Aristotle on small initial error

In De Caelo Aristotle formulates a thought that has repercussions far beyond the particular context in which it is uttered:

σχεδὸν γὰρ αὓτη πασῶν ἀρχὴ τῶν ἐναντιώσεων τοῖς ἀποφηναμένοις τι περὶ τῆς ὃλης φύσεως καὶ γέγονε καὶ γένοιτ΄ ἄν, εἴπερ καὶ τὸ μικρὸν παραβῆναι τῆς ἀληθείας ἀφισταμένοις γίνεται πόρρω μυριοπλάσιον. (De Caelo I v. 271b7-10)

"This, one could say, is what has been and will [continue to] be the source of all the contradictions among those pronouncing on beings as a whole, since going astray from the truth just a little initially becomes multiplied ten-thousandfold standing far [from the starting-point]." (De Caelo I v. 271b7-10)

The crucial word here is ἀρχὴ, which can be rendered in various ways in English as 'principle', 'starting-point', 'source', 'beginning', especially in the sense that the ἀρχὴ governs what proceeds from it. In the quote, this meaning is apparent because a little error in the principle results in major deviations from the 'truth' further down the line.

In Metaphysics Book Delta, Aristotle provides a definition of the multiple meanings of ἀρχὴ, pointing out that they are all a kind of ὄθεν, i.e. a 'whence'.

Most frequently, this 'whence' is taken in a chronological sense as the source whence something develops in linear time, but this is a restriction that suits the lazy thinking apparent everywhere, for instance, in evolutionary theory.

Or the 'whence' is taken as the starting-point, i.e. the premises, of an argument that proceeds by logical deduction. If your premises are false, so formal logic proclaims, then your conclusions will be fallacious. This is a favourite way of proceeding in mainstream philosophy and elsewhere: arguing for your position.

A deeper signification of ἀρχὴ, however, is the starting-point for thinking through phenomena conceptually. Such a path of thinking has to respect a sequence of thinking in which the concepts of more fundamental and elementary phenomena precede and provide the basis for thinking, i.e. forming adequate concepts for, more complex phenomena with more determinations. This is often called, as in Hegel, moving from the most abstract, simple phenomena step by step to the more concrete phenomena whose concepts require the concepts of simpler phenomena to be properly articulated.

One crucial case in point, which to date philosophical thinking has obstinately ignored, is whether a concept of time must precede the phenomena of movement and change to grasp them adequately, or whether time is a concept that presupposes the phenomenon of movement. The traditional conception of time is the latter: that time is a number lifted off movement by counting it. Hence Aristotle's famous formula for time:
"Time is the number of movement with regard to before and after." 
(ὀ χρόνος ἀριθμὸς κινήσεως κατὰ τὸ πρότερον καὶ ὕστερον ἐστιν. Phys. IV xi 219b2; cf. also De Caelo I ix. 279a15)

Elsewhere, in Aristotle's "before and after" & quantum gravity, I have pointed out that this definition of time is viciously circular. There has to be a deeper, more elementary conception of time on which this definition is based in order to account for the "before and after" in it. I won't repeat here what I have said before.

Here it is important to see that the path on which thinking thinks through phenomena makes a very big difference, and if you choose the wrong starting-point you will end up, down the road, very far from the truth of the phenomena themselves, no matter how clever your theories are. The truth here is not the conclusion of a valid logical argument, but an insight into how the phenomena show, disclose themselves of themselves if you care to look closely enough and do not obscure their self-disclosure with theoretical constructions.

Western thinking's starting with a linear conception of time counted off movement has had enormous repercussions for the trajectory of Western history as a whole, for without it there would be no science, no ἐπιστήμη, and therefore also no global technoscience. We would be on another historical path altogether on which science with its absolute will to efficient power would assume its proper, more modest place.

Further reading A Question of Time.