Many of us recall the sense of wonder we felt upon learning that there is no biggest number; for some of us, that wonder has never quite gone away. It is obvious that, given any counting number, one can be added to it to give a larger number. But the implication that there is no limit to this process is perplexing.

## Archive for July, 2014

### Invention or Discovery?

Published July 24, 2014 Occasional ClosedTags: Analysis, Logic, Number Theory, Social attitudes

Is mathematics invented or discovered? As many great mathematicians have considered this question without fully resolving it, there is little likelihood that I can provide a complete answer here. But let me pose a possible answer in the form of a conjecture:

** Conjecture:*** Definitions are invented. Theorems are discovered.*

The goal is to prove this conjecture, or to refute it. Below, some arguments in support of the conjecture are presented. Continue reading ‘Invention or Discovery?’

*That’s Maths* in *The Irish Times* this week (TM049, or Search for “thatsmaths” at irishtimes.com) is about applications of computer aided design and computer aided manufacture to making dental crowns.

### Gauss’s Great Triangle and the Shape of Space

Published July 10, 2014 Occasional ClosedTags: Gauss, Geophysics, Maps

In the 1820s Carl Friedrich Gauss carried out a surveying experiment to measure the sum of the three angles of a large triangle. Euclidean geometry tells us that this sum is always 180º or two right angles. But Gauss himself had discovered other geometries, which he called non-Euclidean. In these, the three angles of a triangle may add up to more than two right angles, or to less.

Continue reading ‘Gauss’s Great Triangle and the Shape of Space’

This week, *That’s Maths* in *The Irish Times* (TM048: Search for “thatsmaths” at irishtimes.com) is about the beauty of mathematics.