The Ancient Greeks and the
Foundations of Mathematics
1.1 Pythagoras
1.1.1 Introduction to Pythagorean Ideas
Pythagoras (569–500 B.C.E.) was both a person and a society (i.e., the
Pythagoreans). He was also a political figure and a mystic. He was
special in his time because, among other reasons, he involved women as
equals in his activities. One critic characterized the man as “one tenth
of him genius, nine-tenths sheer fudge.” Pythagoras died, according to
legend, in the flames of his own school fired by political and religious
bigots who stirred up the masses to protest against the enlightenment
which Pythagoras sought to bring them.
As with many figures from ancient times, there is little specific that
we know about Pythagoras’s life. We know a little about his ideas and
his school, and we sketch some of these here.
The Pythagorean society was intensely mathematical in nature, but
it was also quasi-religious. Among its tenets (according to [RUS]) were:
• To abstain from beans.
• Not to pick up what has fallen.
• Not to touch a white cock.
• Not to break bread.
• Not to step over a crossbar.
• Not to stir the fire with iron.
• Not to eat from a whole loaf.
• Not to pluck a garland.
• Not to sit on a quart measure.
• Not to eat the heart.
• Not to walk on highways.
• Not to let swallows share one’s roof.
• When the pot is taken off the fire, not to leave the mark
of it in the ashes, but to stir them together.
• Not to look in a mirror beside a light.
• When you rise from the bedclothes, roll them together
and smooth out the impress of the body.
The Pythagoreans embodied a passionate spirit that is remarkable
to our eyes:
Bless us, divine Number, thou who generatest gods
and men.
and
Number rules the universe.
The Pythagoreans are remembered for two monumental contributions
to mathematics. The first of these was to establish the importance
of, and the necessity for, proofs in mathematics: that mathematical
statements, especially geometric statements, must be established by
way of rigorous proof. Prior to Pythagoras, the ideas of geometry were
generally rules of thumb that were derived empirically, merely from observation
and (occasionally) measurement. Pythagoras also introduced
the idea that a great body of mathematics (such as geometry) could be
derived from a small number of postulates. The second great contribution
was the discovery of, and proof of, the fact that not all numbers are
commensurate. More precisely, the Greeks prior to Pythagoras believed
with a profound and deeply held passion that everything was built on
the whole numbers. Fractions arise in a concrete manner: as ratios of
the sides of triangles (and are thus commensurable—this antiquated terminology
has today been replaced by the word “rational”)—see Figure
1.1.
Pythagoras proved the result that we now call the Pythagorean theorem.
It says that the legs a, b and hypotenuse c of a right triangle (Figure
1.2) are related by the formula
a2 + b2 = c2 . (?)
This theorem has perhaps more proofs than any other result in
mathematics—over fifty altogether. And in fact it is one of the most
ancient mathematical results. There is evidence that the Babylonians
and the Chinese knew this theorem nearly 1000 years before Pythagoras.
In fact one proof of the Pythagorean theorem was devised by President
James Garfield. We now provide one of the simplest and most
classical arguments. Refer to Figure 1.3.
Foundations of Mathematics
1.1 Pythagoras
1.1.1 Introduction to Pythagorean Ideas
Pythagoras (569–500 B.C.E.) was both a person and a society (i.e., the
Pythagoreans). He was also a political figure and a mystic. He was
special in his time because, among other reasons, he involved women as
equals in his activities. One critic characterized the man as “one tenth
of him genius, nine-tenths sheer fudge.” Pythagoras died, according to
legend, in the flames of his own school fired by political and religious
bigots who stirred up the masses to protest against the enlightenment
which Pythagoras sought to bring them.
As with many figures from ancient times, there is little specific that
we know about Pythagoras’s life. We know a little about his ideas and
his school, and we sketch some of these here.
The Pythagorean society was intensely mathematical in nature, but
it was also quasi-religious. Among its tenets (according to [RUS]) were:
• To abstain from beans.
• Not to pick up what has fallen.
• Not to touch a white cock.
• Not to break bread.
• Not to step over a crossbar.
• Not to stir the fire with iron.
• Not to eat from a whole loaf.
• Not to pluck a garland.
• Not to sit on a quart measure.
• Not to eat the heart.
• Not to walk on highways.
• Not to let swallows share one’s roof.
• When the pot is taken off the fire, not to leave the mark
of it in the ashes, but to stir them together.
• Not to look in a mirror beside a light.
• When you rise from the bedclothes, roll them together
and smooth out the impress of the body.
The Pythagoreans embodied a passionate spirit that is remarkable
to our eyes:
Bless us, divine Number, thou who generatest gods
and men.
and
Number rules the universe.
The Pythagoreans are remembered for two monumental contributions
to mathematics. The first of these was to establish the importance
of, and the necessity for, proofs in mathematics: that mathematical
statements, especially geometric statements, must be established by
way of rigorous proof. Prior to Pythagoras, the ideas of geometry were
generally rules of thumb that were derived empirically, merely from observation
and (occasionally) measurement. Pythagoras also introduced
the idea that a great body of mathematics (such as geometry) could be
derived from a small number of postulates. The second great contribution
was the discovery of, and proof of, the fact that not all numbers are
commensurate. More precisely, the Greeks prior to Pythagoras believed
with a profound and deeply held passion that everything was built on
the whole numbers. Fractions arise in a concrete manner: as ratios of
the sides of triangles (and are thus commensurable—this antiquated terminology
has today been replaced by the word “rational”)—see Figure
1.1.
It says that the legs a, b and hypotenuse c of a right triangle (Figure
1.2) are related by the formula
a2 + b2 = c2 . (?)
This theorem has perhaps more proofs than any other result in
mathematics—over fifty altogether. And in fact it is one of the most
ancient mathematical results. There is evidence that the Babylonians
and the Chinese knew this theorem nearly 1000 years before Pythagoras.
In fact one proof of the Pythagorean theorem was devised by President
James Garfield. We now provide one of the simplest and most
classical arguments. Refer to Figure 1.3.
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