Reductio ad absurdum

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Reductio ad absurdum, a Latin expression that literally means 'reduction to absurdity', is one of the logical methods of proof most used in mathematics to demonstrate the validity (or invalidity) of categorical propositions.

It starts by assuming as hypothetical the truth or falsity of the thesis of the proposition to be demonstrated and, through a concatenation of valid logical inferences, it is intended to reach a logical contradiction, an absurdity. If a contradiction is reached, it is concluded that the starting hypothesis (which had been assumed to be true at the beginning) must be false (or vice versa).

To prove the invalidity of a proposition, it is assumed as a starting point that the proposition is true. If the final derivation is a contradiction, it is concluded that the original proposition is false and the argument is invalid.

This method is also known as proof by contradiction or ad absurdum proof. Part of the basis is compliance with the principle of exclusion of intermediates: a proposition that cannot be false is necessarily true, and a proposition that cannot be true is necessarily false.

Its use in mathematics

The proof by reduction to absurdity is a type of argument widely used in mathematical proofs.

It consists of demonstrating that a mathematical proposition is true, proving that if it were not, it would lead to a contradiction, for which reason it would be true.

To obtain a valid test it must be proved that, given a proposition ${displaystyle P,!}$ , «No. ${displaystyle P,!}$ » implies a false property in the mathematical system used. The danger is the logical fallacy of the argumentation by ignorance, by which it is proved that "No. ${displaystyle P,!}$ » implies a property ${displaystyle Q,!}$ it seems false but that such falsehood has not really been proved.

A classic example of this fallacy is the false proof of a fifth postulate of Euclid from the previous ones. Because when these proofs were established there was no Geometry other than Euclidean, they seemed correct. After the appearance of other geometries it was shown that the system was incorrect. For a more in-depth explanation of these fallacies see Mathematical Thought: from Ancient to Modern Times, by Morris Kline.

Although in mathematical proofs this method is used with great freedom, not all schools of mathematical thought accept the reductio ad absurdum as universally valid. In schools like that of intuitionism, the law of exclusion of intermediates is not accepted as valid. From this point of view there is a very significant difference between demonstrating that by means of a real example of a "something" that exists it would be absurd to demonstrate its non-existence.

Examples

There is no minimum rational number greater than zero

Suppose you want to prove a statement P. The procedure consists of demonstrating that assuming the falsehood of P as true (ie P negated) leads to a logical contradiction. So P should not be false. Therefore it has to be true.

For example, consider the proposition “there is no minimal rational number greater than zero”. In a reduction to absurdity we would begin by assuming the opposite and our thesis would be: there exists a minimum rational number greater than zero: r₀.

Now let's take x = r₀/2. Therefore x is a rational number greater than zero, and x<r₀. This is absurd, since it contradicts the initial hypothesis that r₀ was the minimum rational number. Therefore it must be concluded that the proposition assumed to be true: “there is a minimum rational number greater than zero” is false.

It is not unusual to use this type of reasoning with propositions such as the one stated, about the non-existence of a certain mathematical element. It is assumed that this element exists and it is proved that this leads to a contradiction. Therefore that object does not exist.

Are there infinitely many prime numbers?

There are numerous proofs that there are infinitely many prime numbers, the first of which there is evidence is from Euclid, where it is demonstrated by means of Reductio ad absurdum in Proposition 20 from Book IX of Elements (There are more prime numbers than any proposed number of prime numbers).

From assuming that the truth is the opposite, therefore our thesis would remain: «The prime numbers are finite»Then we have $n$ prime numbers that would be ${displaystyle P=p_{1},p_{2},...,p_{n}}$ .

Then the following number is now taken:

${displaystyle m=p_{1}cdot p_{2}cdot...cdot p_{n}+1}$

We have to ${displaystyle m}$ is the product of all prime numbers plus 1, and ${displaystyle m}$ is not a prime number, because it is not on the previous list, then by definition ${displaystyle m}$ is a composite number and must be divisible by some prime number.

If we make the division between any prime number on the list ${displaystyle P=p_{1},p_{2},...,p_{n}}$ , we get the rest 1, so there must be at least another prime number that is not on that list.

Then we arrive at a contradiction of our thesis «Prime numbers are finite» which is false, so there are infinitely many prime numbers.

The square root of 2 is irrational

An example is the proof that the square root of 2 is an irrational number. The initial statement (our thesis) is the opposite, that is, that: "the square root of 2 is a rational number".

Being a rational number, let's express it as $p/q$ ; then it would remain:

{displaystyle {sqrt {2}}={frac {p}{q}}}

{displaystyle p,qin mathrm {Z} /qneq 0}

(where) p and q They're integers, q is different from 0).

Without loss of generality, it can be assumed that p and q are positive (if both were negative, it would suffice to multiply them by -1) and that they are prime to each other, that is, they do not share any common factor (since if there were common factors, we can simplify them and be left with the resulting irreducible fraction). Now we raise both members to the square:

{displaystyle 2={frac {p^{2}}{q^{2}}}}

Multiplying on both sides by ${displaystyle q^{2},!}$ You got it.

{displaystyle 2q^{2}=p^{2},!}

The expression ${displaystyle 2q^{2},!}$ It's a number pair, so ${displaystyle p,!}$ also is pair (if not, ${displaystyle p^{2},!}$ it would not be even, and equality could not be fulfilled.

Sea ${displaystyle p=2n,!}$ Where $n ,!$ It's an integer. Replacement, the expression would remain:

{displaystyle 2q^{2}=(2n)^{2}=4n^{2},!}

We can simplify by dividing both parts by two and obtain that

{displaystyle q^{2}=2n^{2},!}

By the same reasoning before, where ${displaystyle 2n^{2},!}$ It's a number pair, ${displaystyle q^{2},!}$ It's a pair, too. ${displaystyle q,!}$ .

Like ${displaystyle p,!}$ and ${displaystyle q,!}$ are pairs, they have at least one common factor, ${displaystyle 2,!}$ . This contradicts the previous assumption that the numbers ${displaystyle p,!}$ and ${displaystyle q,!}$ they had no factors in common. Like this choice ${displaystyle p,!}$ and ${displaystyle q,!}$ was done without loss of generality and subsequent reasoning is correct, it implies that the initial premise that ${sqrt {2}}$ It was rational. It's false.
Later. ${sqrt {2}}$ It's irrational. Q.E.D.

Logic

In symbolic logic, the reduction to absurdity is expressed as follows:

Yeah.

{displaystyle Scap {neg P}vdash F}

then.

{displaystyle Svdash P}

In this representation, P is the statement to be proved, and S is a series of previous statements taken to be true. For example, the axioms of the theory in which work has been done or the previous theorems already demonstrated. Consider the negation of P in conjunction with S. If this leads to a contradiction F it can be concluded that S necessarily leads to P.

In the words of G. H. Hardy: «The reduction to the absurd, which Euclid loved so much, is one of the best weapons of Mathematics. It is a much better gambit than any of those in chess: a chess player can offer the sacrifice of a pawn or another piece, but a mathematician offers the game.

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