Cirrus Haven

In the desert
I saw a creature, naked, bestial,
Who, squatting upon the ground,
Held his heart in his hands,
And ate of it.
I said, “Is it good, friend?”
“It is bitter—bitter,” he answered;

“But I like it
“Because it is bitter,
“And because it is my heart.”

Stephen Crane, In the Desert

Academic Publications

Anti-Specker properties in constructive reverse mathematics

Ph.D. thesis. University of Canterbury, 2013.

Constructive reverse mathematics is a programme in which non- and semi-constructive principles are classified in accordance with which other principles they imply or are implied by, relative to the framework of Bishop-style constructive mathematics. One such principle that has come under focus in recent years is an antithesis of Specker's theorem (that theorem being a characteristic result of Russian recursive mathematics): this so-called anti-Specker property is intuitionistically valid, and of considerable utility in proving results of real and complex analysis.

We introduce several new weakenings of the anti-Specker property and explore their role in constructive reverse mathematics, identifying implication relationships that they stand in to other notable principles. These include, but are not limited to: variations upon Brouwer's fan theorem, certain compactness properties, and so-called zero-stability properties. We also give similar classification results for principles arising directly from Specker's theorem itself, and present new, direct proofs of related fan-theoretic results.

We investigate how anti-Specker properties, alongside power-series-based arguments, enable us to recover information about the structure of holomorphic functions: in particular, they allow us to streamline a sequence of maximum-modulus theorems.

Figure 2.2 depicts some key relationships between the common properties of constructive reverse mathematics, and is available as a stand-alone document here.

Musical Compositions

The Firmament Turns, Unheeding (2026)

For flute & viola; 3 minutes
[score & parts]