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Etymologists tend to categorize the probability of theories under formulaic labels. These range from "uncertain" over "tentative" or "not convincing" to "established", "accepted" or "nonsense". P values (or anything else to that effect) are not used, as far as I know, although chance similarities are argued for on quantitative grounds. Stochastic methods are used at least in recent computative mass comparison studies. So quantifying results is possible in principle, it's just not commonly done, although historic linguistics is full of unknowns. The basic question is: Is that unscientific? In the past it was not feasable, maybe, but does it need to continue like that? The underlying question is: How uncertain are most "established" etymologies, e.g. hundred < PGmc *hundaradą < *hundą < PIE *ḱm̥tóm?

Perhaps accademia.SE would be a better place to ask, if the question is tangential to other fields, e.g. history. But the question is specificly about the hindrences in etymology.

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    The use of statistics, notably P-value, is possible only in 2 conditions: a wide sample and your population has to follow a normal law. These conditions are not reunited in historical linguistics. – amegnunsen Nov 24 '18 at 12:58
  • @amegnunsen, exactly, that's the question. You imply that no metric space can be defined in which words for daughter from IE languages are normally distributed around the PIE reconstruction. Why? Even if p value was a loose analogy, maybe instead a bayesian tree for the dependency of one theory on another could be useful - you'd just estimate chances for a start. The power of a theory is it's ability to predict e.g. cognates and phonetic correspondances, but the more pressing need comes from the difficulty to account for semantic development. – vectory Nov 25 '18 at 20:23
  • You can use statistics even though you have a short sample (via non parametric statistics) if you are able to have a dependent and independent variables whose the values are numerical. And the distribution of these values is concentrated around a central value (mean, median, ...) which will be the "representative" of your sample. If you can manage to have those from historical linguistic data, so it is great. But, I will be agreeably surprised if you can. – amegnunsen Nov 25 '18 at 22:39
  • However, when you are talking about bayesian probability, it is nothing to do with statistics. This latter deals with data ("real" and measured variables), whereas for the former, data are unreal, they are supposed. – amegnunsen Nov 25 '18 at 22:41
  • @amegnunsen, I meg nunsens, too ;D nvm. The independent variable would be the meaning (described in English). Word Vectors are used to encode sense in a multidimensional space, coded numerical; while speach production is algorithmic, and algorithms can be encoded numerically. The dependent value is any comparanda to compare to. E.g. En. "gorse" and Ger. "Gerste" are close by, then a probable PIE root is the central element (meaning bushy). Testing the predictive power of the theory, we find Ger. "garstig". If we wouldn't find more, the null hypothesis "not related" would be stronger. No? – vectory Nov 26 '18 at 7:42
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You may want to look at D. Ringe's On Calculating the Factor of Chance in Language Comparison, which lays out some of the problems. I believe that uncontrolled variables are the greatest impediment to subjecting word-relatedness questions to valid statistical testing. Moreover, the idea that one could ever compute a p-value that a given word of a modern language is the result of ... (what?) from a hypothesized earlier word is a conceptual abuse of what a p-value is. It is mathematically possible to cook data so that a program will return a p-value: the question is, what reality does the number represent?

As an example of an uncontrolled variable, we might conjecture that bandanna derives from the same PIE root as bind, and could enter both words in a PIE-to-English table of pairs that would also contain 'father; hound; wheel' and corresponding PIE roots. In doing this, we will have failed to control the fact that 'bind, father, hound, wheel' are words transmitted Genetically through (West) Germanic into modern English, but 'bandanna' is recently borrowed from Hindi. Uncertainty is created because etymological claims usually mean "following this transmission path", yet "bandanna" does not follow that path (instead, the path is via Indic then recently by contact between Indic and English).

Another factor that has to be controlled in constructing etymologies is meaning. When doing etymologies in Bantu (a relatively shallow time depth proto-language), the meanings of words from a given root are usually the same, so various descendants of *bón- mean "see". However, some of the words mean "find" or "acquire", thus you have to allow imprecision in semantics, to allow including the entire set of words derived through normal historical transmission from Bantu *bón-. We impose regularity-of-correspondence conditions on our etymological n-tuples, when it comes to the segments of the words – a Bantu word of the form cimb- does not go into the etymological database along with puk- just because the words mean "dig" (no set of historical changes turns puk- into cimb-). Rigorous semantic laws mapping classes of meanings of proto-languages to daughter languages are uncommon (to say the least – non-existent, as far as I know).

The claim that English "path" is a descendant of a PIE root that resulted in Latin pōns, Greek πόντος, OPrus. pintis, and O. Armenian hun would be excluded on the grounds of sound-change irregularity (the predicted regular outcome is something like "find"), but we can supplement the rules for admitting examples into the etymological database by positing (as we obviously must) that not all transmission paths are from parent to child – there are also borrowings. We might conjecture that English "path" is borrowed from some other language, such as Proto-Iranian. While this is very reasonable, it introduces an element of uncertainty. Subjectively, the uncertainty is not enough to render some alternative more plausible.

On the philosophical front, your question asks the fundamental epistemological question, "what is 'certainty'?; what is a theory of knowledge?". In history and philosophy of science, you are asking "what is 'probability'?" (there are different combinatoric, frequency and evidentiary senses). In statistics, where the really heavy lifting would be done, the problem is conceptualizing the question in a statistically valid way, taking care to understand what "the population" is, what the hypotheses would be, how independent variables are coded, what a "random sample" is, and what a p-value is. The concept of p-value simply is not applicable to the hypothesis that "hundred" derives through normal transmission from IE *ḱm̥tóm.

  • So take the word ‘path’. In context a path is for walking a certain route mostly more often and with others using it. So it can be associated with walking and legs. Hence words like foot and podo are cognate although they mean different things. The foot is lifted and this motion is found in cognates for sky like Ancient Egyptian ‘pt’ and in relation to height; its inverted anaphone ‘top’. Russian Nebo and German Oben are cognates with inverted spellings and cognates of Bein/been (Germanic words for ‘leg’. Leg is an inverted cognate for French ‘ciel’. – Ajagar Nov 24 '18 at 22:09
  • It is impossible for mathematics to apply the different values and the historical linguistic proof uses an invented language (PIE) in its model, which makes the whole science speculative. The model is based only on written evidence but this evidence is given more worth than it should get. Because words are first written down and attested in language A and then B and C, does not proof the word cane from source A. Sound changes could have reversed from C through B to A, while the invention of writing advanced the other way. – Ajagar Nov 24 '18 at 22:12
  • Hundred: it is 10 x 10 or the fingers on the hands times the fingers of the hands. Or the hands raised by the hands. Hundred is cognate with Arabic phrase; أنا أثير أيدي /'ana 'uthir 'aydi/ – Ajagar Nov 24 '18 at 22:36
  • That answer is not convincing, the conclusion, "... p value simply is not applicable" comes as a non-sequitur. In all fairness, I did give nothing to go on conceptually. I don't know what I expected. The question was suggestive, so I should not be dissappointed that you ignored the basic question. – vectory Nov 25 '18 at 16:43
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    @Ajagar: please keep your comments focused on the question, mainly. It's distracting. It's irritatingly excessive and at the same time not detailed enough to be complete. I can hardly see where your ideas are coming from. If leg is to be cognate to ligament, then a connection to cloth, cover (Ger. "Kleidung") and by extension sky (Fr. "ciel"; not from *kel-), and to path ~ Ger. "Pfad", thread ~ Ger. "Faden" is not something I can rule out or confidently agree with. But your ad-hoc allusion to height, the "inverted anaphone" or the Arabic ("I raised my hands") you better had supported with math – vectory Nov 25 '18 at 22:48

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