In using the backward and/or forward processes to prove that a hypothesis is true, we always assume that the hypothesis is true. Fallacy?

Question

It is probably worthwhile to state that I am self-learning how to reason about mathematical proofs. As will probably be obvious, I am not experienced and am just seeking to learn and have my thoughts clarified.

The forward process involves deriving from the statement A (hypothesis), which you assume to be true, some other statement, A1, that you know is true as a result of A being true. The statement derived from A must be directed towards linking up with (the conclusion) statement B.

The backward process starts by asking, "How can I conclude that the statement B is true?". However, you must ask this question in an abstract way, in order to clear away irrelevant details, which may otherwise hinder your ability to solve the problem; this allows you to focus on those elements of the problem that are significant/relevant. Eventually, you must make use of the assumption that A (the hypothesis) is true.

More on the forward/backward method: http://www.faculty.fairfield.edu/ssawin/371/notes/proofs.pdf.

In using the backward and/or forward processes to prove that a hypothesis is true, we always work under the assumption that the hypothesis is true. However, is this not a fallacy? If we are attempting to prove something, we do not yet know for certain if it is true or false. Indeed, there are many problems, such as those in number theory, where a conjecture initially seems obviously true but is actually false. If we always worked under the initial assumption that the conjecture is true, we would never arrive at a solution!

For Instance, "If the right triangle $XYZ$ with sides of lengths $x$ and $y$ and hypotenuse of length $z$ has an area of $\dfrac{z^2}{4}$, then the triangle $XYZ$ is isosceles.". The hypothesis (A) would be, "The right triangle $XYZ$ with sides of length $x$ and $y$ and hypotenuse of length $z$ has an area of $\dfrac{z^2}{4}$.". In this case, the hypothesis is true. However, it would not be difficult to come up with a scenario where the hypothesis was false.

Is this not a fallacy? If not, what is the logic/reasoning behind this method? Obviously, conjectures are often proven to be false, so there must be some other reasoning behind using the forward/backward processes?

Thank you.

Answer 1

When you say

In using the backward and/or forward processes to prove that a hypothesis is > true, we always work under the assumption that the hypothesis is true.

Edit: this is not, in fact, what the OP meant. A proper answer is below.

I assume you mean "conclusion" (i.e. $B$, not $A$) both times! (We are trying to derive a maybe-true-conclusion from hypotheses-we-know-to-be-true.)

Of course, as you note, if your conclusion is not true, then neither the forward nor backward process will ever find a proof! However this is not a "fallacy", as you claim: a fallacy is when you have flawed logic. Finding a mathematical proof is not (yet?) an algorithmic process, so we can't really talk of the forward or backward processing "terminating" in any fixed way.

In practice, when you're solving a problem, you'll probably have a guess as to whether $B$ is true or not. If neither your forward or backward processes are making any progress, you'll start to doubt your guess, and maybe try showing the opposite of what you were trying to prove originally, i.e. "$B$ is false". Often these feed back into each other: realising why a particular proof fails might give you a clue as to where to look for a counterexample, or vice versa.

In short, when you're applying the forward/backward method, you are assuming the conclusion $B$ is true and trying to find a way to prove it. This is not a fallacy because you of course won't be using the fact that $B$ is true in your proof!

---

Edit: an attempt to answer the OP's actual question.

If you're wondering whether your hypotheses are true, that's a very different question which doesn't have anything to do with the forward/backward method.

Whenever you're proving something, you always have to have some background assumptions. Some of these are obvious (e.g. in proving the statement "if $X$ then $Y$" then you're assuming $X$ to be true and trying to show $Y$; more formally, this is via the "deduction theorem" of mathematical logic) and some are less obvious (e.g. if I know "$X$ implies $Y$ and I know $X$, then I'm allowed to deduce $Y$; that is, assumptions about how logic itself works).

In maths, you will in practice be working from a set of results that have already been proved by other people, and trying to prove something new. In theory, there are a set of axioms (e.g. ZFC) that you've taken as the basis for your mathematics, and everything else has to be proved from only those axioms.

In your example, if $A$ were not true then it would of course be fallacious to claim that $B$ were true because $A \Rightarrow B$. However, it's still perfectly valid to say "suppose $A$ were true, then $B$ would be true. Therefore we conclude $A \Rightarrow B$". In fact, this is often what we use to show that $A$ is not true! (e.g. if $B$ is something we know to be obviously false.) This is called "proof by contradiction".