## Archive for March, 2022

### Infinitesimals: vanishingly small but not quite zero

Abraham Robinson (1918-1974)  and his book, first published in 1966.

A few weeks ago, I wrote about  Hyperreals and Nonstandard Analysis , promising to revisit the topic. Here comes round two.

### The Chromatic Number of the Plane

To introduce the problem in the title, we begin with a quotation from the Foreword, written by Branko  Grünbaum, to the book by Alexander Soifer (2009): The Mathematical Coloring Book: Mathematics of Coloring and the Colorful Life of its Creators:

If each point of the plane is to be given a color, how many colors do we need if every two points at unit distance are to receive distinct colors?

About 70 years ago it was shown that the least number of colours needed for such a colouring is one of 4, 5, 6 and 7. But which of these is the correct number? Despite efforts by many very clever people, some of whom had solved problems that appeared to be much harder, no advance has been made to narrow the gap

${4\le\chi\le 7}$.

### The Improbability Principle and the Seanad Election

A by-election for the Seanad Éireann Dublin University constituency, arising from the election of Ivana Bacik to Dáil Éireann, is in progress. There are seventeen candidates, eight men and nine women. Examining the ballot paper, I immediately noticed an imbalance: the top three candidates, and seven of the top ten, are men. The last six candidates listed are all women. Is there a conspiracy, or could such a lopsided distribution be a matter of pure chance?

To avoid bias, the names on the ballot paper are always listed in alphabetical order. We may assume that the name of a randomly chosen candidate is equally likely to appear at any of the positions on the list; with 17 candidates, there about 6% chance for each of the 17 positions; the distribution for a single candidate is uniform. However, when several candidates are grouped, the distribution is more complicated  [TM231 or search for “thatsmaths” at irishtimes.com].
Following the invention of calculus, serious concerns persisted about the mathematical integrity of the method of infinitesimals. Leibniz made liberal use of infinitesimals, with great effect, but his reasoning was felt to lack rigour. The Irish bishop George Berkeley criticised the assumptions underlying calculus, and his objections were not properly addressed for several centuries. In the 1800s, Bolzano, Cauchy and Weierstrass developed the ${\varepsilon}$${\delta}$ definition of limits and continuity, which allowed derivatives and integrals to be defined without recourse to infinitesimal quantities. Continue reading ‘Hyperreals and Nonstandard Analysis’