Fallacies: Fallacies And Fallacies (Mathematical ones)
Fallacies???: Fallacies??? Here we are talking about the fallacies in mathematics and not about the philosophical fallacy. Fallacy is an argument which seems to be decisive, but in reality is not. Like 2+2=5 can be proved and it seems decisive but in reality 2+2=4. Not anything else!!
Need an example? It’s here: Need an example? It’s here 0/0 = ½ Really????? 0/0 = 100-100/100-100 0/0 = 10(10-10)/10²-10² 0/0 = 10(10-10)/(10+10)(10-10) [a²-b²=(a+ b )(a-b)] 0/0 = 10/10+10 0/0 = 10/20 = ½
PowerPoint Presentation: Hahaha. But this isn’t true. Francois Englert or Peter Higgs?
PowerPoint Presentation: We know that sir We were just telling them about fallacies Then what’s the flaw
The flaw: The flaw The flaw lies in line 4. The identity [a²-b²=(a+b)(a-b)] only works when ‘a’ and ‘b’ are 2 different numbers. Finding flaws in mathematical fallacies (sometimes known as howler) is FUN.
Let’s try one more(It is useful): Let’s try one more(It is useful) To prove : π = 3 Let x = ( π + 3 )/ 2 Multiply by 2: 2x = π + 3 Multiply by ( π -3 ): 2x( π - 3) = ( π + 3 )( π - 3) Simplify: 2 π x - 6x = π ² - 9 Rearrange terms: 9-6x = π ² - 2 π x (I feel like something is wrong around here) Add x ² : 9-6x + x ² = π ² - 2 π x + x ² Factor: ( 3-x) ² = ( π - x) ² Take the square root: 3-x = π - x Add x: 3 = π
What’s the flaw here(I gave you the hint) Just think upon it I won’t tell everything: What’s the flaw here(I gave you the hint) Just think upon it I won’t tell everything
Fallacies. Are they new??: Fallacies. Are they new?? Nopes. They aren’t. They are as old as the human mind. Fallacies do not mean proving a wrong thing. A fallacy is an invalid argument with poor reasoning. Making up of a mindset without any logic may be called fallacial thinking. Only a true logic can remove a fallacy. Earlier we had a fallacy that the earth is flat but Magellan proved that the earth is round(and of course the satellite pictures prove that). They(humans in 13 th century)gave the logic that if the ground is flat, then the Earth is flat. One more example – My friend from USA is black so all Americans are black!!!
Fallacies with units: Fallacies with units 10 paisa is equal to 1 ⁄ 10 of a rupee . If we square both numbers, we get 100 paisa = 1 ⁄ 100 of a rupee. But 1 ⁄ 100 of a rupee is just 1 paisa! Even less!!! Well, we didn't treat the units properly. If we square 10 paisa, the units are not paisa and rupee, but (paisa)² and (rupee)² . The conversion factor between square rupees and square paisa is 10,000, not 100, so the paradox vanishes.
PowerPoint Presentation: I must take care of the units as well Or I’ll end up making A fallacy
Paradox: Paradox A paradox is a statement that goes against our intuition but may be true , or a statement that is or appears to be self-contradictory . Many mathematical paradoxes fall into one of two categories: either they result from the counter-intuitive properties of infinity, or are a result of self-reference.
Zeno’s bisection paradox (A runner can never reach the end of a racecourse in a finite time.): Zeno’s bisection paradox (A runner can never reach the end of a racecourse in a finite time .) Statement: Reason: 1. The runner must first pass the point ½ located halfway between himself and the finish line before he can finish the race. ½ is between the runner and the finish line. 2. It will take a finite time to reach the point ½. It is a finite distance from the start (1) to ½. 3. Once reached, there is another halfway point ½ which the runner must reach before he can finish. The remaining interval is divided in half. 4. There are an infinite number of such halfway points which the runner must reach. Each of these points will take a finite time. Statements 1, 2, and 3 can be repeated an infinite number of times. 5 The total time for the race is infinite. The sum of an infinite series of finite terms is infinite.
FLAW????: FLAW???? The problem with this reasoning is in step 5. The sum of an infinite series of finite terms is not necessarily infinite. Some infinite series, such as the harmonic series, are infinite, but not this series. The sum of the series ½ + ¼ + 1/8 + ... is equal to 1. The total time is finite because each step is done in half as much time as the previous step. Many of the ancient Greeks had problems with infinite concepts like this.
Then what is the difference between fallacy and paradox??: Then what is the difference between fallacy and paradox?? A paradox is something that seems to be false, but may be true or may not be. Example: There are just as many even integers as there are integers. A fallacy is something that seems to be true, but is false. Example: Every infinite set has the same number of elements .
PowerPoint Presentation: IS IT OVER????????
PowerPoint Presentation: I THINK SOMETHING IS LEFT …
Ah! That flaw in Pi = 3: Ah! That flaw in Pi = 3 The problem here is that, in order to get to the step in red, there is an implicit square root taken of both sides of the equation. However, there are potentially two roots to any square; e.g. x ² = 9 has possible solutions x= +3 or x=-3. The ‘proof’ above assumes that the positive root is correct, which leads to the erroneous answer π =3. If one looks at the negative root, one finds that 3-x = x- π , which leads right back to the starting equation x = ( π +3 )/2.
PowerPoint Presentation: Any questions in your curious mind??