Paraconsistency and Evolvability

Reflections on the unreasonable effectiveness of mathematics and its implications for Artificial Intelligence.

Aidan Rocke https://github.com/AidanRocke
10-23-2022

When the facts change, I change my mind. What do you do, sir?-Keynes

It may come as a shock to future generations of mathematicians to discover that no set of axioms sufficiently powerful to express number theory may be put on a consistent basis. The main difficulty, as they will discover, lies with the Law of Excluded Middle and the Principle of Mathematical Induction which are indissociable. However, this will require a revolution in our way of thinking as there are no technical fixes.

Thus, the meta-objective of mathematics shall shift from determining which propositions are true to determining which axioms are necessary and sufficient. It appears inevitable in turn that we shall approach mathematics from a systems perspective in order to maximize its evolvability. Two clear examples of such a radical shift in perspective include Grothendieck’s development of the Theory of Motives and the development of Geometric Algebra pioneered by Hestenes, Doran, Lasenby and Gull.

Perhaps this evolution in mathematical thought may come as less of a surprise if we view mathematics as a natural ontology that emerges from the scientific mind. The inconsistency of mathematics would appear to make the unreasonable effectiveness of mathematics in the natural sciences a lot more unreasonable, if not for our eventual acceptance of the fact that mathematics emerges from the consistent application of Occam’s razor. Few mathematicians understood this better than Kolmogorov, whose theory of Algorithmic Probability established rigorous foundations for Occam’s razor. Moreover, this identification of Occam’s razor with the Natural Sciences would represent the zenith of this research programme if it did not force the gaze to turn inwards and inquire about the nature of the human mind itself.

If life is viewed as an imperfect and incomplete information game then inconsistent logics may confer an evolutionary advantage to humans as they may be essential for cognitive flexibility. In lieu of a premature conclusion, I believe these ideas must be developed further as they have important implications for the development of strong AI and the future course of human evolution.

References:

  1. Rocke (2022, Oct. 6). Kepler Lounge: Kolmogorov’s theory of Algorithmic Probability. Retrieved from keplerlounge.com

  2. Akama, Seiki (ed.), 2016, Towards Paraconsistent Engineering (Intelligent Systems Reference Library 110), Dordrecht: Springer. doi:10.1007/978-3-319-40418-9

  3. Priest, G., Routley, R. & Norman, J. eds. (1989). Paraconsistent Logic: Essays on the Inconsistent. Philosophia Verlag.

  4. Rocke (2022, Jan. 5). Kepler Lounge: Revisiting the unreasonable effectiveness of mathematics. Retrieved from keplerlounge.com

  5. J. Hintikka and G. Sandu, 2009, “Game-Theoretical Semantics” in Keith Allan (ed.) Concise Encyclopedia of Semantics, Elsevier, ISBN 0-08095-968-7, pp. 341–343

  6. Anthony Lasenby, Chris Doran, and Stephen Gull. Gravity, Gauge Theories and Geometric Algebra. Arxiv. 2004.

  7. Leslie G. Valiant. Evolvability. Journal of the ACM. 2009.

Citation

For attribution, please cite this work as

Rocke (2022, Oct. 23). Kepler Lounge: Paraconsistency and Evolvability. Retrieved from keplerlounge.com

BibTeX citation

@misc{rocke2022paraconsistency,
  author = {Rocke, Aidan},
  title = {Kepler Lounge: Paraconsistency and Evolvability},
  url = {keplerlounge.com},
  year = {2022}
}