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Accessible proof

Courant and Robbins provide an accessible proof. —Preceding unsigned comment added by 198.144.199.xxx (talk • contribs) 30 August 2001

First proved by Bol?

According to Lyusternik Convex Figures and Polyhedra, the theorem was first proved by a Lettish mathematician named Bol. No references are provided. Anyone know what this is about?--192.35.35.36 00:08, 18 Feb 2005 (UTC)

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"Constructive proof"?

The article sais "The first algorithm to construct a fixed point was proposed by H. Scarf." and also "Kellogg, Li, and Yorke turned Hirsch's proof into a constructive proof by observing that..."
I'm wondering if it's indeed a constructive proof, since Brower's theorem for one dimension is equivalent to intermediate value theorem, which does not admit a constructive proof.
See for example this discussion in MathOverflow. Nachi (talk) 16:58, 4 February 2018 (UTC)[reply]

This is a symptom of different people using "constructive" to mean different things. The paper by Kellog, Li, and York really is titled "A Constructive Proof of the Brouwer Fixed-Point Theorem and Computational Results". But they are working in numerical analysis, not in constructive mathematics. So perhaps all that they mean by 'constructive proof' is that their proof can be used to obtain a numerical algorithm to approximate a fixed point. I am not completely sure what they mean by constructive, though, as I look at their paper. They also assume that the map is not only continuous, but twice differentiable. In the sense of many branches of constructive mathematics, it is known that the fixed point theorem implies nonconstructive principles such as LLPO, and so the fixed point theorem is not constructive in the sense of those branches. — Carl (CBM · talk) 17:25, 4 February 2018 (UTC)[reply]
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