Domain basis: Difference between revisions

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The behavior of these are not yet well-described. For instance, is it best to represent them as matrices? If a basis element is <math>π/ɸ</math>, does the vinculum allow us to treat the two irrational numbers as separate basis elements (where the basis elements are expanded to include not only prime numbers)? Perhaps, but as far as this author understands, this hasn't been pinned down yet. And so, for example, irrational numbers are not supported yet in the RTT library in Wolfram Language.

~~= Terminology: domain basis vs. subgroup =~~

In some other RTT writings on the wiki, the term "subgroup", is used in various senses. Sometimes it seems to be used to refer to nonstandard domain bases. Other times it seems to be used as a synonym for doman. Sometimes it seems to be used for both at once. In any case, this article breaks from that terminology, considering it to be confusing and unhelpful, for a number of reasons.

~~== Mixing terminology from different mathematical fields ==~~

The term "subgroup basis" mixes mathematical terminology from different mathematical fields: "[[Wikipedia:Subgroup|subgroup]]" comes from [[Wikipedia:Group_theory|group theory]], while "[[Wikipedia:Linear_basis|basis]]" comes from [[Wikipedia:Linear_algebra|linear algebra]]. The equivalent term for "subgroup" in linear algebra is "[[Wikipedia:Linear_subspace|subspace]]", and the equivalent term for "basis" in group theory is "[[Wikipedia:Minimal_generating_set|minimal generating set]]". So the consistent terminology would be to either call something a "subspace basis" (such as a "nullspace basis") or to call it a "subgroup minimal generating set". So "subgroup basis" is already problematic as an inconsistent mix of different fields' terminology.

~~== Simple vs. advanced math: linear algebra vs. group theory ==~~

Regarding the choice between these two internally consistent versions of this term, then, this article prefers "subspace basis". 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.

"Subgroup" and "subspace" are indeed analogous, but due to differences between group theory and linear algebra, they are not completely synonymous. Essentially, group theory takes some of the convenient assumptions which we rely on when doing linear algebra and sets them aside. Doing so can be powerful, and 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.

~~== Generic math terms vs. specialized application terms ==~~

~~For the previous two reasons — consistency, and simplicity — choosing "subspace basis" over "subgroup basis" would be preferable. But this article thinks it can do even better.~~

Setting aside the specialized use it has taken on in these RTT writings, a subgroup (or subspace) in the general mathematical sense is just a generic mathematical structure, like a matrix or vector. 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. Just like how we call certain objects represented by matrices "mappings" and certain objects represented by vectors "intervals", this article prefers using a specialized term for this RTT object — one that cannot be confused with a generic mathematical structure.

A common need when dealing with domains is determining whether they are subspaces of other domains (as discussed in [[Cross-domain temperament merging#Interval subspaces as subspaces of other domains]]). If the name for the specialized RTT object was simply "subspace" instead of "domain", then each use of the word "subspace" could be unclear whether it was referring to the specialized RTT object or to the generic mathematical structure. Communicating about such things would become terribly confusing (as it is at present, in existing writings that use the term "subgroup" in both senses).

~~== A fresh start re: not excluding standard prime-limit domains ==~~

"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.

= Domain basis operations =

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 The behavior of these are not yet well-described. For instance, is it best to represent them as matrices? If a basis element is <math>π/ɸ</math>, does the vinculum allow us to treat the two irrational numbers as separate basis elements (where the basis elements are expanded to include not only prime numbers)? Perhaps, but as far as this author understands, this hasn't been pinned down yet. And so, for example, irrational numbers are not supported yet in the RTT library in Wolfram Language.
-= Terminology: domain basis vs. subgroup =
-In some other RTT writings on the wiki, the term "subgroup", is used in various senses. Sometimes it seems to be used to refer to nonstandard domain bases. Other times it seems to be used as a synonym for doman. Sometimes it seems to be used for both at once. In any case, this article breaks from that terminology, considering it to be confusing and unhelpful, for a number of reasons.
-== Mixing terminology from different mathematical fields ==
-The term "subgroup basis" mixes mathematical terminology from different mathematical fields: "[[Wikipedia:Subgroup|subgroup]]" comes from [[Wikipedia:Group_theory|group theory]], while "[[Wikipedia:Linear_basis|basis]]" comes from [[Wikipedia:Linear_algebra|linear algebra]]. The equivalent term for "subgroup" in linear algebra is "[[Wikipedia:Linear_subspace|subspace]]", and the equivalent term for "basis" in group theory is "[[Wikipedia:Minimal_generating_set|minimal generating set]]". So the consistent terminology would be to either call something a "subspace basis" (such as a "nullspace basis") or to call it a "subgroup minimal generating set". So "subgroup basis" is already problematic as an inconsistent mix of different fields' terminology.
-== Simple vs. advanced math: linear algebra vs. group theory ==
-Regarding the choice between these two internally consistent versions of this term, then, this article prefers "subspace basis". 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.
-"Subgroup" and "subspace" are indeed analogous, but due to differences between group theory and linear algebra, they are not completely synonymous. Essentially, group theory takes some of the convenient assumptions which we rely on when doing linear algebra and sets them aside. Doing so can be powerful, and 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.
-== Generic math terms vs. specialized application terms ==
-For the previous two reasons — consistency, and simplicity — choosing "subspace basis" over "subgroup basis" would be preferable. But this article thinks it can do even better.
-Setting aside the specialized use it has taken on in these RTT writings, a subgroup (or subspace) in the general mathematical sense is just a generic mathematical structure, like a matrix or vector. 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. Just like how we call certain objects represented by matrices "mappings" and certain objects represented by vectors "intervals", this article prefers using a specialized term for this RTT object — one that cannot be confused with a generic mathematical structure.
-A common need when dealing with domains is determining whether they are subspaces of other domains (as discussed in [[Cross-domain temperament merging#Interval subspaces as subspaces of other domains]]). If the name for the specialized RTT object was simply "subspace" instead of "domain", then each use of the word "subspace" could be unclear whether it was referring to the specialized RTT object or to the generic mathematical structure. Communicating about such things would become terribly confusing (as it is at present, in existing writings that use the term "subgroup" in both senses).
-== A fresh start re: not excluding standard prime-limit domains ==
-"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.
 = Domain basis operations =