Degrees of Infinity

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).

Georg Cantor (1845 – 1918) around 1870 (left) and in later life (right).

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Invention or Discovery?

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?’

Digital Dentistry

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].

High-precision digitally-driven mill carving a dental crown from a solid ceramic block
[photo from www.sirona.com].

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Gauss’s Great Triangle and the Shape of Space

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

10 Deutschmark currency note (front)

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

Beauty is the First Test

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 a  patchwork quilt by accomplished textile artist Janice Gunner.

Indra’s Indigo, detail of patchwork quilt by textile artist Janice Gunner.

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Balancing a Pencil

Does quantum mechanics matter at everyday scales? It would be very surprising if quantum effects were to be manifest in a macroscopic system. This has been claimed for the problem of balancing a pencil on its tip. But the behaviour of a tipping pencil can be explained in purely classical terms.

A pencil balanced on its point. Is there a trick? Yes: see below.

A pencil balanced on its point. Is there a trick? See below.

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When did Hammurabi reign?

The consequences of the Earth’s changing climate may be very grave. It is essential to understand past climate change so that we can anticipate future changes. This week, That’s Maths in The Irish Times ( TM047 ) is about the chronology of the Middle East. Surprisingly, this has important implications for our understanding of climate change.

Left:  Image of Hammurabi in the US Congress. Right: Part of an inscription of the Code of Hammurabi.

Left: Image of Hammurabi in the US Congress.
Right: Part of an inscription of the Code of Hammurabi.

Continue reading ‘When did Hammurabi reign?’


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