Software and Math
Ari4 thoughts on software:
Software is best seen as a service, not as a product.
Open source is better, all else being equal.
Reasonable profits are good.
Complexity is an enemy.
A spreadsheet is often a database table lacking a primary key.
It is best to keep data processing close to the data.
Correlation does not imply causation.
Isomorphism does not imply computability.
If the goals above keep conflicting, take a break!
Freyd, Kechris, Scedrov, Vaananen and others on categories, computability, logic, and model theory. Noted 15 March 2019.
Broad category theoretic study of history and recent results in computability theory. Section 3 includes equivalence classes of proofs; 3.8 introduces topoi, exploring relationships between them and formal languages. Freyd gluing techniques are introduced in 4.4. Classical type theory and non-standard type theory may require consideration in separate categories. (1/20)
Category theory applied to statistical analysis, results in Python and R. (noted 6/20) Cartesian closed categories are central in this analysis. A strict distinction between 'data type' and 'set' is observed. An ontology is a cartesian closed category with implicit (type and subtype) conversion, given by a finite presentation. Statistical models become viewable as models in mathematical logic, a 'structuralist conception of statistics.' If arbitrary morphism equations are allowed, the word problem becomes undecidable. Connections with lambda calculus and functional computing are noted. Limitations of automation are noted. Non-convergent statistical algorithms may pose issues for defining morphisms. Adjunctions and isomorphism classes are suggested for future work. 'At its most successful, ideas that were once the exclusive province of philosophy are transformed into actionable scientific methodology.' Includes extensive bibliography.
Evidence for connections between logic and (machine) learning, noted 6/20. The old tension between deduction and induction is still active. Includes summaries of recent issues in continuous vs discrete, infinite vs finite, and deterministic vs non-deterministic procedures.
Other items of current interest.
Benford's Law (2019)
Probability Logic, a recent effort. (2019)
Algorithms for Beauty?
Logic: interesting and useful recent completeness results. IMO K is a pretty strong assumption in a constructivist framework. The proofs themselves rely extensively on non-constructive elementary logic. Some things just never get old! (2019)
Amazing game-theoretic approach to welfare distribution. It helps illuminate several problems with socialism: the large economy is usually not decomposable into tractable sub-economies. Envy-free solutions in general contradict full decomposability. Adverse selection (IMO in Joseph Stiglitz’ sense) is still an issue. (2019)
Envy: I would suggest that we cannot defeat it by using the closed preference sets assumption. In economics this would imply that every situation of resource consumption is zero-sum. Also, as at the end of this paper, the importance of individual leadership is minimized. (1/20)
God as Ultrafilter? LOL! I can't help remembering Elizabeth Anscombe echoing Kant about the existence of God: ‘That is not the sort of thing you prove!’ I like to imagine her debating St. Anselm of Canterbury and St. Thomas Aquinas about this. With Karl Barth and Kurt Goedel in the audience taking copious notes? At any rate, this remarkable paper shows interesting and helpful results about reducibility of modal logic and reasoning to first order calculus. (12/19)
Structure Theory: recent results in 2019.
Recent results on orthostatic intolerance (med x stat). (2019)
Intuitionistic math: good summary and recent results. Also a constructivist re-presentation of work of Brouwer, Skolem and Lowenheim. (2019)
Intuitionistic logic: more history. Although it may have a dubious metaphysical provenance, it seems to help enforce very conservative procedures for managing computer memory.
Intuitionism and Constructivism, noted 3/2020.
Intuitionistic Math and Logic, Brouwer vs. Russell. A good history of the foundations of constructivism. Choice sequences and spreads replace sets. Intuitionistic/constructivist equality is undecidable. Many-oneness is the best basis for mathematical activity. Verification that a mathematical object has a property requires a construction. Homomorphisms are noted between finitely branching spreads (fans) and important structures of finite model theory. All embeddable in a topos? These authors conclude that these concepts can be valuable extensions of classical math.
Constructive logical analysis of differential equations. Singularity in the sphere = Brouwer's Fixed Point?
Infinitesimals turn up in unexpected places. Cauchy as a proto-constructivist?
This short paper suggests roots of non-standard analysis in Leibniz and Aristotle.
AI, noted 3/2020.
Kolmogorov complexity of computer file generators is not computable, shown by reduction arguments. Heuristic workarounds are discussed. All seem to support the claim that most AI adds to code bloat without corresponding real world payoff.
Logic, noted 5/2020.
Beyond Sets With Atoms: Definability in First-Order Logic.
This recent approach to Hilbert's Tenth Problem (1900), how to formalize the undecidability of solvability of Diophantine equations, includes reference to a library of undecidable problems.
For engineering, interesting categories have morphisms and objects whose costs can be minimized.
Peter P. Chen
References and free eBooks:
Teleology in Logic. You can almost see Prof. Serre's mind work!