Mathematical fallacies | Wikipedia audio article
Channel: wikipedia tts
Date: 07/10/2019

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Mathematical_fallacy


00:00:41 1 Howlers
00:01:22 2 Division by zero
00:02:03 3 Multivalued functions
00:02:44 4 Calculus
00:03:05 5 Power and root
00:03:46 5.1 Positive and negative roots
00:04:07 5.2 Square roots of negative numbers
00:04:48 5.3 Complex exponents
00:05:50 6 Geometry
00:06:31 6.1 Fallacy of the isosceles triangle
00:07:54 7 Proof by induction
00:08:56 8 See also
00:09:16 9 Notes
00:10:18 10 References
00:10:59 11 External links
00:11:40 Fallacy of the isosceles triangle
00:12:22 ∠OQA
00:12:42 ∠QAO; AO
00:13:03 ∠CQO
00:13:24 OC (hypotenuse); RO
00:13:44 AQ, RB
00:14:05 AR + RB
00:14:46 BC and AC
00:15:27 Proof by induction
00:16:08 1 (i.e. one horse is a group where all the horses are the same colour). Thus, by induction, N horses are the same colour for any positive integer N. i.e. all horses are the same colour.The fallacy in this proof arises in line 3. For N
00:16:50 0 horses in common, and thus are not necessarily the same colour as each other, so the group of N + 1
00:17:31 1. The basis case is correct, but the induction step has a fundamental flaw. If we were additionally given the fact that any two horses shared the same colour, we could correctly induct from the base case of N
00:18:12 See also



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SUMMARY
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In mathematics, certain kinds of mistaken proof are often exhibited, and sometimes collected, as illustrations of a concept of mathematical fallacy. There is a distinction between a simple mistake and a mathematical fallacy in a proof: a mistake in a proof leads to an invalid proof just in the same way, but in the best-known examples of mathematical fallacies, there is some concealment in the presentation of the proof. For example, the reason validity fails may be 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.  Therefore, these fallacies, for pedagogic reasons, usually take the form of spurious proofs of obvious contradictions. Although the proofs are flawed, the errors, usually by design, are comparatively subtle, or designed to show that certain steps are conditional, and are not applicable in the cases that are the exceptions to the rules.
The traditional way of presenting a mathematical fallacy is to give an invalid step of deduction mixed in with valid steps, so that the meaning of fallacy is here slightly different from the logical fallacy. The latter applies normally to a form of argument that is not a genuine rule of logic, where 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 (such as the introduction of Pasch's axiom of Euclidean geometry and the five colour theorem of graph theory). Pseudaria, an ancient lost book of false proofs, is attributed to Euclid.Mathematical fallacies exist in many branches of mathematics.  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.  Well-known fallacies also exist in elementary Euclidean geometry and calculus.