An old mistake and a new version (or: Hilbert, Poincare, and us)

by Orr Shalit

[Update June 28, 2014: This post originally included stories about Poincare and Hilbert making some mistakes. At some point after posting this I realised how unfair it is to talk about somebody else’s mistake (even if it is Hilbert and Poincare) without giving precise references. Instead of deleting the stories, I’ll insert some comments where I think I am unfair. Sorry!]

I was recently forced to reflect on mistakes in mathematics. The reason was that my collaborators and I discovered a mistake in an old paper (16 years old), which forced us to make a significant revision to two of our papers.

A young student of mathematics may consider a paper which contains a mistake to be a complete disaster. (By “mistake” I don’t mean a gap – some step that is not sufficiently well justified (where “sufficiently well” can be a source of great controversy). By “mistake” I mean a false claim). But it turns out that mistakes are inevitable. A paper that contains a mistake is a terrible headache, indeed, but not a disaster.

Arveson once told me: “Everybody makes mistakes. And I mean EVERYBODY”. And he was right. There are two well known stories about Hilbert and Poincare which I’d like to repeat for the reader’s entertainment, and also to make myself feel better before telling you about the mistake my collaborators and I overlooked.

First story: [I think I first read the story about Hilbert in Rota’s “Ten Lesson’s I wish I had been Taught” (lesson 6)]: When a new set of Hilbert’s collected papers was prepared (for his birthday, the story tells), it was discovered that the papers were full of mistakes and could not be published as they were. A young and promising mathematician (Olga Taussky-Todd) worked for three years to correct (almost) all the mistakes. Finally, when the new volume of collected (and corrected) papers was presented to Hilbert, he did not notice any change. What is the moral here? One moral, I suppose, is that even Hilbert made mistakes (hence we are all allowed to). The second is that many mistakes — say, the type of mistakes Hilbert would make — are not fatal: if the mistakes are planted in healthy garden, they can be weeded out and replaced by true alternatives, often-times leaving the important corollaries intact.

[Update June 28: A reference to Rota’s “10 Lessons” is not good enough, and neither is reference to the Wikipedia article on Olga Taussky-Todd, which in turn references Rota’s “Indiscrete Thoughts”, where “10 Lessons” appear.]

Second story: actually two stories, about Poincare. Poincare made two very important mistakes! First mistake: in 1888 Poincare submitted a manuscript to Acta Mathematica – as part of a competition in honour of the King of Norway and Sweden – in which, among other things (for example inventing the field of dynamical systems), he claimed that the solutions of the 3-body problem (restricted to the plane) are stable (meaning roughly that the inhabitants of a solar system with a sun and two planets can rest assured that the planets in their solar system will continue orbiting more or less as they do forever, without collapsing to the sun or diverging to infinity). After winning the competition, and after the paper was published (and probably in part due to the assistant editor of Acta, Edvard Phragmen, asking Poincare for numerous clarifications during the editorial process), Poincare discovered that his manuscript had a serious error in it. Poincare corrected his mistake, inventing Chaos while he was at it.

[Update June 28: This story is well documented. I learned it from Donal Oshea’s book “The Poincare Conjecture: In Search of the Shape of the Universe” , but it is easy to find online references, too].

Second mistake: In 1900 Poincare claimed that if the homology of a compact 3 manifold is trivial, then it is homeomorphic to a sphere. He himself found out his mistake, and provided a counterexample. In order to show that his example is indeed a counter example he had to invent a new topological invariant: the fundamental group. He computed the fundamental group of his example and saw that it is different from the one of the sphere. But this led him to ask: if a closed manifold has a trivial fundamental group, must it be homeomorphic to the 3-sphere? This is known as the Poincare conjecture, of course, and the rest is history.

[Update June 28: Here I should have given a reference of where exactly Poincare claimed that trivial homology implies a space is a sphere. I don’t know it (it probably also appears in Oshea’s book)]

The moral here? I don’t know. But it is nice to add that after making his first mistake, Poincare and Mittag-Leffler (the editor) set a good example by recalling all published editions and replacing them with a new and correct version.

So that’s what I’ll try to imitate now.

Ken Davidson, Michael Hartz and I recently uploaded a new version of our paper “Multipliers of embedded discs” to the arxiv (I announced this paper without giving any details at the end this previous post). In the comments we wrote:

…the earlier version relied on a result of Davidson and Pitts that the fibre of the maximal ideal space of the multiplier algebra over a point in the open ball consists only of point evaluation. This result fails for $d = \infty$ , and has necessitated some changes…

The comment says it all. (On the bottom of page 6 on the new arxiv version we give precise coordinates of the old result we relied on, and a counter example in the $d = \infty$ case). We also had to submit a corrigendum since the paper already appeared online in the journal Complex Analysis and Operator Theory. The discovery that the old result of Davidson and Pitts is only valid for $d<\infty$ also forced us (now “us” means Chris Ramsey, Ken Davidson and myself) to revise another paper of ours, “Operator algebras for analytic varieties” (the paper is still in press, and will be updated on the arxiv after we know exactly how the final journal version looks like). Here I will explain briefly the problem.

Our paper is another in a series that deals with what I call “the isomorphism problem for complete Pick algebras”. (see this previous post for a short overview what this is). Recall that every (irreducible) complete Pick multiplier algebra is isometrically isomorphic to an algebra of the kind $\mathcal{M}\big|_V = \{f\big|_V : f \in \mathcal{M}_d\}$ , where $\mathcal{M}_d$ is the space of multipliers on Drury-Arveson space, and $V \subseteq \mathbb{B}_d$ is a variety (zero set of multipliers) in the unit ball of dimension $d$ (see this survey for background on Drury-Arveson space. BTW – I have to update the survey as well, since it cites the old result). The main problem in this series of papers is to classify the algebras of the form $\mathcal{M}_V$ by the varieties $V$ . Naively, one would like to have the following theorem $\mathcal{M}_V$ is isomorphic to $\mathcal{M}_W$ if and only if $V$ and $W$ are biholomorphic. In reality such a clean theorem working in both directions is false, and even for one direction one needs to add various hypotheses (on the varieties or on the biholomorphism) to make it work.

As is pretty common in the study of commutative algebras, the key to obtaining such a result is understanding the maximal ideal space. Indeed, if $\phi : \mathcal{M}_V \rightarrow \mathcal{M}_W$ is an isomorphism, and if $\rho$ is a multiplicative linear functional on $\mathcal{M}_W$ , then the adjoint map $\phi^*$ maps $\rho$ to a multiplicative linear functional $\phi^* (\rho) = \rho \circ \phi$ on $\mathcal{M}_V$ . So the maximal ideal spaces $\Delta_V$ and $\Delta_W$ of $\mathcal{M}_V$ and $\mathcal{M}_W$ are homeomorphic.

Unfortunately, the maximal ideal spaces of the algebras $\mathcal{M}_V$ are quite pathological and hard to understand. But the picture is simplified by projecting the maximal ideal space into the closed unit ball $\overline{\mathbb{B}_d}$ : if $\rho$ is a multiplicative linear functional on $\mathcal{M}_V$ , then we denote by $\pi(\rho)$ the point $(\rho(Z_1), \rho(Z_2), \ldots)$ (here $Z_1, Z_2, \ldots$ denote the coordinate functions). It is well known that this map is not one-to-one, so $\Delta_V$ can not be identified with a subset of the ball.

There is a tractable part of these maximal ideal spaces, which can be identified with $V$ as follows. If $v \in V$ , then we have a well defined multiplicative linear functional $\rho_v : f \mapsto f(v)$ . These evaluation functionals are precisely the wot continuous functional on $\mathcal{M}_V$ . In some special classes of varieties (for example when $V$ and $W$ are biholomorphic images of a disc) we were able to show that if an isomorphism $\phi : \mathcal{M}_V \rightarrow W$ sends point evaluations to point evaluations, then it is induced by composition with a bihlomorphism between the varieties (so in particular the varieties are biholomorphic).

Previously we believed in that the following two assertions are true:

For every $v \in V$ , $\rho_v$ is the unique point in $\pi^{-1}(v)$ .
For every $x \notin V$ , the fibre $\pi^{-1}(x)$ is empty.

Both these “facts” would follow from the old result of Davidson and Pitts, both are true when $d < \infty$ , but both fail when $d = \infty$ . We previously relied on these facts to show that isomorphisms take point evaluations to point evaluations, and the discovery that they are false forced us to work much harder to save our main results. Further results can be found in the paper.

7 Comments to “An old mistake and a new version (or: Hilbert, Poincare, and us)”

Eliahu Levy says:

June 24, 2014 at 6:28 pm

I recall from memory another mistake of Poincare, this time with some “philosophical” significance. Poincare claimed that there cannot be a model for Peano’s axioms for the natural numbers without the Induction Axiom which does not automatically satisfy this axiom too. (Such a model are two copies of the natural numbers, one preceding the other – i.e. the ordinal $\omega+\omega$, otherwise put, the set of natural numbers with the successor function defined as adding 2, not 1.) I do not recall now the subsequent history regarding Poincare and this mistake. (Poincare was an ardent supporter of the supremacy of intuition over axioms and formalism in mathematics.)

- Eliahu Levy says:
  
  June 24, 2014 at 9:07 pm
  
  Me again. There is a better counterexample to Poincare’s claim: the natural numbers plus a cycle (or several cycles). Here only 0
  does not have a predecessor, of course different members have
  different successors, and the Induction Axiom does not hold.
  
  - Orr Shalit says:
    
    June 28, 2014 at 6:37 pm
    
    Hello Eliahu,
    Thanks for the comments!
    I was going to ask you for a reference, since it is not clear to me if Poincare really claimed what you say he claimed. But then I realised that I didn’t give any reference to the mistakes I said Poincare made. Big mistake! I’ll put some correction up.
Eliahu Levy says:

July 4, 2014 at 7:39 pm

Poincare’s claim comes from his celebrated book for laypersons: La Science et l’hypothese (see Wikipedia for links where one has it free online).

There, In the last paragraphs of the chapter “Non-euclidean Geometries” he discusses the nature of axioms, and he distinguishes between axioms in geometry and in arithmetics. While in geometry, where there are several geometries – euclidean ans non-euclidean, axioms are just conventions – if you wish, disguised definitions of the system, in arithmetics, and especially the Induction Axiom, they are, a-la Kant, synthetic a priori, meaning that one cannot conceive them being false:

“Pour s’en convaincre, qu’on prenne un véritable jugement synthétique a priori, par exemple celui-ci, dont nous avons vu au chapitre premier le rôle prépondérant :

Si un théorème est vrai pour le nombre 1, si on a démontré qu’il est vrai de n+1, pourvu qu’il le soit de n, il sera vrai de tous les nombres entiers positifs.

Qu’on essaie ensuite de s’y soustraire et de fonder, en niant cette proposition, une fausse arithmétique analogue à la géométrie non euclidienne, — on n’y pourra pas parvenir.”

And he emphasizes, that had the axioms of geometry also been synthetic a priori, there would be no non-euclidean geometry.

- Orr Shalit says:
  
  July 10, 2014 at 11:01 am
  
  Thank you Eliahu!
  
The isomorphism problem: update | Noncommutative Analysis says:

July 10, 2014 at 10:56 am

[…] second paper that was affected by a discovery of a mistake in the literature, which I told about in the previous post. Luckily, we were able to save all the results in that paper, but had to work a a little harder […]

Eliahu Levy says:

August 3, 2014 at 3:27 pm

Further about Poincare’s claim about (not im)possible models where the Induction Axiom does not hold.

Benefiting from the insights that mathematical logic gained in the century that passed, we can easily construct systems with the holding of every first-order statement about the natural numbers (the statement must speak only about natural numbers, not (infinite) sets of them), but which are not isomorphic to the natural numbers – just take an ultrapower of $\mathbb{N}$, say the set of sequences of natural numbers modulo equivalence with respect to a certain ultrafilter.

In this ultrapower, the successor function satisfies, of course, the axioms of Peano except the Induction Axiom (these are first-order statements about numbers). It satisfies Induction if we interpret it as: “For every first-order property, if it holds for $0$ and holds for $n+1$ provided it holds for $n$, then it holds for all $n$”, but not as “For any subset of $\mathbb{n}$…” Indeed, the ultrapower contains as a proper subset a system isomorphic to $\mathbb{n}$ (the set of constant sequences modulo the ultrafilter).

Of course, one cannot blame Poincare for being “unaware” of these achievements of the 20th century.

Noncommutative Analysis