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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.
Georg Cantor (1845 – 1918) around 1870 (left) and in later life (right).
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.
![High-precision digitally-driven mill carving a dental crown from a solid ceramic block [photo from www.sirona.com].](https://thatsmaths.files.wordpress.com/2014/07/cerec-mill-detail.jpg)
High-precision digitally-driven mill carving a dental crown from a solid ceramic block
[photo from www.sirona.com].
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.
10 Deutschmark currency note (front)
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.
Indra’s Indigo, detail of patchwork quilt by textile artist Janice Gunner.
ThatsMaths 