Popper's falsification expressed in Bayesian terms

Popper's idea that there is only falsification and no such thing as confirmation turns out to be incorrect. Bayes' Theorem shows that falsification is very strong evidence compared to confirmation, but falsification is still probabilistic in nature; it is not governed by fundamentally different rules from confirmation, as Popper argued.

Eliezer S. Yudkowsky, An Intuitive Explanation of Bayes' Theorem.

Chance and free will

To defend free will was not, of course, to insist on the operation of pure chance. Most thinkers still saw objective chance as virtually impossible, and few were prepared to identify the rational will with it. But at least the connotations of chance had changed. Previously it had seemed impiuous to allow chance a role in the world, as if God did not attend to every sparrow. Now chance stood for the incompleteness of mechanical law, for the possibility of non-material causation.

Gerd Gigerenzer et al., The Empire of Chance, section 2.6. This joins the defense of free will laid down by K. Popper as I noticed in this post and this note. See also this post as well as the work on incompleteness by Gregory Chaitlin.

View on Einstein on the determinism of nature

Max Born, in his book Physics in my generation, relates Einstein's view on the determinism of nature:

"His conviction seems always to have been, and still is today, that the ultimate laws of nature are causal and deterministic, that probability is used to cover our ignorance if we have to do with numerous particles, and that only the vastness of this ignorance pushes statistics into the forefront."

Max Born, Physics in my generation, Einstein's statistical theories.

Now, nobody says that Einstein was always right. Indeed, the development of quantum mechanics went in the opposite direction to Einstein's view.

The probabilistic basis of thought

McIntyre (2007) argues that the process of thought, biology tells us, is not deterministic as the Platonic ideal has taught us, but is rather the result of a probabilistic mental calculation wired into our brains:

"[A]t the most fundamental levels - and I mean fundamental biologically as well as mathematically - there is no such thing as deterministic thinking. Our very thought processes, including mathematically thought processes, are fundamentally and inherently probabilistic."

Biologically, the reason is the following:

"The ubiquitous protein molecules called allosteric enzymes are logic elements. But they interact in massively-parallel information-processing «circuits» whose very «wiring» is probabilistic, indeed stochastic. Brownian motion - thermal fluctuation on picosecond timescales - connects those logic elements together in a fundamentally noisy way."

McIntyre pursues by saying that

"That of course is why, given the mechanical strengths of chemical bonds including hydrogen bonds, life can exist only in a rather narrow temperature range."

Interesting.

How then, Platonic, perfect, optimal, symmetric ideas or geometry can result from such noisy thinking process? One answer for this question may come from some ideas put recently by Mumford (2000) and, posthumously by Jaynes (2003) in a recent book. Those ideas are that mathematical reasoning, among which probabilistic reasoning, can be proved to be the result of a well-posed probabilistic theory

"the very foundations of mathematics should be reformulated on a stochastic basis"

according to Mumford.

McIntyre goes on by showing that, starting with weak, self-consistent assumptions, the whole basis of probability theory can be deduced. He also insists on the "conditioning statements" that are systematically undermined in the classic teaching of probability. Those statements are the a priori knowledge available to the observer and that has to be taken into account in the calculation of the probability of an event. For example, the statement "some roads are closed by the rain" could be a conditioning statement to calculate what is the most probable course taken by a FedEx truck. It appears that the explicit and careful description of those statements are essential to make the probability theory cleared of any subjectivity. McIntyre goes so far to state that in the classic debate between "frequentists" and "Bayesians", although the former claim that they do not add any subjective knowledge to the probabilistic calculation, they actually do with the additive information being in their case often implicit and/or unconscious.

McIntyre finally explains that the nearly perfect shape observed in Nature, the aerodynamical shapes of a wing or a fish, the roundish shapes of trunk or flowers, are not the result of some innate knowledge of perfect, Platonic forms but is rather the result of an optimization problem based on statistical inference. Many visual examples exist that illustrates that our brain can indeed perform such statistical inference: the brain can for instance guess that a man is walking just by the knowledge of the motions of several points located on the man's body.

McIntyre concludes, maybe surprisingly reminiscent of some post-modernists, that the concept of an absolute truth is dangerous, behind which there is often or always some kind of implicit knowledge or information taken for granted and not put forward explicitly.

For more, see

Jaynes, E. T. (2003), Probability theory: The logic of science, edited by G. Larry Bretthorst, Cambridge, University Press, 727 pp.
McIntyre, M. E. (2007), On thinking probabilistically, Proceedings Hawaiian Winter Workshop, University of Hawaii at Manoa, 172 pp.
Mumford, D. (2000), The dawning of the age of stochasticity, in Mathematics: Frontiers and perspectives, edited by V. I. Arnol'd, M. Atiyah, P. Lax and, B. Mazur, Providence, RI, Amer. Math. Soc., 460 pp.