Diagrams and Proofs in Analysis

Abstract

This article discusses the role of diagrams in mathematical reasoning in the light of a case study in analysis. In the example presented certain combinatorial expressions were first found by using diagrams. In the published proofs the pictures were replaced by reasoning about permutation groups. This article argues that, even though the diagrams are not present in the published papers, they still play a role in the formulation of the proofs. It is shown that they play a role in concept formation as well as representations of proofs. In addition we note that ‘visualization’ is used in two different ways. In the first sense ‘visualization’ denotes our inner mental pictures, which enable us to see that a certain fact holds, whereas in the other sense ‘visualization’ denotes a diagram or representation of something.

Acknowledgements

Versions of this paper were presented at the 13th International Congress of Logic, Methodology and Philosophy of Science in Beijing, summer 2007, and at the Conference Foundations of Formal Sciences VII in Brussels, October 2008. I wish to thank the two anonymous referees of this journal for their helpful comments.

Notes

[1] The use of diagrams is also referred to in the debate between realists and anti‐realists. Brown (1999) claims that diagrams provide our access to ‘Plato’s heaven’, whereas Sherry (2009) argues that a realist view is problematic in some uses of diagrams.

[2] The idea is to define notions from classical probability theory into corresponding notions in operator algebra. For example, suppose there is given an algebra and a state on this algebra, ( ,φ). One then defines the notions of, for example, random variables (the elements of ), their distribution (the value of φ(a^p ), p ∊ ℕ) and independence. The concept of independence is translated into a notion of freeness, hence the name ‘free probability theory’.

[3] Haagerup and Thorbj⊘rnsen (1999) develop certain techniques that were later used to prove a 1978 conjecture stating that Ext(C∗(F ₂)) is not a group. In 1973 notions from algebraic topology were introduced to the field of C∗‐algebras, which led to the question whether Ext(A) would always be a group. In 1978 Anderson proved that it was possible to find a (rather complicated) C∗‐algebra A involving a projection where Ext(A) is not a group. Haagerup’s and Thorbj⊘rnsen’s result is that this projection is not needed.

[4] More precisely, a Gaussian Random Matrix GRM(m, n, σ ²) is an m × n‐matrix, whose entries b(i, j) are complex random variables. Re(b(i, j)) and Im(b(i, j)) have distribution N(0, σ ²/2) (in this article it is N(0, 1)). Furthermore the real and imaginary parts of the entries form a family of 2mn independent random variables.

[5] The procedure in Haagerup and Thorbj⊘rnsen (1999) is first to perform the actual calculation of taking the trace of the product of the GRM. This results in a sum of products of (complex valued) random variables. From this expression, it is then possible to read off certain conditions for the expectation of each term in the sum being different from zero. Working on these conditions and introducing the equivalence relation leads to the final result.