
Turing's Euclidean Stroke of Genius - soundsop
http://syntax.wikidot.com/blog:11
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antiform
To most mathematicians, valuing intuition is not a new idea. While one of the
most distinguishing qualities of mathematics is its rigor, to focus on this
quality misses the big picture. Henri Poincare, arguably the greatest 20th
century mathematician, said it best: "It is by logic we prove, it is by
intuition we invent." This theme is echoed in many of his writings about
creativity in mathematics.

Turing's argument is beautiful and elegant, undoubtedly one of the gems of
modern mathematics. However, the reason you don't see the intuition behind
Turing's argument taught is that there is too much material to cover in a
general theoretical computer science course. You need to be able to master the
known material, to stand on the shoulders of giants, to make new advances in
the discipline. This may not necessarily be the best way to encourage
creativity and insight in computer science, but the hope is that interested
students will delve deeper (like the author) and discover insights for
themselves.

Lastly, at least among many mathematicians, Turing may be up there with Euclid
among the demigods of mathematics, but there are those that lie still higher,
people like Gauss and Euler who have yet to be matched in terms of
mathematical output and impact.

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pg
I don't think the greatness of Euclid (or more precisely, he and the many
unnamed predecessors of whose work he was the compiler) was that he came up
with especially user-friendly axioms. What other axioms would someone who was
less skillful at coming up with user-friendly axioms have produced instead?

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antiform
Perhaps somebody would have discovered hyperbolic or other non-Euclidean
geometries before Lobachevsky? Through the lens of modern mathematics, non-
Euclidean geometry seems like such a simple extension/modification of
Euclidean geometry, yet it was only explored and developed in detail in the
early 19th century.

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dangoldin
Well it took more than a thousand years for the belief of a spherical Earth to
be accepted. It's hard to imagine a new type of geometry when you believe the
Earth is flat.

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pg
Actually in ancient times they knew the world was spherical. It's debatable
how many people really thought it was flat even in the darkest of the dark
ages.

~~~
dangoldin
Hmm. You learn something new every day.

Turns out that you can actually measure the Earth's circumference pretty
easily with some basic geometry. You can see the distance at which a ship
starts to disappear on the horizon and are able to get a relation of distance
to angle drop.

It's odd how they teach that the round Earth theory started with Copernicus.

Let me try another stab at figuring out why non Euclidean geometry took so
long to arise: I'm under the impression that a large portion of Greek geometry
was based on construction using a compass and protractor. Seeing that it is
difficult to use those instruments on spherical or hyperbolic planes they
never went beyond Euclidian geometry.

~~~
antiform
On top of that, you don't even an ocean to measure the Earth's circumference.
Eratosthenes of Cyrene (born in the 3rd century BC) calculated the
circumference of the earth fairly accurately using the angle of elevation of
the Sun measured from two different cities. Note that implies that a round
earth was common knowledge to the Greeks even at this date.

