Dr. Dennis Sullivan is fond of saying that the most important equation in mathematics is

\[\hspace{50pt} \partial \partial = 0, \hspace{45pt} (1)\]

where \(\partial\) is the boundary operator in the chain complex of chain groups on a topological space.

Indeed, it is precisely due to (1) that homology exists. One can interpret homology as algebraically probing more information from (1): which quantities are \(\partial\)-ed to zero despite not arising as \(\partial\)s of something else. This leads to the geometric interpretation of cycles in homology as being equivalence classes of closed manifolds.

But there is more to (1)—it bears resemblance to an idea that dwells right at the heart of mathematics. The man whose name is inextricably linked to this story is L.E.J. Brouwer.

 

Brouwer and Intuitionism

The story of Brouwer is a curious one. In his twenties, he proved a number of famous theorems in topology, including the invariance of domain and the Brouwer fixed point theorem, which is one of the great-grandparents of the field of economics.

Later in life, however, Brouwer grew to reject his old work due to a fundamental philosophical disagreement with the rules of logic. This is partially about the law of the excluded middle, which dates back in written accounts to Aristotle.

Proof by contradiction is commonplace in mathematics: if you want to prove \(P\), assume \(P\) does not happen, and reach a contradiction. Due to this contradiction, \(P\) not happening does not happen. And now, to Brouwer’s distaste, one takes the liberty of concluding that \(P\) must happen.

Mathematically, one can write vaguely the related idea

\[\hspace{50pt} NN = 0, \hspace{40pt} (2)\]

where \(N\) is the negation operator on a statement. Negating twice is the trivial action on a statement, leaving its truth value unchanged. Note the resemblance to (1).

This is one of those things that seems somewhat absurd to question. But it is interesting to note that there are mathematical contexts where one does not conform to the spirit of (1) and (2).

 

The Devil in the Details

In Sheaves in Geometry and Logic, MacLane and Moerdijk motivate the definitions that lead to the algebraic structure of sheaves with a powerful and simple example from basic topology. Topology, which Brouwer ironically abandoned.

Thus, as observed first by Stone and Tarski, the algebra of open sets is not Boolean, but instead follows the rules of the intuitionistic propositional calculus.

The axioms for topology are set up in the spirit of Brouwer. It is a fact that the negation operator for sets, the complement, is not faithful in the world of open sets. The complement of an open set is always closed, and need not be open. Thus, if one restricts to the world of open sets, the natural negation operator is to take the interior of the complement. Let us label this operator \(J\).

Interestingly enough, doing this twice is not the trivial action.

Consider \(U\), which is the complement of the Cantor Set on \([0,1]\). \(U\) is open. Its negation within the world of open sets is the interior of the Cantor Set, which is empty. The interior of the complement of the empty set is \([0,1]\), which is not what we started with—the complement of the Cantor Set.

That is,

\[\hspace{50pt} JJ \neq 0, \hspace{40pt} (3)\]

Doing \(J\) twice is not the trivial action, to the delight of Brouwer. So the devilish (3), which is practically an axiom for topology, is capable of creating a rich and deep mathematical world, just like its angelic counterparts (1) and (2).

 

Heyting Algebras

The formalism of this intuitionistic logic is found in the Heyting algebras. Let us start by insisting that we want to live in a world with modus ponens. This is the rule that if \(P\) implies \(Q\), and \(P\) is true, then \(Q\) is true.

Within this world, Heyting algebras provide the formalism for the weakest logical systems. This is contrast to Boolean algebras, which are the ones that describe the traditional laws of logic with excluded middle and elimination of double negations.

 

Duality

This motif’s universality is evident when one notices that it recurs in one of math’s most important themes: duality.

It is well-known that the dual of the dual of a finite dimensional vector space is canonically isomorphic to the original vector space. That is, (1) and (2) occur. However, with infinite-dimensional vector spaces, this is no longer always the case. (3) occurs.

In the classical Galois theory, the same pattern emerges, with the first train departing under stronger conditions (the extension being Galois), and the second under weaker conditions.

When does doing something twice lead to triviality, and what does it imply? When is the enemy of your enemy your friend?

 

Well, whether \(2 = 0\), as in (2), or \(2 \neq 0\), as in (3), there is plenty to discover. And there is no doubt that in either case, I will continue to abuse notation.

But now the thematic question—what if neither holds? That puts us in Godel, Escher, Bach territory. I don’t go to such dangerous places, I had enough of that growing up on the mean streets of Dasarahalli in suburban Bangalore.