Maths resources


“Am I a constructive mathematician?”, by Andrej Bauer

“Exploring mathematical objects from custom-tailored mathematical universes”, by Ingo Blechschmidt

“Five stages of accepting constructive mathematics”, by Andrej Bauer

“relation between type theory and category theory”, on the nLab

“The Explanatory Value of Category Theory”, by Ellen Lehet

“What is applied category theory?”, by Tai-Danae Bradley

Mathematical constraints on society/politics

Arrow's impossibility theorem ... states that when voters have three or more distinct alternatives (options), no ranked voting electoral system can convert the ranked preferences of individuals into a community-wide (complete and transitive) ranking while also meeting the specified set of criteria: unrestricted domain, non-dictatorship, Pareto efficiency, and independence of irrelevant alternatives.

— Wikipedia: ‘Arrow's Impossibility Theorem’

Gibbard's theorem ... states that for any deterministic process of collective decision, at least one of the following three properties must hold:

1. The process is dictatorial, i.e. there exists a distinguished agent who can impose the outcome;

2. The process limits the possible outcomes to two options only;

3. The process is open to strategic voting: once an agent has identified their preferences, it is possible that they have no action at their disposal that best defends these preferences irrespective of the other agents' actions.

— Wikipedia: ‘Gibbard's theorem’

Sen's paradox ... shows that no means of aggregating individual preferences into a single, social choice, can simultaneously fulfill the following, seemingly mild conditions:

1. The unrestrictedness condition, or U: every possible ranking of each individual's preferences and all outcomes of every possible voting rule will be considered equally,

2. The Pareto condition, or P: if everybody individually likes some choice better at the same time, the society in its voting rule as a whole likes it better as well, and

3. Liberalism, or L (from which the theorem derives its gist): all individuals in a society must have at least one possibility of choosing differently, so that the social choice under a given voting rule changes as well. That is, as an individual liberal, anyone can exert their freedom of choice at least in some decision with tangible results.

— Wikipedia: ‘Sen's paradox’

[A]ggregating judgments with majority voting can result in self-contradictory judgments ... Philosopher Philip Pettit believes the discursive dilemma makes it impossible to make simple statements about the beliefs of a collective.

Wikipedia: ‘Discursive dilemma’

Category-theoretic ecology

Work relevant, or potentially relevant, to using category theory in ecology:

“The representation of biological systems from the standpoint of the theory of categories”, by Robert Rosen (1958) [PDF]

“The ecosystem as an algebraic category”, by B.S. Niven (1988) [PDF]

“The Algebra of Open and Interconnected Systems”, by Brendan Fong (2016) [abstract + link to PDF]

“Behavioural Mereology”, by Brendan Fong, David Myers and David Spivak (2018) [abstract + link to PDF]

“Symmetric Monoidal Categories: A Rosetta Stone”, by John Baez (2021?) [PDF of slides]

“Reformalizing the notion of autonomy as closure through category theory as an arrow-first mathematics”, by Ryuzo Hirota, Hayato Saigo and Shigeru Taguchi (2023) [abstract + link to PDF]

“A Categorical Framework for Quantifying Emergent Effects in Network Topology”, by Johnny Jingze Li, Sebastian Prado Guerra, Kalyan Basu, Gabriel A. Silva (2023) [abstract + link to PDF]

“An Abstract Category of Dynamical Systems”, by James Schmidt (2024) [abstract + link to PDF]

“Axiomatic phylogenetics”, by Vladimir Turaev (2024) [abstract + link to PDF]