Proposal: ma'o + PA

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This is a brief proposal concerning the interpretation of constructs of form "ma'o PA* te'u", particularly as constant functions.

Summary

Proposal: "ma'o PA* te'u" is a grammatically valid construction which refers to a constant function, the domain set of which is implicit or specified elsewhere, such that the image of the domain set under this function is the singleton set which contains exactly the number to which the PA* string refers after interpretation/numeric evaluation (confer, for example, Proposal: distinguishing digits from numbers).

Introduction/Discussion

According to the CLL: "There is a potential semantic ambiguity in “ma'o fy. [te'u]” if “fy.” is already in use as a variable: it comes to mean “the function whose value is always ‘f’”.".

The CLL goes on to say that in practice this will not arise frequently. I am not sure of that claim, but it is explicitly addressed (in my opinion appropriately), so it is all good.

One thing which is not explicitly addressed, but which is hinted at, is the outcome of a "ma'o PA* te'u" construction. Even without the aforementioned rule for "ma'o BY* te'u" in the case of the string belonging to BY* referencing a defined number (in the CLL, called a "variable" - terminology of dubious merit, but that is of no matter here), one would think that the result should be a constant function which everywhere takes the value of PA* once the latter is interpreted as a number. That is indeed my proposal here. The domain set should be specified in the same way that it would be specified for any function in mekso, particularly those which are expressed via "ma'o".

Examples, Justification, and Words of Caution

For example: "ma'o re te'u" is the function which is defined on some set such that it identically evaluates to 2 (id est: no matter what element of the domain set is submitted, the output is 2). Technically, outside of this domain set, the function is undefined, but it can be easily extended as necessary so as to remain constant over its newly extended domain set and, in particular, attaining only the value 2.

A generic constant function can be specified via "ma'o cy te'u" (where "cy" refers to some element of a known set of numbers) or, perhaps even more preferred, "ma'o xo'e(i) te'u".

Caution must be maintained, however. This would mean that "ma'o te'o te'u" is the constant function which evaluates to the number <math>\mathbb{e}</math> on its domain set. It would NOT be the exponential function with natural base, commonly denoted by "exp" or "<math>\mathbb{e}^\square</math>". Likewise, "ma'o pa te'u" is the constant function which evaluates to the number 1 on its domain set while "ma'o no te'u" is the constant function which evaluates to the nunber 0 on its domain set, rather than identity functions (such as the multiplicative linear transformation which is represented by the identity matrix), although they can be closely related if not identical in some mathematical structures. I think that this is an okay sacrifice to make because it keeps the interpretation consistent and predictable for all PA*. It is also utile. Moreover, it seems like a natural pair for the rule mentioned in the CLL - one which would be guessed even if the CLL rule were not prescribed. In order to be clear: With the CLL rule aforementioned, this is a natural extension/generalization/analog to make; however, even without it, one would easily and independently guess that this proposal is the result of such constructs. Finally, each of these alternative options ('exp', identity functions, etc.) deserve either their own words (such as "te'o'a" for 'exp'), can be built from existing words (something similar to "te'o te'a" would do for 'exp', modulo some nuance and grammatical issues), or could be referenced by a word which generically means "identity element in structure ___" (which should have at least one word); in other words, the functionality can and will be supported in other ways.

Being able to easily convert to constant functions is nice and should technically be done quite often. For example, the derivative of the number 2 does not exist because the differentiation operator is defined on only some subset of (only) functions (say, differentiable and real-valued ones), and necessarily not the set of real numbers in particular, but the number 2 is just a real number and not a function. But if we convert, then it is okay: "li sa'o ma'o re du li ma'o no" (note the inclusion of both "ma'o"'s) is perfectly acceptable (syntactically, mathematically, and semantically; it is also true); the exclusion of at least one of the "ma'o"'s makes the statement unacceptable in a multitude of ways.

Potential Issues which may Newly Arise

  • How do strings which consist of a mix of PA and BY words behave in this context? For example, what does "ma'o by re cy te'u" mean? The resolution of this issue is dependent on the resolution of other proposals concerning mixed strings.

Generalizations and Further Questions

  • If a constant nonscalar tensor (or the representation thereof), for example, is converted via "ma'o", should the output be a nonscalar tensor-valued function such that it attains the original tensor as output on its entire domain set? (lai .krtisfranks. is currently of the opinion that it should when the statement is syntactically acceptable).

Contributors

This article was written by the following people. All ideas and opinions are theirs. First-person personal pronouns refer to them.

  • lai .krtisfranks.