality.”
Curiosity was his motivation for learning physics. … Curiosity was also, in his own mind, the greatest reason for his accomplishments.
Source: Slavov blog, Aug 2014
Fermi, E (1934). An attempt of a theory of beta radiation. Z. phys, 88(161), 10.
Nature Editors: It contained speculations too remote from reality to be of interest to the reader
Thaler, R. (1980). Toward a positive theory of consumer choice. Journal of Economic Behavior & Organization, 1(1), 39-60.
Richard Thaler: Toward a Positive Theory of Consumer Choice was rejected by six or seven major journals
Shechtman, D., Blech, I., Gratias, D., & Cahn, J. W. (1984). Metallic phase with long-range orientational order and no translational symmetry. Physical Review Letters, 53(20), 1951.
Dan Shechtman: It was rejected on the grounds that it will not interest physicists
Source: NYTimes, Mar 2000
In a career of unquenchable creativity, he has always kept moving, searching for new and different problems to tackle, founding a new field as soon as the old one seemed stale.
A ceaseless flow of ideas, some successful, some less so, is one of Dr. Brenner’s traits. Another is his gift of spellbinding listeners with his latest enthusiasm. He speaks with distinctive English diction and in perfectly constructed sentences that often end with a joke.
”Sydney Brenner is probably the cleverest and most articulate of the founding fathers,” said Dr. Norton Zinder of Rockefeller University, referring to the creators of modern biology. ”He is the most enjoyable company. He overflows with ideas, some of which are occasionally even useful.”
Wikiquote, date indeterminate
Source: SMA, Aug 2007
I have always been interested in twists of words. I think it is something that you can make life amusing with, while also saying quite important things.
Everybody would like to be an innovator, because they believe innovation is what gives them the edge. You need a lot of conditions to be satisfied for innovation. Some are personality driven, in that they depend on individuals. If you notice, the number of major discoveries in Science has remained constant since the 17th century, even though the number of scientists keeps on increasing.
to allow innovation, you cannot have this. You need people to step out and do things that have not been done before. I mean, if you know the answer, why bother to do it at all?
the subject was “The Casino Fund”. The idea was that everybody who gives money for research takes out 1% or 0.5% and puts it into the Casino Fund – and forgets about it. Who manages the Casino Fund? You give it to successful ‘gamblers’ – people like me [laughs] who can hand it out, and people who have a nose for people and projects. And this is with the full expectation that most of the money will ‘disappear’. But when you do this, all the people in the business will say: “Oh no, we can’t have that because how do we ensure payback?” So I said: “Let’s make it 0.1%.”
But even when I tell them to put 0.1% into this “Casino Fund”, they still would not. Even if they think this might lead to the most successful breakthroughs but yet they are not prepared to do it themselves, to put their money where their mouth is!
You can say to these investors – concentrate on the other 99% of the research funds and do not bother with the 1% in the Casino Fund. But then all the academics will say: “We must have peer review.”
Now, peer review is regression to the mean, and the mean is mediocre. If you have peer review alone, it means you are always going to select the conventional, middle of the road activities – you are thus not going to gamble on big ideas and big breakthroughs.
These days when people write a research grant, it has been said that half of their proposed research has already been done, so they somewhat know the answer already when they submit for a research grant application. That is how a lot of people escape the constraints of the grant funding system. But it is very hard on the younger researchers, because they do not have a reserve of data accumulated or capital which they can invest in future results, and so they would stand less chance of being successfully funded. But some of what is going on in this research grantsmanship is absolutely ridiculous.
I think the most important people now who are funding research are the charities, like the Gates Foundation. These organisations also would like to drive innovation, but because they use all the same people in the scientific community, it is more or less going to be conventional. Basically all you have to do is to separate the nutcases from the real research.
at the moment, Singapore goes too much on written records, achievements, and examination results. The big thing about doing Science is motivation. In fact, I think, one really needs to pick the right people to do Science. I feel very strongly, and I have often said so before, that I am very suspicious of people who obtain First Class Honours degrees. They would satisfy me more if they could have gotten a Second Class if they had really tried harder [laughs]!
Because I think motivation to do research is much more important than aiming to get the top grades. Everybody just wants to get top marks these days, and publishing papers in the journals are all about journal impact factors, which is another form of achieving top marks. I think this is nonsense.
When you look for a successful scientist, you go for the truly motivated individual because Science is still a very personal thing.
I think there is now a greater lack of communication between scientists. There are now so many journals and such a large body of scientific literature that we are losing communication between the various scientific fields. People working in one part of their own fields may have no idea what is going on in another’s field. So one of the problems of modern society is actually how to turn data into usable knowledge, because all we have got is plenty of data on everything.
I gave an interview here to the Singapore press and they asked me: “Is there anything else Singapore needs for success?” I said: “Yes, I don’t think the people here are cheeky enough!” And the reporter asked me how we could teach people to be cheeky, which was ridiculous! What I meant by “cheeky” was to question – question authority and question things in a productive way. And you do not get innovation if you are just doing things according to the rules.
I think the American PhD produces, for the average person, an overall much more competent scientist, whereas the British PhD allows people much more freedom to get on with the job of scientific inquiry.
I just think that in Britain it is a different way of doing things and asking questions. People are not so, how shall I say, organised.