Ramble, Part 34: Dysfunctional Notation
Oct. 13th, 2007 05:52 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
stoutfellowThe use of the notation y=f(x) to represent a functional relationship between two variables is deeply entrenched in modern mathematics, and the notation has proven extremely convenient for many purposes. Nonetheless, there are problems with it, and it is possible that some superior notation could be devised. More under the cut.
There is a longstanding tradition of using the notation f(x) ambiguously: on the one hand, it represents the function, the relationship between the two variables; on the other, it represents the value of the function - that is, the second of the two variables. Those of us who become mathematicians learn fairly soon to recognize which meaning is intended, but the fact of the ambiguity still niggles. There's been an increasing tendency to refer to the function as f and restrict f(x) to mean the value of the function, but it isn't universal, especially at lower levels.
Another problem arises from the very important operation of composition, whereby two functions f and g are applied in succession. Traditional notation places a small circle between the names of the functions, but in many contexts simple juxtaposition is used; that is, gf(x)=g(f(x)). This presents a bit of a psychological problem, in that functions compose right-to-left - it is the rightmost function that is applied first. This collides with the expectation of left-to-right reading, and students often stumble on it. There was a movement a few decades ago to write functions on the right - xf instead of f(x) - which would correct the problem, but it didn't gain much traction and seems to be defunct.
Finally, and most subtly: when one writes, say, f(x)=x2, is this an assertion regarding all possible xs, or only a specific x? That is, is x actually the name of a number (or whatever), or is it a dummy, standing in for every possible number (or whatever)? Let me clarify. In the symbol-processing language Mathematica, if you write f[x]=x2, then, whenever you write f[x], the program will interpret it as meaning x2, but if you write, say, f[y], it will not know how to interpret it: y is not x. To define the general squaring function, you must write f[x_]=x2; the underscore alerts the program that the x is not the name of some specific (though unidentified) number, but something generic. This sort of "slot variable" appears frequently in mathematics, and, again, most of us accustom ourselves to it, but students do struggle with it.
None of these, I think, are major problems, but in my experience they do cause difficulties for students. Good notation should make it easier to think, and should not cast up unnecessary roadblocks. Perhaps we mathematicians can take a cue from Mathematica, as may be happening with the equals sign, but I suspect it will take a long time to take hold, and longer still to percolate down to the lower-level classes where it might be most valuable.
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There is a longstanding tradition of using the notation f(x) ambiguously: on the one hand, it represents the function, the relationship between the two variables; on the other, it represents the value of the function - that is, the second of the two variables. Those of us who become mathematicians learn fairly soon to recognize which meaning is intended, but the fact of the ambiguity still niggles. There's been an increasing tendency to refer to the function as f and restrict f(x) to mean the value of the function, but it isn't universal, especially at lower levels.
Another problem arises from the very important operation of composition, whereby two functions f and g are applied in succession. Traditional notation places a small circle between the names of the functions, but in many contexts simple juxtaposition is used; that is, gf(x)=g(f(x)). This presents a bit of a psychological problem, in that functions compose right-to-left - it is the rightmost function that is applied first. This collides with the expectation of left-to-right reading, and students often stumble on it. There was a movement a few decades ago to write functions on the right - xf instead of f(x) - which would correct the problem, but it didn't gain much traction and seems to be defunct.
Finally, and most subtly: when one writes, say, f(x)=x2, is this an assertion regarding all possible xs, or only a specific x? That is, is x actually the name of a number (or whatever), or is it a dummy, standing in for every possible number (or whatever)? Let me clarify. In the symbol-processing language Mathematica, if you write f[x]=x2, then, whenever you write f[x], the program will interpret it as meaning x2, but if you write, say, f[y], it will not know how to interpret it: y is not x. To define the general squaring function, you must write f[x_]=x2; the underscore alerts the program that the x is not the name of some specific (though unidentified) number, but something generic. This sort of "slot variable" appears frequently in mathematics, and, again, most of us accustom ourselves to it, but students do struggle with it.
None of these, I think, are major problems, but in my experience they do cause difficulties for students. Good notation should make it easier to think, and should not cast up unnecessary roadblocks. Perhaps we mathematicians can take a cue from Mathematica, as may be happening with the equals sign, but I suspect it will take a long time to take hold, and longer still to percolate down to the lower-level classes where it might be most valuable.
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