**C. The Existence of Equilibrium**
There can be little question that the ultimate “crown jewel” of economic theory consists of the proofs of the existence of economic equilibrium, whether game theoretic or Walrasian general competitive. Central to these has been the use of various fixed-point theorems, with the first to be used being the original one, that due to L.E.J. Brouwer [9] and the second being its close cousin, that due to Kakutani [35]. In both cases the application was first made in game theory and then later to general Walrasian equilibrium. It was von Neumann [46] who initiated this exercise in proving the existence of minimax equilibria for mixed strategies in certain games. He would follow this by using it again to provide the first proof of the existence of a competitive equilibrium [47], although one in the rather particular form of a balanced-growth path. Following in the path of von Neumann, it would be John Nash [44] who would first apply the Kakutani variant [35] to prove the existence of equilibrium for non-cooperative games, which inspired Arrow and Debreu to use it in their proof of the existence of Walrasian general equilibrium [2].
While Velupillai discusses contraction theorems and the Schauder fixed point theorem, most of his focus is on the foundational Brouwer fixed point theorem. I shall focus my intention on it, especially given the contradictions and issues arising from Brouwer’s role in inventing intuitionist mathematics [8], which was in conflict with the methods he used (and those used by others since) to prove his most celebrated theorem.^{7} Velupillai discusses the generally non-constructive nature of two different approaches to proving Brouwer’s fixed point theorem.
One that is used most frequently in mathematical economics (e.g. Scarf [58]) relies fundamentally upon the Bolzano-Weierstrass theorem that essentially states that every bounded sequence contains a convergent sequence.^{8} Ultimately these proofs end up selecting such a convergent sequence that conveniently ends up approaching the fixed point. However, this involves invocation of strong versions of the Axiom of Choice to find this sequence.^{9} To add further to the problems with the Bolzano-Weierstrass theorem from the constructivist perspective, especially its intuitionist variant, the theorem also relies on proof by reductio ad absurdum, that is the law of the excluded middle, the very idea the rejection of which lies at the heart of the intuitionist philosophy {Dummett [25]).
The other main approach to proving the theorem is that of Brouwer himself [9]. As noted by Velupillai, Brouwer went at this in a highly indirect way that ultimately relied upon the logical equivalence between a proposition and its contrapositive and the law of double negation. So, first Brouwer showed that given a map of the disk onto itself with no fixed points there exists a continuous retraction of the disk to its boundary. Then he showed its contrapositive that if there is no continuous retraction of the disk to its boundary then there is no continuous map of the disk to itself without a fixed point. When Brouwer developed intuitionism he fully understood how this proof did not correspond with it.
Brouwer’s intuitionism involved a strong form of constructivism very much in the tradition of Kronecker who had debated with Cantor in the 19^{th} century. While the law of the excluded middle was to be disavowed, the Axiom of Infinity was also denied, although potential infinity was allowed (that there is no upper limit to the natural numbers, even if there is no meaning to their aggregate constituting an infinite set). Many modern constructivists are more willing to allow the Axiom of Infinity, but then draw the line at higher levels or logics based on the existence of higher levels of infinity, the proof of whose existence by Cantor involved reduction ad absurdum. However, in contrast to the computable emphasis of many constructivists, Brouwer’s concerns were ultimately more philosophical and even mystical [11]. His emphasis on possibly allowing for something to be both true and false can lead to the sort of transcendental perspective that is often argued to arise upon contemplating the apparently absurd Zen koans,^{10} (as well as possibly Marxist dialectics) even as it stands aside from the usual conflict between Aristotelianism (which asserts the Law of the Excluded Middle) and Platonism (which is rejected by all the constructivists).
However, some theories thought to be related to intuitionism may simply involve probabilistic statements regarding degrees of truth, which may themselves still be declared to be true or false, as in fuzzy set theory [76]^{11} or Boolean algebra [13]. In any case, while many view intuitionism as philosophically attractive, the formalism of Hilbert would dominate the leading mathematics journals from the 1920s on, leaving Brouwer and his main follower, Heyting [32], somewhat isolated until Kleene [39] and then Dummett [25] would come to their defense.
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