In developing a way of life which distinguishes the Pythagoreans from the rest of the world, Pythagoras clearly distinguishes himself from the Milesians and their nature philosophy, as K&R point out: but the distinction is not quite so clear-cut -- Heraclitus too begins philosophy as a way of life, insofar as he includes a concern with a certain kind of wisdom.
In any case, Marias' comment is also helpful here:
We have in the Pythagorean school a first clear example of philosophy understood as a way of life. The problem of the self-sufficient life issues in a special discipline consisting in contemplation. There appears here the theme of freedom, of self-reliance.
"The Mystical Side of Pythagoras' Teaching"
Includes the notion of the transmigration of the soul; story of Pythagoras recognizing a soul in a whipped dog.
More generally establishes the kinship of all living things. This kinship follows from the notion that souls qua eternal can be reincarnated in a variety of living things: the suggestion is that the process is cyclical (--> source of Nietzsche's notion of eternal return). To put it still differently: the kinship of all living things is a "biological" expression of an emphasis on unity apparent in physical matters.
P. further established rules of abstinence and prohibitions, including a "philosophically-inspired vegetarianism."
A Pythagorean vocabulary:
kosmos (an orderliness found in the arrangement of the universe)
schole ("leisure" --> "school")
mania ("orgy") --> "enthusiasm" (en-theos) --> sophos
philosophia (includes notions of freedom and self-reliance)
mathematikos (fond of learning)
bios theoretikos (the theoretical life) -- see above]
"By contemplating the principle of order revealed in the universe -- and especially in the regular movements of the heavenly bodies -- and by assimilating himself to that orderliness, man himself was progressively purified until he eventually escaped from the cycle of birth and attained immortality."
In other words, what we might characterize as a religious sort of salvation is centrally dependent upon a "scientific" understanding of the central order of nature, and an "ethic" (from ethos, habit, rule or pattern of behavior) based on that "scientific" understanding which seeks to replicate the cosmic order in the life of the individual.
(This complementary attitude towards what we might call "faith" and "reason" will reappear in the early Middle Ages, and make possible both the recovery of the ancient Greek and Roman developments in science, mathematics, technology, and philosophy , especially as expanded and refined in the Muslim world - and the development of these knowledges and disciplines into the foundations of modern natural science. The sense of opposition between "faith" and "reason" emerges in the West primarily post-Augustine (4th ct. C.E.) through the Dark Ages, and again with Cartesian dualism (17th ct. C.E.) and some strands of modern Protestantism [especially 19th ct. North American Fundamentalism].)
1) establishes an ultimate dualism between Limit and the Unlimited;
2) establishes the equation/identity of things with numbers.
More specifically, it is probable that Pythagoras discovered that the chief musical intervals are expressible in simple numerical ratios of the first four integers, i.e.:
Octave -- 2:1
Fifth -- 3:2
Fourth -- 4/3
This discovery, coupled with the discovery/invention of a mathematical order to the universe itself (familiar to us since Anaximander), leads to the venerable notion of "the harmony of the spheres." As Julian Marias paraphrases it: since the distances of the planets correspond approximately to the musical intervals -- then every/ star emits a note, all the notes together comprise the harmony of the spheres, a celestial music. We do not hear it because it is constant and without variation.
While we may be tempted to dismiss such a notion, note that this vision provided a foundation for such "modern" figures as:
a) Copernicus (who follows the Pythagorean astronomer Ecphantus in affirming the rotation of the earth), and
b) Kepler (who diligently searched for over 10 years to find the Pythagorean harmonies -- discovering the three laws of planetary motion in the process). [We will also hear the computer realization of Kepler's version of the Harmonia Mundi when we explore modern philosophy and natural science.]
More broadly, as K&R put it:
If the musical scale depends simply upon the imposition of definite proportions on the indefinite continuum of sound between high and low, might not the same principles, Limit and the Unlimited, underlie the whole universe? If numbers alone are sufficient to explain the "consonances," might not everything else be likewise expressible as a number of a proportion?
Moreover, since the first four integers contain the whole secret of the musical scale, their sum, the number 10 or the Decad, might well "seem to embrace," as Aristotle puts it, "the whole nature of number," and so come to be regarded, as it certainly was, with veneration. As well, the first four integers generate the three maior figures beyond the point (cf. Speusippus, Kirk & Raven, pp. 253ff.).
Also attributed to Pythagoras - the Pythagorean theorem, with its corrollary, the incommensurability of the diagonal and the side of a square. Revealing this secret cost one poor student his life, it is said.
[For those who are really with it: the experience and conception of a harmony (=connection in the face of difference) avoids the conflict implicitly raised by Anaximander (dualism) and Anaximenes (monism) -- and between Parmenides (dualism) and Heraclitus (monism)]