I recently finished the first draft of the last of my books based on my 2010 blog posts (and also my Google buzzes), entitled “Compactness and contradiction“. The PDF of this draft is available here. This is a somewhat assorted (and lightly edited) collection of posts (and buzzes), though concentrating in the areas of analysis (both standard and nonstandard), logic, and group theory. As always, comments and corrections are welcome.

### Recent Comments

Maths student on A corollary of the ergodic… | |

Maths student on A corollary of the ergodic… | |

Maths student on A corollary of the ergodic… | |

Hari on Analysis I | |

Alex M Kimuya on A geometric proof of the impos… | |

Chetan on Which universities should one… | |

My CAREER PLANS afte… on Does one have to be a genius t… | |

STUDY MOTIVATION at… on Does one have to be a genius t… | |

Aritra Bhattacharya on A corollary of the ergodic… | |

Alex M Kimuya on A geometric proof of the impos… | |

burrobert on A corollary of the ergodic… | |

Terence Tao on Polymath15, ninth thread: goin… | |

Terence Tao on 246C notes 2: Circle packings,… | |

Terence Tao on Ultraproducts as a Bridge Betw… | |

Kitty Sasa on Compressed sensing and single-… |

### Articles by others

- Gene Weingarten – Pearls before breakfast
- Isaac Asimov – The relativity of wrong
- Jonah Lehrer – Don't! – the secret of self-control
- Julianne Dalcanton – The cult of genius
- Nassim Taleb – The fourth quadrant: a map of the limits of statistics
- Paul Graham – What You'll Wish You'd Known
- Po Bronson – How not to talk to your kids
- Scott Aaronson – Ten signs a claimed mathematical proof is wrong
- Timothy Gowers – Elsevier — my part in its downfall
- Timothy Gowers – The two cultures of mathematics
- William Thurston – On proof and progress in mathematics

### Diversions

- Abstruse Goose
- Assembler
- BoxCar2D
- Factcheck.org
- FiveThirtyEight
- Gapminder
- Literally Unbelievable
- Planarity
- PolitiFact
- Quite Interesting
- snopes
- Strange maps
- Television tropes and idioms
- The Economist
- The Onion
- The Straight Dope
- This American Life on the financial crisis I
- This American Life on the financial crisis II
- What if? (xkcd)
- xkcd

### Mathematics

- 0xDE
- A Mind for Madness
- A Portion of the Book
- Absolutely useless
- Alex Sisto
- AMS blogs
- AMS Graduate Student Blog
- Analysis & PDE
- Analysis & PDE Conferences
- Annoying Precision
- Area 777
- Ars Mathematica
- ATLAS of Finite Group Representations
- Automorphic forum
- Avzel's journal
- Blog on Math Blogs
- Blog On Mathematical Journals
- blogderbeweise
- Bubbles Bad; Ripples Good
- Cédric Villani
- Climbing Mount Bourbaki
- Coloquio Oleis
- Combinatorics and more
- Compressed sensing resources
- Computational Complexity
- Concrete nonsense
- David Mumford's blog
- Delta epsilons
- DispersiveWiki
- Disquisitiones Mathematicae
- Embûches tissues
- Emmanuel Kowalski’s blog
- Encyclopedia of Mathematics
- Equatorial Mathematics
- fff
- Floer Homology
- Frank Morgan’s blog
- Gérard Besson's Blog
- Gödel’s Lost Letter and P=NP
- Geometric Group Theory
- Geometry and the imagination
- Geometry Bulletin Board
- Girl's Angle
- God Plays Dice
- Good Math, Bad Math
- Graduated Understanding
- Hydrobates
- I Woke Up In A Strange Place
- Igor Pak's blog
- Images des mathématiques
- In theory
- James Colliander's Blog
- Jérôme Buzzi’s Mathematical Ramblings
- Joel David Hamkins
- Journal of the American Mathematical Society
- Kill Math
- Le Petit Chercheur Illustré
- Lemma Meringue
- Lewko's blog
- Libres pensées d’un mathématicien ordinaire
- LMFDB – L-functions and modular forms database
- LMS blogs page
- London number theory
- Low Dimensional Topology
- M-Phi
- MAA MinuteMath
- Mark Sapir's blog
- Math Overflow
- Mathbabe
- Mathblogging
- Mathematical musings
- Mathematics Illuminated
- Mathematics in Australia
- Mathematics Jobs Wiki
- Mathematics Stack Exchange
- Mathematics under the Microscope
- Mathematics without apologies
- Mathlog
- MathOnline
- Mathtube
- Matt Baker's Math Blog
- Mixedmath
- Motivic stuff
- Much ado about nothing
- Multiple Choice Quiz Wiki
- nLab
- Noncommutative geometry blog
- Nonlocal equations wiki
- Nuit-blanche
- Number theory web
- outofprintmath
- Pattern of Ideas
- PDE blog
- Pengfei Zhang's blog
- Persiflage
- Peter Cameron's Blog
- Phillipe LeFloch's blog
- ProofWiki
- Quomodocumque
- Random Math
- Reasonable Deviations
- Regularize
- Rigorous Trivialities
- Roots of unity
- Secret Blogging Seminar
- Selected Papers Network
- Sergei Denisov's blog
- Short, Fat Matrices
- Shtetl-Optimized
- Shuanglin's Blog
- Since it is not…
- Sketches of topology
- Soft questions
- Stacks Project Blog
- SymOmega
- tcs math
- TeX, LaTeX, and friends
- The accidental mathematician
- The capacity to be alone
- The Cost of Knowledge
- The Everything Seminar
- The Geomblog
- The L-function and modular forms database
- The Mathematics Literature Project
- The n-Category Café
- The n-geometry cafe
- The On-Line Blog of Integer Sequences
- The polylogblog
- The polymath blog
- The polymath wiki
- The Tricki
- The twofold gaze
- The Unapologetic Mathematician
- The value of the variable
- The World Digital Mathematical Library
- Theoretical Computer Science – StackExchange
- Tim Gowers’ blog
- Tim Gowers’ mathematical discussions
- Todd and Vishal’s blog
- Van Vu's blog
- Vaughn Climenhaga
- Vieux Girondin
- Visual Insight
- Vivatsgasse 7
- Williams College Math/Stat Blog
- Windows on Theory
- Wiskundemeisjes
- XOR’s hammer
- Yufei Zhao's blog
- Zhenghe's Blog

### Selected articles

- A cheap version of nonstandard analysis
- A review of probability theory
- American Academy of Arts and Sciences speech
- Amplification, arbitrage, and the tensor power trick
- An airport-inspired puzzle
- Benford's law, Zipf's law, and the Pareto distribution
- Compressed sensing and single-pixel cameras
- Einstein’s derivation of E=mc^2
- On multiple choice questions in mathematics
- Quantum mechanics and Tomb Raider
- Real analysis problem solving strategies
- Sailing into the wind, or faster than the wind
- Simons lectures on structure and randomness
- Small samples, and the margin of error
- Soft analysis, hard analysis, and the finite convergence principle
- The blue-eyed islanders puzzle
- The cosmic distance ladder
- The federal budget, rescaled
- Ultrafilters, non-standard analysis, and epsilon management
- What is a gauge?
- What is good mathematics?
- Why global regularity for Navier-Stokes is hard

### Software

### The sciences

### Top Posts

- A corollary of the ergodic theorem
- Career advice
- Does one have to be a genius to do maths?
- The Euler-Maclaurin formula, Bernoulli numbers, the zeta function, and real-variable analytic continuation
- Books
- Which universities should one apply to?
- On writing
- About
- Polymath15, ninth thread: going below 0.22?
- A geometric proof of the impossibility of angle trisection by straightedge and compass

### Archives

- July 2018 (2)
- June 2018 (1)
- May 2018 (4)
- April 2018 (4)
- March 2018 (5)
- February 2018 (4)
- January 2018 (5)
- December 2017 (5)
- November 2017 (3)
- October 2017 (4)
- September 2017 (4)
- August 2017 (5)
- July 2017 (5)
- June 2017 (1)
- May 2017 (3)
- April 2017 (2)
- March 2017 (3)
- February 2017 (1)
- January 2017 (2)
- December 2016 (2)
- November 2016 (2)
- October 2016 (5)
- September 2016 (4)
- August 2016 (4)
- July 2016 (1)
- June 2016 (3)
- May 2016 (5)
- April 2016 (2)
- March 2016 (6)
- February 2016 (2)
- January 2016 (1)
- December 2015 (4)
- November 2015 (6)
- October 2015 (5)
- September 2015 (5)
- August 2015 (4)
- July 2015 (7)
- June 2015 (1)
- May 2015 (5)
- April 2015 (4)
- March 2015 (3)
- February 2015 (4)
- January 2015 (4)
- December 2014 (6)
- November 2014 (5)
- October 2014 (4)
- September 2014 (3)
- August 2014 (4)
- July 2014 (5)
- June 2014 (5)
- May 2014 (5)
- April 2014 (2)
- March 2014 (4)
- February 2014 (5)
- January 2014 (4)
- December 2013 (4)
- November 2013 (5)
- October 2013 (4)
- September 2013 (5)
- August 2013 (1)
- July 2013 (7)
- June 2013 (12)
- May 2013 (4)
- April 2013 (2)
- March 2013 (2)
- February 2013 (6)
- January 2013 (1)
- December 2012 (4)
- November 2012 (7)
- October 2012 (6)
- September 2012 (4)
- August 2012 (3)
- July 2012 (4)
- June 2012 (3)
- May 2012 (3)
- April 2012 (4)
- March 2012 (5)
- February 2012 (5)
- January 2012 (4)
- December 2011 (8)
- November 2011 (8)
- October 2011 (7)
- September 2011 (6)
- August 2011 (8)
- July 2011 (9)
- June 2011 (8)
- May 2011 (11)
- April 2011 (3)
- March 2011 (10)
- February 2011 (3)
- January 2011 (5)
- December 2010 (5)
- November 2010 (6)
- October 2010 (9)
- September 2010 (9)
- August 2010 (3)
- July 2010 (4)
- June 2010 (8)
- May 2010 (8)
- April 2010 (8)
- March 2010 (8)
- February 2010 (10)
- January 2010 (12)
- December 2009 (11)
- November 2009 (8)
- October 2009 (15)
- September 2009 (6)
- August 2009 (13)
- July 2009 (10)
- June 2009 (11)
- May 2009 (9)
- April 2009 (11)
- March 2009 (14)
- February 2009 (13)
- January 2009 (18)
- December 2008 (8)
- November 2008 (9)
- October 2008 (10)
- September 2008 (5)
- August 2008 (6)
- July 2008 (7)
- June 2008 (8)
- May 2008 (11)
- April 2008 (12)
- March 2008 (12)
- February 2008 (13)
- January 2008 (17)
- December 2007 (10)
- November 2007 (9)
- October 2007 (9)
- September 2007 (7)
- August 2007 (9)
- July 2007 (9)
- June 2007 (6)
- May 2007 (10)
- April 2007 (11)
- March 2007 (9)
- February 2007 (4)

### Categories

- expository (267)
- tricks (10)

- guest blog (10)
- Mathematics (731)
- math.AC (6)
- math.AG (38)
- math.AP (95)
- math.AT (17)
- math.CA (150)
- math.CO (165)
- math.CT (6)
- math.CV (26)
- math.DG (35)
- math.DS (76)
- math.FA (23)
- math.GM (9)
- math.GN (21)
- math.GR (85)
- math.GT (15)
- math.HO (10)
- math.IT (11)
- math.LO (46)
- math.MG (41)
- math.MP (25)
- math.NA (19)
- math.NT (159)
- math.OA (19)
- math.PR (95)
- math.QA (5)
- math.RA (35)
- math.RT (21)
- math.SG (4)
- math.SP (47)
- math.ST (6)

- non-technical (140)
- admin (43)
- advertising (30)
- diversions (4)
- media (12)
- journals (3)

- obituary (10)

- opinion (29)
- paper (185)
- question (116)
- polymath (81)

- talk (64)
- DLS (19)

- teaching (164)
- 245A – Real analysis (11)
- 245B – Real analysis (21)
- 245C – Real analysis (6)
- 246A – complex analysis (9)
- 246C – complex analysis (5)
- 254A – analytic prime number theory (16)
- 254A – ergodic theory (18)
- 254A – Hilbert's fifth problem (12)
- 254A – random matrices (14)
- 254B – expansion in groups (8)
- 254B – Higher order Fourier analysis (9)
- 275A – probability theory (6)
- 285G – poincare conjecture (20)
- Logic reading seminar (8)

- travel (25)
- Uncategorized (1)

additive combinatorics
approximate groups
arithmetic progressions
Ben Green
Cauchy-Schwarz
Cayley graphs
central limit theorem
Chowla conjecture
circular law
compressed sensing
correspondence principle
divisor function
eigenvalues
Elias Stein
Emmanuel Breuillard
entropy
equidistribution
ergodic theory
expander graphs
exponential sums
finite fields
Fourier transform
Four Moment Theorem
Freiman's theorem
Gowers uniformity norm
Gowers uniformity norms
graph theory
Gromov's theorem
GUE
Hilbert's fifth problem
inverse conjecture
Kakeya conjecture
Lie algebras
Lie groups
Liouville function
Littlewood-Offord problem
Mobius function
moment method
multiple recurrence
Navier-Stokes equations
nilpotent groups
nilsequences
nonstandard analysis
parity problem
politics
polymath1
polymath8
Polymath15
polynomial method
polynomials
prime gaps
prime numbers
prime number theorem
project heatwave
pseudorandomness
random matrices
randomness
random walks
Ratner's theorem
regularity lemma
Ricci flow
Riemann zeta function
Schrodinger equation
sieve theory
spectral theorem
structure
Szemeredi's theorem
Tamar Ziegler
UCLA
ultrafilters
ultraproducts
universality
Van Vu
wave maps
Yitang Zhang

### The Polymath Blog

- Polymath proposal: finding simpler unit distance graphs of chromatic number 5 10 April, 2018
- A new polymath proposal (related to the Riemann Hypothesis) over Tao’s blog 26 January, 2018
- Spontaneous Polymath 14 – A success! 26 January, 2018
- Polymath 13 – a success! 22 August, 2017
- Non-transitive Dice over Gowers’s Blog 15 May, 2017
- Rota’s Basis Conjecture: Polymath 12, post 3 5 May, 2017
- Rota’s Basis Conjecture: Polymath 12 6 March, 2017
- Blog theme changed 28 February, 2017
- Rota’s Basis Conjecture: Polymath 12? 23 February, 2017
- MO Polymath question: Summary of Proposals 13 August, 2016

## 10 comments

Comments feed for this article

1 July, 2011 at 2:27 pm

Bo JacobyFirst of all: Thank you for giving me this book. I look forward to studying it.

I looked at the Contents, and I would like to tell you about (my) Ordinal Fractions. They are effective for numbering chapters and paragraphs and anything else. They are too elementary to catch the attention of mathematicians.

The digit zero has two meanings. In ‘1770’ zero means ‘nothing’, and in ‘the 1770s’ zero means ‘everything’.

In order to avoid misunderstandings, zero should mean only one thing, and in an ordinal fraction, zero means ‘everything’.

There are only 9 one digit numbers left when zero doesn’t count. So this is not decimal, it is most nonal.

There is no dot between the chapter number and the section number. When there are more than 9 sections in a chapter the sections are numbered with two (nonzero) digits.

An ordinal fraction is something like ‘the third fourth’, because ‘the third’ is an ordinal number and ‘a fourth’ is a fraction. Like this:

00 whole

01 odd fourths

02 even fourths

10 first half

11 first fourth

12 second fourth

20 second half

21 first fourth

22 second fourth

The five ordinal fraction relations are these:

1 The first half is equal to the first half: 10=10

2 The first half is part of the whole: 1011

4 The first half is parallel with the second half: 10><20

5 The first half intersects the odd fourths: 1001

Ordinal fractions are and’ed like this:

10+10=10

10+00=10

10+11=11

10+20=Ø

10+01=11

10+Ø=Ø

Ø is the empty set or the impossible condition.

Ø is the improper ordinal fraction.

So your table of contents may look like below,

Note that 2122<2100, meaning that the section on circular arguments are part of chapter on logic and foundations.

Yours truly, Bo.

0000 Compactness and contradiction

1100 Preface

1200 A remark on notation

1300 Acknowledgments

2100 Logic and foundations

2111 Material implication

2112 Errors in mathematical proofs

2113 Mathematical strength

2114 Stable implications

2115 Notational conventions

2121 Abstraction

2122 Circular arguments

2123 The classical number systems

2124 Round numbers

2125 The \no self-defeating object" argument, revisited

2131 The \no self-defeating object" argument, and the vagueness paradox

2132 A computational perspective on set theory

2200 Group theory

2211 Torsors

2212 Active and passive transformations

2213 Cayley graphs and the geometry of groups

2214 Group extensions

2215 A proof of Gromov's theorem

2300 Analysis

2311 Orders of magnitude, and tropical geometry

2312 Descriptive set theory vs. Lebesgue set theory

2313 Complex analysis vs. real analysis

2314 Sharp inequalities

2315 Implied constants and asymptotic notation

2321 Brownian snowflakes

2322 The Euler-Maclaurin formula, Bernoulli numbers, the zeta function, and real-variable analytic continuation

2323 Finitary consequences of the invariant subspace problem

2324 The Guth-Katz result on the Erd}os distance problem

2325 The Bourgain-Guth method for proving restriction theorems

2400 Nonstandard analysis

2411 Real numbers, nonstandard real numbers, and nite precision arithmetic

2412 Nonstandard analysis as algebraic analysis

2413 Compactness and contradiction: the correspondence principle in ergodic theory

2414 Nonstandard analysis as a completion of standard analysis

2425 Concentration compactness via nonstandard analysis

2500 Partial dierential equations

2511 Quasilinear well-posedness

2512 A type diagram for function spaces

2513 Amplitude-frequency dynamics for semilinear dispersive equations

2514 The Euler-Arnold equation

2600 Miscellaneous

2611 Multiplicity of perspective

2612 Memorisation vs. derivation

2613 Coordinates

2614 Spatial scales

2615 Averaging

2621 What colour is the sun?

2622 Zeno's paradoxes and induction

2623 Jevons' paradox

2624 Bayesian probability

2625 Best, worst, and average-case analysis

2631 Duality

2632 Open and closed conditions

3100 Bibliography

3200 Index

1 July, 2011 at 8:58 pm

Matthew N. PetersenPg. 18, you have “Gdel’s incompleteness theorem”, rather than “Gödel’s…”.

1 July, 2011 at 9:07 pm

Matthew N. PetersenSame difficulty on p. 29

[Corrected, thanks – T.]1 July, 2011 at 11:30 pm

gowersA quick typo: in the statement of Theorem 4.3.11 you write “Szemerdi”. An easy exercise in reverse engineering suggests, in the light of the “Gdel” typo mentioned above, that something more general has gone wrong …

[Ah, some of the text I incorporated to the book contained accents which LaTeX then refused to recognise. I think the problem is fixed now and will be corrected in the next revision of the ms – T.]4 July, 2011 at 12:53 pm

GregCouple of typos right at the start. On page 2, in statement (8) you have “Is A, then B” which looks like it should be “If A, then B”. Lower down, penultimate paragraph (just before the footnote 1) your implication arrows seem to have gone wrong (-> rather than -> for A \implies B and B \implies A)

Otherwise, this looks good so far!

[Thanks, this will be corrected in the next version of the ms. -T.]4 July, 2011 at 1:12 pm

GregOne more before bed: on page 9, top paragraph, you have “the two are not symmetric, because in both cases is bounding an unknown quantity by a known quantity.” The “is” in the second clause is missing its subject — we’re not told _what_ is bounding the unknown quantity.

–g

[Thanks, this will be corrected in the next version of the ms. -T.]4 July, 2011 at 2:25 pm

JohnPage 80: “algberaic” -> “algebraic”

[Thanks, this will be corrected in the next version of the ms. -T.]5 July, 2011 at 12:11 pm

diego.maldona@gmail.comDear Terry, thank you for posting the book. It’s always insightful to read your unifying take on so many, seemingly disconnected, Math topics. Below are a few typos I found.

p.13, line 8, misplaced comma

p.20, line 16, “doesnt”

p.22, line -13, “and its proof is” should be “and its proof are”

p.22, last line, extra semicolon

p.23, “If B was an element” should read “If B were an element”

p.29. The sentence starting with “However…” goes on for ten and a half lines and there is a “then” that’s not quite working out.

p.29, first footnote, I believe “boolean” should be “Boolean” (as “Bayesian” reads “Bayesian”). “boolean” also appears in other parts of the manuscript. Same issue with “gaussian”; but, apparently, “abelian” is ok.

p.30, line 8, “that A was ﬁnite” should be “that A were ﬁnite” or “that A is ﬁnite”

p.48, line 8, “translation” should be “translations”

p.110, There are several “Erdos” and “Holder” (without the umlaut).

p.110, line 11, “considing”

p.192, lines -5, -4, the parameter “p” should be in math font

In Section 1.11, after discussing Richard’s paradox (p.32), you could mention that if a reader runs into a genie willing to grant three wishes, the instruction “I wish for 1000 wishes” won’t have any effect, since that’s not a wish, but a meta-wish. This situation could also be phrased as a paradox (writing, for instance, that the genie will grant exactly 3 wishes).

Regarding paradoxes, it will be illuminating to have your take on the “surprise test paradox”

http://en.wikipedia.org/wiki/Unexpected_hanging_paradox

http://arxiv.org/abs/math/9903160

I loved the “digital-images analogy” when illustrating the concepts of extension and quotient.

[Thanks for the corrections! They will be incorporated in the next revision of the ms. The surprise test paradox is discussed on this blog at https://terrytao.wordpress.com/2011/05/19/epistemic-logic-temporal-epistemic-logic-and-the-blue-eyed-islander-puzzle-lower-bound/ -T.]9 July, 2011 at 11:51 pm

Hemant VermaHi All,

There are related mathematics and algorithms problem here, for those who love mathematics / algorithm problem solviing Mathalon , after all whats in mathematics without problems.

-Hemant

26 September, 2016 at 11:13 am

AnonymousDear Terry!

What’s the point of thinking of an implication A –> B as asserting that B is at least as true as A in order to understand the disjunction elimination? In know that “A or B” has the biggest truth value of the two truth values of A and B. Thus if C is at least as true as A and also at least as true as B, then “A or B” is at least as true as C (since “A or B” has the same truth value as A or as B). But why does this shed light on the disjunction elimination?