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ADDED LATER: One should also distinguish between the "one-time costs" of a notation (e.g., the difficulty of learning the notation and avoiding standard pitfalls with that notation, or the amount of mathematical argument needed to verify that the notation is well-defined and compatible with other existing notations), with the "recurring costs" that are incurred with each use of the notation. The desiderata listed above are all in the "recurring" category, but the "one-time" costs are also a significant consideration if one is only using the mathematics from the given field $X$ on a casual basis rather than a full-time one. In particular, it can make sense to offer "simplified" notational systems to casual users of, say, linear algebra even if there are more "natural" notational systems (scoring more highly on the desiderata listed above) that become more desirable to switch to if one intends to use linear algebra heavily on a regular basis.

ADDED LATER: One should also distinguish between the "one-time costs" of a notation (e.g., the difficulty of learning the notation and avoiding standard pitfalls with that notation, or the amount of mathematical argument needed to verify that the notation is well-defined and compatible with other existing notations), with the "recurring costs" that are incurred with each use of the notation. The desiderata listed above are primarily concerned with lowering the "recurring costs", but the "one-time costs" are also a significant consideration if one is only using the mathematics from the given field $X$ on a casual basis rather than a full-time one. In particular, it can make sense to offer "simplified" notational systems to casual users of, say, linear algebra even if there are more "natural" notational systems (scoring more highly on the desiderata listed above) that become more desirable to switch to if one intends to use linear algebra heavily on a regular basis.

To evaluate these sorts of qualities, one has to look at the entire field $X$ as a whole; the quality of notation cannot be evaluated in a purely pointwise fashion by inspecting the notation $\mathrm{Notation}^{-1}(C)$ used for a single mathematical concept $C$ in $X$. In particular, it is perfectly permissible to have many different notations $\mathrm{Notation}_1^{-1}(C), \mathrm{Notation}_2^{-1}(C), \dots$ for a single concept $C$, each designed for use in a different field $X_1, X_2, \dots$ of mathematics. (In some cases, such as with the metrics of quality in desiderata 1 and 7, it is not even enough to look at the entire notational system $\mathrm{Notation}$, but also its relationship with the other notational systems $\widetilde{\mathrm{Notation}}$ that are currently in popular use in the mathematical community, in order to assess the suitability of use of that notational system.)

(See also Section 2 of Thurston's "Proof and progress in mathematics", in which the notion of derivative is deconstructed in a fashion somewhat similar to the way the notion of inner product is here.)

To evaluate these sorts of qualities, one has to look at the entire field $X$ as a whole; the quality of notation cannot be evaluated in a purely pointwise fashion by inspecting the notation $\mathrm{Notation}^{-1}(C)$ used for a single mathematical concept $C$ in $X$. In particular, it is perfectly permissible to have many different notations $\mathrm{Notation}_1^{-1}(C), \mathrm{Notation}_2^{-1}(C), \dots$ for a single concept $C$, each designed for use in a different field $X_1, X_2, \dots$ of mathematics. (In some cases, such as with the metrics of quality in desiderata 1 and 7, it is not even enough to look at the entire notational system $\mathrm{Notation}$; one must also consider its relationship with the other notational systems $\widetilde{\mathrm{Notation}}$ that are currently in popular use in the mathematical community, in order to assess the suitability of use of that notational system.)

(See also Section 2 of Thurston's "Proof and progress in mathematics", in which the notion of derivative is deconstructed in a fashion somewhat similar to the way the notion of inner product is here.)

ADDED LATER: One should also distinguish between the "one-time costs" of a notation (e.g., the difficulty of learning the notation and avoiding standard pitfalls with that notation, or the amount of mathematical argument needed to verify that the notation is well-defined and compatible with other existing notations), with the "recurring costs" that are incurred with each use of the notation. The desiderata listed above are all in the "recurring" category, but the "one-time" costs are also a significant consideration if one is only using the mathematics from the given field $X$ on a casual basis rather than a full-time one. In particular, it can make sense to offer "simplified" notational systems to casual users of, say, linear algebra even if there are more "natural" notational systems (scoring more highly on the desiderata listed above) that become more desirable to switch to if one intends to use linear algebra heavily on a regular basis.

1. (Unambiguity) Every well-formed expression in the notation should have a unique mathematical interpretation in $X$. (Related to this, one should strive to minimize the possible confusion between an interpretation of an expression using the given notation $\mathrm{Notation}$, and the interpretation using a popular competing notation $\widetilde{\mathrm{Notation}}$.)
2. (Expressiveness) Conversely, every mathematical concept or object in $X$ should be describable in at least one way using the notation.
3. (Preservation of quality, I) Every "natural" concept in $X$ should be easily expressible using the notation.
4. (Preservation of quality, II) Every "unnatural" concept on $X$ should be difficult to express using the notation. [In particular, it is possible for a notational system to be too expressive to be suitable for a given application domain.] Contrapositively, expressions that look clean and natural in the notation system ought to correspond to natural objects or concepts in $X$.
5. (Error correction/detection) Typos in a well-formed expression should create an expression that is easily corrected (or at least detected) to recover the original intended meaning (or a small perturbation thereof).
6. (Suggestiveness, I) Concepts that are "similar" in $X$ should have similar expressions in the notation, and conversely.
7. (Suggestiveness, II) The calculus of formal manipulation in $\mathrm{Notation}$ should resemble the calculus of formal manipulation in other notational systems $\widetilde{\mathrm{Notation}}$ that mathematicians in $X$ are already familiar with.
8. (Transformation) "Natural" transformation of mathematical concepts in $X$ (e.g., change of coordinates, or associativity of multiplication) should correspond to "natural" manipulation of their symbolic counterparts in the notation; similarly, application of standard results in $X$ should correspond to a clean and powerful calculus in the notational system. [In particularly good notation, the converse is also true: formal manipulation in the notation in a "natural" fashion can lead to discovering new ways to "naturally" transform the mathematical objects themselves.]
9. etc.

1. (Unambiguity) Every well-formed expression in the notation should have a unique mathematical interpretation in $X$. (Related to this, one should strive to minimize the possible confusion between an interpretation of an expression using the given notation $\mathrm{Notation}$, and the interpretation using a popular competing notation $\widetilde{\mathrm{Notation}}$.)
2. (Expressiveness) Conversely, every mathematical concept or object in $X$ should be describable in at least one way using the notation.
3. (Preservation of quality, I) Every "natural" concept in $X$ should be easily expressible using the notation.
4. (Preservation of quality, II) Every "unnatural" concept in $X$ should be difficult to express using the notation. [In particular, it is possible for a notational system to be too expressive to be suitable for a given application domain.] Contrapositively, expressions that look clean and natural in the notation system ought to correspond to natural objects or concepts in $X$.
5. (Error correction/detection) Typos in a well-formed expression should create an expression that is easily corrected (or at least detected) to recover the original intended meaning (or a small perturbation thereof).
6. (Suggestiveness, I) Concepts that are "similar" in $X$ should have similar expressions in the notation, and conversely.
7. (Suggestiveness, II) The calculus of formal manipulation in $\mathrm{Notation}$ should resemble the calculus of formal manipulation in other notational systems $\widetilde{\mathrm{Notation}}$ that mathematicians in $X$ are already familiar with.
8. (Transformation) "Natural" transformation of mathematical concepts in $X$ (e.g., change of coordinates, or associativity of multiplication) should correspond to "natural" manipulation of their symbolic counterparts in the notation; similarly, application of standard results in $X$ should correspond to a clean and powerful calculus in the notational system. [In particularly good notation, the converse is also true: formal manipulation in the notation in a "natural" fashion can lead to discovering new ways to "naturally" transform the mathematical objects themselves.]
9. etc.