 In this video, I wanna continue our discussion of logical fallacies. This will be the third in a trilogy of videos we've been talking about logical fallacies so far. And in this video, I wanna focus on logical fallacies that are actually very tempting to mathematical writing. In the previous one in our trilogy, we talked about fallacies that exist because of human emotion and human bias, basically when we're irrational, we might make a fallacy like that. Let's now use our rational minds, okay? And talk about what are some logical fallacies that happen there because even if we're being rational, even if we're being reasonable, our arguments could still be invalid, they could still be unsounded. So we have to be careful about those. So the first fallacy we're gonna talk about here is commonly referred to as the appeal to ignorance. An appeal to ignorance is any type of argument that assumes something is true because it hasn't been proven false or conversely, you assume it's false because it hasn't been proven true yet, okay? An example of that would be something like, nobody has proven that this photo is not bigfoot, so it must be bigfoot, right? No one has proven the statement false, therefore the statement must be true. This appeal to ignorance, we don't have information yet, like there's a statement, a statement is either true or false, but we might not know the truth value of the statement. And we make the argument that like, well, if it was false, someone would have proven it false by now. Or if it was true, someone would have proven it true by now, therefore, it hasn't happened. In mathematics, we fall into the temptation of the appeal to ignorance when we consider conjectures. There exist many conjectures that have existed for hundreds of years that have been left unproven. At the timing of this video recording, the Riemann hypothesis, for example, is a conjecture that is still unproven. Most mathematicians, I believe, would argue that the Riemann hypothesis is probably true. We don't say certainly it's true because no proof has been provided to it yet, but the thought process is that it's probably true, among other reasons, there's lots of reasons you could argue here, but you might argue that if it were false, that means there would exist a counter-example, and if a counter-example existed, we probably would have found it by now, okay? That itself is not a proof that the Riemann hypothesis is true. That gives us evidence to think it probably true, but that itself is not the proof, right? The fact that we haven't found a counter-example doesn't mean that counter-examples don't exist. Sometimes counter-examples can be quite elusive and can hide from mathematicians for centuries as well. In this lecture series, we talked about perfect numbers and we asked about odd perfect numbers. Do there exist an odd perfect number? Well, the mathematical community, for those part, believes that odd perfect numbers don't exist. And one of the reasons to support that claim is that if an odd perfect number exists, it would have to be gigantic. We've searched and searched for all the small possibilities and they don't exist. So if they exist, they have to be huge. And what then causes it to happen eventually if it hasn't happened already? That's reason to think it might not be true that that is the existence of odd primes, odd perfect numbers. There's a lot of odd primes, don't get me wrong. But that it's not a proof. So you don't wanna say the statement is true or false because of the lack of counter-examples. One should be cautious. Now, this can help us make a judgment. We can make a statement like it's probably true because we have improvement in false otherwise. But that itself, to say that it's true because of the lack of evidence, is a fallacy. Another one that we should be very cautious about is what's commonly referred to as the false dilemma. Some people refer to as limited choice. The false dilemma argument falsely claims that an argument is either or. That is, there's like two options. There's option A, there's option B. And then typically what people say is like, well, it's not option B, therefore it's option A. But it turns out there could be other possibilities. There could be option C, option D. So even if we rule out option B, that doesn't necessarily point to option A, there could be other possibilities as well. So imagine the following. Either those lights in the sky were aliens or an airplane. I looked at the manifest at the airport. There are no airplanes scheduled for tonight. So it must be aliens. This is an example of a false dilemma. They're claiming that they saw something in the sky. It could only be an airplane or an alien. They ruled out the possibility of airplanes. Therefore it has to be aliens. Well, could there be other possibilities? Maybe there's airplanes that weren't scheduled flying last night. Maybe it was like a stealth plane or maybe it was a satellite. I don't know, maybe it's a comet. There's a lot of things that could be that maybe it wasn't aliens. So the false dilemma means that we didn't consider every possibility as we start ruling them out. In mathematics, we have to be cautious about this too. We'll talk about this actually in the very next lecture in our lecture series, this idea of proving things by cases. If you wanna prove a statement by considering cases, you have to make sure that you look at every possible case. You have to exhaust all of the cases. If you miss even one case that could actually make your statement be false and thus it could invalidate your proof. So we have to make sure we consider all the cases because if we forget one case, we are falling into this false dilemma because we've limited the choices beyond what is acceptable. All right, the next one has a very fun Latin name, post hoc ergo proctor hoc for which you can obviously understand why this is often abbreviated as the post hoc argument. The post hoc argument claims that because two things are related or we might use the word that two things are correlated, perhaps because they happened in sequence, one thing happened and then the next thing happened, we think that there is some type of causation, some cause and effect relationship between them. Sometimes this is called the false cause argument because maybe you don't speak Latin. An example of this would be something like the following. Today, I wore a red shirt and my football team won. I need to wear a red shirt every time they play to make sure they keep on winning. This type of superstition is a post hoc argument. The idea is you're supposing that, well, I wore a shirt and then they won, you're supposing that there's some type of causality relationship between the red shirt and the winning and perhaps because there was another time where you didn't wear a red shirt and they lost, right? There could be some correlation between it perhaps, the football team, they see you wearing the red shirt, the red shirt could be, of course, the team colors and they saw you wearing it in the stands and that gave them the energy because they're like the audience wants us to win, the fans want us to win and that gave the football team the encouragement to try harder. Things like that happen, psychology does affect sports a lot, not wearing the red shirt, maybe the audience didn't give the football team enough cheer and they got taken over by the yips, they lost, maybe there's some relationship there but probably not. If you're just, if you're watching like the NFL and you're at home, no one cares what you wear, that won't make any bearing on the game whatsoever but because the things happen in sequence, you could falsely assume that they're related. The second example actually was given to me by a psychology professor when I was an undergraduance, one that stuck with me all the time and basically the situation's the following, during the months for which when ice cream sales are at the highest, drowning deaths are also at the highest. Therefore, eating ice cream makes people drown. That is another example of this post-hoc fallacy. The idea is these two things are correlated. I mean, there are statistics that you could then reference to verify this correlation as ice cream sales go up, swimming deaths also go up as well but perhaps you can see how these things are related. I'm gonna have to go like Phineas and Ferb on you right now. The relation is summer, okay? Because at least in the United States here, summer, right? It gets hot during the summer and so people look for opportunities to cool down. One way to cool down during hot summer days is eat some ice cream. That's how I like to do it. Another way to cool down during hot summer days is to go swimming. So as the temperature gets hotter, more people buy ice cream, more people go swimming, okay? There's more people swimming but when there's more people swimming, then more people are gonna unfortunately drown, right? If there's like a 0.1% chance that if you go swimming today, you're gonna drown. I have no idea, I just made up that statistic. Don't freak out about swimming right now. It's fairly safe for the most part but tragedies do happen once in a while. If there's a certain percentage of people who go swimming, they're gonna drown. If you increase the number of people swimming, then the percentage stays the same but the number of deaths will increase as well. These are related to each other because these events happen because of summer but ice cream does not cause, does not cause the deaths in the water. It's not like people got an ice cream cramp and then they drowned from that. Maybe that happened to one person but that's not what's happening here. And so we have to also be cautious about this in mathematics. This is actually particularly a problem in statistical reasoning and statistics and I should say this is not statistics false. When people misuse data, when they misinterpret data and don't follow the laws of statistics, they might be thinking that, oh, there's a correlation between these things, therefore there's a cause. So one should be very cautious about that in the mathematical sciences that we don't force a causality when only we have a correlation. Speaking of statistics, another fallacy we can refer to is the gambler's fallacy. The gambler's fallacy is the belief that the probability for an outcome after a series of outcomes is not the same as the probability of a single outcome. Take for the following example here. Imagine you have someone who's like at Las Vegas betting on a roulette table or something. They might say something like the following. I bet on black the last three times and I lost. I'm gonna bet on black one more time because the roulette table is due for a black. And you can actually understand where this fallacy might come from. In fact, this fallacy is actually grounded within mathematical or statistical reasoning. There's the so-called law of large numbers. Which basically says, without going through all the details of it, the law of large numbers tells us that if we have a random experiment and we were to repeat that experiment over and over and over again, then if you look at the ratio between successes and failures, which in this case you could think of like if black comes up, that's considered a success for our gambler here. And anything else coming up, which for a roulette table that would be red, anything else would come up would be a failure. The law of large numbers is telling us that the ratio, the percentage of wins versus the everything, right? So if you take your percentage of wins divided by the total, the total now outcomes, this percentage as the number of trials goes towards infinity, if n is the number of trials, this percentage will converge towards the probability of the event. So if you take a roulette table for example, as you spin and the ball moves around the wheel, every spot is either black or red. And so there is a 50% chance of black coming up every time you spin that wheel. Our gambler here has noticed that the last three times he played, he didn't get black. So if you look at zero blacks out of three outcomes, that's zero percent, okay? But it should be 50%. Now if the experiment were to repeat over and over and over again, this law, the statistical law says that this ratio should converge towards 50%. So that's why the gambler feels like he's due, right? How is this number gonna get closer to 50% unless it somehow pops up another black? And so you could think with that reasoning in mind, like there is actually some reasoning there that's based upon valid mathematics and statistics. The problem of the fallacy, the logical fallacy here, or maybe I should say it's a statistical fallacy, is that the next experiment, the next time you spin the roulette wheel, it's an independent random variable there. It's an independent experiment. The previous amounts, the previous experiments you conducted have no bearing on the current one. So the next time you spin it, it is still a 50-50 chance you're gonna get black or red. There's no guarantee that the next spin is gonna be black. This information right here has no effect. It doesn't change the probability. So this is the gambler's fallacy that the outcome of a series somehow changes the probability of the next outcome. And that's not true. Past experience for an independent random variable has no bearing on what the next random assignment is gonna be. All right, then the last one I wanna mention in this video here, this is the one I see the most when I actually have to read students' mathematical proofs. I mean sure, proofs will have like mathematically incorrect statements or maybe they have logical gaps, they're incomplete, maybe it's poorly written or unclear to read. There's always issues there. But if we're talking about like logical fallacies, the one that shows up the most often in my experience for mathematical students in a class like Math 3120 would be circular reasoning. This is an argument that relies on the conclusion being true for the premise to be true. That is to say, somewhere you're assuming, you're assuming the conclusion to be true or something logically equivalent to it and then use that to argue why it's true. Basically, you have the proof that P implies P. Excuse me, P implies P. Well, sure, if P is true, then P implies P, right? If P is true, then this conditional statement is true. Now, of course, if P is false, the conditional is vacuously true. And so this right here is an example of a totality, that it's always true regardless of the assignment of the P. And so if you're trying to prove P implies P, then of course that's a true statement. But what we are probably trying to prove, you're probably trying to prove something like Q implies P, but you've erroneously changed it to be P implies P, which of course this is true, you can't fight that, but it's like that's not this statement right here, which we're trying to prove. And so an example of circular reasoning that I really, really love, because I've actually heard this said by students before, something like the following, I shouldn't have gotten a C in that class. I'm an A student. I want you to realize how this is an example of circular reasoning. If you were in fact a A student, then you wouldn't have got a C. So the fact that you got a C means that you're not an A student, right? Of course, what they're trying to say here is that they've received A's in the past and therefore that means they should get A's in the future, in which case if they were to reframe it, I would then go back and talk about the gambler's fallacy there that past experience doesn't tell you what the next experiment's gonna be. Now, yes, of course I should clarify when this is mostly a joke right now, but of course assigning grains is not a random experiment, but nonetheless, if you are in fact an A student, you probably aren't gonna be arguing with your professor on why you're getting a C because A students get A's. There's circular reasoning built into it. We have to be cautious that we don't hide an equivalent form of the conclusion inside of our proof because then we didn't prove anything, nothing valid at least. And so that's gonna bring us to the end of lecture 14. Thanks for watching. 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