> We are preparing a Millennium timeline covering maths since the year 1000.
>
> We end by examining the big ideas in contemporary mathematics, and even
It will be more realistic to say "some big ideas", not "THE big ideas".
> more tantalisingly we sneak a preview at what mathematics might important
> a century from now, in the year 2100.
Lagrange thought that mathematics was practically finished at his age.
This shows how futile it is to try to predict what will happen in a
hundred years time. I cannot take such a preview seriously.
> January: Maths around the year 1000 (bi-biography)
> Al-B and Al-K (Al-Biruni and Al-Khwarizmi) were two prominent Arab
> mathematicians from Khwarizm (Khorezmskaya), the region south of the
> Aral sea. They were both central figures in developing the decimal
> number system that we use today, and in bringing it to the West.
The Arab writing mathematicians not only developed decimal notation, they
also gave irrational numbers equal rights in the realm of Number. And
they developed the language, though not yet the notation, of algebra. One
of the influential persons in both areas was Omar Khayyam, known in the
west more as a poet (I consider that an important point; too many people
still believe that mathematicians have to be dry and uninteresting).
> ...
> May: Maths around the year 1400
> Chaucer's "Treatise on the astrolabe"
On the one hand, it's nice to note that Chaucer was interested in
mathematics. On the other hand, astrolabes were used mostly in medieval
times and in the Moslim world. To connect them with the name of Chaucer
of all people seems strange.
> June: Maths around the year 1500
> Invention of + and - signs etc (Cajori)
Here one can mention that negative numbers and the rules to calculate
with them were introduced in India centuries earlier.
> The Treviso arithmetic (1478): reprinted in Smith: Source Book
> Pacioli 1494
During the 16th century cubic and quartic equations were solved
algebraically, by Ferro, Tartaglia, Cardano, and Ferrari. May note the
strange history and bitter controversies
around these discoveries. They in turn brought about the discovery of
imaginary and complex numbers, by Cardano and Bombelli.
> July: Maths around the year 1600 (bi-biography)
> In the year 1600, Rene Descartes (1596-1650) was aged about four,
> and Pierre de Fermat (1601-1665) was yet to be born. Some thirty years
> later, these tow children were to astonish the world by simultaneously
> coming upon the same BIG idea - namely how to bring algebra and
> geometry under one common framework.
>
> OR .... Galileo (1564-1642)
> Napier (1550-1617)
And Buergi (1552-1632)
> Kepler (1571-1630) begins assisting Brahe (1546-1601)
>
> August: Maths around the year 1700
> Newton (1642-1727)
And Leibnitz (1646-1716). Time and again we see the same great idea
developed by more than one great person. The discoverers are, however,
usually not great enough to overcome petty arguments about priority.
> Stress continuities with Descartes.
>
> September: Maths around the year 1800 (bi-biography? Gauss & Sophie
> Germain)
> Gauss (1777-1855): 17-gon 1796
> Gaussian integers: a + bi where a and b are ordinary integers.
> A Gaussian integer is prime if it cannot be expressed as the product
> of two other Gaussian integers (excepting the units 1, i, -1 and -i).
> Links with Fermat's last theorem (Katz: 585-591
Gauss proved in 1798 the fundamental theorem of algebra, so, perhaps
mysteriously, numbers introduced to solve quadratic equations suffice to
solve all algebraic equations.
>
> October: Maths around the year 1900
Cantor introduced infinite numbers, again causing a controversy:
Poincare's disease is Hilbert's paradise.
> Hilbert lists 23 problems (JB has copy)
> Again - link with FLT?
>
> November: Maths around the year 2000
> Super-strings?
>
The advances in electronics during the second half of the 20th century
brought about a situation whereas mathematical theories formerly thought
to be the height of purity and inapplicability, say elliptic curves,
became important for rapid international communication. May be compared
with the importance of Riemannian geometry in relativity.
>
> December: Maths around the year 2100
Cannot predict that, but each of us has some favourite problems he would
like to see solved. I may mention the infinitude of perfect numbers and
the (in)existence of odd perfect numbers, these being perhaps the oldest
open problems in mathematics, going back to Pythagorean times. On the
less elementary side, I would like to see the theory of the finite simple
groups made really simple. A non-mathematical question that may have much
impact on mathematics (and on its applications, such as cryptography) is
whether quantum computers can really be made to work.
Avinoam Mann