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Posts tagged with Pythagoras

In 518 BC, Pythagoras journeyed west … interview with the … ruler Leon of Phlius while both were attending the Olympic Games. Prince Leon was most impressed by Pythagoras’ range of knowledge and asked in which of the arts he was most proficient.

Pythagoras replied that, rather than being proficient in any art, he regarded himself as being a “philosopher”. Prince Leon had never heard this term before—it had been newly coined by Pythagoras—and asked for an explanation.

Pythagoras said: “Life may well be compared with these pan-Grecian Games. For in the vast crowd assembled here, some are led on by the hopes and ambitions of fame and glory, while others are attracted by the gain of buying and selling, with mere views of profit and wealth. But among them there are a few, and they are by far the best, whose aim is neither applause nor profit, but who come merely as spectators through curiosity, to observe what is done, and to see in what manner things are carried on here.

“It is the same with life. Some are slaves to glory, power
and domination; others to money. But the finest type of man gives himself up to discovering the meaning and purpose of life itself. He, taking no account of anything else, earnestly looks into the nature of things. This is the man I call a lover of wisdom, that is, a philosopher. As it is most honourable to be an onlooker without making any acquisition, so in life, the contemplation of all things and the quest to know them greatly exceed every other pursuit.”

free translation by A.H. Louie of anecdote recorded in Marcus Tullius Cicero (c. 45 BC) Tusculanae Questiones - Liber Quintus, III: 8-9

What a self-serving view! I regard my own addiction to philosophy as base and irrational. I can think of many ways in which it’s made my life worse and none in which it’s provided a benefit for anyone else.

Yet it’s not at all infrequent I read philosophers claiming they’re better than other people—without proof, of course.

(And if you try to take the route of philosophy → natural philosophy → technology: well first you’re avoiding the question of whether pure thought is per se valuable, and second neither Thomas Newcomen nor Robert Conrad were natural philosophers.)

(Source: ontoslink.com)




Much learning does not teach understanding; else it would have taught Hesiod and Pythagoras, and again Xenophanes and Hecataeus.

Heraclitus (who died ∼475 BC)

via University of David




Everybody knows that

(referring to the sides of a right triangle). That formula tells you the straight-line distance on a flat plane. Say I walked three blocks east in a grid-style city and then four blocks north.  Then I’ve traveled

blocks i.e. 5 blocks, as the crow flies.  I could also go one block down to the sub-sub-…-sub-basement and then the same rule would apply in 3-D, as the mole burrows



blocks north-east-down.

You see the same formula—weird, right?—in statistical calculations of standard deviation:


Each of the |dev|'s is a data point's deviation from the center (middle, barycenter, mean) of all such observations.

So standard deviation is a Pythagorean concept — in a way, situating your n data points on the corner of an n-dimensional box and then calculating a hypotenuse.  If that doesn’t make you wary of standard deviation as the only, absolute measure of variability … well, it should.

But so, what’s up with all the 2's?  Higher mathematics doesn’t use numbers! Replace each of the ^2's with a ^p for any power, and you’ve got ∞ new, valid, measures of distance, called L_p norms. This symbolic change gives rise to some mind-expanding imagination weapons, including the much-hyped non-Euclidean geometry (p ≠ 2, Cthulhu fans).

p=1 corresponds to city-block distance — (not in Boston, Prague, or Edinburgh because the streets don’t connect squarely – in some imaginary flat, square Lattice City).

p=2 corresponds to Navy distance on a flat lake. If you sail or fly across the Atlantic, the curvature of the Earth starts to make a difference and p ≠ 2.
p=4 could measure financial volatility in a way that penalizes kurtosis. (It’s still a two-way measure, though. Better would be p=5 or another high, odd number, to get a quasimetric.)
These new conceptions of “distance” make more sense of physical reality:
General relativity (which gave us GPS’s and satellites) requires Riemannian geometry, since spacetime is curved.
Euclidean geometry (p=2) fails if you’re Magellan, since the Earth’s surface is curved.
And we wouldn’t have the atom bomb without the even more brain-wrinkling geometry of nuclear physics.
If you want to tolerate noise or slight differences among data, Lp norms let you treat similar things as the same while still maintaining différence among quite different things.
If you want to use mathematics on things besides physics – like text mining, psychology, chess, racism, self-versus-other, financial time series, cluster analysis, terms of trade, marketing data, voting, bargaining, morality, functional spaces, utility theory, strategic arms races – you’ve got to be aware of the distance measure in your relevant space.

Everybody knows that

a² + b² = c²

(referring to the sides of a right triangle). That formula tells you the straight-line distance on a flat plane. Say I walked three blocks east in a grid-style city and then four blocks north. Then I’ve traveled

sqrt( 3² + 4²)

blocks i.e. 5 blocks, as the crow flies. I could also go one block down to the sub-sub-…-sub-basement and then the same rule would apply in 3-D, as the mole burrows

sqrt( 3² + 4² + 1²) = sqrt(26)

blocks north-east-down.

You see the same formula—weird, right?—in statistical calculations of standard deviation:

sqrt( dev&sub1; ²  +  dev&sub2; ²  +  dev&sub3; ² + ...) = ST DEV

Each of the |dev|'s is a data point's deviation from the center (middle, barycenter, mean) of all such observations.

So standard deviation is a Pythagorean conceptin a way, situating your n data points on the corner of an n-dimensional box and then calculating a hypotenuse. If that doesn’t make you wary of standard deviation as the only, absolute measure of variability … well, it should.

But so, what’s up with all the 2's? Higher mathematics doesn’t use numbers! Replace each of the ^2's with a ^p for any power, and you’ve got ∞ new, valid, measures of distance, called L_p norms. This symbolic change gives rise to some mind-expanding imagination weapons, including the much-hyped non-Euclidean geometry (p ≠ 2, Cthulhu fans).

p=1 corresponds to city-block distance — (not in Boston, Prague, or Edinburgh because the streets don’t connect squarely – in some imaginary flat, square Lattice City).

p=2 corresponds to Navy distance on a flat lake. If you sail or fly across the Atlantic, the curvature of the Earth starts to make a difference and p ≠ 2.

p=4 could measure financial volatility in a way that penalizes kurtosis. (It’s still a two-way measure, though. Better would be p=5 or another high, odd number, to get a quasimetric.)

These new conceptions of “distance” make more sense of physical reality:

  • General relativity (which gave us GPS’s and satellites) requires Riemannian geometry, since spacetime is curved.
  • Euclidean geometry (p=2) fails if you’re Magellan, since the Earth’s surface is curved.
  • And we wouldn’t have the atom bomb without the even more brain-wrinkling geometry of nuclear physics.
  • If you want to tolerate noise or slight differences among data, Lp norms let you treat similar things as the same while still maintaining différence among quite different things.

If you want to use mathematics on things besides physics – like text mining, psychology, chess, racism, self-versus-other, financial time series, cluster analysis, terms of trade, marketing data, voting, bargaining, morality, functional spaces, utility theory, strategic arms races – you’ve got to be aware of the distance measure in your relevant space.


hi-res




Pythagorean Theorem
This is how I first really understood the Pythagorean Theorem.
The outer circle looks just a little bit larger than the inner circle. But actually, its area is twice as large.
Kind of like the difference between medium and large soda cups, or how a tiny house still requires kind of a lot of timber, for how much air it encloses. If you buy a slightly wider pizza or cake it will serve proportionally more people; and if an inverse-square force (sound, radio power, light brightness) expands a little bit more it will lose a lot of its energy.
Ideas involved here:
scaling properties of squared quantities(gravitational force, skin, paint, loudness, brightness)
circumcircle & incircle
√2
This is also how I first really understood √2, now my favourite number.

Pythagorean Theorem

This is how I first really understood the Pythagorean Theorem.

The outer circle looks just a little bit larger than the inner circle. But actually, its area is twice as large.

Kind of like the difference between medium and large soda cups, or how a tiny house still requires kind of a lot of timber, for how much air it encloses. If you buy a slightly wider pizza or cake it will serve proportionally more people; and if an inverse-square force (sound, radio power, light brightness) expands a little bit more it will lose a lot of its energy.

Ideas involved here:

  • scaling properties of squared quantities
    (gravitational force, skin, paint, loudness, brightness)
  • circumcircle & incircle
  • √2

This is also how I first really understood √2, now my favourite number.


hi-res




Pythagoras
visual proofs of the Pythagorean Theorem, a² + b² = c²

Pythagoras

visual proofs of the Pythagorean Theorem, a² + b² = c²


proof 3

proof 2

proof 4


hi-res