What are some examples of narrowly missed discoveries in the history of mathematics?

What are the examples of some mathematicians coming very close to a very promising theory or a correct proof of a big conjecture but not making or missing the last step?

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Shizhuo- The original title of this post was so unhelpful, I went in and changed it myself (if you don't like what I chose, of course, you're free to change it to something else). When choosing a title, imagine yourself in the position of someone reading on the first page. If you read "Can you tell the similar phenomenon in the history of mathematics?" would you have any clue as to what the question was about other than something about the history of mathematics? –  Ben Webster Mar 25 '10 at 17:27
Thank you for changing the title! –  Shizhuo Zhang Mar 25 '10 at 18:39
With Ben's title change, I at least now understand what the question intends to ask. But the actual question in the text I still don't understand. Maybe since this is CW I should just change it, but I can't think of a way to make it a significantly better question. –  Theo Johnson-Freyd Mar 25 '10 at 19:24
I've removed the "motivational" part of the question, since it doesn't seem particularly helpful or correspond with the answers already given. –  Victor Protsak Jun 20 '10 at 16:48

Freeman Dyson discusses a few examples of this in his article Missed Opportunities. One that I thought was particularly striking was that mathematicians could have discovered special relativity decades before Einstein just by staring at Maxwell's equations hard enough, and also on the basis that the representation theory of the Poincare group is simpler than the representation theory of the Galilean group.

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I always thought that Poincare did discover special relativity years before Einstein. His discovery just wasn't noticed our understood well enough. –  Johannes Hahn Mar 25 '10 at 20:07
My understanding is that Lorentz did not understand the point of the formal time parameter he introduced. He regarded it as a computational convenience, whereas Einstein's contribution was to think of it as "real" time. –  Qiaochu Yuan Mar 25 '10 at 23:52
Yes, but the point is that Einstein basically had no mathematical work left to do, but he did not cite any of the results that he basically lifted from Lorentz and Poincaré. –  Harry Gindi Mar 26 '10 at 0:33
Did he lift those results, or rediscover them independently? They're not obvious but not that deep either. –  Mark Meckes Mar 26 '10 at 13:46
And let's not forget Minkowski in connection with special relativity. As for representation theory of the Poincare and Galilean group, that's wishful thinking in the extreme: the conceptual development of representation theory to the point that we can compare them took a lot of highly nontrivial effort, even if we tend to forget it nowadays. –  Victor Protsak Jun 20 '10 at 16:09

Emil L. Post was very close to proving Gödel's incompleteness theorem, and the existence of algorithmically unsolvable problems in the early 1920s. He realized that one could enumerate all algorithms, and hence obtain an unsolvable problem by diagonalization. Moreover, the "problem" can be viewed as a computable list of questions $Q_1,Q_2,Q_3,\ldots$ for which the sequence of answers (yes or no) is not computable. It follows that there cannot be any complete formal system that proves all true sentences of the form "The answer to $Q_i$ is yes" or "The answer to $Q_i$ is no," because this would solve the unsolvable problem.

But then Post was stuck because he needed to formalize the notion of computation. He had in fact (an equivalent of) the right definition, but logicians were not ready for a definition of computation, and did not believe there was such a thing until the Turing machine concept came along in 1936. Gödel avoided this problem when he proved his theorem (1930) by proving incompleteness of a particular system (Principia Mathematica).

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Church's $\lambda$-calculus is essentially simultaneous to Turing's machines, though, IIRC? –  Mariano Suárez-Alvarez Mar 25 '10 at 4:33
Yes, it is, and of course it is "Church's thesis" that lambda-calculus captures the concept of computability. However, logicians weren't convinced that the concept of computability was formalizable until the Turing machine arrived and Gödel threw his weight behind it. –  John Stillwell Mar 25 '10 at 4:50
I've had a conversation with John Conway in which he mentioned that von Neumann was very frustrated that he missed Gödel's incompleteness theorem. –  Victor Protsak Jun 21 '10 at 3:03
Von Neumann certainly would have been frustrated to have missed Gödel's second incompleteness theorem -- the one about the unprovability of consistency. He pointed it out to Gödel in a letter before Gödel had published it, but Gödel still got full credit for the theorem. –  John Stillwell Jun 21 '10 at 3:55

Archimedes and the Integral calculus?

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How can you almost discover calculus without knowing what a function is? –  Franz Lemmermeyer Jun 20 '10 at 6:45
@Franz Yeah,RIGHT?My question exactly! Archimedes almost discovering integration is like saying Newton almost discovered general relativity except for some erronous base assumptions and the absence of multilinear algebra. –  Andrew L Jun 20 '10 at 7:13
To Franz Lemmermeyer: you know, I'm not really sure Newton and Leibniz knew what functions are either. –  Zsbán Ambrus Jun 20 '10 at 11:57
Here is what David Mumford says about a proposition of Archimedes (cf. EMS Newsletter, Dec 08): "...he [Archimedes] is evaluating a Riemann sum of $\int_0^\theta\sin(\phi)d\phi$, /.../ No historian will convince me that his idea is not that same of mine when looking at this mathematical proposition." It seems that Archimedes did indeed discover some of the basic ideas of calculus, expressed in a geometric language. –  A Stasinski Jun 20 '10 at 14:19
I'm not a historian, but I'm pretty sure that both derivatives and integrals were studied in some form before Newton and Leibniz. Their accomplishments were 1. discovering the fundamental theorem of calculus, and 2. giving a systematic treatment of derivatives and integrals. –  Andy Putman Jun 20 '10 at 18:15

In the book The Scientists by John Gribbin, he mentions that, in his search for the theory of general relativity, Einstein apparently wrote down a correct equation that would have led him to correctly discovering the rest of the equations for general relativity very quickly. But, he did not see the equation for what it was and ran down the wrong path for two entire years before coming back to the correct equation. Here's the quote from the book:

"Einstein himself is often presented as the prime example of someone who did great things alone, without the need for a community. This myth was fostered, perhaps even deliberately, by those who have conspired to shape our memory of him. Many of us were told a story of a man who invented general relativity out of his own head, as an act of pure individual creation, serene in his contemplation of the absolute as the First World War raged around him.

It is a wonderful story, and it has inspired generations of us to wander with unkempt hair and no socks around shrines like Princeton and Cambridge, imagining that if we focus our thoughts on the right question we could be the next great scientific icon. But this is far from what happened. Recently my partner and I were lucky enough to be shown pages from the actual notebook in which Einstein invented general relativity, while it was being prepared for publication by a group of historians working in Berlin. As working physicists it was clear to us right away what was happening: the man was confused and lost - very lost. But he was also a very good physicist (though not, of course, in the sense of the mythical saint who could perceive truth directly). In that notebook we could see a very good physicist exercising the same skills and strategies, the mastery of which made Richard Feynman such a great physicist. Einstein knew what to do when he was lost: open his notebook and attempt some calculation that might shed some light on the problem.

So we turned the pages with anticipation. But still he gets nowhere. What does a good physicist do then? He talks with his friends. All of a sudden a name is scrawled on the page: 'Grossman!!!' It seems that his friend has told Einstein about something called the curvature tensor. This is the mathematical structure that Einstein had been seeking, and is now understood to be the key to relativity theory.

Actually I was rather pleased to see that Einstein had not been able to invent the curvature tensor on his own. Some of the books from which I had learned relativity had seemed to imply that any competent student should be able to derive the curvature tensor given the principles Einstein was working with. At the time I had had my doubts, and it was reassuring to see that the only person who had ever actually faced the problem without being able to look up the answer had not been able to solve it. Einstein had to ask a friend who knew the right mathematics.

The textbooks go on to say that once one understand the curvature tensor, one is very close to Einstein's theory of gravity. The questions Einstein is asking should lead him to invent the theory in half a page. There are only two steps to take, and one can see from this notebook that Einstein has all the ingredients. But could he do it? Apparently not. He starts out promisingly, then he makes a mistake. To explain why his mistake is not a mistake he invents a very clever argument. With falling hearts, we, reading the notebook, recognize his argument as one that was held up to us as an example of how not to think about the problem. As good students of the subject we know that the agument being used by Einstein is not only wrong but absurd, but no one told us it was Einstein himself who invented it. By the end of the notebook he has convinced himself of the truth of a theory that we, with more experience of this kind of stuff than he or anyone could have had at the time, can see is not even mathematically consistent. Still, he convinced himself and several others of its promise, and for the next two years they pursued this wrong theory. Actually the right equation was written down, almost accidentally, on one page of the notebook we looked at it. But Einstein failed to recognize it for what it was, and only after following a false trail for two years did he find his way back to it. When he did, it was questions his good friends asked him that finally made him see where he had gone wrong."

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This is very interesting but not really a near miss. You explain that general relativity is in fact a near near-miss. –  Roland Bacher Jun 21 '10 at 16:09

In their attempts of justifying Euclid's fifth postulate, Girolamo Saccheri and Johann Heinrich Lambert proved many theorems of non-Euclidean geometry. However, they were so convinced that the fifth postulate must be true that they stopped short of actually discovering the new geometry.

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Schubert came extremely close to discovering the JSJ-decomposition of 3-manifolds in his paper "Knoten und Vollringe" (1953). With a little more work, one could turn Schubert's paper into something equivalent to the JSJ-decomposition applied to knot and link complements in $S^3$. That would have allowed people to conjecture the JSJ-decomposition for 3-manifolds, around 20 years earlier than it was.

It's interesting to speculate whether or not the connection between 3-manifold theory and hyperbolic geometry would have been made much earlier, as some of the ingredients were already in place -- Seifert-Weber space, and the Gieseking manifold, but I do not think people knew finite-volume hyperbolic manifolds to be atoroidal until much later.

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A decade after Schubert, Waldhausen also came close to the JSJ decomposition, in his work on graph manifolds for example. It almost seems as if Seifert back in the 1930s could have discovered the JSJ decomposition. –  Allen Hatcher Jun 20 '10 at 17:33

Leibniz was extremely close to 'discovering' modal logic. He definitely understood the difference between intension and extension, and knew about valuations as functions from possible worlds. And I don't mean that one can "extract" this from his writings by reading hard between the lines -- it is quite explicit, and in many manuscripts. He did not, of course, have the formal tools in hand needed to formalize these ideas, but then again no one did until the work of Frege after 1880 and Russell after 1900. That's a 200 year gap!

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And determinants, too! (Found in his manuscripts and published in the 19th century.) –  Victor Protsak Jun 20 '10 at 16:14

Fulkerson came extremely close to a proof of the Perfect Graph Conjecture eventually proved by Lovasz. He was stuck on one relatively easy lemma, which he had apparently become convinced was false.

I don't know a reliable source for this, but I have heard that after learning that the conjecture was proven, Fulkerson went to his office and finished his own proof later that day.

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One of the important conjectures in set theory in the 60s was that strongly inaccessible cardinals are measurable. This was disproved by Tarski in 1960. Soon after this Erdős and Hajnal realized that they were very close to Tarski's result in their paper "On the structure of set mappings" which appeared in 1958.

Another miss was in the paper of Erdős, Hajnal, and Milner ("On sets of almost disjoint subsets of a set") from 1968 where they went pretty close to discovering Silver's famous theorem on GCH at singular cardinals of uncountable cofinality.

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In his final scientific work, the Two New Sciences, Galileo almost discovered Cantor's theory of infinite cardinalities ...

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I fundamentally disagree with this statement. It's like claiming that Jules Verne almost put a man on the moon. –  Franz Lemmermeyer Mar 25 '10 at 19:36
Perhaps you would prefer the wording that I use when I teach a first course in set theory? "In his final scientific work, the Two New Sciences, Galileo blew his chance of becoming a famous set theorist." –  Simon Thomas Mar 26 '10 at 3:02

A famous example is that John Conway, who fathered the concept of a Skein relation, didn't discover the Jones (or HOMFLYPT) polynomials. By just searching for knot invariants defined via Skein relations, he would doubtlessly have found them, and the history of low-dimensional topology would have looked quite different.

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As mentioned in Categories for the Working Mathematician, Bourbaki nearly had the right definition of an adjunction 10-15 years before the proper definition was formulated by Dan Kan. In fact, Bourbaki actually proved the Special Adjoint functor theorem (cf. Categories for the Working Mathematician - notes at the end of the chapter on Adjunctions)

Edit: To explain a little more (from Mac Lane), Bourbaki had figured out most of these notions, but since they had not formulated the general adjoint functor theorem, only the special adjoint functor theorem, they restricted their definition of a "universal construction" to those adjunctions for which the SAFT holds. I find it pretty impressive that they were able to do all of this without the benefit of a formal framework for category theory.

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Bourbaki is not an actual person. Who exactly had the right definition before the proper definition was formulated? –  Rune Mar 25 '10 at 18:39
A useful piece of information with regards to contacting him is that Serre is not dead. –  Mariano Suárez-Alvarez Mar 25 '10 at 19:51
@fpqc, just a little correction: Looking at the notes at the end of the section on limits in Mac Lane, I see that Bourbaki formulated some sort of solution set for the construction of universal arrows (books.google.co.il/…). At least in Mac Lane (p. 130), SAFT does not include a solution set condition (as opposed to the AFT). Also, I think that there is no reference to the AFT/SAFT in the notes at the end of the section on adjunctions (pp. 107--108). –  user2734 Mar 25 '10 at 20:11
Rune- The Bourbakistes. –  Ben Webster Mar 25 '10 at 21:30
@fpqc: Thank you for your suggestion. It seems that such editing requires deleting half of the post, and I feel uncomfortable to make such a massive change. I also have some ideas of what should be added: The fact that Bourbaki was looking for something too general, and the final verdict given by Mac Lane: "good general theory does not search for the maximum generality, but for the right generality." I think that it would not be appropriate for someone other than the OP to make such a large change in the answer. [Just to put things in context, I really like your answer and I have +1'ed it] –  user2734 Mar 25 '10 at 22:48

Issac Barrow very nearly had the fundamental theorum of calculus a generation before Newton and Liebniz,but he failed to see the relationship between the quadrature problem and the fluxion problem,despite making huge contributions to the development of the theory of limits in both.

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When I took a course in set theory, I was told that in the early 1920's, Thoralf Skolem essentially proved what is now Gödel's Completeness Theorem, but was unaware of its significance because the concept of completeness was not understood fully at the time.

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I might be wrong but I thought that what he proved was the compactness theorem. –  Tran Chieu Minh Mar 26 '10 at 10:03
I think you are right. They are equivalent, though. –  Ketil Tveiten Mar 26 '10 at 16:42
Ketil Tveiten: I don't think they're equivalent. Compactness theorem can be stated and proved in a simple way without defining a formal proof system at all. In that case, the theorem claims that if any finite subset of a set of formulas has a model satisfying it, then there's a model satisfying the whole set. –  Zsbán Ambrus Jun 20 '10 at 12:02
They are deducible from eachother. That usually means they are equivalent, no? –  Ketil Tveiten Jun 21 '10 at 11:15
See the discussion in mathoverflow.net/questions/9309/… –  Dan Petersen Jun 21 '10 at 14:35