- QAR (Quasi-Penultimate Accent Rule)There are many other changes I propose but they extend beyond this particular topic. I really should get off my lazy arse and put this in a pdf for you because quite frankly, it's tiresome to repeat this over and over to explain myself.
- Rhotacization of word-final *-n
- Dephonemicization of labialized dentals
- Reduction of unstressed *a to schwa *ə and supershort schwa *ᵊ
- Laryngeal Vocalization (in some environments)
- Syncope (with Paradigmatic Strengthening & a-Epenthesis)
- Vowel Shift
In PIE, the word for 'four' is reconstructed as *kʷetwóres while 'eight' is *h₁oḱtóu (*h₁ is most likely a glottal stop /ʔ/). I admit it would seem strange that those two words could be related but then again there are stranger things that happen. Let's explore a possibility.
Based on my theory so far, *kʷetwóres should predict Mid IE (MIE) *kʷatʷáras. This is because Paradigmatic Strengthening during Syncope prevented the first *a preceding stress from eroding in order to avoid overly obscure alternations between different case forms. A system in a language can only handle so much obscurity before levelling kicks in. This corollary to Syncope helps explain the otherwise mysterious vowel change in *pedós 'of the foot' (< *pod- 'foot'). So this logically implies that during the event of Syncope, both *kʷatʷáras with animate plural ending *-as and forms in *kʷátʷar- with stress on the initial syllable and without plural ending coexisted. Reconstructing further back, MIE *kʷátʷar must be the more ancient form but this has the look of a heteroclitic, a class of inanimate nouns that alternated between *r and *n. Inanimate marking would make sense for a number word if it was perceived as a word for an abstract grouping regardless of the animacy of the entities grouped. This then would imply early Mid IE *kʷátʷan before Rhotacization, the change of word-final *-n to *-r, thereby explaining the odd r/n alternation in heteroclitic stems (e.g. PIE *yḗkʷr̥ 'liver' versus *yekʷ(e)n-ós ~ *yekʷén-s 'of the liver') . Ultimately then, since this *-an is a type of inanimate suffix, *kʷátʷa- must be the most original pre-IE root form for '4' recoverable from this brainstorming exercise.
As for 'eight', a-Epenthesis could help solve its etymology since an MIE form *kʷatʷáxʷa should in theory first change to *kʷatwáxʷa (via Dephonemicization of Labialized Dentals) then *kʷᵊtwawᵊ (via Reduction and Laryngeal Vocalization). Without a-Epenthesis we'd get a bizarre result with a crowded onset of three phonemes in the first syllable: **kʷtwau. However a-Epenthesis solves this catastrophe by easing pronunciation during the growing pains of Syncope by adding *a in the first available spot from left to right that will repair the awkward syllabics. So here, a prothetic position is most efficient which yields a pronounceable *ʔaktwáu, eventually becoming *h₁oḱtou via Vowel Shift. This explains the suspicious relationships between other pairs like *h₂awi- 'bird'/*o-h₂uy-om- 'egg' and *h₁eḱwo- 'horse'/*o-h₁ḱu- 'swift' that plague the minds of other IEists who have trouble explaining the unsemantic "o-prefix".
So with early MIE *kʷátʷan '4' and *kʷatʷáxʷa '8', the etymological relationship between the two numerals would then be clear. That is, the word '8' would have been formed in Old IE from *kʷatʷa- '4' and the inanimate collective *-xʷa, signifying literally 'fours'. If PIE's words for '3', '6' and '7' are Semitic loans as well, this means that mindful Nostraticists are logically only left with '1', '2', '4', '5', '9' and '10' to use in their tentative claims.
(Mar 22 2008) Eeek! A mistake: "This then would imply early Mid IE *kʷátwan before Rhotacization, [...]"! That should read *kʷátʷan with superscript w after the t. I also repaired the error in "while 'eight' is *h₁októu (*h₁ is most likely a glottal stop /ʔ/)". That should show k with a diacritic: *h₁oḱtóu. I follow traditional notation on my blog despite the issues I've mentioned concerning PIE phonetics and despite the unlikelihood that "palatal k" was actually palatal at all. (Alas, if I use my own notation, IEists and paleoglot hobbyists are likely to get confused so I'll stay conservative here to keep life simple.)