Radically elementary probability theory
by Nelson, Edward · first published 1987
9mentions
6recommended by
7HN commenters
2011–2025first – last mentioned
86% recommend14% neutral
Why HN recommends it
A short, original treatment of probability using finite sums and internal set theory instead of measure theory. Worth reading for anyone who knows measure-theoretic probability or wants a finite, elementary view.
Summarised by Claude from recommending comments
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…unlimited sequences often are "long enough" to incorporate all the interesting behavior of infinite sequences. 0. Alain Robert's Nonstandard Analysis is a good starting point. 1. See his short book Radically Elementary Probability Theory. I love this book, and didn't much like probability theory before reading it.
on “What are the 'real numbers', really?” ↗…against a smooth function (i.e., a convolution) But these excuses are not rigorously justified, even at the graduate level, in my experience. * Separately from that, I wonder if OP has ever seen the book Radically Elementary Probability Theory, by Edward Nelson (https://web.math.princeton.edu/~nelson/books/rept.pdf). It uses nonstandard analysis to get around a lot of the (elegant) fussiness of measure theory. The preface alone is fun to read.
on “Derivative at a Discontinuity” ↗Radically Elementary Probability Theory (REPT above) is an excellent book. A very thin volume that anyone who labored through measure-theoretic probability would enjoy at least scanning. It is fascinating and original, like the proof mechanism…
on “Peano arithmetic is probably inconsistent” ↗…works, and we don't need to talk about sigma-algebras to talk about conditional probabilities or expectations. A wacky alternative implementation of probability theory can be found in Nelson's book "Radically Elementary Probability Theory" - I highly recommend it as a way to see how little the underlying framework matters.
on “Probability 101, the intuition behind martingales and solving problems with them” ↗Side comment: anyone interested in calculus via infinitesimals may also be interested in taking a look at Radically Elementary Probability Theory by Ed Nelson: https://web.math.princeton.edu/~nelson/books/rept.pdf
on “Multiplicative Infinitesimals” ↗…teaching about numbers methodologically aligned with how we actually measure and use them. I have personally been working on building various finite equivalents to familiar math. I recommend anyone to read Radically Elementary Probability Theory by Nelson to get a better sense of how to do finite math, at least at the theoretical level. Once again, on a practical level to do with directly computing quantities, we've only ever done finite math.
on “Mathematical secrets of ancient tablet unlocked after nearly a century of study (2017)” ↗…require [1], it just undoes some historical assumptions made in [1] and, like everything else by McLarty, is extraordinarily well-written) 3. Goldblatt's "Lectures on the Hyperreals"[3] 4. Nelson's "Radically Elementary Probability Theory"[4] 5. Tao's "Ultraproducts as a Bridge Between Discrete and Continuous Analysis"[5] 6. Some canonical machine learning text, like Murphy[6] or Bishop[7] 7. Koller/Friedman's "Probabilistic Graphical…
on “Physics, Topology, Logic and Computation: A Rosetta Stone (2009) [pdf]” ↗…http://en.wikipedia.org/wiki/Synthetic_differential_geometry 2. http://mathforum.org/dr.math/faq/analysis_hyperreals.html#co... 3. http://en.wikipedia.org/wiki/Internal_set_theory 4. See Edward Nelson, Radically Elementary Probability theory, available at http://www.math.princeton.edu/~nelson/books/rept.pdf [PDF] 5. Yes, this is Gödel's first theorem: http://en.wikipedia.org/wiki/Gödel%27s_incompleteness_theore...
on “Intuitionistic mathematics for physics” ↗Interesting that you say that. I've skimmed but have been meaning to properly read Nelson's: Radically Elementary probability theory http://www.math.princeton.edu/~nelson/books/rept.pdf and http://www.stat.umn.edu/geyer/nsa/. They do away with that problem all together as well as infinite constructions (replaced by hyperfinite) by…
on “What does 0^0 equal? Why do mathematicians and high school teachers disagree?” ↗