Mathematical fallacy

For example, the reason why validity fails may be attributed to a division by zero that is hidden by algebraic notation.

There is a certain quality of the mathematical fallacy: as typically presented, it leads not only to an absurd result, but does so in a crafty or clever way.

[1] Therefore, these fallacies, for pedagogic reasons, usually take the form of spurious proofs of obvious contradictions.

The latter usually applies to a form of argument that does not comply with the valid inference rules of logic, whereas the problematic mathematical step is typically a correct rule applied with a tacit wrong assumption.

Beyond pedagogy, the resolution of a fallacy can lead to deeper insights into a subject (e.g., the introduction of Pasch's axiom of Euclidean geometry,[2] the five colour theorem of graph theory).

Pseudaria, an ancient lost book of false proofs, is attributed to Euclid.

In elementary algebra, typical examples may involve a step where division by zero is performed, where a root is incorrectly extracted or, more generally, where different values of a multiple valued function are equated.

[4][5] Examples exist of mathematically correct results derived by incorrect lines of reasoning.

Such an argument, however true the conclusion appears to be, is mathematically invalid and is commonly known as a howler.

[note 1] Another classical example of a howler is proving the Cayley–Hamilton theorem by simply substituting the scalar variables of the characteristic polynomial with the matrix.

Bogus proofs, calculations, or derivations constructed to produce a correct result in spite of incorrect logic or operations were termed "howlers" by Edwin Maxwell.

The problem is that antiderivatives are only defined up to a constant and shifting them by 1 or indeed any number is allowed.

Since the difference between two values of a constant function vanishes, the same definite integral appears on both sides of the equation.

Proof: The fallacy is in the second to last line, where the square root of both sides is taken: a2 = b2 only implies a = b if a and b have the same sign, which is not the case here.

In this case, it implies that a = –b, so the equation should read which, by adding ⁠9/2⁠ on both sides, correctly reduces to 5 = 5.

Another example illustrating the danger of taking the square root of both sides of an equation involves the following fundamental identity[9] which holds as a consequence of the Pythagorean theorem.

[10] In the above fallacy, the square root that allowed the second equation to be deduced from the first is valid only when cos x is positive.

Invalid proofs utilizing powers and roots are often of the following kind: The fallacy is that the rule

[11] Alternatively, imaginary roots are obfuscated in the following: The error here lies in the incorrect usage of multiple-valued functions.

If this property is not recognized, then errors such as the following can result: The error here is that the rule of multiplying exponents as when going to the third line does not apply unmodified with complex exponents, even if when putting both sides to the power i only the principal value is chosen.

Many mathematical fallacies in geometry arise from using an additive equality involving oriented quantities (such as adding vectors along a given line or adding oriented angles in the plane) to a valid identity, but which fixes only the absolute value of (one of) these quantities.

This quantity is then incorporated into the equation with the wrong orientation, so as to produce an absurd conclusion.

This wrong orientation is usually suggested implicitly by supplying an imprecise diagram of the situation, where relative positions of points or lines are chosen in a way that is actually impossible under the hypotheses of the argument, but non-obviously so.

In general, such a fallacy is easy to expose by drawing a precise picture of the situation, in which some relative positions will be different from those in the provided diagram.

In fact, O always lies on the circumcircle of the △ABC (except for isosceles and equilateral triangles where AO and OD coincide).

Furthermore, it can be shown that, if AB is longer than AC, then R will lie within AB, while Q will lie outside of AC, and vice versa (in fact, any diagram drawn with sufficiently accurate instruments will verify the above two facts).

Because of this, AB is still AR + RB, but AC is actually AQ − QC; and thus the lengths are not necessarily the same.

The basis case is correct, but the induction step has a fundamental flaw.