Scott, I had to reread your email several times to make sure I wasn't missing something. Usually I agree with your take on things, but this posting seems wrong to me, both pedagogically and historically. Maybe if I had a better understanding of where you're coming from I could write a different reply, but anyway, here goes: I'm thinking about teaching the problem solving process in mathematics, and
have run into a curious question: can one ask a mathematical question purely in mathematical notation?
I could write Fermat's Last Theorem as a quantified proposition that uses only mathematical symbols and then stick a question mark at the end of it: (\exists n \in N) (\exists x,y,z \in Q) (n > 2 \and xyz \neq 0 \and x^n + y^n = z^n) ? Would that count? I guess your answer would be "no". Mine would be "yes", since I'm pretty sure I've seen researchers do it in seminars. It seems to me that your question hinges on how broadly one defines "mathematical notation". People with different definitions will have different definitions, so unless there's a right definition of the phrase "mathematical notation" there's no right answer to your question. Some definitional issues have right answers; for instance, some students are puzzled by the fact that 0 is an even number, but one can usually convince them that they're using the wrong definition of "even number". Ditto for "Is a square a rectangle?" But it's unclear to me whether "mathematical notation" is like "even number" and "rectangle" in that respect. One difference is that "even number" and "rectangle" are inherently mathematical concepts, whereas the term "mathematical notation" is part mathematical, part historical-social. Mathematical notation is an interface (and a storage device) that was designed by, and for, people, so it's a social construct. Sometimes we have to draw a definitional line even when there's no God-given line, in questions like "When does human life begin and end?" In such cases, it's helpful to respond with the counter-question "For what purpose?" Which brings me back to the question of why you're asking the question in the first place. Is it about how we teach math, or is it about how we can be sure mathematical assertions are true?
I believe the answer is no — mathematical questions always require human language in addition to mathematical notation. Problem statements are therefore always extramathematical in nature.
This seems to conflate mathematics with notational (formulaic) mathematics. But most of mathematical notation is a fairly recent innovation. Math, even algebra, was rhetorical until a few centuries ago. Fermat's statement of FLT was made up entirely of words (including the very human word "however").
In practice, school kids frequently see problems stated in forms like: 13+78 = ___, which means "what is the sum of thirteen and seventy eight?" But that involves a nonmathematical symbol (the blank), AND ends up misleading kids to think that the equals sign means "the answer is". Very sloppy.
I agree that the way we teach the meaning of the equal sign is often deficient. Teachers talk a lot about this. (Good teachers like James Tanton, anyway.)
We can do better.
What might "doing better" look like? Do you want a typology of questions, for starters? Would it include a punctuation mark for open-ended questions that begin "Is there anything interesting to be said about ..." or "Are there any patterns governing ..."? I would argue that if we want to engage students, we need fewer symbols and more words. Symbols are great for stripping away distracting aspects of a problem so we can focus on pure form, but part of what gets stripped away is why we might care about the problem in the first place. Incidentally: In the classroom, my favorite questions are the ones my students ask, and my favorite subterfuges are the ones that lead them to ask the very questions I want to answer! This is an extension of something that has always bothered me: if
mathematics is so rigorous, then how come it is conducted in a mish mosh of English and formal notation? The practical reason for mixing in human language is clear…like a computer program, a formal mathematical proof is more readable if it is annotated with comments. But too much reliance on human language means that formal proofs are not checkable by computer — a weird situation at best. I suppose that makes me a formalist, or a computer scientist…I've certainly got the latter bias because I don't trust anything I can't program.
It's definitely a problem that so little of mathematics has been checked by computer. We agree there! But would a system for having computers check a human-designed proof require a question mark? All we need is a command that says "Check this proof for errors!" But even a computer with superhuman proof-checking powers would have no curiosity and hence no need for question marks; we humans are the ones that ask the questions. So when you write "Mathematical questions always require human language", part of me wants to say "Yes, and that's a good thing!" Thanks for your question, Scott, and I apologize for misunderstanding you, as I'm sure I must have done in several spots. Jim Propp