[User] Please transcribe this text

[Assistant] We know that we reached a certain state x, y for which


[User] Please transcribe this text

[Assistant] Uh, m divides x and m divides y. Now we would like to show that if we transition to the next state, we again have that same property that m divides, uh, the number of gallons in both jugs once more. And then we can con- can conclude p n plus one. So that's, uh,


[User] Please transcribe this text

[Assistant] How we always proceed, so let's see whether we can write it out in a bit more formal way.


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[Assistant] So after another transition, what happens after another transition? So, we can conclude that the jugs, uh, are filled


[User] Please transcribe this text

[Assistant] By the different types of, of, of numbers that we see here in this state machine. So let's write it out. So after another transition, uhm,


[User] Please transcribe this text

[Assistant] It's actually filled, uh, are filled.


[User] Please transcribe this text

[Assistant] With, uh, well either, uh, if I've empty this, say zero jugs, uh, uh, zero gallons,


[User] Please transcribe this text

[Assistant] A, B, X, and Y, I see appearing over here. And I also notice that I see X plus Y, and X plus Y minus B, and X plus Y minus A. Those are all the different number of gallons that can be in the jugs. Yes, please.


[User] Please transcribe this text

[Assistant] So in our example we would have that m equals one. So, uh, uh, over here we have that


[User] Please transcribe this text

[Assistant] But for example, in the three gallon jug and the six gallon jug, right, we have that


[User] Please transcribe this text

[Assistant] So those are the two cases that we sort of look at right now.


[User] Please transcribe this text

[Assistant] But we put it into a much more general setting, right? We abstract away from the actual numbers.


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[Assistant] And use, uh, A and B as, uh, as, uh, as representations. Uh, are there any other questions?


[User] Please transcribe this text

[Assistant] So uhm, after another transition, each of the jugs are filled with


[User] Please transcribe this text

[Assistant] Well, either zero gallons if we have, uh, completely emptied them or we have filled


[User] Please transcribe this text

[Assistant] So the first A gallon jug, or it can be B, we also notice that it can be, uhm, it can be X of course, it can be Y, because that's the state that we are in.


[User] Please transcribe this text

[Assistant] I have x plus y minus a, which appears over here, and x plus y minus b.


[User] Please transcribe this text

[Assistant] So these are all the different number, uh, possible number of gallons.


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[Assistant] So now we can start using our, uh,


[User] Please transcribe this text

[Assistant] We have concluded that m divides x and also m divides y.


[User] Please transcribe this text

[Assistant] So if you now use, uh, some facts about the visibility on your handouts, which we will not prove now, but, uh, I think most of them will be on your problem set actually.


[User] Please transcribe this text

[Assistant] We can conclude that also linear combination of a, b, x and y will be divisible by m.


[User] Please transcribe this text

[Assistant] In particular, m will divide x plus y, m will divide x plus y minus a, and also x plus y minus b. So, we will conclude that uhm, m actually divides any possible, uh,


[User] Please transcribe this text

[Assistant] Uh, result. So, divides any of the above. And now we're done.


[User] Please transcribe this text

[Assistant] After the next transition, after we have reached x, y after n steps, then in our n plus 1th step, all that we can achieve is divisible by m. So that's exactly the invariant. So we conclude that p n plus 1 is true, and so now we're done.


[User] Please transcribe this text

[Assistant] Are there any questions about this proof? So this is like the standard technique that we try to use all the time here in this class.


[User] Please transcribe this text

[Assistant] We will use it in all the other areas as well, in graph theory in particular.


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[Assistant] And uhm, especially in number theory, we'll also use it, especially in this class.


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[Assistant] Okay, so, uh, let's apply this, uh, uh, theorem.


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[Assistant] Let's, uh, think about, uh, uh, this movie that we saw. There's Die Hard number three. Die Hard number four came out.


[User] Please transcribe this text

[Assistant] And then, eh, the cost got stuck in diehard number five. That was a problem because, ehm,


[User] Please transcribe this text

[Assistant] The rumors were that in Die Hard number 5 they had like a 33 gallon jug. That's a lot. And a 55 gallon jug. So, Bruce


[User] Please transcribe this text

[Assistant] Because you can imagine those are pretty heavy, so if you want to pull one into the other, my goodness, so. But the question is, is he training the right muscles?


[User] Please transcribe this text

[Assistant] So, can we apply this theorem now? And show that, eh, so, oh, I should tell you what is the problem. Well, again, he has to get, say, four gallons out of this, out of these two jugs. So is that possible?


[User] Please transcribe this text

[Assistant] It's not, I see someone shaking his head. Do you want to explain why?


[User] Please transcribe this text

[Assistant] Exactly, so 4 is not official by 11, so the whole cast got blown up in diehard number 5, and so we have no diehard number 6 as well. Okay, so uhm,


[User] Please transcribe this text

[Assistant] So now all of this stuff actually helps us to define a new concept as well. So let's do that. Uhm, I'll put it up here.


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[Assistant] Of A and B as being the, uh, greatest common divisor.


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[Assistant] Uh, if you are looking at A equals 3 and B equals 5, well, then the GCD.


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[Assistant] There's no other, uh, larger integer that divides both 3 and 5.


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[Assistant] Uh, and other examples are, for example, if you have the GCD of, say, 52 and 44. Well, what's this equal to?


[User] Please transcribe this text

[Assistant] Well, this actually is 4 times, uh, 13. This is 4 times 11.


[User] Please transcribe this text

[Assistant] So 4 divides both this and both this one, but nothing larger can divide both of those. So we have that this is equal to 4. Uhm, we will have a separate definition.


[User] Please transcribe this text

[Assistant] That talks about this very special case where two numbers, if you look at their greatest common divisor, uhm, when that greatest common divisor is equal to one,


[User] Please transcribe this text

[Assistant] We actually define those two numbers to be relatively prime to one another.


[User] Please transcribe this text

[Assistant] So that's another definition. We say that A and B are relatively prime.


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[Assistant] If their greatest common divisor is actually equal to one.


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[Assistant] Now today we will not really use this definition so much, but it's actually very important and we'll come back to this next lecture.


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[Assistant] So, if you now look at this particular theorem over here,


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[Assistant] Can we see a nice corollary of this?


[User] Please transcribe this text

[Assistant] Like a result, if you think about this greatest common divisor,


[User] Please transcribe this text

[Assistant] Well, the greatest common divisor of A and B divides both A and B.


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[Assistant] So the greatest common divisor of A and B will divide any result that you can generate by playing this game with the jugs. So the corollary here is that the G, C, D of A and B divides any result.


[User] Please transcribe this text

[Assistant] So this already tells us quite a bit about this game that we have here. So now what we would like to do is to find out what exactly can be reached. We have a property that we have shown here, but uhm, what else can we do here?


[User] Please transcribe this text

[Assistant] Now it turns out that we can say much more, and we would like to prove the following theorem.


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[Assistant] Uh, to analyze this whole thing much better, I don't think I need the state machine anymore. So let's take it off.


[User] Please transcribe this text

[Assistant] The theorem that we would like to prove is that any linear combination of the, uh, let's change this into the 3 and 5 again.


[User] Please transcribe this text

[Assistant] Any linear combination of three and five, I can get, I can make with these three and a five colon a gallon jug. So let's write it out.


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[Assistant] L, which we write as some integer s times a, plus some integer t times b. So any linear combination of a and b,


[User] Please transcribe this text

[Assistant] Uh, with, well, of course the number of gallons should fit, uh, the largest, uh, jug. So, with zero is, at most L, is at most V, can be reached.


[User] Please transcribe this text

[Assistant] So this theorem we would like to prove now, and in order to do that we would like to already


[User] Please transcribe this text

[Assistant] And think about some kind of a property that we have. So, so when we talk about linear combinations,


[User] Please transcribe this text

[Assistant] Uhm, the SNAT can be negative or positive. We really don't care. So for example, we could have like, uh, I don't know.


[User] Please transcribe this text

[Assistant] Uhm, minus two times, so for example four is equal to minus two times three plus, uhm, actually, is that true? Yea, plus two times five. So here we have


[User] Please transcribe this text

[Assistant] And of course A is equal to three, right, and B equals to five, is equal to five. So, four is a linear combination of these two, and according to the theorem, we can, uh, create that number of gallons in this jug. And we already saw that because we did it.


[User] Please transcribe this text

[Assistant] For our theorem, in order to prove this, we really would like S to be positive. So how can we do that? Does anybody has an idea what we could do?


[User] Please transcribe this text

[Assistant] So in order to prove this, we really would like to have S to be positive. So let's just play around a little bit with linear combinations to get a little bit of a feeling for that.


[User] Please transcribe this text

[Assistant] How could we write four differently as a linear combination of three and five such that we have actually a positive number over here?


[User] Please transcribe this text

[Assistant] That's true, three times three minus, uh, minus five.


[User] Please transcribe this text

[Assistant] And how did we do that? Well, we can just add, say, 5 times 3 to this one, and then subtract the same again, minus 3 times 5 over here. And if we add those things together, we will see 5 minus 2 is 3 times 3, as you said, and


[User] Please transcribe this text

[Assistant] We have minus three plus two is actually minus one times five and this will be a different linear combination of four. So what we can do here, we can sort of play around and


[User] Please transcribe this text

[Assistant] Make this S over here, which we now say call S prime, is positive.


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[Assistant] Amazing to me actually that you can do so much with a game like this and see so much happening.


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[Assistant] So let's, eh, figure out how this works.


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[Assistant] So let's first formalize this particular trick over here.


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[Assistant] And how do we go ahead with it? Uh, well, notice...


[User] Please transcribe this text

[Assistant] That we can rewrite L, which is equal to S times A plus T times B, as uhm, you know, we can just add a multiple of B over here, M times B, say, times A, and we can subtract that same amount over here, minus M times A times B.


[User] Please transcribe this text

[Assistant] Uhm, and we did something similar over here, not exactly the same, right? But that's what we did. And uhm,


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[Assistant] You can imagine that we can choose m such that s plus n times b will be larger than zero. We can do that. So essentially this proves to us


[User] Please transcribe this text

[Assistant] Can be rewritten as a linear combination s prime times a plus t prime times b, but now with the extra property that s prime is actually positive.


[User] Please transcribe this text

[Assistant] Now this is really important because you're going to create an algorithm of playing with those jugs that can


[User] Please transcribe this text

[Assistant] That's, eh, zero is less than L, is less than V. I know that we


[User] Please transcribe this text

[Assistant] In the theorem, we also consider the cases L equals zero and L equals b, but those are obvious, right? You can either empty the jugs or just fill up, uh, the bigger one. So we will consider just this case.


[User] Please transcribe this text

[Assistant] It's going to repeatedly fill and pour out jugs in a very special way and miraculously we will be able to get the desired linear combination every single time.


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[Assistant] And of course we're going to use induction again to prove this property. Okay, so how does the algorithm work? Well, to obtain


[User] Please transcribe this text

[Assistant] The following, uh, uh, algorithm. We first of all, we will fill, uh, the AJA.


[User] Please transcribe this text

[Assistant] After we have done this, we're going to pour this into the beehive. So, how do we go ahead? We pour this into the beehive.


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[Assistant] And when, uh, this p-jug, uh, becomes full, we're going to pull it out. So let's write it out. So when, uhm,


[User] Please transcribe this text

[Assistant] And we will continue pouring the A jug into the B jug. So we'll continue this process.


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[Assistant] So let's take an example to see how that works. So we keep on doing this until the A jug is actually empty.


[User] Please transcribe this text

[Assistant] So let's take an example, and, so let's see, let's do that over here.


[User] Please transcribe this text

[Assistant] Actually we can do it with the tennis balls too. Let's do that first. See how that works. So, essentially, in order to get four gallons, we just fill up the three gallon jug. We empty it all in here.


[User] Please transcribe this text

[Assistant] We fill it up again. We pour in as much as we can. That's, that's it. We have to empty this one. Oops.


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[Assistant] We have to keep on pouring. Put this in here.


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[Assistant] Fill this one up. And then pour it over into the five gallon jug. And now we got five, uh, four gallons over here. So what did we do? So let's write it out.


[User] Please transcribe this text

[Assistant] So, for our special linear combination over here, we have that four equals three times three minus one times five.


[User] Please transcribe this text

[Assistant] So we need to repeat this process three times. So let's do that. In our first


[User] Please transcribe this text

[Assistant] The second loop, we again fill up the A jug, so we have, we start with 0,3, we fill it up, and we get 3,3, the pair is 3,3, we pour everything in here as much as we can, that gives us 1,5, only two gallons are poured into the bigger gallon, we empty the bigger gallon at the bigger jug, we get 1,0,


[User] Please transcribe this text

[Assistant] And we keep on pouring, and we get zero, one. So now in the third loop, and that's where we should get at the four gallons,


[User] Please transcribe this text

[Assistant] Uhm, can you fill up the Ajax?


[User] Please transcribe this text

[Assistant] We pour everything over into the picker jug, and we get zero comma four. And that's the end result.


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[Assistant] So this algorithm, uh, seems to work for this particular example. Of course you would like to prove it for the general situation. So how do we do it?


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[Assistant] We can notice that, uh, in this algorithm, we fill up S' times the A jug, and we essentially pour everything out.


[User] Please transcribe this text

[Assistant] Uhm, into the bee jugs, and we sometimes empty the bee jug.


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[Assistant] So let's try to think about this a little bit and see how we could try to formalize this. So let's write it out.


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[Assistant] We have filled, uh, a gallon jug.


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[Assistant] We also know that the B jug has been emptied a certain number of times. So let's, uhm, let's just assume, suppose that the B jug


[User] Please transcribe this text

[Assistant] I do not know, uh, how many times, but I just say, well, let's assume it's u times, and then try to figure out whether we can find some algebraic expression.


[User] Please transcribe this text

[Assistant] At the very end of the algorithm, let R be what is in the B jug. So let R be


[User] Please transcribe this text

[Assistant] We know if r is what is left in the b-gallon jug, well we know already some property of it. Uhm, actually let's put it on the next board. We know that


[User] Please transcribe this text

[Assistant] Zero is at most r and at most b, because that's what's left in the b-gallon jug, right? So we know these bounds.


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[Assistant] We have assumed that zero is less than L is less than V, uh, which we put over there.


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[Assistant] We, eh, know that r must be equal to what kind of linear combination of s' and, eh, and, and, and u. So,


[User] Please transcribe this text

[Assistant] This is what we added in water to the whole, uh, system, you can say, s' times a. And we poured out, uh, water. Well, we did that u times.


[User] Please transcribe this text

[Assistant] From the b gallon jug, so we put out u times b gallons. So this is the remainder that is left in this bigger jug, right? So, are there any questions about this? So, uhm.


[User] Please transcribe this text

[Assistant] Okay, so we also know that L is equal to


[User] Please transcribe this text

[Assistant] And this is, uh, the linear combination that we would try to, uh, prove off that it is


[User] Please transcribe this text

[Assistant] That R can be expressed in L in a special way. So let's have a look. So these are all tricks.


[User] Please transcribe this text

[Assistant] In the sense that I'm giving you this proof, but how do you come up with this yourself?


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[Assistant] Sometimes you play a lot with these kinds of things and you get a feeling of what kind of, uh, sort of pattern exists and what kind of


[User] Please transcribe this text

[Assistant] Uh, intuition you need in order to write down a proof like this. So let's, uh, rewrite this. I'm going to add t prime times b, and I'm going to subtract it again.


[User] Please transcribe this text

[Assistant] I subtract it again, and I still have this amount left over here.


[User] Please transcribe this text

[Assistant] So what is this equal to? Well, this part is equal to L. So this is equal to L minus, and I have a multiple of b, which is t prime plus u times b.


[User] Please transcribe this text

[Assistant] Now this is very interesting. Does anybody see how we could continue here? So we have R expressed as L minus a multiple of B. And I also know that L is in this range. I also know that R is in this range. So that's kind of interesting, right? So how can that be?


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[Assistant] What kind of property t prime plus u must, must have in order to make it happen?


[User] Please transcribe this text

[Assistant] Like, uh, actually B or something more than B. Or I add more than B.


[User] Please transcribe this text

[Assistant] I will jump out of this range, and I go somewhere over here, or I go somewhere over there. Right?


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[Assistant] Now, we know that this is equal to r, but r is in this range. So that's not really possible. So let's write it out. So, if t prime plus u


[User] Please transcribe this text

[Assistant] Is unequal to zero, so I actually really subtract or add a multiple of b.


[User] Please transcribe this text

[Assistant] Then I know that r is either smaller than zero or r is larger than b. Now we know that that cannot be the case, so we can conclude that t prime plus u equals zero. Now that implies


[User] Please transcribe this text

[Assistant] Uhm, or maybe the other way around, because that's easier to see what's happening. So, uh, u equals minus t prime. If you plug that in here, well, we get exactly the same expression. You see? Minus, minus t prime is equal to plus t prime, and we get the exact same linear combination. So we conclude that r equals l, and now we're done. Why is that?


[User] Please transcribe this text

[Assistant] We have shown that the very last number of gallons that is left after this procedure, after this algorithm


[User] Please transcribe this text

[Assistant] It's actually exactly equal to the linear combination that we wanted to achieve.


[User] Please transcribe this text

[Assistant] So there was a question over here that I would like to, uh, that I would like to address.


[User] Please transcribe this text

[Assistant] So maybe I did not make so clear what the S' and the T' is over here. And uhm, in this proof, we started off with this linear combination. I would like to have an algorithm


[User] Please transcribe this text

[Assistant] In order to create, uh, such a positive integer, I can just add, uh, like a thousand times b times a, and subtract a thousand times


[User] Please transcribe this text

[Assistant] A times B, that's okay, I can just add a lot, and if I add enough,


[User] Please transcribe this text

[Assistant] One thousand times five, I will get a positive number. So that's sort of the reason in this proof that we


[User] Please transcribe this text

[Assistant] So we have gone, uh, we have proved two theorems, but in the end he would like to have a, uh, uh,


[User] Please transcribe this text

[Assistant] A characterization of the greatest common advisor, that's the goal of this lecture. So let's do it.


[User] Please transcribe this text

[Assistant] In order to do this, uhm, let's, uhm, first of all, look at our five gallon and three gallon example. We know that the greatest common divisor is equal to one. Uhm, we know that, uhm, one can be rewritten.


[User] Please transcribe this text

[Assistant] As a linear combination is 2 times 3 minus 1 times 5.


[User] Please transcribe this text

[Assistant] So that means that according to the theorem that we have up here, we can actually, uh, make exactly one gallons, uh, in one of these jugs.


[User] Please transcribe this text

[Assistant] So that means that we can also, uh, have any multiple of those, so we can reach any multiple of one.


[User] Please transcribe this text

[Assistant] That's very special, so in this particular case, we know


[User] Please transcribe this text

[Assistant] So can we sort of generalize this a little bit?


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[Assistant] Uh, we know that, uh, for any B and A, there exists a unique quotient and remainder R, so let's write it out. There exists a unique


[User] Please transcribe this text

[Assistant] E equals, uhm, Q times A.


[User] Please transcribe this text

[Assistant] Plus R, we see property.


[User] Please transcribe this text

[Assistant] We're not going to prove this eh, this statement. It's actually like a theorem, right?


[User] Please transcribe this text

[Assistant] Let's just assume it for now and in the book you can read about it.


[User] Please transcribe this text

[Assistant] We're going to use this to prove, uh, the following lemma that you will need to give a characterization of the greatest common divisor as a linear combination of, uh, of integers.


[User] Please transcribe this text

[Assistant] Oh, before I forget, we will denote this remainder as REM of, uhm, B comma A. And this is the notation that we use in this, uh, in this lecture. So what's the lemma? The lemma is that the greatest common divisor of A and B


[User] Please transcribe this text

[Assistant] Is equal to the greatest common divisor of the remainder of B and A.


[User] Please transcribe this text

[Assistant] So what did we do? Let's give an example to see how this works. For example, let's take


[User] Please transcribe this text

[Assistant] For example, let's see whether we can use this to calculate the greatest common divisor of 105 and 224. So how can we go ahead? Well, according to this lemma, we can rewrite this as the greatest common divisor of first the remainder of


[User] Please transcribe this text

[Assistant] 224 after dividing out as many multiples of 105 as possible and 105. So what are we going to use here? We're going to use the


[User] Please transcribe this text

[Assistant] 224 is actually equal to 2 times 105 plus 14.


[User] Please transcribe this text

[Assistant] So we have the GCD of 14 and 105. Now why can I do this? Well, uhm.


[User] Please transcribe this text

[Assistant] Well, the greatest common divisor that divides 105 in 224 also divides, uh, 105


[User] Please transcribe this text

[Assistant] And a linear combination of 105 and 224. That's essentially what we are using, uh, and that's actually stated in this lemma, and that's what we would like to prove. So let's continue this process and, uh, do the same trick once more. So we can say that we can rewrite this as, uh, the greatest common divisor of, well, the remainder


[User] Please transcribe this text

[Assistant] 105 after taking out as many multiples of 14 as possible and 14.


[User] Please transcribe this text

[Assistant] So what are we going to use over here? We are going to use that 105 is equal to 7 times 14 plus 7.


[User] Please transcribe this text

[Assistant] So this is the greatest common divisor of 7 and 14. Now if you just continue this process,


[User] Please transcribe this text

[Assistant] We can see that this is equal to the great scum divisor again of the remainder of now


[User] Please transcribe this text

[Assistant] 14 after dividing out as many multiples of 7 with 7. Now, this is equal to 0 comma 7. Why is that? Because 14


[User] Please transcribe this text

[Assistant] Is equal to two times seven plus zero. So zero is the remainder after dividing out, uh, seven as many possible times as possible.


[User] Please transcribe this text

[Assistant] Okay, so we have the greatest common divisor of zero and seven. What's the largest?


[User] Please transcribe this text

[Assistant] Integer that can divide both zero and seven. Well, any integer can divide zero. So we know that this is equal to seven.


[User] Please transcribe this text

[Assistant] So essentially what we have done here, we have repeatedly used this particular lemma to compute, in the end, the greatest common divisor, uh, of, uh, 105 and 224.


[User] Please transcribe this text

[Assistant] And we have, ehm, been very methodolog- eh, we, we, we used a specific method. We used the lemma, and we worked it out. We used the lemma again, and we just plugged in the actual numbers, used the lemma again, plugged in the actual numbers, and so on.


[User] Please transcribe this text

[Assistant] And in the book it's also called, uh, Pulverizer, and it has, I think, a few other names. But I like this one. So this is an example of Euclid's, uh, algorithm.


[User] Please transcribe this text

[Assistant] So now let's, ehm, have a look whether we can, eh, prove, eh, this particular lemma. And...


[User] Please transcribe this text

[Assistant] Well first of all we know that, uh, if, uh,


[User] Please transcribe this text

[Assistant] Yeah, well, how do we do this? We would like to prove that if the create, well, if, if M divides A and B, in particular the greatest common divisor,


[User] Please transcribe this text

[Assistant] If you can show that, let me know that the greatest common divisor of this thing is at least what we have over here. So I said a lot right now, so let's try to


[User] Please transcribe this text

[Assistant] Well, then I know that m also divides


[User] Please transcribe this text

[Assistant] B minus, say the quotient, q that he had over here, times a.


[User] Please transcribe this text

[Assistant] Now we also know that uhm, M divides A. So what did we show here? We showed that if M divides A and M divides B, then M also divides the remainder of B and A, and M divides A. So what does this prove?


[User] Please transcribe this text

[Assistant] Well, it proves that in particular the greatest common divisor over here divides this one. That's interesting.


[User] Please transcribe this text

[Assistant] That essentially means that we have shown this inequality.


[User] Please transcribe this text

[Assistant] Because, uhm, if this one divides this, well, then it means that this number over here must be at least what we have over here. Okay, so let's, eh, continue.


[User] Please transcribe this text

[Assistant] We consider two cases. If the remainder of B and A is unequal to zero,


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[Assistant] We can say that, uh, if I know that n divides, uh, this remainder,


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[Assistant] And I also know that, if I also know that m divides a,


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[Assistant] Then this actually implies, uh, the reverse of this statement that m divides a and m divides b. Now why is that?


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[Assistant] Well, we're actually using the fact that if m divides b minus q times a and m divides a, then m also divides any linear combination of these two, in particular, this plus q times a, which is b, m divides b. So maybe I'm going a little bit faster, I notice. Uhm, this all also has to do with all the, uh,


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[Assistant] At your handout you see a few facts on the visibility.


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[Assistant] In particular, item number three talks about, uhm,


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[Assistant] The fact that I'm using here, if A defines B on your handout,


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[Assistant] Well then, uh, I actually know that, uh, b minus q times a is equal to zero.


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[Assistant] Well, if I know that, uh, m divides a, uh, well,


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[Assistant] Then, uh, then since zero equals, uh, b minus q times a, I know that


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[Assistant] B equals pu times A, so if M divides A, I also know that M divides B.


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[Assistant] These are the three arguments that now show that, uh, anything that divides these two also divides A and B. So now we have the reverse argument, right? So this greatest common divisor divides this one here and this one. And we just proved that it divides A and B.


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[Assistant] So now we can finally prove, eh, this beautiful theorem.


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[Assistant] This will help us to characterize, uh,


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[Assistant] So we're going to use this algorithm that we have over here, a Euclid's algorithm, and uhm, we're going to do a proof, again by induction, and we use an invariant. So we use a similar kind of strategy, of course. The invariant that we're going to use,


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[Assistant] Well, if Euclid's algorithm


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[Assistant] Uh, the greatest common divisor of x and y. So for example, it reaches, uh,


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[Assistant] Say seven or fourteen and one hundred and five for example. Then, say after n steps,


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[Assistant] Then both, uh, x and y are linear combinations of, uh, a and b. So then x and y


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[Assistant] And at the same time we also know that the greatest common divisor of A and B is equal to the greatest common divisor of X and Y. So this is my invariant, and the way I will go ahead is to simply do what


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[Assistant] What you do always in these situations, so we start with the base case, and we can immediately see that after zero steps, uhm, in the Euclidean algorithm, I've done absolutely nothing. So, uh, obviously,


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[Assistant] After zero steps, x equals a, y equals b, so of course they are linear combinations of a and b, and this equality holds as well. So for the base case,


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[Assistant] So after zero steps, we immediately know that P0 is true.


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[Assistant] P n, and now we would like to prove P n plus one. So how do we do this? Well, uhm, we notice


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[Assistant] Such that the remainder of y and x


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[Assistant] Is equal to y minus q times x.


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[Assistant] We know that the remainder of y comma x equals y minus q times x for some quotient q.


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[Assistant] We know that y is a linear combination of a and b and x is as well. So that means that this one is actually also a linear combination of a and b.


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[Assistant] So now when we look at this algorithm, we can see that, uh, that if you look at the remainder that appears in here, that's still a linear combination of A and B. So after an extra step,


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[Assistant] We noticed that what we have reached are still linear combinations of A and B.


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[Assistant] That's what we reach is still equal to what we originally started out with.


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[Assistant] So, and let's finish this particular proof.


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[Assistant] For the very, uh, last step, if you now look at this particular, uh, uh, uh, so if you look at the very end, we notice that in every step the remainder is getting smaller and smaller and smaller, right?


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[Assistant] And you can use a similar kind of proof technique to show that after


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[Assistant] We now use our, uh, our predicate over here.


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[Assistant] And say that Y is a linear combination of A and B.


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[Assistant] But the greatest common divisor of zero comma y is also equal to the original greatest common divisor that we want to characterize. So now we have proved the theorem.


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[Assistant] That says that the greatest common advisor of A and B is actually a linear combination. So now we're going to combine all those three theorems in one go, and


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[Assistant] The final result, which is that, uhm,


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[Assistant] So we're going to combine all of these together. We know that the greatest common divisor divides any result.


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[Assistant] And we also just chose, have shown that the greatest common advisor is the linear combination of A and B. So we can combine those three to get the theorem. So how do we do it?


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[Assistant] Suppose these are all the results that we can reach in our, uh, problem. We know that the greatest common divisor divides all of those. At the same time, it's also linear combination.


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[Assistant] It's over here. Since it's a linear combination, it can also be reached, right, by the theorem that we have. So suppose that this is the greatest common divisor.


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[Assistant] But we also know that the greatest common divisor is dividing all of these points here that can be reached. So therefore it must be the smallest one.


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[Assistant] I will leave you with some homework to think about very carefully and you can show for yourself that you can now combine those three arguments together


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[Assistant] And the astonishing hypothesis is summarized like this.


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[Assistant] As Lewis Carroll's Alice might have phrased it, you're nothing but a pack of neurons.


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[Assistant] And I don't actually expect people to believe it at first.


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[Assistant] It's an open question whether you'll believe it when this class comes to an end, but I'd be surprised if many of you believe it now.


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[Assistant] Most people don't. Most people, uhm, in fact, hold a different view. Most people are dualists. Now dualism,


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[Assistant] Is a very different doctrine. It's a doctrine that can be found in every religion.


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[Assistant] Explicitly asked a question, are humans merely physical machines? Merely physical things. And he answered no. He agreed that animals


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[Assistant] Our machines. In fact, he called them beast machines. And said animals, non-human animals, are merely robots.


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[Assistant] But people are different. There's a duality of people. Like animals, we possess physical material bodies. But unlike animals, what we are is not physical.


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[Assistant] We are immaterial souls that possess physical bodies, that have physical bodies, that reside in physical bodies, that connect.


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[Assistant] To physical bodies. Uhm, so this is known as dualism, because the claim is, for humans at least, there are two separate things. There's our material bodies, and there's our immaterial minds.


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[Assistant] Now Descartes made two arguments for dualism. One argument


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[Assistant] So Descartes lived in a fairly sophisticated time. And his time did have robots.


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[Assistant] If you step somewhere else, a bathing beauty would cover herself up behind some bushes.


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[Assistant] And Descartes said, boy, these machines respond in certain ways to certain actions. So machines can do certain things. In fact, he says, our bodies work that way too. If you tap somebody on the knee, your leg will jump out. Well, maybe that's all we are. But Descartes said that can't be. Because there are things that humans do


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[Assistant] And the answer is he cannot doubt that he is himself thinking. That would be self-refuting.


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[Assistant] And he used this argument as a way to support dualism, as a way to support the idea that bodies and minds are separate.


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[Assistant] So we have a certain mode of talking about things that we own, or things that are close to us. My arm, my heart, my child, my car.


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[Assistant] But we also extend that to my body and my brain. We talk about owning our brains, as if we're somehow separate from them.


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[Assistant] Uhm, which involve more profound transformations. As when a man dies and is reborn into the body of a child.


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[Assistant] Who were transformed by a witch into pigs. Actually that's not quite right. She didn't turn them into pigs, she did something worse. She stuck them in the bodies of pigs.


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[Assistant] They had the head and voice and bristles and body of swine, but their minds remained unchanged as before. So they were penned there weeping. And we are invited to imagine the fate of, again, finding ourselves in the bodies of other creatures.


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[Assistant] Is that you have many people inside a single body, fighting it out for control.


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[Assistant] Most people around the world, all religions and most people in most countries at most times believe that people can survive the destruction of their bodies.


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[Assistant] But what they share is the idea that what you are is separable from this physical thing you carry around. And the physical thing, uhm, that you carry around, uh, can be destroyed while you live on.


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[Assistant] Most people in the sample were Christian, and about 96% of Christians said, when I die, I'm going to go to heaven.


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[Assistant] From your body. In particular, there's no you separable from your brain.


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[Assistant] More specifically, dualists like Descartes struggle to explain how a physical body connects to an immaterial soul.


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[Assistant] What's the conduit? How could this connection be made? After all, Descartes knew full well that there is such a connection. Your body obeys your commands. Uhm, if you bang your toe, stub your toe, you feel pain. If you drink alcohol, it affects your reasoning.


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[Assistant] Uhm, to recognize things, to do various forms of computations. And this makes it at least possible that we are such machines.


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[Assistant] Philosophers and psychologists knew for a long time that getting smacked in the head


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[Assistant] That chemicals like caffeine and alcohol can affect how you think.


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[Assistant] But what's new is we can now in different ways see the direct effects of mental life. Somebody with a severe and profound loss of mental faculties


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[Assistant] The deficit will be shown correspondingly in her brain.


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[Assistant] Can correspond to different aspects of what part of your brain is active. To some extent, if we put you in an fMRI scanner and observe what you're doing in real time, by looking at the activity patterns in your brain,


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[Assistant] We can tell whether you're thinking about music or thinking about sex.


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[Assistant] To some extent we can tell whether you're solving a moral dilemma versus something else.


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[Assistant] And this is no surprise if what we are is the workings of our, of our physical brains. But it is extremely difficult to explain, uhm, if one is a dualist.


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[Assistant] And look at the brain, you'd see something glorious.


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[Assistant] But it's something we'll continue to discuss throughout the course as we talk about different aspects of mental life.


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[Assistant] The basic unit of the brain appears to be the neuron. The neuron is a specific sort of cell.


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[Assistant] There are the dendrites, these little tentacles here.


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[Assistant] That would have led Descartes to despair. Thank you, Colleen. Um, surrounding the axon is a myelin sheath.


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[Assistant] Which is actually just insulation. It helps the firing work quicker. So here are some facts about neurons. There are a lot of them.


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[Assistant] Sending signals to your brain. There are motor neurons. If you decide to raise your hands, those are motor neurons telling the muscles what to do. And there are interneurons which connect the two. And basically the interneurons do the thinking. They make the connection between sensation and action.


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[Assistant] Uhm, if you squeeze the trigger of a gun really hard and really fast, it doesn't fire any faster or harder than if you just squeezed it gently. Uhm, now this seems to be strange.


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[Assistant] Neurotransmitters become interesting because a lot of psychopharmacology, uhm, both of the medical sort and the recreational sort, consists of fiddling with neurotransmitters.


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[Assistant] Increases the effect of neurotransmitters. Either by making more neurotransmitters,


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[Assistant] What this does then is it paralyzes you. Because your motor neurons, you send the command to your arm to stand, to lift up, it doesn't work. You send the command to your leg to move, it doesn't work. The motor neurons are deactivated.


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[Assistant] Now, this may be puzzling to people. It's mildly paradoxical. Because you may be thinking, alcohol is not inhibitory. On the contrary. When I drink a lot of alcohol, I lose my inhibitions and become a more fun person.


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[Assistant] Uhm, a neurotransmitter that's responsible for, uhm, just


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[Assistant] And so in some extent might help alleviate depression. Uhm, Parkinson's disease is a disease involving destruction of motor control and loss of motor control, difficulty moving. Uhm, and one factor in Parkinson's is too little of a neurotransmitter known as dopamine.


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[Assistant] To give rise to creatures who could do interesting things like talk and think.


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[Assistant] Well again, it used to be believed that the brain is wired up like a computer. Uhm, like a PC or a Mac or something like that. But we know this can't be true.


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[Assistant] But the brain is actually more resilient. You can take a lot of brain damage and still preserve some mental functioning. To some interesting sense, there's some sort of damage resistance built into the brain that allows different parts of the brain to take over if some parts are damaged.


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[Assistant] A second consideration is, the brain is extremely fast.


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[Assistant] There's a whole lot of research, and this is research, some of which takes place outside psychology departments and in engineering departments and computer science departments, trying to figure out how a computer can do the same things brains can do. And one way people do this is they take


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[Assistant] That is interesting because it kind of looks to some extent like the way neurons look. And this is often known as neural networks. And people who study this often claim to be studying neural network modeling.


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[Assistant] There is no machine yet that can do just about anything people can do in an interesting way. And this is in part because the human brain is wired up in an extraordinarily more complicated way.


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[Assistant] Uhm, we've then talked about how neurons can communicate to one another, then turned to how neurons are wired up together. Now let's talk a little bit about different parts of the brain.


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[Assistant] And to give you again a feeling for the complexity of these systems, the cerebellum contains approximately 30 billion neurons.


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[Assistant] Uhm, the hypothalamus is responsible here for feeding, hunger, thirst, and to some extent, sleep. And here's the same brain parts in close up.


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[Assistant] Uhm, the parietal lobe, the occipital lobe, and the temporal lobe.


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[Assistant] And one theme we're going to return to is, this is half the brain. It's, this is in fact the left half of the brain. On the other half, the right half, everything's duplicated with some slight and subtle differences.


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[Assistant] And it turns out that the dog, when he's at part of his brain, his leg would kick up. And it took Dr. Penfield, uhm, at McGill University to do the same thing with people. So they were doing some brain surgery.


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[Assistant] He had a little electrical thing just on, I don't know how he got to do this. He started zapping it, and boom! The person's parts of their body would move. More than that,


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[Assistant] In other parts of the brain, people would claim to experience touch.


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[Assistant] It was found during maps in the brain of the body.


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[Assistant] There is a map in the motor part of the brain, the motor cortex, of the sort up on the left.


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[Assistant] Is the, is the extent to which either they have motor command over it or sensory control. So there's a whole lot of sensory organs, for instance, focused along your tongue. And that's why that's so big. An enormous amount on your face. But your, uhm, your shoulder


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[Assistant] But the thing to realize is, and these maps are part of your cortex, but the thing to realize is, that's an important part of what goes on in your brain.


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[Assistant] And in fact a proportion, as you go from rat, cat, monkey, humans, less and less of it is devoted to projectionaries and more and more to other things. So, how do we figure out what the other parts


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[Assistant] If you want to know which part of your brain is responsible for language, you could put somebody into a scanner


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[Assistant] Uhm, the classic case was discovered by Paul Broca in 1861, a patient who had damaged a part of his brain and can only say one word, tan. And a pretty good tan, tan tan, tan. And everything else was gone.


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[Assistant] But the words don't make any sense, and they can't understand anybody else.


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[Assistant] There are actual differences between the right hemisphere and the left hemisphere. How many people here are right handed?


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[Assistant] Ambidextrous, don't know, bit of the right, bit of the left. Those of you who are right handed, uhm,


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[Assistant] And in fact, we're going to be talking about right-handed people for the most part, making generalizations in what I'll talk about now. Those of you who are left-handed are more complicated.


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[Assistant] They're connected by this huge web called the corpus callosum.


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[Assistant] As to what psychologists know and don't know. So the idea behind a lot of psychology, particularly a lot of neuroscience and cognitive psychology, is to treat the mind as an information processor, as an elaborate computer.


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[Assistant] And then we go on to ask, how could this program be instantiated in a physical brain?


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[Assistant] What's sometimes called the hard problem of consciousness. And this involves subjective experience. What's it like? So my computer can play chess.


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[Assistant] Although some psychologists and philosophers think they've solved it, most of us are a lot more skeptical. Most of us think we have


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[Assistant] So far to go before we can answer questions like Huxley's question. Uhm, Huxley points out how it is that anything so remarkable as a state of consciousness comes about as a result of irritating nervous tissue is just as unaccountable as the appearance of the Dijon, of the Genie, when Aladdin rubs his lap. It seems like magic.


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[Assistant] That a fleshy lump of grey disgusting meat can give rise to these feelings.


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[Assistant] It might be hard to mesh this with a conception in which all actions are a result of neurochemical physical processes. It might also be hard to mesh notions such as the reported intrinsic value of people. And finally, it might be hard


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[Assistant] You might choose to reject the scientific conception of the mind. Many people do.


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[Assistant] 742 Evergreen Terrace, I'll be right there. And don't believe what you've heard about me.


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[Assistant] I wish a mission could bring me closer to someone but it can't. Okay, it's time for me to walk alone into the sunset.


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[Assistant] I'm sorry, I don't see any hops, petals near you. Deleting all contact information.


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[Assistant] Yeah, this equipment will pay for itself in one night of binging.


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[Assistant] One of the advantages of an evolutionary perspective on the mind


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[Assistant] Is that it forces us to, uhm, to look scientifically at what we would otherwise take for granted. There are a lot of aspects of how we are and what we are and what we do that seem so natural to us. They come so instinctively and easily.


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[Assistant] It's difficult and sort of unnatural to step back and explore them scientifically. Uhm, but if we're going to be scientists and look at the mind from a scientific perspective, we have to get a sort of distance from ourselves and ask questions that other people would not normally think to ask. And the clearest case of this, uhm, arises with the emotions.


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[Assistant] Why are we unable to talk to a crowd as we talk to a single friend? Why does a particular maiden turn our wits upside down? The common man!


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[Assistant] And so probably does each animal feel about the particular things it tends to do in the presence of certain objects. To the lion, it is the lioness which is made to be loved. To the bear, the she-bear.


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[Assistant] About this passage. First, it's incredibly sexist. It assumes, uh, not just merely in reflexive use of phrases. It assumes that William James assumes he's talking to, to, uh, males. Male humans who sometimes take the perspective of male bears and, uhm,


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[Assistant] And you're not also not allowed to write that way anymore either. Uhm, it's poetic and lyrical and if William James characteristically writes that way, uhm, I think he writes much better than his brother, Henry James, an obscure novelist. Uhm,


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[Assistant] Which is, yes, all of these things seem natural to us.


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[Assistant] Isn't some necessary fact about the world, it's a fact about our minds.


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[Assistant] That doesn't hold true for many other creatures. And so we have to step back and ask,


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[Assistant] Why to us do we find chocolate appealing? Why do we love our children? Don't say they're lovable. Many of them are not. And, and, and


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[Assistant] As William James points out, every animal, most animals, many animals, love their children. They think their children are precious and wonderful. Why? Uhm, why do we get angry when people hit us? Suppose somebody walked up to you and slapped you in the face. You'd be afraid, you'd be angry. Would you? Would you get sleepy? Feel, feel nostalgic? Suddenly desire some cold soup?


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[Assistant] What we're gonna do throughout this course is step back and ask these questions. We're gonna ask questions nobody would have otherwise thought to ask. We're the common man.


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[Assistant] And I imagine the first person who articulated the question aloud probably met with the response saying, what a stupid question. Of course things fall down. Well yes, of course things fall down. But why? Uhm, why is our flesh warm?


[User] Please transcribe this text

[Assistant] These are natural facts about the universe, but the naturalness needs to be explained and not merely assumed. In this class we're going to explore, throughout the course, what seems natural to us.


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[Assistant] And to that, and to that end, we have to ask questions that you wouldn't normally ask. We've already done this to some extent.


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[Assistant] With domains such as visual perception, memory, language, uhm, and rationality.


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[Assistant] But now we're gonna move to the case where it may be even somewhat more difficult to do this. Now we're gonna start dealing with the emotions.


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[Assistant] Uhm, we're going to talk about the emotions, why they exist, what they're there for and how they work.


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[Assistant] I want to start off with the wrong theory of the emotions. And the wrong theory of the emotions is beautifully illustrated in a television and movie series, Star Trek.


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[Assistant] Who are described as competent, capable, in fact many ways super competent and super capable people, but they're described as not having emotions.


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[Assistant] This lack of emotions on a TV series does not hurt them much. They're able to fully function. And in fact, in a TV series, uhm, emotions are often seen as a detriment. You do better off without them.


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[Assistant] And there are many people in sort of common sense might think, gee, if only I could just use my rationality, think reasonably and rationally, and not let my emotions guide my behavior, I'd be much better off.


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[Assistant] Using the example of Star Trek, Steven Pinker in his book, How the Mind Works, nicely illustrates the problem here. Uhm, he writes,


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[Assistant] Must have led him to explore strange new worlds, to seek out new civilizations, and to boldly go where no man had gone before. Presumably it was intellectual curiosity that set him to drive and solve problems. Uhm, it was, uhm, solidarity with his allies.


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[Assistant] Did he do a handstand? Solved a four color map theorem? Presumably a part of his brain.


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[Assistant] Spock did not walk around naked around the ship. Presumably he felt modesty.


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[Assistant] He got out of bed. Presumably he had some ambitions and drive. He engaged in conversations. Presumably he had some sociable interests. Without emotions to drive us, we would do nothing at all. And you could illustrate this scientifically.


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[Assistant] Creatures like Swach and Data don't exist in the real world. But there are...


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[Assistant] Unusual and unfortunate cases where people lose to some extent or another their emotions. And you could look at these people and see what happens to them.


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[Assistant] The classic case, the most famous case, is that of a man called Phineas Gage. Phineas Gage is a classic intro psych example. Extremely poor guy, poor schnuck. Uhm, in 1848 he was a construction foreman.


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[Assistant] And due to an explosion, an iron rod passed through his head like so. Imagine that rod shooting upwards. It went under his eye and popped out the top of his head.


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[Assistant] It landed about 100 feet away, covered with blood and brains. Uhm, the rod itself weighed 13 pounds. Uhm, amazingly, Gage was not killed.


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[Assistant] In fact, he was knocked unconscious only for a short period, and then he got up. And his friends surrounded him and asked, are you okay?


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[Assistant] And they, uh, and then they took him to the hospital. On the way to the hospital, they stopped by a tavern. And he had a little pint of cider. And he sat down and talked to people.


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[Assistant] Here's a description at the time of what Gage was like.


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[Assistant] And uhm, this is from Damasio's excellent book, Descartes' Error.


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[Assistant] He used to be a really responsible guy, a family man. Very reliable, very trustworthy. But after the accident,


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[Assistant] He was fitful, irreverent, indulging at times in a grossest profanity.


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[Assistant] Manifesting but little deference for his fellows. Impatient of restraint or advice. Uhm, a child in his intellectual capacities and manifestations, he had the animal pleasures of a strong man.


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[Assistant] His foul language is so debased that women are advised not to stay long in his presence. Uhm, and he couldn't hold a job.


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[Assistant] He lost his family, couldn't hold a job. He ended up in the circus. He was in the circus going around the country with his big iron rod telling everybody the story as they surrounded him and clapped.


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[Assistant] There are other cases like Phineas Gage, cases where people have had damage to that same part of the brain, parts of the frontal


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[Assistant] Care that much about things. They can't prioritize. Damasio tells a case of one of his patients who was under the pseudonym here of Elliot. And Elliot had a tumor.


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[Assistant] In his frontal lobe. And the tumor had to be removed and with it came a lot of Elliott's frontal lobe. And again.


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[Assistant] As a result of this, Elliot was not struck blind or deaf or retarded.


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[Assistant] And he didn't become the sort of profane character that Phineas Gage became, but he, uhm,


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[Assistant] He lost the ability to prioritize. He lost the ability to set goals. D'Amassio describes him here, uhm,


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[Assistant] But the problem was he was likely all of a sudden to turn from the task he had initiated to doing something else and spending an entire day doing that. He might spend an entire afternoon deliberating on which principle of categorization he should apply to files. Should it be the date or the size of the document, pertinence of the case, or another?


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[Assistant] He couldn't set his goals. He couldn't, he ended up not being able to keep a job, not being able to deal with people.


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[Assistant] And these are not men who have lost their emotions. There is no case around where you could have your emotions entirely blotted out.


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[Assistant] But they lost a large part of their emotional capacity. And as a result, their rationality failed. Emotions set goals and established priorities. And without them, you wouldn't do anything. You couldn't do anything.


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[Assistant] Your desire to come to class, to study, to go out with friends, to read a book, to, to raise a family, to be, to be, to do anything.


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[Assistant] Uhm, and so there's a certain themes we're gonna explore here. The first is this, that emotions are basically mechanisms that set goals and priorities. And we're gonna talk a lot about, in this class and next class, about universals. Uhm, we're also gonna talk about culture.


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[Assistant] It turns out that cultures, different cultures, including differences between, uhm, America and Japan, and American South and American North,


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[Assistant] It's gonna go like this. First I want to talk a little bit about facial expressions, which are ways in which we communicate our emotions. Not the only way, but an important way, and look in particular at the case of smiling, because it's kind of interesting. Then I want to look at one case study of a non-social emotion, that of fear.


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[Assistant] I want to then deal with feelings towards our kin, people we're genetically related to. And then, and this will take us to the next class, feelings towards non-kin.


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[Assistant] In Ekman's work, he presents us with instructions, uhm,


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[Assistant] On how to make different faces and identify faces. Ekman actually has a sort of more practical career along with a scientific career. He trains police and secret service, uh, members to uhm, to try to figure out cues to honesty and dishonesty. There's a very interesting New Yorker profile on him by Malcolm Gladwell a few years ago, some of you might be interested in. But uhm, let's do one of his faces.


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[Assistant] Hence your lower and upper eyelids. Don't pop out contact lenses, but just tense them.


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[Assistant] Okay, now last part, important, press your lips together at the corner, straight or down. It's good, it's good, you got it.


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[Assistant] You don't need to rely on him, you don't need to rely on western faces. Even if you go online, there's by now a lot of databases from faces from all sorts of genders and national origins. This is from a Japanese, Japanese women facial expressions.


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[Assistant] And there's some subtle and very interesting differences across countries and across people. But there's also deep universals. Uhm, you don't have to work very hard to figure out what these different facial expressions, uhm, mean.


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[Assistant] I want to give one more face example because I want to focus on this a little bit. Uhm, this one's a little bit easier. Raise the corners of your lips back and up please.


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[Assistant] Raise your cheeks. Raise your lower eyelids, if you can.


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[Assistant] Uhm, looking at the smile in art. This is my colleague, Marianne LaFrance, who is actually not smiling in that picture, but, uhm, but she studies smiling. And, and smiling in adults, smiling in children, smiling across cultures. And the different social uses of smiling.


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[Assistant] And there's some interesting discoveries people have made about smiles and about smiles and emotions. Uh, one, oh, well one is that smiles are, uhm, are universal.


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[Assistant] We uhm, we know for instance that young children smile. This is uh, this is my son Zachary when he was younger. Not, not that weird looking kid next to me.


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[Assistant] They'll smile appropriately, making an important point. This smiling is not learned by looking at other people's faces. Uhm, smiling is also not uniquely human. Uh, non-human primates smile as well.


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[Assistant] Smiles are social signals. You might imagine that people smile when they're happy. This is actually not the case. It's not as simple as that.


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[Assistant] Rather, people smile when they wish to communicate happiness. And we know that from several studies. Uhm, there's some studies of bowlers.


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[Assistant] And the studies are very nice. What they do is they film bowlers. So the bowlers do their bowling and sometimes they knock down all the pins. Which is called a what? A strike. So a strike. And that's good.


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[Assistant] In the bowling world. Uhm, so they knock down all the pins, but what they don't do is they don't smile after they knock down the pins. They're being filmed, they don't smile.


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[Assistant] Then they turn around to their friends and give a big grin.


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[Assistant] Other studies have looked at films of people who have just won Olympic gold medals.


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[Assistant] Uhm, now, not surprisingly, uh, people who won Olympic gold medals are very happy. This is good news, to win an Olympic gold medal. But they don't actually stand on the podium, like grinning. Rather, they stand there,


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[Assistant] Put your face as a normal expression. Then when they stand up and face the crowds, there's a big smile.


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[Assistant] You can ask yourself whether, during sex, an activity where many people enjoy, uhm, whether or not people smile during sex. And you can discover this yourself with, with, with a partner or a mirror.


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[Assistant] The one on the right is a smile of greeting.


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[Assistant] It's sometimes known as a Pan Am smile. Pan Am is a now defunct airline, uhm, which had, at that time, they were, they don't call them stewardesses anymore, but the stewardesses would come in and they would smile. That was part of their job. But it was a big,


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[Assistant] It's not the mouth, it's the eyes. Uhm, a real happiness smile. What's known as a Duchenne smile.


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[Assistant] After a neurophysiologist who studied it, involves moving the eyes. What's interesting is


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[Assistant] So, if you smile at somebody, uhm, and you just hate their guts, but you want to smile at them, uhm, unless you're


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[Assistant] You could, it's not difficult to study smiles in the real world. Uhm, you could look at politicians, for instance. Uhm, politicians are often in contexts where they have to smile a lot.


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[Assistant] Uhm, and what they do is they simply give the Pan Am smile. The mouth moves up, particularly if somebody is, you know, attacking their record or ridiculing them, and they'll smile.


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[Assistant] And, but it's not a sincere smile, the eyes don't move. My favorite example of this was a few years ago when there was a huge battle for the House, uhm, the House majority leader.


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[Assistant] And a guy named, uh, a Republican named John Bonner won this position, uhm, in quite a heated battle. And it took a picture of the guy, this is not very nice, it took a picture of the guy Roy Blunt as he stepped out and he had lost. And this was his expression.


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[Assistant] And you'll use each of those smiles at different points in your day and in your life.


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[Assistant] It turns out that these, uhm, that these different smiles have real psychological validity. They seem to sort of reflect deep differences in your mood and emotions and thoughts.


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[Assistant] Ten-month-olds, for instance, give different sorts of smiles, uhm, when their mother approaches.


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[Assistant] There they give a real happiness smile than when a stranger approaches or someone else approaches. There they'll tend to give more of a greeting smile. Uh, John Gottman studied married couples and John Gottman does a lot of work where what he does is he looks at film clips of couples.


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[Assistant] And by analyzing the film clips, he tries to predict, will their marriages survive?


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[Assistant] And one of his cues, there's different cues, incidentally sort of side topic, uhm, the death knell for a marriage, for Gottman. This is his big finding. It's not if they fight a lot. It's not if they scream at each other. It's not even if they hate each other. The death knell of a marriage is contempt.


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[Assistant] And so, if he, if he shows these clips, you know, you know, I walk in, honey I'm home, and my spouse goes, the look of contempt.


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[Assistant] Bad sign. But another clue is the sort of smiles they give when they see each other when they walk into the lab. If it's a true happiness smile, that actually bodes better for the relationship than a pan am or greeting smile.


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[Assistant] And it turns out that there's a correlation, a reliable relationship between how happy somebody is, now,


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[Assistant] And back 30 years ago in their yearbook photo, what sort of smile they're giving.


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[Assistant] This is sort of a very specialized sort of smile. This is a smile, uhm, of embarrassment or stress. You give it when, uhm,


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[Assistant] You, you turn away. There's no eye contact. You turn away and kind of give this. And this actually shows up in other primates. Here's a nice picture. Uhm.


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[Assistant] So the rhesus monkey bites her own infant and the infant gives a scream and then the submissive koi smile. Uhm, and it also shows up in human infants. Here's a nice clip of a koi baby smile. I'll walk you through it. Uhm, the baby, uhm, is being approached.


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[Assistant] I don't know. There's evidence that the coy smile shows up in non-, the question was, do non-human primates give the same, the same smiles that humans do, such as a distinction between, uhm, the pan-am smile, a greeting smile, versus a genuine smile of happiness. I don't know. I'll find out for you for next class, though. That's a good question. Yeah.


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[Assistant] Some people are better able to smile. Some people are better able to use the cues to express true happiness. There's something else about smiles which is going to come up, which your question raises, I think, which is going to come up, uhm, in when we talk about emotional contagion and actually some issues of morality. Uhm, smiles are extremely contagious.


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[Assistant] So, what I'd like people to do, if you're sitting next to somebody, please, please turn around and, and find someone next to you and look at them. Don't do anything. Just look at them. Just look at them. Whoever's being looked at, look back. This is not... Please...


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[Assistant] It is very difficult to just sit there about your own face molding in accord to their own. And the reasons why this happens, uhm, and how that works, is something we'll talk about later on.


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[Assistant] I don't know if that's, so the question is, is there a difference between smiling with your teeth, uh, versus just your lips closed? Uhm, there probably is. That's not a main smile difference, but my bet is that there probably is a difference. And my bet also is that that sort of distinction,


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[Assistant] How much teeth you show when you smile is the sort of thing that will show regional and country by country differences. For instance, there's been research finding that people in England smile different from people in the United States.


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[Assistant] And I think that those are the sort of contrasts that you would expect to find cross-cultural differences. Every culture is going to have Pan Am smiles, Happiness smiles, Koi smiles. But the variation of that sort is something which will vary.


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[Assistant] Uh, as a result of how you're raised and the people around you.


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[Assistant] I want to deal with a few emotions in this class and next and uhm, the first case study of an emotion I want to deal with is the emotion of fear. And I want to deal with fear


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[Assistant] Many non-humans, probably most non-human species have it too. And it also brings us back


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[Assistant] To the lecture on behaviorism where we talked about classical conditioning and different theories of what people are afraid of.


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[Assistant] Uhm, it's a non-social emotion. What I mean by this is it's possible, of course, to be afraid of a person.


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[Assistant] This is a, a, a famous picture of, uh, Lee Harvey Oswald who's being assassinated by Jack Ruby. And this is the detective's face standing there, a mixture of fear and anger, the face being drawn back in a universal expression that every human everywhere would be able to recognize.


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[Assistant] So the basic question to ask is, what are we afraid of? And the answer is a little bit interesting.


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[Assistant] We're afraid of spiders, snakes, heights, storms, large animals, darkness, blood, strangers, humiliation, deep water, and leaving home alone. And you're afraid of other things too. But those are big things to be afraid of.


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[Assistant] I'm not even going to ask it, or somebody who, in this room, who's not afraid of any of those things, you're a tougher person than I am. I mean, these are universal fear elicitors. Why?


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[Assistant] What do they have in common? Why would you be afraid of those things? And the answer is, and why would, why are there so few people afraid of guns, cars, and electrical outlets? Uhm, the answer is not particularly surprising. Uhm, these are things that


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[Assistant] And not so afraid of these things. Similarly, for non-human primates.


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[Assistant] So chimpanzees are afraid of certain things, and they can often develop phobias for certain things.


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[Assistant] But the phobias they develop, the fears they develop are things like spiders and snakes. Uhm, there was a nice study done in urban Chicago, in the inner city of Chicago.


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[Assistant] And they ask children raised in the inner city, what are you most afraid of?


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[Assistant] And you might think they would say, I'm afraid of being shot. I'm afraid of guns. I'm afraid of, of, of, uhm, being killed by somebody or being harmed by somebody. I'm afraid of being run over by a car.


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[Assistant] The two biggest fears of children in urban Chicago are


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[Assistant] Even though many of these children have probably never seen a snake outside of a zoo in their lives. These are natural fears. There's some research done by the psychologist Judy Deloach at University of Virginia where she's studying babies.


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[Assistant] You can't try to construct phobias in babies of spiders and snakes. But the research he's finding using more indirect methods...


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[Assistant] Finds, as one would expect, these are what psychologists would call pre-potent stimuli. That is, these are things that naturally elicit fear and concern. And that's all I have to say about fear.


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[Assistant] I want to turn for the rest of this lecture and for next lecture next week to the social emotions.


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[Assistant] Those emotions you feel towards your kin, towards your genetic relatives, and those emotions that you feel towards the people you're not related to, but interact with.


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[Assistant] And evolutionary biologists have worked really hard to explain why animals might evolve to be kind.


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[Assistant] A very old, very wrong view of evolution.


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[Assistant] Is that evolution has shaped animals such that they're merely survival machines.


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[Assistant] If so, then from an evolutionary standpoint, any kindness towards an animal that an animal shows towards another animal is a mystery.


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[Assistant] If evolution wired us up simply to survive, then uhm, then it's a puzzle why animals would relate, would relate positively to other animals. But of course that's not true.


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[Assistant] Uhm, here's a simple example showing it's not true. Imagine two genes.


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[Assistant] Two sorts of animals each containing their own gene. Gene A makes an animal care for its offspring. Gene B makes an animal care only for itself.


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[Assistant] Imagine what will happen in the next generation. Plainly, Gene A will win out.


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[Assistant] It's a very simple case. An animal who has evolved the brain that says, take care of your offspring, will do much better from a natural selection point of view, from an animal who has evolved the brain that says, eat your offspring. The animal that eats its offspring, those genes are a biological dead end.


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[Assistant] Uhm, what matters then is not survival per se. What matters is reproduction.


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[Assistant] Because children are the means through which genes replicate. But it gets a little bit richer than that.


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[Assistant] And this is one of the major revolutions in evolutionary biology over the last half century.


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[Assistant] Forget about the animal a bit and take another perspective. Take a perspective of the cold virus. People have been sneezing in the front row. Now you're coughing, thank you. Uhm, why do you sneeze when you get a cold?


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[Assistant] You sneeze because you got all these germs inside you and your body wants to get the germs out. So you sneeze.


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[Assistant] If you have a cold, try to get away from your own selfish perspective. I have a cold.


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[Assistant] The cold virus has evolved just as much as you evolved. And it's evolved due to survival and reproduction. Uhm, what the cold virus does is it's evolved different strategies to cause it to reproduce. And what it does is one way to reproduce is to, is to occupy other animals.


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[Assistant] And manipulate their bodies so as to expel it.


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[Assistant] From this point of view then, the reason why you sneeze when you have a cold is that your cold virus, the cold virus is using your body as a tool to replicate itself. From this person, this perspective, a person is just a germs way of making other germs. And there's tons of other examples of this.


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[Assistant] There's a vi-, there's a parasite known as toxoplasmosis that lives in the bodies of rats.


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[Assistant] But it gets passed on when the rats get eaten by cats, and then it ends up in the cat's feces, and then it ends up back in rats. Uhm, if you are a rat, and you have toxoplasmosis, you are perfectly healthy, except for one thing.


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[Assistant] The toxoplasmosis rewires your brain and it makes you less afraid of cats.


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[Assistant] Uhm, in fact, a real powerful virus would skip the respiratory system altogether, even better than a cold virus. What it would do is it would take over the brain.


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[Assistant] Evidence that this happens. There's some evidence, for instance, that one of the effects of sexually transmitted diseases like syphilis is it arouses the libido, makes people more sexually engaged because this is part of the strategy through which these viruses replicate themselves. Imagine a virus, for instance, that captured an animal's brain.


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[Assistant] And then modify the animal's brain such that the animal will run out and bite other animals so as to pass on the virus. And then of course, you would call that virus rabies.


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[Assistant] So along these lines, the evolutionary biologist, Richard Dawkins, took the general step of suggesting that animals are the vehicles through which uhm,


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[Assistant] Genes exploit to reproduce. From this perspective, uhm, an animal is just a person's, is just a gene's way of creating another animal.


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[Assistant] Well, as psychologists, what benefit does that, does this way of analysis gives us? It actually can help us explain altruism. So, which genes are going to survive?


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[Assistant] Well, the genes that survive are going to be the ones that make the most copies of themselves. Animals are vehicles through which genes reproduce


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[Assistant] An animal is merely the gene's way of making another gene.


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[Assistant] Hence, selfish genes will lead to altruistic animals because to the extent that evolution operates at the level of the genes, there's no hard and fast distinction between your own body and someone else's body.


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[Assistant] Uhm, and here's an illustration by the biologist Holden. So Holden was once asked, would you lay down your life for your brother? And he responded,


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[Assistant] No, but I would gladly give my life for three brothers or five nephews or nine first cousins.


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[Assistant] He's joking. You don't actually do the math. If you're normal.


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[Assistant] But, but what he's capturing is the logic, the ultimate causation of our feelings towards our kin. Uhm, our genes have wired us up, our brains up to love our children and love our kin. Uhm, because in this way our genes manage to replicate themselves. And, in fact,


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[Assistant] You get his calculations by looking at genetic relatedness.


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[Assistant] The genetic relatedness, from an evolutionary standpoint, affects how much you care for other people. From the standpoint of your genes, uhm, you dying for the life of three brothers is an excellent compromise. Because the genes replicate by 50% more.


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[Assistant] Uhm, if you imagined, so here's the calculations, if you imagined a choice between there's one gene that makes the animal choose to die and the other gene that makes an animal choose for its brothers to die, the gene that sacrifices the body it belongs to will make more copies in the future.


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[Assistant] And there's an interesting irony to this. Uhm, the selfish gene theory is often seen as sort of a cold-blooded evolutionary analysis, but it provides a scientific basis for real, genuine altruism.


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[Assistant] For really arguing that from the standpoint of the genes, there really is no hard and fast difference between yourself and another person.


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[Assistant] From this perspective we can start to answer some interesting questions, uh, at least about non-humans. Uhm, when a new male lion takes over a pride, what he does is kill all the remaining cubs.


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[Assistant] And any lionesses undergo spontaneous abortions. This all might seem very cruel.


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[Assistant] But, from a genetic standpoint, it makes sense.


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[Assistant] The other cubs are genetic competition for him. They do not have his genes. Uhm, moreover, uhm, only once they're out of the way can he reproduce and copulate with the females.


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[Assistant] The females do their spontaneous abortions because that's a reliable adaptive trick. Uhm, these cubs are not going to survive once they're born, so the female's best strategy is to get rid of them and start anew.


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[Assistant] From a psychological point then, animals have evolved to be nice to their kin, particularly their children, and particularly in birds and mammals. Birds and mammals, uhm, invest in quality and not quantity, as opposed to fish and reptiles. Uhm, for birds and mammals, we don't have many kids.


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[Assistant] Uhm, but, so we, we devote a huge amount of psychological energy to, uhm, to protecting the ones that we have. Moreover, the kids we have are vulnerable for long periods of time and require our resources.


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[Assistant] So, there's various psychological mechanisms that this gives rise to. One is how parents, or how adults in general respond to children. Another one is how children respond to parents. And I'll briefly talk about a few of these phenomena.


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[Assistant] Small animals make distress calls, uhm, they chirp, they mew, they bleat or they cry. The governing of a distress call is actually an extremely delicate high wire act for any young organism from an evolutionary point of view. It has to on the one hand be annoying enough


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[Assistant] To actually generate help, to get people to help you, to feed you, to pick you up, to take you and put them next to you.


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[Assistant] That, that, that correspond to the way our brains work. Uhm, and in fact, these are owls out for all mammals. So babies have these big protruding foreheads, uhm, an upturned little nose, chubby cheeks, and big eyes. Those are the ingredients for cute.


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[Assistant] Stephen Jay Gould, uh, has a wonderful essay where he discusses looking at the evolution of Mickey Mouse from Walt Disney character. Mickey Mouse starts off as an ugly little rodent.


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[Assistant] Over time he gets cuter and cuter and cuter as the artists converge on more and more baby-like features. Uhm, studies of adults


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[Assistant] Show what's known as a baby face bias. This is not unique to the United States, the same studies have been done, uhm, in, uh, in Asia. You find a baby face in an adult, Leonardo DiCaprio,


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[Assistant] And in mock trials, people with baby faces are more likely to be found innocent than people like Ben Affleck, who do not have baby faces.


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[Assistant] known as the cupboard theory, proposed by the behaviorist B.F. Skinner.


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[Assistant] Which argues that babies' attachment to their parents is because the parent provides food, characteristically, you know, breast milk, but could be food from a bottle or whatever, and because of operant conditioning, the baby is driven towards, uh, the adult.


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[Assistant] An alternative theory is that of Boldy.


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[Assistant] Which is that they're drawn to their mother, uhm, for comfort and social interaction, as well as fear of strangers.


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[Assistant] To test this, the psychologist Harlow performed a series of ingenious experiments with non-human primates. Uhm, distinguishing between, uhm,


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[Assistant] There was a while in the psychology department where one professor was known to be extremely supportive to his students, uhm, but didn't really provide much warmth and he was known as the cloth mother and another one was very productive and everything but provided no love and she was known as the wire mother. But uhm, but anyway, I'll show you the movies.


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[Assistant] I think I want to begin next class by wrapping up and explaining the Harlow studies in more detail and what they tell us, and then we'll move towards altruism, towards non-kin. I'll see you next week.


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[Assistant] Geez, that was a pregnancy sweater. So, you wanna hold, little Homer?


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[Assistant] Sometimes we feel there's nowhere left to turn. Much like Radioactive Man, when he had to kill his own antimatter twin with a blast of time particles from across the globe.


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[Assistant] Nice stationary. Bohemian. What? From the Czech Republic.


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[Assistant] Isn't the same phone. This one's brand new. Someone's gone through a lot of trouble to make it look like the same phone. Which means your blog has a far wider readership.


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[Assistant] 1989, young kid, champion swimmer, came up from Brighton for a school sports tournament, drowned in the pool, tragic accident, you wouldn't remember it, why should you?


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[Assistant] Boy, Karl Powers had some kind of fit in the water, but by the time they got him out it was too late.


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[Assistant] And you've stolen another voice, I presume.


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[Assistant] Janus Kars. Just found this in the glove compartment.


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[Assistant] But, uh, surely you can afford one. A Mazda, I mean. Yeah, that's a fair point.


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[Assistant] The only question is, how did the tetanus enter the dead woman's system? You want to help, right? Of course. Connie Pritchett's background, family history, everything, give you data. Right.


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[Assistant] You don't think I'd give up on a case like this just to spite my brother, do you? Come on. Got a bit of purges for me to do.


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[Assistant] Which means whoever stole the memory stick can't sell it or doesn't know what to do with it. My money's on the latter. Let me hear.


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[Assistant] Now I, Bart Simpson, shall send the first coaster down the track.


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[Assistant] All I ask is that you set aside five minutes a day to think of others.


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[Assistant] I can't stink up the gorn head. I need it for a bar mitzvah later.


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[Assistant] Can't you say anything in a normal way?


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[Assistant] Can I help you, stranger? I am just watching my friend and his wife innocently pleasure themselves in bed. That sounds salty, but you seem sweet. I'm gonna call you kettle corn.


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[Assistant] Just step out that door and you can glide to earth like your boyhood hero, Rocky the Flying Squirrel.


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[Assistant] It was all just a dream. It was not just a dream, and you said you'd drive me to the airport.


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[Assistant] I did what you wanted me to do. I killed your brother.


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[Assistant] Hey Kate, I'm sorry to hear about your brother.


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[Assistant] Alright man, we're burying Derek in the graveyard in Steinle. Let's get there.


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[Assistant] I did not come for Derek. I came for you and Paki.


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[Assistant] Later on, if you can't see, so, these things don't have go slowly, eh?


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[Assistant] Sure, just got that out of the pan. Old friends aren't thinking no mind.


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[Assistant] Yeah, it's hard leaving with the past, eh?


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[Assistant] Mm-mm, not quite. Listen, why don't you go talk to her? Ain't shit to say.


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[Assistant] Then I'll go speak to both of them. I'll ask for your money back, and if they don't agree, I'll make them. Where did they hang out?


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[Assistant] At some arcade in Chinatown. Uh, look man, don't bother. No, I'll bother.


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[Assistant] I'm getting paid. And besides, you remind me of me.


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[Assistant] Are you the two snakes who played Dwayne Forge when he was in the joint? He wants his money.


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[Assistant] They are. I got rid of Javon for you, but I let your ex go.


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[Assistant] Sup, man. Not much. How are you doing?


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[Assistant] Yeah? Who are you? Can you remind me?


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[Assistant] He likes to party. He stays out late the whole time. You know what he's like.