Today's guest is about as San Franciscan as it's possible to be as you see I'll see in a few moments Now you I asked this question even though you told me a little while ago Are you a native San Franciscan? Yes indeed? I was born here.  What about your parents were they born here also my mother was my father I don't think was Now do I I'm trying to recall from our original conversation whether or not you have been interested in your genealogy at all Do you do you have a record way back of your family or or does it end at your grandparents or great grandparents? Do you happen to have that record? Well, my daughter has done some researching on my mother's family and They came from they were scotch Irish. Yes. Well, well your mother's name. What was there? What was their family name Ferguson Ferguson? I see yes, that's clutch. Yes and they came my grandmother Came to the United States after the potato famine in Ireland. Oh, yes, father was a linen merchant and There was really I don't think they were ever actually hungry, but there was so much Difficulty at that time that she came to New York as a young girl about 18 Approximately what year would that be? Oh That must have been about 1840 about 1840. Oh, yes. Yeah now Did your mother tell you any stories now? Oh, of life F here in San Francisco Let's say during the time before you were born. Did she impress you with the circumstances and Your own mother now. Oh, yes. Yes indeed. She did She was born on Lincoln Hill. Oh on Lincoln Hill. Yes Yes, and as a little girl she played down among the boats on the beaches of Lincoln Hill because Lincoln Hill was like Telegraph Hill it spread out into the into the bay a nice sandy beach So that she was a real San Francisco and in that era to see that was when Rincon Hill was Really a hill. Yes, that's right. I kind of a Crescent the harbour was a little question between Telegraph Hill and Lincoln Hill Oh, it was a sort of a crescent in there and as they as they made the streets for San Francisco by San Francisco was one great big sand dune Covered with lupin bushes and as they began to make the street They would haul some of the sand and dump it into the Crescent of the bay until they finally had it pretty well filled up. I Recall now seeing an old map whereby San Francisco Bay came up as far as First Street or so didn't it I think came up almost as far as Montgomery It did it did on the north side of market. Yes. Yes, and of course we have a street named front Street So so that was each street and yes and and beat street that that is right So that this has all been filled in there and of course recently when they were building the building this ship They found there is evidence of that of course by the way where in San Francisco Were you born you yet? What where were you born? Your mother was born on Rincon Hill. Where were you born? Well, I was actually born in San Mateo. They were down there for the summer vacation. Oh, yes You you did however your parents had a residence in the city at the time Yes, it was being built at that time on Broadway and the Visadero. Oh, yes. Is that particular building still standing? In fact, I think it's alias or son who owns it now. Oh, is that right? You mean the former mayor Joe Alioto's son? Yes, that's right. Oh, yes, that's interesting. Well, it is being lived in then and has been kept in good repair Yes, I have found interestingly enough that some of the folks born in San Francisco have their their original homes Of course have gone the way of of all buildings in that they have been taken down for reconstruction or some other thing now Then do you do you have any recollections of your very first entrance into a San Francisco school or your your study? I happen to know some of the part of the answer, but I want you to answer it if you will You did you attend the San Francisco public schools at all? No, but the schools were not very good in those days And I wait my sister and myself had a had just the governess who taught us reading and writing Oh, yes, you told me a little bit about this and I thought it was a good story And and I'd like to have you repeat it as much as you can remember or would like to Because do you remember some of the things? Let's let's say for example that your father had to do with helping this governess or teacher that you had To begin to take in additional students and become an established school. Is that not correct? Well, it wasn't this teacher that the first I see has the governess to begin with it was governance to begin with But later on I did go to Miss West school in those days And I was in the sixth grade and my teacher was Miss Burke Miss Catherine Del Mar Burke and I was very fond of her and She became acquainted with my mother because I she taught me at school And she said she wanted to found her own school someday So my mother lent us some money and encouraged her to do so and that was the beginning of Burke school That's an interesting story and you were there firsthand and and perhaps one of Two or three of the first students. Is that correct? Yes one of six one of six and three little sisters Well, they were in different grades, of course, so that she was teaching at that time all of the studies necessary for Several different grades. Is that right? Yes, well, we're all about the same grades who started the school So she taught us reading writing and arithmetic Did you go at an early hour to that school? Did you for example, did you arrive at school at 8 o'clock or 9 o'clock? What time in the morning 8 30 used to go to school and then then you had your normal recesses I presume the same as a public school would have had or did you have that sort of a break just went in the morning We didn't go in the afternoon. There's no afternoon session. I see you do you do you feel that that you were given? Adequate knowledge or adequate lessons were they were they hard lessons? Do you remember that they were difficult for you? Oh, I don't know that they were difficult but we certainly were well educated miss Burke was a brilliant has a brilliant mind and a Very dramatic way of teaching. She could get us into into a state of tears or laughter at a moment's notice And I remember that when we were studying Greek history and she said the The Greek fleet sailed from the pirate Piraeus never to return and she had it all in tears Oh, is that right? This this is a talent all in itself. Yes, it is. Yes The other thing I think about can I go back to your tutor when you were first tutored before you went to miss Burke school? Did you how much time did you spend with the tutor? Did the tutor come to your home? Yes, I see so that they came perhaps for half a day We are is that is that so that it was not a full-day session But they again would have the time to give you the individual instruction that was certainly most helpful Even though I haven't had the opportunity or never did have reason although I I have to Uncertain courses in chemistry. I remember in school that I did go to an instructor for tutoring So that I I do have some semblance of knowledge on what that is. Well, then you were taught in your home you went to miss Burke school and Your assignments were effective certainly because you have been successful throughout your life and Were you given a diploma or certificate of proficiency? What was the status of learning at that time? Did they I don't remember you don't remember you don't have it today then I presume But at least you felt that you you had adequate knowledge Yeah, and I wondered about that today, of course, you can get a certificate for most most anything you do and But then then you did follow through and you did attend college am I correct? Yes I went the first year of college. It was the first World War and then I married this and gave up college and got married because my Fiancé was going off to France. Oh, yes. Well, then that must have been in 1916 or 17 right? Yes, that's right in 19 say and and so that you were left here and of course I Feel certain he did return and or were you married before he went? Oh, he did. Oh, I see so that he was there and did he see action your husband see action No, he didn't he didn't I see Yes, of course this I remember such things, but I'm afraid many of our listeners will not so I shouldn't talk about things that that They wouldn't associate with Properly well then during the war. Did you take up some kind of work? Did you did you actually work for pay? Let's say for at some firm or a hospital or no We all did volunteer work Rolling bandages and things of that kind but not being paid for it. You do remember of course the flu epidemic I presume. Yes, I had it. Did you have it severely? Did you have a severe case of it? Yes, and my mother performed a cesarean on me. She was a surgeon doctor. Oh, is that right?  Yes while you were pregnant. Is that right? Yes, that's right and she performed and your mother did this Yes, that's right. This is exciting because I thought there was some deference Given a strange doctor has compared to a relative in such instances, but that is not the case Well, she was the best in the city in that I see a line I see and so you couldn't very well Or no one would make any attempt to change that that that's that's interesting and there's another interesting thing Is my son is a surgeon and he one time operated on to on me for a hernia So I think he says I'm the first woman who's ever been operated on by her mother and her son That is interesting, isn't it? Yeah. Well, let's see now. We didn't get you to college I want I want to know about your college days Well, I was only there you see a part of the first year I see then then you you were married And you did have you you had no reason then to return or you didn't return to to college in any in any event Well, that's what I wasn't clear and in the short discussion I had with you before I didn't I didn't pick up that kind of a detail well now I want to get into the thing that you have done Over a great many years and I'd like to know if you'll tell me how you became interested in promoting the ideas the idea of open spaces in San Francisco or I'm sure that your interest is not limited to San Francisco if I assume correctly Yes, well, it's a very natural thing you live in a city that's only it's in the building and where we lived on Broadway and Isidore Street There were lupin bushes and sand dunes as far as the eye could see almost So you saw that the city begin to fill up with people and after a certain point the filling up process was Was anything but Welcome you wanted the open spaces you wanted something left Was this personal now this is your own feelings were that you were being surrounded is that right? Yes, but I wasn't the only one with almost every San Francisco We're grown up in the city felt like the crowding of people into the city after the First World War So that your interest in in working toward open spaces is not wasn't new it originated almost by yourself with other Neighbors, that's right, and and you continued it on for the balance of your life so far. Well, it isn't about you Look, pardon me. That's the wrong he's writing for it for your life from that time to till today Well, this is exciting. We'll follow that through for me Will you because I want I want you to tell everything you can about your open space interest and what you have done about it and To promote it and how you believe it will be beneficial Well in the 30s during the Rosefeld era I became interested in civic affairs and we we did a lot about public housing at that time and in the course of that City planning became very interesting to us and in the late 30s I And other young students from the University of California Did a great deal to establish the city planning department in San Francisco. It hadn't existed before so city planning Was my great interest from that point on city planning and open space as part of the balance of density of a city as against Urban sprawl out into the country in the destruction of farmland Would you believe that if my arithmetic is correct? This has been a matter of 60 years that you have been working in this direction more or less. Yes You know you are given credit for effectively working for the passage of proposition J in 1974 Now I presume you had lots of help of course and but that was to establish open spaces Will you tell us a little bit about what that how you worked on that what you did to promote it? And it was passed of course, so yes, it was a charter amendment which permitted the city to Set aside something like three million a year for 15 years to buy the last hilltops the last shoreline the last empty spaces in San Francisco so that we would have some sort of a balance of of our community of dense city living as against open space for parks Tell me how has it worked out has it been has it operated satisfactorily has the city performed Yes, on proposition J. Yes We worked for about two years until proposition 13 was passed And we have we have purchased all the last hilltops in San Francisco. They all as And it's quite interesting Those last hilltops amounted to seven acres and we have paid for those seven acres $1,700,000 Is that right? This shows how little the word gets around even though you have a successful operation I was not aware that the last hilltops have been purchased yes as a Preservation activity by the city. Yes, that's right. That is exciting And I now begin to understand a little bit of why you have a hilltop named after you well we The proposition J. That was passed in 1974 Made it a charter amendment to put aside these three million dollars to buy the last hilltops for 15 years Now there's a hat has the money to do it sure it it takes that and you had to have the cooperation of the city And of course when it's it this was a charter amendment, right? Yes, and this voted by made it possible for the board of supervisors to To approve the money and and then it became it became a fact that that's wonderful Yes, they didn't even have to approve it. It was just a part of the law. It was just a part of the law Yes, and it was passed by an initiative. I You mentioned too that you've got a new program that you will be Expounding on may I say that word expounding be talking about let's say and that is that the San Francisco Procedure may already be declared surplus by the federal government. Perhaps it is to be shortly and You have some ideas about that. Is that right? Well, I heard that room in the other day and following it through the next day. I found it that Plan has been sent back to Congress for Restudying so I don't think it's an immediate problem. Thank goodness. It has been in the news enough Times recently that it eventually may come to pass. However, yes, it could be but I was also assured that if it ever did become so it would be part of the GG NRA the The natural resource Recreation area. Oh, yes. This was that was that what they call a reliable source that told you this sometimes They say things that are not necessarily going to come to pass. Well, I rang up the fort nation headquarters of the GG NRA and they said that it certainly wouldn't be sold to private enterprise At any rate, there'll be part of that Program. Well, this this makes sense. It's a beautiful piece of land Prime as far as all of the Western United States is concerned. I feel I feel certainly now the thing I'd like to have you tell us Is to explain a little bit on how you go about getting support for your open space plans Particularly in San Francisco, but any place in California that are interested in an open space program. What might you suggest they do? to Get interest. Well, I think the first thing is to form a coalition of the already existing conservation groups in that area Whatever they may be. How many of them would you say are in San Francisco? There must be a good number. We have about 18 on our list. There are 18 in San Francisco Yes, and then you call a meeting of some kind with representatives from all of those organizations and you label for them a Proposition of some kind that we must defend our open spaces our hilltops and our shoreline and Keep them for the public. Yes. Now those people are all thinking citizens who have considered What is good and followed it through in that way and you must then have the legal talent necessarily to occasionally Check on the legality or to work on the legality of certain properties. Is that correct? That's correct But there are always plenty of lawyers around to instruct you. Yes, I should say that is right There are plenty around I what I had thought was that you might have some volunteer lawyers in your organization that that would Possibly give you set you right on such things. Well, it always is so too Let's say that I wanted to do something positive to promote open spaces in San Francisco. What would I do? I talked to you first. That's what I know I would do But but what would what would anyone listening do who may not? Wish to either call or contact you in other words There there there is a piece of property that looks like it should be preserved What what would what would you suggest I do about it? Well at the present time there's a program emanating from the natural resource agency It's a state agency and Hewley Johnson is the head of it And he was the president of the trust for public land before he was appointed by Brown Oh, yes to this position so he's a great conservationist and he has a program of forestration in the cities of planting trees in cities right now and Setting aside certain land for vegetable gardens in the city now the organization to which he was a president before he became the natural resource agency director has a program of urban a little urban trust whereby Volunteers in certain neighborhoods can work in a framework. There's legal a little trust and they Develop Say the vacant lots in their neighborhood and plant them in vegetable gardens and the trust for public land tries to get that Private land into public land. Oh, yes by purchase or some other way. Oh, yes I'm going to ask you the only questionable Question of today, you know, San Francisco has a limited space. Let me ask you this question if I may Isn't it more important to have housing and buildings than more open spaces? Well for the economics of it, let's say I Think you simply have to have parks for people the more crowded they become the more they need those parks and open spaces What about the the economics of it? Have you have you delved into that? Is there something that I might find to read telling us about Economics and that sort of thing well, of course is a land is owned by the city. They don't pay taxes on it. Truly. There is a loss So there are taxes, but we haven't so many Vacant lots that would make any appreciable difference It wouldn't you think then that that the open spaces wherever we can get them is They're limited is what I know you believe this incidentally But I wanted to ask you the question and then I want to ask you another question What do you think do you think these open spaces are going to continue to exist through the year 2000 and beyond? It all depends if they're made into a park They probably will if they become under the under the public Control yes, and under the control of organizations such as those you belong to you believe that they will then they will Then the title will run with the land, you know What would you like to say do you know do you would you believe we're running short on time? No, I would like to have you make a statement about open spaces in the next O15 or 20 seconds. What would you say what you might like to say in the interest of open spaces? well, I think we should consider the the open spaces surrounding our cities as just as important as as The land where the tall the high-rise buildings go We need on the outside of our cities a fringe of open space for recreation and for For beauty and for certain a farmland that would be close into the cities things of that kind our time has run out and Nearly every guest which makes me wish we could arrange a repeat And we'll try to think about this at a future time, and I hope you will if we give you a call as I did this time We'll get together because we haven't covered the whole thing by any means. Thank you Dorothy Erskine I'm Truman ball speaking for the San Francisco history room and city guides at the main library in the Civic Center I'll be back in a week at this same time with a guest who will have a story to tell that will help us to Understand San Francisco better than ever So one more time I truly do. Thank you for listening. Good day


Most pressing issue in the world in the next year is re-engaging people in the labor force who want to be there and creating resources to do that.


Recall that a statement is either true or it's false. It's one or the other, it can't actually be both. Now, as we try to write proofs of statements, we sometimes struggle to prove that something is true. Like we think it's true, and really, really struggled to prove it's true. It turns out what you could do instead, instead of proving it's true, you could pretend that it's actually a false statement and see where that takes you. Like for me personally, I like to play lots of logic games, like Sudoku for example, that's a fun logic game you can play.  So what you can do is you can play where it's like, I'm not gonna guess, what if it's one? And then it's like, okay, if that's a one, that's a three, and if that's a three, this is a two, and if that's a two, this is a four, and if that's a four, this is a five, and if that's a five, this is a three. And then you're like, oh wait, you know, in Sudoku you can't repeat the same number in the same row or column. And so the fact that I got two threes in the same column means that this is not a possible solution. So then I have to follow back my logic to here when it's like, oh, it wasn't one, and therefore it must have been the other possibility, it must have been two. This is what an example of what we call a proof by contradiction, or also referred to as reductio on absurdum. And a Latin phrase often abbreviated as R-A-A, which just translates as to reduce to the absurd. That is, you take an assumption and if it leads to something absurd, or in this case, if it leads to a contradiction, then that means your assumption was wrong. If you assume a statement's false and get a contradiction for which the king of all contradictions will be you have some statement Q and you've also proven it's negation. The statement Q and not Q is a contradiction. It is always false. No matter what the statement Q is, this is always false. If you can prove a contradiction after you make some false statement, then that means the statement actually was false. And that's how one proves by contradiction. It's a really nice technique. Now, if you're not sure how all of that actually gives you a valid proof argument, consider the following. Let's take any old statement P and I wanna show you that the following is logically equivalent to it. Now the statement P is logically equivalent to P or false. Remember how a false statement, how an or statement works here. Or statements are true if any portion of the statement is true. Now, if you take P or false, then oring a false will never make it true. The only way that this statement can be true is if P is true. If P is true, then P is true. So P and P or false are logically equivalent to each other. Now, of course, I can replace the P with not not P. If you take a double negation, you get back the original statement not not P is equivalent to P and therefore P or false is the same thing as not not P or false. And then this one is equivalent to not P implies false. Here I'm using the fact that a conditional statement implies B is equal to not A or B. Remember if the hypothesis is false, it's vacuously true. And if the conclusion is true, it's trivially true. Therefore a conditional will be true if the hypothesis is false or if the conclusion is true. So thus we see that not not P or false is the same thing as not P implies false. And so thus when you put these together, not P implies false is logically equivalent to P. Therefore if this statement is true, then this statement is true. And conversely, if this statement is true, this statement is true. Now in the latter case, not P implies false, that's a conditional statement. We typically would prove that to be true using the method of direct proof, which direct proof means that you're going to assume the hypothesis, so you assume not P and then you go about trying to prove the conclusion. Well, the conclusion is false. So that is we're looking for a contradiction. You're looking for two statements that conflict with each other, which typically like I said earlier, this will manifest itself in the form that you prove some statement Q, you also prove it's negation and hence Q and not Q. This is a contradiction. So this is what we're looking for. Sometimes people draw a contradiction as these implication arrows, these logical arrows pointing towards each other. So it's like there's a conflict there. And so a lot of proof by contradiction is a valid proof technique. Since if you can prove this to be true, which again, the way you do that is you assume not P and then you find a contradiction, this then shows that P is true. And sometimes it can be easier to prove something by contradiction than it is to prove it directly. Let me provide you some examples of that. So perhaps one of the most classic proofs by contradiction there exists is the proof of the infinitude of the prime numbers. And this goes back to the days of Euclid in the book, The Elements. Euclid proved that there are infinitely many prime numbers. And the way that you can do it is by contradiction. So suppose to the contrary that there are only finitely many prime numbers. So this phrase right here, suppose to the contrary, this is the author telling the reader that I'm gonna prove something by contradiction. I'm gonna make an assumption opposite to what I want to be true in order to derive a contradiction. You say this to the reader so you know what's going on but also so the reader knows that you know what's going on as the writer, right? You're not just coming up with false statements, you know it's false. And so you're telling it, this is a false statement because it'll give us a contradiction that's gonna be coming forward. And so the reader knows to look for a contradiction because you're telling us suppose to the contrary such and such or you could say for the sake of contradiction such and such. Some people like to say by way of contradiction such and such like this BWOC by way of contradiction. You should indicate to the reader that you're going to prove something by contradiction. All right, so suppose to the contrary that there are only finitely many primes and so let's list them, there's a first prime, there's a second prime, there's a third prime, there's a fourth prime up to the nth prime. So these are the finite primes that exist in many primes. Now with these in many primes, I'm gonna make a new number. I'm gonna make the number A, which is the product of every single prime exactly once and then you add one to it, okay? So I want to then show that this number has to have a prime divisor. And we actually proved this previously in a different lecture using induction that every, well actually you strong induction if I'm being more accurate, every positive integer greater than or equal to two has a prime divisor as two is clearly one of the divisors. A has to be larger than two, so therefore it has a prime divisor. Now without the loss of generality, let's assume that the prime divisor of A is P1. Am I listed these primes? I didn't say they were in ascending order, like I didn't say that the first prime I listed was the smallest prime and the second was the next prime. I'm just saying here's my primes. So without the loss of generality let's assume that P1 is a prime divisor of A. It has a prime divisor, it's one of these n primes, let's call P1. So we have that P1 divides A because A has a prime divisor. But also if you take the product, P1 times P2 times P3 times P4 times up to Pn, this is actually what's known as a primorial. It's kind of an amalgamation of the word factorial and primes because a factorial is the product of all integers, one upward. So the primorial is actually the product of all the consecutive primes from the first prime to the second prime all the way up to the nth prime, okay? Anyways, clearly the primorial product there is divisible by P1. I mean, I can see P1 as one of the factors, clearly, okay? So if P1 divides A and P1 divides the primorial, that means that P1 also divides their difference. It divides, P1 divides A minus the product P1, P2 up to Pn. But this product, excuse me, this difference, A minus the product of primes is equal to one. So by Lemma 194, which we saw previously in this lecture series, it told us that if a prime, well, not even as a prime, if you have a common divisor between two numbers, then it'll also divide the difference of those two numbers. So P1 has to divide one. But wait a second, one doesn't have a prime divisor. So for one reason, we now have that one has a prime divisor, but for other reasons, because honestly all primes have to be larger than one, one doesn't have a prime divisor. This is a contradiction. One cannot have a prime divisor and also not have a prime divisor. That's a contradiction. And that's the contradiction we were looking for. So because we contradicted our original hypothesis that there are finitely many primes, the negation actually must be true. Therefore, there are infinitely many primes. That's a pretty cool result there. It's a very nice proof by contradiction. There exists dozens and dozens of proofs showing that there's infinitely many primes. But I honestly think that the simplest of proofs, in this case was probably the first one, which was Euclid's proof by contradiction, that there are infinitely many prime numbers. Let's look at another classic proof by contradiction because one can actually use a contradiction to show that an element doesn't belong to a set. For example, how would you show that one half does not belong to the set of integers? Because to be in the set of integers, there's a certain definition there, and then you have to show that one half doesn't satisfy it. That can sometimes be difficult. And so what you could do is you can do, by way of contradiction, you could suppose that one half is inside the integers. You can then make the argument. There is an argument that can be made here. I'm not gonna supply it at the moment. This is just an example. But you can make the argument that one half is between zero and one. That's one statement. But then you can also make the argument that there is no integer between zero and one. Be aware that with the integers, the integers by construction are discrete objects. That is there are gaps between the integers. And so in particular, there's a gap zero and one. One is the successor of zero with regard to the integers and there's no integers between it. If one half was an integer, it would have to be an integer between zero and one. That's a contradiction. And so therefore, one half cannot be an integer. Now, one half is clearly a rational number because it is a ratio of integers. It's one over two, but it's not an integer. So this is an example of a fraction which gives you a rational number which is not integral. That's approved by contradiction even if I didn't supply all the details to it. One that I will supply all the details to it is another classic infamous theorem here. The real number, the square root of two is irrational. That is to say we're gonna show that the square root of two is a element of the real numbers, but it's not an element of the rational numbers. So more specifically, the square root of two is not a rational number. The fact that it's a real number is not in question at all. But every real number is either rational or irrational. To be rational, you have to express the number as a fraction of integers. So if I think that the square root of two is not a rational number, I have to show that it's not ever a fraction of integers. But there's a whole lot of fractions out there. How do I know it's not a fraction of integers? I could try a bunch of fractions like, oh, the square root of two doesn't equal three over two. It doesn't equal five over seven. It doesn't equal 11 over 13. Those are all true statements, but I can't just list a bunch of examples and expect that to work. In order to show that there's no fraction that works, contradiction actually becomes an effective method to do it. Because instead, what I'm gonna do is suppose to the contrary that the square root of two is a fraction of integers and then derive a contradiction from that. That'll show there exists no such fraction because if there was a fraction, we would get absurdity. So let's then see that proof. For the sake of contradiction, suppose that the square root of two is a rational number. And if it's a rational number, there exists integers p and q such that the square root of two equals p over q. And sure, we can assume that q is not zero. We don't really have to worry about that right now. Instead, what we're gonna do is we're gonna assume that this fraction p over q is in lowest terms. So we say that a fraction is in lowest terms when the GCD between the numerator and the denominator is equal to one. Now, one can actually provide an argument, a proof that every rational number can be written in lowest terms. This basically comes down to an argument using the well ordering principle. For which if you take a rational number you take all the possible fractions that represent that rational number. For example, if you take the rational number one half, you would have things like one half, two fourths, three sixths, et cetera. Take all of the possible fractions you can use to represent a rational number and look at the set of positive denominators that can be used there. That's a set of natural numbers because they have to be positive. And so by the well ordering principle, there exists a minimal element inside of that, call that minimum element q. There has to be some companion to q, call it p. I then claim that that fraction is the lowest terms for that rational number. And I'll leave it as an exercise to the viewer here to provide the details of that. That the well ordering principle guarantees that every fraction can be written in lowest terms. Okay? So we're just gonna go forward with that fact established. Like I said, I'll leave it up to the viewers to prove that here. So the square root of two can be written as a fraction in lowest terms. Now, by definition, what is the square root of two? The square root of two is a real number such that when you square that real number, you get two. So therefore, if I take this equation, square both sides, the left hand side becomes two because that's what the square root of two is. And then on the right hand side, we're gonna square p over q. And now when you square a fraction, you can square the numerator, you can square the denominator. And so we get two equals p squared over q squared. If you cleared the denominators, I can make this into an equation of integers. Two q squared is equal to p squared in that situation. Now notice the left hand side. The left hand side is clearly divisible by two because I can see the factor of two there. Two divides the left hand side, two q squared. Therefore, two has to also divide the right hand side, p squared. Now by Euclid's lemma, remember Euclid's lemma tells you that if a prime divides a product AB, then the prime has to also divide one of the factors. It divides a or divides b. I don't necessarily know which one it is, but Euclid's lemma guarantees that. Again, this is the same Euclid. Euclid in the elements proved that the square root of two is irrational. And remember, irrational numbers, these were controversial numbers, like there's legend goes that the first person to prove that the square root of two was irrational belonged to the school of Pythagoras. And the Pythagoreans had him drowned because all numbers have to be expressible and irrational numbers by definition can't be expressed. And therefore this was heresy and the man was executed for this. Whether such a legend is true or not, I can't attest to, but it definitely did cause the Pythagorean school in ancient Greece a lot of angst, this conversation of irrational numbers. And so they kind of were believed to not be existent. Years later, of course, we can accept this proof by contradiction that in fact it is an irrational number here. But anyways, by Euclid's lemma here, which we haven't proven Euclid's lemma, we'll actually prove it in the very next lecture. So this won't be a circular argument. We can prove Euclid's lemma without anything we've proven so far. So take a look at that in the next lecture here, lecture 24. Anyways, by Euclid's lemma, since two divides p squared, it has to divide p or it has to divide p. But if that's your options there, two divides p or two divides p, that's the same option. So we can actually conclude from that, that two divides p, okay? Now, if two divides p, that means four divides p squared because each of the p's in p squared offers a two. And therefore, since p squared is the right-hand side, if four divides the right-hand side, four has to divide the left-hand side as well. So that is four divides two q squared. Now, there's a two there and four is two times two. So that accounts for one of the twos. We could divide two from both sides there, but we still have to produce another factor of two for which the only thing remaining is two has to divide q squared. But then, again, by Euclid's lemma, if two divides q squared, it must have been that two divides q. And so I want you to see what we've now done here. We've now argued that two divides p, but two also divides q. But by definition of being in lowest terms, the numerator and divisor, their greatest common divisor has to be one, but we found a common divisor two, which is bigger than one. This would contradict the fact that p and q is in lowest terms. And since the well-ordered principle guarantees that every fraction has lowest terms, the square root of two, its fraction can't be written in lowest terms, which actually contradicts the fact that there is no such fraction. That is to say that the square root of two is actually not a rational number. And you could modify this proof to show that any square root of a prime number is in fact irrational. And honestly, if you take the square root of any number that's not a perfect square, such as a prime number, this same argument applies in that situation there. Now, of course, if you take a number that's not a perfect square that's a little bit, you have to be careful that you don't make a circular argument there. But for example, you can show that the square of six is not a rational number by using a similar argument that we have done right here. So these two proofs illustrate how powerful a proof by contradiction can be. And we'll use this proof technique in our future study of advanced mathematics.


Here's our atmosphere, one of the great global commons. You're using it right now, breathing in and breathing out, and we use it collectively as a sewer, dumping our waste and our pollution into it, hoping it'll just go somewhere else. And for most of human history, that's been a good assumption, there haven't been too many of us around, but now there are 7 billion of us, and the problems are accumulating.  We've had 20 years now of UNFCCC conferences, and over those 20 years, emissions have continued rising and rising, interrupted only by the financial crisis. So one question is, are there other approaches or new ways of thinking about this?  Here in blue, you can see the hole in the ozone layer emerging from 1979 to 89 on the top, and stabilizing from 2006 to 10 on the bottom. And we got to grips with this, because the Montreal Protocol used the forces of trade and of technology to grapple and to bring together a creative coalition of business and governments. So can we cut and paste to climate change? Well, unfortunately not. Climate change is much harder, because greenhouse gases are a pervasive feature of all of our production and consumption, and the incumbent fossil fuel players don't have an easy, quick, technological fix that they can move to, and hence it's a business threat in their fighting climate policy. There are two fundamental things we need to do, in my view. First, it's about money. Clean energy remains more expensive than dirty energy in most places most of the time. And the second is to overcome the free riding problem that we have. Most countries feel that they have an incentive to hold back and wait for others to move first. Now on the first point, there is some very positive news. Research in labs around the world, including at the Oxford Martin School, here are solar perovskites and electric vehicles, have been bringing the cost of clean energy down quite rapidly. But these cost declines are very positive, and there's another positive feature of them, which is that we've learned just last year that the spillovers, the economic spillovers from clean technology are much higher, 40% higher than the spillovers from dirty innovation. So clean ideas beget more ideas in society than dirty ideas do, which is a justification for further public support. And we'll need it. Joint work with economists at Harvard and the LSE shows that we are locked in to a fossil fuel society, much as we are all locked in to using QWERTY keyboards, designed to prevent jams in typewriters. We don't use typewriters now, but we still use the QWERTY keyboard. The fossil fuel system is the same. There are all sorts of lock-in and path dependencies that mean that researchers in the dirty area have a greater payoff than researchers in clean. And we spend in 2012, it'll be less now, between US$600 and US$700 billion a year looking for fossil fuels versus US$350 billion on all forms of mitigation, including renewables and other clean technologies. So there's a big bias in the system. And you may well ask, why are we spending US$600-US$700 billion looking for more fossil fuels when we already have many times more than we can ever afford to burn? This shows the carbon embodied in fossil fuels versus the carbon that we can afford to put in the atmosphere to stay within two degrees. We do not need to be spending money looking for more. So the good news is, the costs of clean are falling fast, but they're not falling fast enough. We will see parity under various assumptions, but they're actually fairly regular. Work at the Oxford Martin School suggests parity at around 2030 at scale deployment. But this is not fast enough for the climate. We need an acceleration. And my view and my proposition is that we need a deal, a creative coalition, an international deal between the key players on renewable energy, research and development and innovation to bring these costs down. This is a first order feature of the problem. And I think such a deal is perhaps more feasible than the deals we're trying to strike, which have to balance ambition, getting people to be ambitious enough, participation, getting the right players in the room, and then compliance, making sure that people and countries live up to their pledges. With research and development, more of the benefits stay at home. And so the free writing problems are not as bad. You might create new industrial clean energy champions. And if we could design a club, like the World Economic Forum, like the G20, where people wanted to join, where the benefits of membership in terms of technological transfer, favorable trade regimes, et cetera, meant that you were better off in than out, and I think we could really trigger a big pulse of support on clean energy innovation that would pull the costs down. And if we did that, we would see the herds of the capital markets shifting the financial flows away from wasteful spending on fossil fuel assets that are just going to be stranded and in towards the newer markets of the future in clean energy. So that's my suggestion for going some way towards helping resolve the great global commons in the atmosphere. Thanks.


right in front of the target. Okay, this might be another one we got to be worried about. Okay, on a DCDM, if a cross shows up right here, or I guess we got to throw in a NIGMA. NIGMA lands right there, right in front of your target, probably safe to say you might want to go, might be another reason for you to go ahead and, TP, you know what I'm saying. Take that profit, no worries. Okay, we've got a down two finger in the middle of the golden ropes on the 29s. Go to verse here, we also have a divergence here. So gotta be careful, we could go up here. We've got a double NIGMA, I'll give it a shot, way behind on the entry on this, but get it anyways, but get it anyways. Now, why did I take that? It was also because I did have high time, and we've touched this a couple of times, we've got cold line, I got two NIGMAs, check the NIGMAs right here. I'll give it a shot. Now, did I go all in? No, I'm gonna throw in the water shirt. I'll grab a little bag.  Press the like button, share, subscribe and comment below with your thoughts and what you've learned.


What's up guys Isaac here, and as many of you know Charlie Murphy passed away the other day and for those that don't know him he's Eddie Murphy's older brother and He had many good comedy skits a lot of good movies. I especially like the ones with Dave Chappelle Anyway Some of his last words were rather interesting and strange how when people are on their deathbed or dying or about dying They seem to give some really good advice Anyway, he wrote want to sleep on Release the past to rest as deeply as possible. What does that mean? I feel like he's talking about how a lot of us tend to Let the past dictate our present life or our future a lot of us carry a little baggage of the past and we don't ever let it go I know people that still get angry over something that happened to them 20 years ago Some some something someone said to them they still Shake up. They still Reimagined it vividly as if it's happening to them right now and they feel the same emotions Matter of fact they get angrier to each other recall it. I know some people are so stuck up in regret and some decision They made they regret to this date. They still feel ashamed.  There's all sorts of things that could have happened But there's one common factor with all these things in the past and that is we have no control over them We have no control over them. So we shouldn't let them dictate our lives We shouldn't replay in our head over and over again. Just tormenting ourselves It destroys the quality of your life and destroys your tranquility What you have to do is take the lessons from the past the lessons you took from every experience They're gonna benefit you and bring them forward you take the lessons and you disregard the actual events You don't have to vividly go over every single event feeling the negative emotions over and over again Just working yourself up take the lessons Bring them forward and focus on the present, which is something you actually have control of which is something you have control of it Because you're in there right now Not being able to let go of the past is a sign of mental weakness It's a sign of not having control over your thoughts a sign of not having control of your mind And it's something that you can train to get rid of Focus on what's in front of you. Don't worry about the past You made your decisions people made their decisions and those decisions have been made the same stone So there's nothing you can do about it Anyway guys, that's the video real nice and short Rest in peace Charlie Murphy one of the greats and leave a comment down below Tell me what you thought his quote was about or what baggage are you holding on from the past? Here we guys peace I'm gonna show you how to be part of enough catching gang.


Hi there today on Typical Books it'll be my top five extreme horror. Now of course this is my top five extreme horror so it's up for debate really honestly and this is one of those weird sub genres of horror where there are a lot of opinions and opinions run wild and it is like does Bizarro fit in? Is it all Splatter Punk? Does it need to be Splatter Punk? Are there things that are too far for even extreme horror fans or is nothing off the table? Of course I have a big affinity for like serial killers and slashers or madness and psychosis not so much supernatural. Now the supernatural debate happens within extreme horror as well. Are there many supernatural titles within extreme horror? Well let's see. Number five was hinted at before and there's some wild cards within this list too okay so this one is a short story by Stephen King called Survivor Type and you may be asking you know how can a 70s old-ass story by Stephen King of all people grandpa horror be in among your extreme picks? Well I'd say read Survivor Type or if you want you can head over to Shutter and check out the most recent creep show iteration that is animated that has a story by Joe Hill and story by Stephen King and this one in particular is harrowing. Survivor Type in particular gave me nightmares. It made me wonder how would this be adapted to screen even at a young age. It made me wonder what goes on in the mind of this man. More than any other Stephen King story because a lot of people do I guess reportedly wonder that about horror writers. I wondered that about Stephen King at that moment and that story has haunted me. I think about it all of the time well not all of the time but a lot of the time and I've thought about it year after year after year it's to see it adapted for screen in any way shape or form was fantastic. The story itself to read it again is still harrowing and it talks about a lot of things that we find extreme. The thoughts, the madness, drug use, and auto cannibalism having to you know sawing the foot off in saw that small scene has gotten nothing on Survivor Type so if you haven't read it I definitely recommend picking up Survivor Type finding it somewhere watching the adaptation of it in the animation. Survivor Type is number five on my extreme list. I'd have to say that it really holds that position as being the first extreme thing I ever read and it left me wanting more. Flash forward way later in my life and Jack Ketchum makes an appearance. This is number four offspring and you could lump in a lot of other Jack Ketchum. It depends on what it is that you count as extreme. The girl next door is most definitely extreme horror and a lot of these have to do with things like cannibalism, insanity, and mutations or just the cruelty. It's really the cruelty I think that really sums up extreme horror for me and you'll find that as a thread going forward in a few of these books that I talk about. Anything by Jack Ketchum really is a study on cruelty and I have a theory of course that he takes a lot of true crime stories and turns them into books like even read it seemed had a hint of a true crime story in there. I don't know so much with offspring though. So before you think that it has to be not supernatural and maybe cannibals and true cruelty to be extreme horror I will say enter my third pick. Here we are at number three with Edward Lee the hunter of the threshold. This is take on some HP Lovecraft and he's also got some more HP if you like in Lucifer's Lottery and his whole house Infernal City Infernal Vet trilogy is fantastic and supernatural and delves into hell and is so crude and extreme and it's sort of edging on bizarro with the humor too. Like I'm not the biggest bizarro fan and it is where that crossover between bizarro and extreme is where I really do like to hang out edging more into the extreme. Edward Lee definitely dances that line really well. Then you get you know that cannibalistic savage cruelty and things like Header very famous for Header he was and I really like his novel The Black Train popularized in the United Kingdom under the title Gast. This is the Black Train. It is not necessarily as extreme as most of Edward Lee's things and nowhere near extreme as the books I'm talking about here today but it does deal with some heavy duty themes. Eurination, urophilia I suppose it would be, golden showers as it were, some pregnancy related things and a cessation of pregnancy related things. So it's not as extreme as the other books but it is a wonderful story and I really enjoyed it very much. I found it extremely dark and he writes cruel women extremely well and it has none of the bizarro humor nor the hellscape of his other books. It's the most rooted I think in real world. So if you like real fairly realistic horror and it's somewhat historical horror here too, then this may be for you. It is quite intense, I will say. Now there are a lot of really insane themes. That's the one thing that hasn't really been mentioned, you know, in the survivor type we have some very one person against themselves insanity going on and it is brutal due to the gross out as Stephen King would say and the brutality of it all and the viscera and the descriptions of that viscera and that sort of thing is what's hard to keep reading. With the Jack Ketchum stuff there is a lot of rape and cruelty and domination. It's very very well written that way and it's harrowing. We get a lot of all of that with Edward Lee and he writes some really degrading stuff and it's a very scatological urology.  He has violence against everything. Violence against everybody. There's violence equally distributed amongst men and women and it is all horrible and vile, rotten and grotesque. I really love Edward Lee. He is the king of Splatterpunk in so many ways. Now it's a real toss up for position number two between Wrath James White. It's The Resurrectionist is the Wrath James White book that I have that ties number two with Ryan Harding and Ryan Harding's genital grinder just a beautiful study in depravity and there's some very dark places here having to do with the sex trade and underground porn snuff films and just some very crazy backwards insanity going on with a degrading depraved death and I really enjoy Ryan Harding's style and I think that there's like a new wave of the Splatterpunk or people trying to keep Splatterpunk undead and he really is one of my favorites as far as the newer crop and not to say totally new he has a lot of workout up there with Edward Lee and Jack Ketchum and all of the other big names as far as extreme horror go and this does not delve into bizarro necessarily. It is straight up horror. There's no supernatural really leanings in this particular collection. It's a very short collection and you'll notice this is from Dead Eye Press. If you're interested in extreme horror and want to get into that bizarro not so much the supernatural usually just depraved human on human insanity. Dead Eye Press is really the place to go. A lot of Harding and Wrath to be found there. Like Wrath James White the person not Wrath the feeling but there is Wrath all kinds of Wrath. And number one extreme thank you girth ennis. I've never read anything quite as insane as crossed. The first editions and some of the books that we have here this is crossed psychopaths still in the plastic so it's all shiny. I took these out of their boards just for this but we've got quite a bit of crossed hanging out in this house. Crossed is something that blew my mind. Way back when when I read Survivor Type I didn't think I could really read something quite as insane and then later on I discovered Jack Ketchum and thought oh my that has been breached. I've never read something quite as insane and then I started reading further into some Edward Lee territory and you could say you know into layman territory as well thought oh wow I've never felt like I've read something so dangerous and then I discovered some of the newer splatter punks and then Wrath James White came into my radar and Ryan Harding of course and I thought oh my god I've reached the zenith of insanity on paper and then I read crossed. A friend of mine that worked at a bookstore was surprised they had crossed and when he saw it there he said oh my gosh you have got to read this. It took me a little bit to get around to it but I could understand why he was surprised that it was allowed in the bookstore and you can find it in the bookstore. I've picked up quite a bit of crossed at black squirrel books here in Ottawa and you can order it of course like a normal book. All of these books are not taboo. They are not on the list of banned materials by any means. They are viable literature in every way shape or form and this is a viable art. I'd say crossed is some of the most beautiful and depraved and fantastical and insane and grotesque horror I've seen on paper. Maybe since Hellraiser. Hellraiser was a very beautiful looking comics and these are very beautiful as well but if you like zero day zombie insanity and things like Hater by David Moody that I was talking about last video my last tag video reminded me quite a bit of crossed and crossed is like that to the nth degree with some really just insane near supernatural depravity going on in crossed. So I really do recommend it especially if you like short form insanity and really good artwork you know definitely look up crossed.  So there you have it my five top extreme with sort of a tie between Raph, James White and Ryan Harding. I hope you enjoyed that. If you've read a lot of these authors let me know if there is some extreme horror out there that I'm missing entirely then let me know in the comments below and have a nooky spooky day.


I was listening to you at the National Fair Breakfast this morning. Yes, that's right. I love your show when you're opening lines. Very good. It's a great privilege for me to be able to hand over this statue to you today, Mr. President. Thank you, Mr. Beffa. His firm, Saint-Germain, have been manufacturing this statue. They've been in the business for 300 years, and they did the whole of mirrors in Versailles for week at dawn. So they go on for the job now. Yes, we go on for some of the people here to think that I was one of three. This is Monsieur de la Moulin, who is the president of the French National Committee for the Statue of Liberty now, and 100 years ago, his great-grandfather launched the whole concept in France. So I understand, I understood that, and we are deeply grateful. I'm so happy to be here to present this wisdom from now on that we have all the 58 governors of the United States, so that the statue would be all over the country. We want every state of the Union to have a replica of the statue of Saint-Germain, giving to you, and this is a very good vessel that I'm kept in the nerve. And you know, she is the incarnation of Mariana. Her past is in all little and big municipal homes of France. I'm very proud that she should be here today. Congratulations. Mr. President, I've got a message, a very warm message from my president for you. And he asked me to bring you a statue. It is a crystal. It is fragile but it represents all the strength and the symbol of the friendship between our two countries and I'll come to that. Well, we are very proud of that. This anniversary is coming up. We celebrate that. We're very pleased that your great-grandfather had the idea, and that you had a hand in one of these. They are symbols of long-time friendship. I think that would be great. Mr. President, we are very pleased that you had the chance to lead me a bunch of others in the White House. Very nice to be here today. Very nice to be here today. And we are hopeful that the President of the United Nations will be able to be here for the anniversary. We're very touched by the invitation. Thank you very much indeed. Thank you. I'm very proud of you. Thank you. And he mentioned that it was done in 2600 sq.m. That's a lot. That's almost an honest thing to do around here sometimes. Well, alright, thank you very much. Thank you. Mr. President, what are you going to do about Haiti, sir? What are you going to do about Haiti? Well, all we can do is keep our eyes open here, and if there's any way we can be helpful, we'll be there. Maybe Dr. Ball here has to come to the United States. To my knowledge, there have been no such requests. Mr. President, do you expect Sharraski to be released? Well, we're hopeful that a great many of those people will be released. Is there a deal in the works? Is there a deal in the works? I'm not going to talk about anything of that kind.