## Money money

I am thoroughly enjoying Jordan Ellenberg’s “How not to be wrong”. This small post is some fun looking at Buffon’s coin puzzle. This shows a sample of twenty coins of varying radius relative to the underlying square. The fair game arises when the radius to length of square side is . The fraction in the boxes are the observed number of coins in the square and the predicted fraction. The value below the box is the ratio of radius to square length.

## Joy in Small Things

I have learned a lot from the Mathematica Stackexchane community. I just crossed the 10 k…(it could easily go down). It has been a trying, stressful period in the everyday life, the small thing brings me some joy. I am not an expert nor a professional, dwell on the simpler level of the question spectrum and I have made some major gaffes (great opportunities to learn). I take each question as an opportunity to learn but learn the most from the creative, sometimes amazing but always inspirational other answers.

Peace to all, and solace to fellow sufferers of the ‘black dog’.

## Real Mathematical Analysis

This is another excellent Springer undergraduate Mathematics textbook. This book is clearly written. The book goes systematically progressively deeper. It covers continuty, differntiability, integration, introduces differential forms and Lebesgue theory. In addition to showing the results, the author shows the “pathological” functions, spaces and points to the wonderful unexpected results, e.g. unmeasurable sets and Banach-Traski paradox. I learned a lot from the Cantor sets.

I should have taken the time to dive into this book but even with my quick read, this book explains the strong foundations of real analysis and calculus that we use in tame, smooth, well behaved domains and provides insights into the world beyond the tame shores I usually play in…”there be dragons out there”.

## A Student’s Guide to Entropy

I have always found Entropy to be a difficult concept. This book is useful in improving understanding. I builds and explores the concept from classical, quantum physical and information theory perspectives. There is very useful discussion of statistical mechanicsand the derivation of the Maxwell-Boltzmann distribution. The relationship between classical and quantum physics results are well illustrated.

## Unknown Quantity

I enjoyed this book. The author has a very entertaining writing style. The book is historical journey from Mesopotamia to modern times tracing the development of algebra within the broader history of Mathematics. The author puts the developments in a historical context and provides insights into the characteristics/personalities (from what is available…little for some in the remote past) of the key figures in the development of algebra.

I am once again inspired to diminish my ignorance of Algebra.

## Probably Approximately Correct

This is a very interesting book. The author presents the case for ecorithms (algorithms, heuristics perhaps) that could explain and ultimately allow quantitative assessment and testable predictions of the mechanisms (and timescale) of evolution and one of its most “mysterious” byproducts consciousness/cognition (I should perhaps not conflate these two).

The author looks at the central problem of evaluating and decision making based on incomplete information, small empirical samples and within biological and physical constraints and be successful. The linear/polynomial time algorithms (using the generalized concept of computation: universal Turing machines) for learning from inputs from external environments in a “theory-less” context could lead to “probably approximately correct” classifications, decisions and actions and be explanatory for evolution and perhaps human learning and human cultural evolution (with the latter having Lamarckian as well Darwinian aspects).

The book explores these matters through the lens of computer science (the author’s expertise). This is a very interesting and instructive perspective. The limits, and similarities and contrasts between computer systems and algorithms was well presented.

I think this book fits nicely with Penrose “Emperor’s New Mind” (which argues for non-algorithmic apsects to consciousness and learning), Kahneman’s “Thinking Fast and Slow” which explores the limitations of human reason (our hard wiring for making fast decisions with limited information but our limitations in statistical and probabilistic reasoning) and Silver’s “The Signal and the Noise”.

I had (and continue) to think hard about the concepts in the book. It is, however, I believe a very refreshing viewpoint to seek to explain the gaps in evolution, cognition and learning. The author ends by looking at issues of artificial intelligence and why this has been more challenging than anticipated and the authors appeal to reason in relation to fears about a ‘Sky-net’ future was very interesting. The integration of external inputs, the central role of learning, the power of inductive reasoning in and the need for composite induction and deductive reasoning (the latter indispensible for what the author calls theory-ful contexts) are all part of the authors rich explanation…it seemed to me “probably approximately correct”.