If you find yourself Pythagoras left no web log, the new main attract out of Pythagorean teachings remains obvious. Number is actually a universal archetype, «the main, source, and you will cause of everything.» More particularly, things are consisting of «the sun and rain regarding number,» that Pythagoreans recognized as the fresh Minimal additionally the Limitless. The fresh new Pythagoreans engaged in the research out of number principle or arithmetic (number in itself), geometry (matter in space), harmonics otherwise tuning principle (matter over time), and you can astronomy (number in dimensions and you will big date). Such meanings coincide really on the modern definition of math just like the «the research off activities in space and you may big date.»
Because the matter and its qualities instance polarity, harmony, and proportion was archetypal prices you to definitely underlie real manifestation, math was found as opposed to conceived, and has now the advantage so you’re able to reflect the essential characteristics off fact, instead of just acting they otherwise discussing it. More over, about Pythagorean look at, because amount try common, it is reasonably divine. If you are modern science is formulated when you look at the experimental terms and conditions, ancient greek language technology try mathematical; and even though math has actually practical software, Pythagoras’s strategy elevated math so you can a survey value seeking above one purely utilitarian finishes whereby it was once employed.
Like this, the fresh new worlds out of «fact» and you may «value» commonly separate domains, however, naturally related
Pythagoras is actually the initial person to label brand new universe good kosmos. The brand new Greek term, the foot of the keyword cosmetic, refers to an equal exposure out of purchase and you may beauty. The new market was an effective cosmos given that phenomena regarding characteristics embody geometrical mode and you may ratio. Such proportions allow what to unfold and setting on the really feminine and you can efficient indicates (that’s a fact off character), also bring about beauty (that’s a value). This is why, new classical Pythagorean metaphor likens the newest world to help you a full time income system as opposed to a keen inanimate host. Plato identifies brand new Pythagorean take a look at better when he describes the newest cosmos once the «one Entire out of wholes» so that as «an individual Way of living Animal hence surrounds all of the traditions pets that will be in it» (Timaeus 33 An effective and you will 30 D).
The fresh Pythagoreans (certainly just who Plato should be measured) recognized a basic matchmaking between proportion as well as the concept out-of justice, in which «every section of the whole gets the best owed
» They thought that one particular characteristics from fairness might possibly be realized through the study of continued geometrical ratio (analogia) and you will from examination of this new statistical ratios of one’s musical measure, in which 
the several extremes of the music scale was bridged due to all types of mathematical ratio. Main into the Greek notion of proportion ‘s the notion of searching for «means» or version of mediation ranging from extremes. For the tuning principle, the new Pythagoreans known the latest arithmetic, mathematical, and you will harmonic function and that underlie the fresh tunes size, together with finest consonances away from sounds, that are mathematical ratios: this new octave (1:2), the ideal 5th (2:3), and also the best 4th (3:4). This new Pythagoreans compared an only and you can really-bought neighborhood to help you a well-updated lyre. While each and every mention holds their individuality, are common proportionally connected with her within the a much bigger whole to make a musical measure, and all of are interdependent regarding their dependence on one another. (Pick Plato, Republic 443 D444). Justice is present in virtually any well-working organism, society-as well as the heart.
From inside the Pythagorean think, amount gives increase so you’re able to proportion, and ratio gives go up so you can harmony. The brand new Greek keyword harmonia mode «fitted together» or «bringing together.» Harmony and you can justice ‘s the results of a beneficial proportion produced reveal, while the kosmos is an equilibrium in which each one of new bits are proportionally likely along with her. While every and each system-including the cosmos-was a good unity, it is harmony that enables the new parts to be hired together with her since the an integral whole. Equilibrium, fairness, and ratio relate with Greek medical theory, since healthy bacteria enjoys a form of dynamic equilibrium where the many aspects collaborate; when shortage of harmony is available, infection often results.