(aleph-null) and its successors. This derived from his development of set theory. The whole
discussion is even more abstruse than Fermat.Physics is mathematical not because we know so much about the physical world, but because we know so little: it is only its mathematical properties that we can discover.
Bertrand Russell, An Outline of Philosophy, p. 163
This is a report on various books intended to provide access to some of the beauty of mathematics to non-mathematicians. Recommendations are always appreciated.
It wasn't the last theorem Fermat proposed—merely the last to be solved. He stated that an + bn = cn for n>2 has no solution for integers a, b, and c and followed it with a note saying "I have a truly marvellous proof of this proposition which this margin is too narrow to contain" (Wikipedia). He never wrote down his solution and others have sought it ever since. A Princeton mathematician, Andrew Wiles, obsessed on the problem and after a number of years—and using modern mathematical techniques not dreamt of for centuries after Fermat—came up with a solution—he thought. But there was yet a stumbling block that took another year to resolve.
The mystery remains, however—what was the "truly marvellous proof"? It certainly wasn't Wiles' method. There have been thousands of invalid proofs—was Fermat's proof only one of those? We'll never know.
Georg Cantor addressed the problem of infinities—how many are there and are some larger than
others. He also invented transfinite numbers, an esoteric quantity annotated as
(aleph-null) and its successors. This derived from his development of set theory. The whole
discussion is even more abstruse than Fermat.
Even in the Renaissance the concept of a mathematical zero did not exist. Roman numerals were still in common use and nearly all reckoning was carried out on an abacus. Yes, one cleared the abacus before starting. But that wasn't zero; it was--well, nothing. It was the starting point, not an annotation. As with anything new at that time, zero was often condemned as an instrument of the devil when Arabic numerals were introduced and its introduction was forced as a placeholder. But it still took some time before a naked zero was accepted as a valid notation and useful in calculation.
Made famous in Dan Brown's The Da Vinci Code, phi is a remarkable number. It is (a+b)/a =
a/b = 
or an irrational number
approximating 1.6180339887…. The simplest calculation of its value is through the equation
(1+√5) / 2. It is called the golden ratio because it is thought to represent a figure of perfect
proportions. The front façade of the Parthenon is supposed to be a golden ratio, with some
pushing and pulling, and many ancient structures show an approximation of it in façades,
porticos, doors, windows, and elsewhere. Some musical compositions use the golden ratio or its
relative the Fibonacci sequence, though as in architecture, some use is more speculative than
actual. It also occurs in nature, most notably in the spiral of the sunflower head. But it also
occurs in leaf veins, stem spirals, and chambered nautilus.