Proof by Contradiction: "Bad Form" or "Finest Weapon"? Reconciling Perspectives

Question

G.H. Hardy famously described proof by contradiction as "one of a mathematician's finest weapons." However, I've encountered claims that some schools of thought consider proof by contradiction to be a less desirable form of proof.

I'm interested in understanding the reasoning behind both perspectives. Could someone elaborate on:

Why proof by contradiction is considered powerful and elegant by some, like Hardy? What are its strengths and advantages compared to other proof techniques? What are the potential criticisms or drawbacks of proof by contradiction? Why might some schools of thought prefer alternative methods? Are there specific areas of mathematics where proof by contradiction is more or less favored? If so, what are the reasons for this preference? How do these differing perspectives impact the practice of mathematics? Do they lead to different approaches or conclusions?

I myself think that proof by contradiction is a placeholder for advanced direct proofs in the future of that proof. For example, historically, The most famous proof of the irrationality of √2 is a proof by contradiction. It assumes √2 is rational and derives a contradiction. This proof is elegant but doesn't offer much insight into why √2 is irrational. Later mathematicians developed direct proofs of the irrationality of √2. These proofs often use properties of continued fractions, providing a more intuitive understanding of why √2 cannot be expressed as a fraction.

I'd appreciate any insights or references that can help me reconcile these seemingly conflicting viewpoints.

Answer 1

One disadvantage of proofs by contradiction is that any intermediate results you derive along the way can't be re-used elsewhere, since they are only valid under the contradiction hypothesis. For this reason, if you're writing a paper whose central argument is a proof by contradiction, it's a good idea to state and prove as many lemmas as you can "outside" of the contradiction argument before applying them in the main proof.

Answer 2

There is some nuance about what is truly considered "proof by contradiction". Consider two cases:

1. You want to prove $\neg P$. So, you assume $P$ and derive a contradiction. You conclude $\neg P$.

2. You want to prove $P$. So, you assume $\neg P$ and derive a contradiction. You conclude $P$.

Are these both instances of proof by contradiction? Maybe, but the second one relies on "double negation elimination" (i.e. that $\neg\neg P \rightarrow P$), while the first does not. So, intuitionist mathematics rejects the second one but embraces the first. Indeed, usually in intuitionist mathematics, $\neg P$ is simply notation for $P \rightarrow \perp$.

I don't have a definite opinion on whether both of these should be called proof by contradiction, or if just the second should be. However, I will say that if "irrational" means "not rational", the argument "assume $\sqrt{2}$ is rational and derive a contradiction" is of the first form and is therefore a valid proof of the statement "$\sqrt{2}$ is irrational" in intuitionist mathematics. (Though it may not be equivalent to "every rational number is not equal to $\sqrt{2}$"; I can't remember the intuitionistic rules for distributing negations over quantifiers.)

Answer 3

For example, historically, The most famous proof of the irrationality of √2 is a proof by contradiction. It assumes √2 is rational and derives a contradiction. This proof is elegant but doesn't offer much insight into why √2 is irrational. Later mathematicians developed direct proofs of the irrationality of √2. These proofs often use properties of continued fractions, providing a more intuitive understanding of why √2 cannot be expressed as a fraction.

This is an excellent example to discuss why I don't like proof by contradiction. It is neither necessary to phrase the classic proof as a proof by contradiction, nor is it necessary to learn anything about continued fractions to write down a direct proof.

Here is a proof equivalent to the classic proof but involving no contradiction: by definition, what it means for a number to be irrational is that it is not equal to any rational number, so we will just prove directly that if $\frac{p}{q}$ is a rational number then it is not equal to $\sqrt{2}$. In fact we will prove the slightly stronger statement that $\frac{p^2}{q^2}$ is not equal to $2$, or equivalently that $p^2$ is not equal to $2q^2$.

The argument is simple: in words, $2$ must divide both the numerator and denominator an even number of times, so it must also divide their quotient an even number of times, but $2$ divides itself only once. In symbols, write $\nu_2(n)$ for the $2$-adic valuation, which is the number of times $2$ divides an integer $n$. Then $\nu_2(mn) = \nu_2(m) + \nu_2(n)$. It follows that $\nu_2(p^2)$ is even, but $\nu_2(2q^2)$ is odd, so

$$p^2 \neq 2q^2$$

for any integers $p, q$. This argument generalizes immediately to the following more general result:

If $n$ is a positive integer such that there exists a prime $p$ such that $\nu_p(n)$ is not divisible by $k$, then $\sqrt[k]{n}$ is irrational. Equivalently, $\sqrt[k]{n}$ is rational iff it is an integer.

Note that once $k \ge 3$ continued fractions no longer offer a route to proving this.

Although it is not technically necessary to state the proof, a natural way to rephrase the proof is the following: the $2$-adic valuation actually naturally extends to rational numbers, by declaring that $\nu_2 \left( \frac{p}{q} \right) = \nu_2(p) - \nu_2(q)$, and by the "logarithmic" property above this is well-defined. Then $\nu_2 \left( \frac{p^2}{q^2} \right)$ is even (possibly negative) but $\nu_2(2) = 1$.

If you stare at this argument a little more you might notice something else: what this argument really appears to be saying is that $\sqrt{2}$ itself has a $2$-adic valuation, namely $\frac{1}{2}$, which is not the $2$-adic valuation of any rational. This is true! And generalizing this leads to some rich ideas in algebraic number theory.

So, what we've learned from the direct argument is that $\sqrt{2}$ is irrational because it has a property, namely having a fractional $2$-adic valuation, or informally "being divisible by $2$ a fractional number of times," that no rational number has. This is obscured by phrasing the proof as a proof by contradiction, because in a proof by contradiction you don't know whether any of the things you're proving are true! In a direct proof you are forced to write down only true statements so you learn much more along the way.

Answer 4

What are the potential criticisms or drawbacks of proof by contradiction?

In the proof of a negative statement, e.g. that there is no greatest prime number, a proof by contradiction is arguably fine, we are in fact providing a counter-example: indeed, even in constructive logic deriving a contradiction is a valid method of (dis-)proof. It is with positive statements, especially of existence, that a proof by contradiction is not much satisfactory, in that it does not find any witness nor provides any way to find one.

Eventually, I think different takes have more to do with one's stance towards classical logic and the idea (in my own words) that, of any proposition, we can unconditionally assume that whether the proposition is true or false is given (namely, that we can do case analysis on any proposition in context).