Although mathematics is famous for its precision and clarity, there are some times when ambiguity creeps in. Here are a few examples. I’ll update this list whenever I encounter new examples.
Consider the two inequalities concerning real numbers: x≠x+1 and x≠2x. Both are “true”, but in different senses. The first is true for every single real number x, but the second is only true if x is nonzero. If someone asks you out-of-context: “In general, is x≠2x?”, you can’t answer the question because you aren’t certain whether they are asking, “Is x always unequal to 2x?”, or whether they’re asking “Are the functions x and 2x unequal?” The answer to the first question is “NO”, because 0 is equal to 2×0. The answer to the second question is “YES”, because if you plot the graphs of “y=x” and “y=2x”, they are definitely different graphs.
The best practice is to be very precise about what is meant, and that goes for equalities as much as inequalities. Thus, it’s true to say: “there exists a real x such that x≠2x”, but it’s false to say “for every real x, x≠2x”.
2. “Generally False”
The ambiguity about inequalities is a special case of the more general “Generally False” ambiguity. Consider the mathematical statement: “In general, a real number has no real square root.” There are two ways to parse this. On the one hand, it might mean: there is no real number with a square root. Or, it could mean, there is at least one real number without a square root. The former reading is obviously false (since, for example, the square root of 4 is 2). The latter reading is true, since, for example, -1 has no real square root.
3. Number or function?
The inequalities ambiguity is also a special case of the “Number or function?” ambiguity. Without any other context, we can’t say whether the expression x+y is a number, or a function. Even if we know it’s a function, we don’t know in how many variables: the most obvious guess is that it’s a binary function in x and y, but it could be a unary function in just x, and y is a constant; or vice versa. In theory, it could even be a ternary function in x,y, and z– it doesn’t explicitly depend on z, but then again, you could rewrite the thing as x+y+0z.
This ambiguity actually causes undergraduates to lose points on quizzes and exams all the time, and not for an altogether trivial reason. Consider the exercise: “Find the equation for the line tangent to the parabola y=x² at the point (3,9).” The solution is to calculate the derivative y’=2x, plug in x=3 to get the slope m=6, and then use point-slope form to get the equation, y-9=6(x-3). A common mistake, however, is to leave the slope as 2x and write the equation as y-9=2x(x-3). Obviously this is wrong, but there is something very subtle going on. The slope definitely is m=2x, and the equation definitely is y-9=m(x-3). So why can’t you just plug in m=2x? It’s an absolutely perfect display of the “number or function?” ambiguity in the wild. When we write m=2x, m is understood as a function, but when we write y-9=m(x-3), m is understood as a number. And that’s why before you plug in for m, you have to actually get a number first.
4. The derivative of x
Here’s a kind of variation on the “number or function?” ambiguity. If someone walks up to you and demands “What’s the derivative of x?”, the most obvious answer is “1″, but this is implicitly assuming they meant the derivative with respect to x. Technically, they could’ve meant the derivative with respect to y, in which case (assuming x and y are independent of each other), x is constant and has derivative zero. It’s also possible, for example, that they meant the derivative with respect to y and that, unknown to you, x=5y, and in that case, the derivative is five! Oh dear.
This partly explains why mathematicians can’t settle on just one notation for the derivative. Everyone loves the prime notation x’, but this only makes sense if the independent variable is understood from context. Thus the more bulky notations which actually spell out the independent variable– in multivariable calculus, the distinction becomes crucial.
5. Plus or minus
The plus-or-minus symbol ± has two different meanings. It can be used to bound the error in an approximation, for example, one might say that the percent of Xamuel.com readers in the world is 90%±1% of the total population, which means it could be anywhere from 89% to 91%. In math, the more common use is to indicate that either a plus sign (+) or a minus sign (-) can be substituted, and either way will give a correct answer. The most familiar example is the quadratic equation, which says that if ax²+bx+c=0, then x must be “one of” (-b±√(b²-4ac))/2a. The expression actually denotes (up to) two possible values of x, since the ± can be either + or – (note that in the special case when the square root is zero, then there’s actually only one possibility for x).
The ambiguity comes when there are multiple ±’s in one expression. For example, what is ±1±1? One possibility is that the ±’s should agree with each other– if one of them is +, then both have to be +. In that case, ±1±1 must be either 2 or -2. The other possibility is that the ±’s are independent, one can be + while the other is -. Then ±1±1 can be -2, 0, or 2. It seems the official tack (at least according to Wikipedia) is the former. One might wonder where on earth multiple ±’s would actually arise in practice: they can come up in the solutions to quartic equations.
6. If FALSE then FALSE
There isn’t actually any ambiguity here, but it can seem ambiguous, and it can certainly confuse people, so I’m including it anyway. In symbolic logic, the if-then operator → operates on two TRUE-FALSE values to produce a third TRUE-FALSE value. The expression A→B is read “A implies B”. If we write A→B=T, this means: the statement ‘A implies B’, is a true statement. In logic, we define F→F=T, in other words, we declare that the statement ‘FALSE implies FALSE’ is true. If this doesn’t seem weird yet, consider a concrete example: “If the moon is made of cheese, then the Earth is flat.” Is this statement true or false? It’s true. People often resist this, on the grounds that the earth isn’t flat.
Here’s one way to understand why logicians declare F→F=T. Consider the statement: “If x=0 then x=0.” This statement is obviously, intuitively true. But it’s actually not one statement, it’s an infinite family of statements, one for every possible value of x. And when we say it’s true, what that really means is that it’s true for every possible x. In particular, if you take x=1, the statement becomes: “If 1=0 then 1=0.” This bizarre statement must be true– if it weren’t, then the more general “If x=0 then x=0″ couldn’t be true. So, “If FALSE then FALSE” really does turn out to be true.
7. Zero to the zeroth
Perhaps the most famous mathematical double-meaning is zero to the zeroth power, 0^0. Recall from grade school the old maxim, “Anything to the 0th power equals 1.” Also recall the well-known fact, “Zero to any nonnegative power equals 0.” Combine these common-sense truths to get an easy proof that 1=0.
The truth is, different authors take different stances on the 0^0 issue. In most math textbooks, the issue never comes up, but when it does, a good author will make sure to explicitly state what they mean by it. The most common definitions are 0^0=1 and 0^0=0, with the former being much more widespread than the latter. Graham, Knuth, and Patashnik give a good argument in a famous textbook why 0^0=1 should be standard. But I have personal experience with other viewpoints– when I published my undergrad math paper (pdf), the referee initially insisted I was wrong to define 0^0=1. I convinced them I was right
8. True Multiplication vs. Formal Multiplication
If f(x)=1/x, what is xf(x)? Is it 1? Or is it equal to 1 everywhere except at x=0, where it’s undefined? Complex analysts are particularly prone to use the former reading.
9. Suppressed parentheses
Certain functions, particularly trigonometric functions like sin and cos, are often written without parentheses: “sin x” instead of “sin(x)”. So what does the expression “sin ab” mean? It can mean either “sin(ab)” or “(sin a)b”. Generally, it’ll mean the former. However, it can sometimes mean the latter! For example, I’m looking at some lecture notes right now which uses implicit differentiation to find the derivative of arcsine: you let y=arcsin x, which means that sin y=x, then you differentiate both sides and get: “cos y dy/dx = 1″. In this context, “cos y dy/dx” means “(cos y)dy/dx”!
I’ll update this list of ambiguities whenever I think of a new one.