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What did the Greeks call the "trial and error" reasoning process?

Bruce Aune's review of Wilson's Peirce's Empiricism: Its Roots and Its Originality claims "The name 'empirici' is in fact traceable to a Greek verb meaning 'to make trial of.'"

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"Trial and error" applied not to a "reasoning process" but to medical practice, and the name of the practice was derived from Greek ἐμπειρία, experience. The inspiration for the approach apparently came from Greek Pyrrhonism, which recommended permanently suspending judgment on how things are in their own nature. This is what Pyrrho of Elis (c. 360 – 270 BC) called ακατάληψία in contrast to Stoic κατάληψις, perceptual grasping of knowledge. As Aune writes in the review:

"He begins with the ancient empirici who were physicians. Unlike their competitors, the dogmatici, the empirici paid little attention to the supposed underlying causes of disease; their concern was principally directed to methods of treatment founded on trial and error... Later empiricists were less concerned specifically with trial and error practice; they instead focused on endorsing only factual claims with an evidential basis in sense experience".

Wikipedia even has an article about the empirici. Perhaps the best known figure of the school is Sextus Empiricus (c. 160-210 AD), whose multi-volume Adversus Mathematicos is one of our major sources on the ancient science and philosophy. Literally, the title translates as Against the Mathematicians, but Greek tà mathémata meant what can be learned and taught, so the commonly used free translation Against the Professors is closer to the original meaning.

But Sextus belongs to the later, more philosophical, generation of empirici. The founders of the school in 3rd century BC, Serapion of Alexandria and Philinus of Cos, were physicians who opposed the "dogmatic" Hippocratic tradition. Their works have not survived, but some fragments are quoted by Galen, Pliny and others.

On the methodology of Hellenistic science in general, such as it was, trial and error included, the standard reference is Russo's Forgotten Revolution.

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The Greeks understood the process of trial and error. You try one way, then you try another. But there is no evidence of which I'm aware that the Greeks elevated trial and error into a methodology of experimentation - a 'reasoning process' explicitly flagged as a proper part of valid scientific inquiry - such as we find in Bacon and later thinkers.

In Aristotle only 'techne' and 'episteme' even brush against this possibility. 'Epagoge', which is generally translated as 'induction', does not have a sense related to induction as any kind of scientific method. (By epagoge I realise, for instance, on seeing one triangle that all triangles are three-sided figures with three internal angles. This isn't experimental science or conducive to it.)

You might find the following account of Aristotle's terminology useful :

https://plato.stanford.edu/entries/episteme-techne/#3

None of this is to say that we do not find experimentation in the work of Greek scientists. Archimedes's work on floating bodies, for instance, appears to have involved a series of experiments systematically conducted. But brilliant scientist though he was, he had no theory of science into which the methodology of experimentation, of trial and error, was an explicitly acknowledged element.

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