User:FloraC/Critique on D&D's terminology: Difference between revisions

<s>First, they argue ''subgroup'' and ''basis'' are inconsistent terms since they are a mix from different mathematical fields.  

The term "subgroup basis" mixes mathematical terminology from different mathematical fields: "subgroup" comes from group theory, while "basis" comes from linear algebra. The equivalent term for "subgroup" in linear algebra is "subspace", and the equivalent term for "basis" in group theory is "minimal generating set".  

That is not correct. The basis is a concept that is used across linear algebra and a particular sector of group theory: the study of free abelian groups. If you look up the definition of the free abelian group, it is simple: a free abelian group is an abelian group that has a basis. Indeed, in RTT, JI is modeled as a free abelian group, rather than a group or module in general. That is why bases are used.  

Since a free abelian group is a group, it naturally has subgroups. It is not a vector space, so we do not speak of subspaces – but that has to do with their next point below. </s>

=== Simplicity ===

They say they prefer linear algebra, the less advanced math theory, to group theory, the more advanced, since they believe the difference between the vector space and the free abelian group is negligible for practical purposes.  

This is because group theory is a relatively obscure and advanced field of mathematics, and this article prefers to leverage terminology from the more well-known and basic field of linear algebra whenever possible. […] some argue that RTT cannot be sufficiently described using only linear algebra. This article, however, prioritizes pedagogy of the basics over any potential considerations arising from such advanced RTT problems.  

Here they admit that ''subgroup'' is technically more correct than ''subspace''. It should be pointed out that their reason seems to be one ''for'' taking the correct terms, not ''against''. Specifically, they draw the comparison between the two math disciplines, and it appears that RTT would be more difficult by taking the correct terms. The problem in their logic is, if "advanced RTT problems" only arise occasionally, the actual difficulty levels should be similar.  

<s>At this point, they further argue for ''domain basis'' than ''subspace basis'': they prefer more specialized tuning terms to general mathematical terms.  

This article prefers to use specialized terminology for objects in our RTT application, so that we can clearly discuss them independently from the mathematical structures that represent them.  

To understand this point of theirs, we must look back at the time before they changed ''interval basis'' to ''domain basis''. Back then, ''interval basis'' was indeed more ''specialized'' for tuning, though not more ''specific'' in its form. Yet that is no longer the case. ''Domain'' is by no means more specialized or specific than ''subgroup'', if not less. So the reason is obsolete. </s>

=== Inclusivity ===

<s>Finally, they argue that ''subgroup'' is often assumed to be nonstandard subgroups, and to exclude the standard type, while ''domain'' is designed to include both standard and nonstandard types.  

"Subgroup" in many typical RTT usages is apparently intended to exclude the standard prime-limit subgroups. This makes it more difficult than necessary to communicate about the standard prime-limit basis, which are still very much subgroups — of the entire space of primes, for one example. So we think this is unnecessary complexity with no clear benefit.  

There is no explicit, clear-cut exclusion of standard subgroups in the definition. Technically, a full prime-limit JI is a subgroup of a larger prime-limit JI. It happens that we have distinct vocabulary for standard subgroups, so the word only tends to appear at the nonstandard type. They also say the term "has taken on a specialized meaning in RTT", which I am again not sure about. For one thing, it has never gained a distinct definition. Still, according to my observation, whether it is meant to include the standard type is up to the context.  

The hope for a term that invariably includes standard subgroups does not hold itself because language does not work like that. So long as prime limits continue to be used, chances are the actual usage of ''domain'' will turn out the same as ''subgroup''. </s>

=== My suggestion ===

Until I do a further research on the original thread. However this part, which I reckon is used to suggest that ''val'' be wrong, does not hold:  

"p-adic valuation" is an obscure term for "prime count", which would be an element of a prime-count vector ("monzo"), not a map ("val").

There is only ''valuation'', not ''p-adic valuation'', and each element is indeed a valuation i.e. how many generators that make the prime.  

The other philosophy embedded in ''uniform map'' and ''integer uniform map'' is a turn of conceptual framing:  

For patent vals and GPVs, patent vals are considered the base case, and GPVs a generalization thereof, whereas for uniform maps and integer uniform maps, uniform maps are considered the base case and integer uniform maps a specialization thereof. There is an argument that uniform maps are the more fundamental and important concept to regular temperament theory and therefore that this framing is the superior of the two.  

Yet think of how we learned division. It is the same cognitive process: we first divide things by integers, and then we learn to extend the divisors to non-integers. That leads to the reason why the base name of ''patent val'' is arguably more friendly to our minds.