What is more fundamental - Mathematics or Logic

 Let’s start this from the beginning… oh…and I do mean THE BEGINNING!

Though this logos is true always, yet men are always unable to understand it when they hear it for the first time as before they have heard it at all - Heraclitus

Aristotle names what we now call ‘pre-socratics’ the physiologoi (φυσιολόγοι) meaning ‘those who discoursed on nature’. Indeed the entire works of Plato and Aristotle are retrospective attempts to address the challenges made by these Philosophers. Several which are important to our question:

1. lógos (λόγος) by Heraclitus (535 – c. 475 BC) that translates into something like ‘a debated accounting of the portions and reason of things‘.
2. kósmos (κόσμος) by Pythagoras (570 – c. 495 BC) that translates into something like ‘a universal complex and orderly arrangement’.

For Pythagorean’s the reality of the kósmos was the Monad or monas (μονάς) this was a ‘singularity‘, ‘aloneness‘ or ‘unit‘ represented by the number One. By extension, all real numbers were seen as manifestations of this unitary identity and therefore the part of an underlying principle behind being.

The plurality of numbers become the second fundamental Pythagorean principle called Dyad (ˈdʌɪad) that can be known as ‘twoness’ or ‘otherness’ and represent the material plurality within the divinity of the Monad. These represent a universe of opposing powers and the subject/object duality. These opposites are only resolved through the third principle of Harmony aka harmonía (ἁρμονία) meaning ‘fit together‘, ‘joining’ and ‘ordinance’.

The essence of Pythagorean harmonía is analogía (ἀναλογία) from aná (ἀνά) meaning in ‘sequence’ with the lógos(λόγος) as the means to ‘portion and account for the reason of things’. Pythagorean’s called those who where most learned in the lógos of shape, quantities and arrangements the mathēmatikḗ tékhnē (μαθηματικός τέχνη).

• Manthanein (μάθημα) meaning ‘what one gets to know’ or ‘knowledgeable learning’
• tikós (τικός) meaning ‘skilled’ or ‘suitably adept’ and ‘proficiency‘.
• tékhnē (τέχνης) meaning ‘abilities’ or ‘a skill’ and ‘a technical trade’

The mathēmatikḗ tékhnē were cosmologists, discussing astrology, music and theology. Because these inquiries were grounded on a creative and rigours approach to ratio, number and geometry, we have appropriated this to mean mathematics per se. While the meaning of mathematics was broader for Pythagorean’s, their notion of proofs were far more restrictive.

In Ancient Greece mathematical proofs were demonstrated through a deductive count called analytikós (ἀναλυτικὰ) from the words analytos (ἀναλυτός) 'solvable' that is in essence analyo (ἀναλύω) meaning to loose. These where physically demonstrated by distribution and count of pebbles called psephos (ψῆφος). Aptitude in this area was known as arithmētikē, from the worlds arithmos (ἀριθμός) meaning ‘distribution‘ or ‘numbered order’ and téchne (τέχνη) meaning ‘abilities‘ or ‘skill‘.

For Pythagoreans the practice of mathematics was deeper than accounting and it’s physical applications. Pythagoras was the first to call him himself a Philosopher (‘lover’ of ‘wisdom’) and for him this meant ‘technical’ ‘proficiency’ in ‘what one gets to know‘ (mathēmatikḗ tékhnē). He believed that catharsis of the psyche was only possible by contemplating the nature of cosmos through the first principles.

For Pythagoreans this was achieved by studious 
mathematical accounting of ratios and distributions that represent the units harmonía with unity, the essence of which isanalogía… the sequence of reasons.