Can Amateurs Solve P=NP?
What is an amateur mathematician, what have they done, and what might they do?
Srinivasa Ramanujan was one of the most remarkable mathematicians of the last century. He discovered countless beautiful theorems in both analysis and number theory. With almost no formal training, he was able to discover results that other mathematicians found amazing, even magical. Sadly he died at the age of 32; one wonders what further great things he could have discovered if he had lived longer.
Today I want to talk about the role that formal training plays in solving open problems in mathematics. When people discuss open problems, sometimes the P=NP question is raised as one that might be solved by an amateur. Is this realistic or not?
I have a read Keith Devlin’s book “The Millennium Problems: The Seven Greatest Unsolved Mathematical Puzzles of Our Time”—of course the count is now six thanks to the brilliant of Grigori Perelman. I recommend it to you, with the proviso that I could only follow the chapters on the Riemann Hypothesis and the P=NP question. The others on the Hodge Conjecture, the Yang—Mills existence and mass gap, the Navier—Stokes existence and smoothness, and the Birch and Swinnerton—Dyer conjecture, were very hard for me to follow. I do not think this is Delvin’s fault—he is an outstanding science writer—I think it is an inherent feature. Problems such as the Hodge Conjecture require, even to just understand their statement, quite a bit of technical background.
In his third chapter, on the P=NP question, Devlin states that perhaps this problem is one that could be solved by an amateur. I have heard this claim before, and I have mixed feelings about it. Of course we cannot predict who will solve any open problem, and that includes the Millennium problems as well as all other open problems. My mixed feeling comes from saying P=NP could be solved by an amateur seems to mean it is “easier” than the other problems. I do not think this is the case, but ranking the difficulty of open problems is probably impossible.
Let’s take a look at what amateur mathematicians have done in the past, and what they might be able to do in the future on these and other open problems.
Who Is An Amateur?
I think that one of key issues is: who is an amateur? Is an amateur someone who is a completely untrained in mathematics, or is an amateur someone who is not an expert in a particular area of mathematics? According to the dictionary, amateur is from the old French and means “lover of.” Thus amateurs often work on problems without any formal reward or pay. Curiously Perelman was essentially working for “free” when he solved the Poincaré Conjecture, but I think no one would consider him an amateur.
Was Ramanujan an amateur? He was basically self taught, yet capable of discovering formulas that amazed even Godfrey Hardy. For example,
was one identity that greatly impressed Hardy.
Ramanujan shared one trait with many amateurs: his “proofs” were often nonstandard and difficult to follow. This is perhaps one of the hardest issues for an amateur mathematician to overcome—the formally trained world of mathematics has certain rules and styles of how to present mathematics. An amateur may use nonstandard notation, may create new definitions for existing concepts, and may re-prove basic facts. For example, Ramanujan’s first published paper was on some new properties of Bernoulli numbers. The journal editor, Narayana Iyengar, noted:
Mr. Ramanujan’s methods were so terse and novel and his presentation so lacking in clearness and precision, that the ordinary [mathematical reader], unaccustomed to such intellectual gymnastics, could hardly follow him.
Some Results of Amateurs
Here is a partial list of some of the results obtained by amateur mathematicians. Some had no formal training, while others had training but worked on mathematics only as a “hobby.”
Thomas Bayes: He was a minister by training, yet he introduced in the 1764 the now famous formula that is named after him:
This formula is the basis of Bayesian analysis. Most of the credit—although not the name—of the consequences of this formula go to others. Quoting Wikipedia:
Bayes was a minor figure in the history of science, who had little or no impact on the early development of statistics; it was the French mathematician Pierre-Simon Laplace who pioneered and popularized what is now called Bayesian probability.
This comment seems a bit harsh to me. I agree more with Bill Bryson’s opinions about its foundational nature expressed here and in the new book Seeing Further of which he is the editor.
Alfred Kempe: In 1879, while he was a barrister, Kempe discovered a “proof” of the Four Color theorem for planar graphs. This stood until 1890 when Percy Heawood discovered Kempe’s proof was flawed. Heawood did use Kempe’s ideas to give a correct proof of the Five Color theorem. In particular, he used Kempe’s notion of chains of alternating colors. Such chains—-now called Kempe chains—play a key role in both current proofs of the Four Color theorem.
Oliver Heaviside: He invented in 1880’s the notion of operator calculus—a theory that can be used to solve linear differential equations. His methods work and give the correct answers. Correct answers or not, operator calculus was not well received by mathematicians. Yes it worked, but there were issues with some of the liberties he took that upset professional mathematicians. Heaviside could solve linear differential equations, yet the “theory” was not sound. Much later, Thomas Bromwich found a correct way to justify the operator calculus of Heaviside.
William Shanks: In 1873 there were no computers, yet he was able to compute to many more places than anyone had previously. He claimed it was correct to places—it was correct to places. Still an incredible feat for his time—he did this work as a hobby when he was not running his boarding house.
Marjorie Rice: She found new ways to tile the plane in pentagons—she did this as a hobby. Doris Schattschneider, a professional mathematician, helped make Rice’s results known to the math community.
Kurt Heegner: He was a radio engineer and published in 1952 a claimed solution of one of the great, then—open problems in algebraic number theory. As with many amateurs his proof was not accepted, due to mistakes in his paper. In 1969 the eminent number theorist Harold Stark solved the problem. Apparently, Stark went back to look once more at Heegner’s paper, and he saw that it was essentially correct. The “errors” were minor and could easily be fixed.
In order for anyone, amateur or not, to be able to solve an open problem they must understand the problem statement. This is almost silly to state, but it is important. It is hard to hit a fuzzy target. One of reasons some think that P=NP is more approachable than many other open problems is that an amateur is more likely to be able to state P=NP correctly, than many other conjectures.
This is the reason so many amateurs have worked on the Four Color Theorem, the P=NP question, Fermat’s Last Theorem, and Graph Isomorphism: they have relatively simple statements. A simple statement does not mean that a problem is easy, but it does mean that anyone can start thinking about it.
An interesting question is which problems have simple statements?
P=NP? This does have a relatively simple statement. See my discussion for my take on the problem statement.
Riemann Hypothesis Jeff Lagarias has a surprisingly “simple” problem that is equivalent to the Riemann Hypothesis. Let . Then, the following is equivalent to the Riemann Hypothesis. For all ,
Hodge Conjecture I do not know how to even state this and the other Millennium problems in a simple way. No doubt this makes them unattractive to non-specialists.
Can Amateurs Help?
Can amateurs help make progress on modern problems? I think that it is possible. Their lack of fear, the ability of thinking out-of-the-box, the ability to think against the conventional wisdom, all make it possible for them to contribute to science. One of the reasons the pros usually get the credit is that amateurs often do not write their work up properly. They often write papers that have errors, and once we see an error we stop reading and skip the rest—Heegner is a prefect example.
It may be useful to look at the type of contributions amateurs have made in the past.
Definitions: Bayes, Kempe, and partially Heaviside’s contribution was in defining new concepts.
Computations: Shanks contribution was certainly a computation.
Examples: Rice’s contribution was in the discovery of new examples.
Proofs: Kempe, Heaviside, and Heegner’s contributions were in proofs.
This suggests there are several ways amateurs can help advance our understanding.
What do you think? Is P=NP the problem most or least likely to be solved by an amateur? Can amateurs still make contributions to mathematics and complexity theory?
A related question: are there relatively simple statements of all the Millennium problems? Such a statement would not necessarily advance their solution, but I would find it interesting if they could be made more accessible. Even experts might be helped by short equivalent statements.