Assuming existence of what doesn't exist!

Question

Some day when I was a young school boy, our math teacher gave us a strange question in an exam. As far as I can remember that question was something like this:

Let $a$ be such that $a^2+a+1=0$ then prove $a$ has the property $P$. (I can't remember that property correctly).

At that time we were not familiar with the notion of imaginary numbers so our world of numbers was restricted to $\mathbb{R}$. As $a^2+a+1=0$ has no root in real numbers I told my teacher that:

Your assumption is false! Are you sure that this is not a typo? Did you mean $a^2+a-1=0$?! If not, the question is trivial because such an $a$ doesn't exist at all!

He told me (with a mysterious smile!):

There is nothing wrong with my question. Just try to imagine existence of what doesn't exist and go forward in the world of imagination!

At that moment I didn't understand his comment properly. I simply wrote: "The assumption is false so the theorem is true!" in my answer to the question.

After a while when I learned more about imaginary numbers, I understood the creative intuition behind such a strange question that:

Some contradictory assumptions can lead us to an expanded intuition using expanding our world with new notions and objects.

Long after that I attended some lectures about Theory of Fields with One Element and the same story repeated. Assuming existence of what doesn't exist and trying to understand its properties! Later I tried to introduce such a phenomena in mathematical thinking to my students. I usually say:

Proving truth of a theorem by proving falsity of its assumption is not fair! Try to find a fair proof!

Question: When a contradictory assumption like existence of an one element field or existence of a number $a$ with $a^2+a+1=0$ can lead us to a richer theory and deeper intuition? What are characteristics of such useful contradictory assumptions? How to teach a math student to distinguish between useful and useless contradictory assumptions? How to teach them to deal with useful contradictory beings and use them to expand their understanding of the nature of the subject?

Answer 1

Whenever we state a theorem in the form "If $P$ then $Q$" we are usually omitting a whole slew of tacit additional hypotheses, namely those that specify what kind of theory we are talking about. What we really mean (but don't bother saying because it is understood from context) is "If $P$ [in the context where $\{ H_1, H_2, \dots H_n \}$ ] then $Q$.

For example, the statement "If $a$ is a number such that $a^2 + a + 1 = 0$, then $Q$" really means "If $a$ is a number [where "number" means a member of $\mathbb{R}$, which satisfies the axioms of a complete ordered field] such that $a^2 + a + 1 = 0$, then $Q$".

When you say "The theorem is true because the assumption is false!" what you mean is that under the hypotheses $\{ H_1, H_2, \dots H_n \}$, $P$ is false, and therefore under those implicit hypotheses $P \implies Q$ is vacuously true.

The sense that this proof is "unfair" stems from the fact that it takes no notice whatsoever of $Q$ and therefore the logical relationship between $P$ and $Q$ -- which is presumably at the heart of the question's intent -- is completely unexamined. What you really want students to do is consider the possibility that under some (not fully specified) weakening of the hypotheses -- say $ \{ H_1', \dots H_m' \}$ -- $P$ might be possible after all, and to then show if that were the case, $Q$ would be true, too.

It seems to me that any question that simply asks "Assume $P$ and prove $Q$" without directly addressing the fact that $P$ is false (under the prevailing set of implicit assumptions that govern the rest of what is going on in the course) is a badly worded question and one that invites an "unfair" proof. If you want students to consider that a change of the "ground rules" could allow for things other than those normally possible, you should say so directly, rather than be mysterious about it. Here is one way you might word such a question:

It is easy to prove that there is no real number $a$ such that $a^2 + a + 1 = 0$. (Before continuing with this paragraph, write such a proof.) But suppose we introduce a new kind of number for which $a^2 + a + 1 = 0$ is true. Such an $a$ would not be a real number; it would have to be some sort of special, non-real number. Without worrying about what exactly such an $a$ would need to be, just assume that it exists. Prove that, under this assumption, every quadratic equation has a solution that can be written in the form $r + sa$, where $r$ and $s$ are ordinary real numbers of the type that we are familiar with, and $a$ is the "special" number satisfying $a^2 + a + 1 = 0$.

Admittedly, this is wordy, and some students might balk at the complexity of the question. But there is nothing wrong with making students read a paragraph of text when doing mathematics! In fact it probably is one of the most under-appreciated tools in a mathematician's skillset, and one that we often do a poor job of teaching our students.

With a question like the above, I don't think you would get any students who simply say "There is no such $a$ so the theorem is vacuously true." You have forestalled that response by acknowledging up front that no such $a$ exists under the normal hypotheses of what a "number" is, and asked students to entertain the possibility of expanding the number system by introducing a new kind of number that would be a solution. Any student who tried to just say "The hypothesis is false, so the theorem is true" is clearly missing the entire point of the question.

Notice that doing this stops rather short of Rory Daulton's suggestion of asking students to actually identify new hypotheses under which $P$ could be true. That might be asking too much of students, and is also sort of missing the point of the question, which presumably is not supposed to be about whether and/or how $P$ could be true, but instead about the relationship between $P$ and $Q$. The formulation I propose simply stipulates that somehow $P$ can happen; just accept that as a working hypothesis and get to work on proving $Q$.

Answer 2

As you wrote, there is no real number $a$ satisfying $a^2+a+1=0$, and this can be proven using the axioms of the real numbers. As you also wrote, there actually is such an $a$ in the complex numbers. However, $a$ and the complex numbers can exist because the complex numbers do not obey all the axioms of the real numbers. To allow the complex numbers you must remove at least one axiom. (In fact, we remove all the ordering axioms.)

So your teacher was not fair in his assignment. The proof that such an $a$ would have property $P$ relies on a somewhat diminished set of axioms, and your teacher did not say so. A particularly bright student could answer, "If we remove these particular axioms then...". A less bright student could still answer by simply forgetting about those axioms or not realizing the contradictions that would ensue by leaving them in.

To answer your question, you could "teach a math student to distinguish between useful and useless contradictory assumptions" by asking them to rigorously show an argument for the contradiction, to clearly specify the axioms used in the argument, and to present a reduced set of axioms that would invalidate that argument. For extra credit, the student could present different, new axioms that would show that such an $a$ exists and would not contradict the other axioms. This of course would be very difficult, but I was given questions similar to this in my "math camp" many years ago. I must admit that I was not good enough to answer such questions.

If you don't point out the different axioms between the real numbers and the complex numbers or other field extensions, you could just confuse the better students. If you are asking for the best balance between rigor and creativity,... that question has consumed better mathematicians than I.