3:42

this is the proof we give.

Â The first part that was actually given to us, that

Â was fine from those definitions, we started with the definitions.

Â Going in the other direction,

Â [UNKNOWN]

Â the definition.

Â So the question is simply, all this really involves is reducing

Â this to the definition and then the argument itself is very simple.

Â that was a slightly more complicated bit of algebra, but not much more.

Â So it's not about the complexity of the algebraic manipulations,

Â you've been able to do that for years I know that.

Â The question is, can you reduce the

Â statement to something coming from the, the definitions.

Â it, it, because in many cases you simply would, this

Â would be so simple compared with most proofs in mathematics.

Â You wouldn't go into these details.

Â But the focus here isn't on the, this stage.

Â Isn't on the, the complexity of the argument.

Â The focus is on the structure of the argument.

Â How are you able to get the logical structure right?

Â And experience tells me that it takes most of us, and it certainly is with me,

Â a long time to get the sense of what do I have to prove in order

Â to prove something.

Â that's not something, that's not cookie cutter.

Â You've got to build up experience in what constitutes a proof.

Â And be able to judge whether a proof is correct or not.

Â 4:57

me jump first of all straight to the to the correct answer.

Â And the correct answer is a.

Â And this says there are people and there are times.

Â That which you can fool those people. So, you can fool some of the people.

Â Some of the time, remember existence quantifies what

Â we use in mathematics to capture the word sum,

Â or at least one.

Â little bit a typical in terms of what

Â the word sum means, in, in, in, Apedel language.

Â Okay, and here the way I've expressed it is there's a person and there is a time.

Â So, so, you cannot fool that person at that time.

Â Okay, there's a person and a time.

Â So, so you cannot fool that person at that time.

Â Which is actually equivalent to saying you can't fool all the people, all the time.

Â Formally, you could take that part and you could

Â re-write it, it's not the case that for all x.

Â And for all t, fx t.

Â It's not the case that you can fool all the people all of the time.

Â Okay?

Â But, but I wrote it this way because we're only asking for the equivalent.

Â I'm not saying which is the closest way of capturing it.

Â I'm just saying which one is equivalent. Okay.

Â let's look at part b. What does this.

Â well if the first part were the same in all

Â of these three, so the distinction is in the second part.

Â So let's see what this one says.

Â Can we express that in English? well this is tricky to express in English

Â because of these American Melanoma Foundation type issues.

Â I mean you could say something like oh, let's say you can't fool everyone let's

Â 6:47

For everyone it's the case that you can't fool them at sometime or other.

Â and really, what we're saying is, is, it's not the case for, for every person.

Â there's a time when you can fool them.

Â Okay. So you can't fool everyone at some time

Â over that. And and I'm, that's sort of, I can't

Â think of a way of saying this that really, that reads

Â well in English.

Â [LAUGH]

Â And yet, it really captures this.

Â but hopefully, it's clear what the thing means.

Â It's not the case for every person there's a time when you can feel them.

Â And, okay, and to me that says the same. But, but, you know, you may interpret that

Â one different because this is one of these thing like American Melanoma Foundation.

Â it's, it's ambiguous, as natural language so often is.

Â Okay, let's see again.

Â And in the case of this one, what does this one say?

Â It's not the case that the reason x number is a t, so you should convert.

Â This one I think is easy to say in English.

Â Okay, it basically says you can never full

Â anyone. Okay.

Â It's not that you cannot find a single person at a single time

Â so that you can fool that person at that time.

Â You can never fool anyone.

Â So this one I'm very happy with. Okay, that, that one really captures it.

Â and the smiley doesn't indicate that I think that's true.

Â It indicates that I think that's a

Â really clean interpretation of what that one means.

Â 8:22

One and number four doesn't arise because we've already found something correct.

Â With questions three, four, and five,

Â which is truth table arguments. That was essentially a revision material.

Â So I won't go through those, those here.

Â let's just take a look at question six which when you first

Â meet it, it looks as though it might be to be deep,

Â especially since we've, we've seen this now after we looked at things

Â like square root of root two and square root of root three.

Â But actually, this one, when you step back and think

Â about what it says, tends out to be very simple.

Â we simply

Â observe that whenever n is a

Â perfect square, in other words.

Â It's a square of some integer, any integer

Â and there infinitely many integers, so there are infinitely many perfect squares.

Â Then of course the square root of n is just k.

Â Which is rational. And that's all there is to that one.

Â And I'm not only is it rational, it, it's a whole number.

Â The point is their infinitely many numbers in for which the

Â square root of n is a whole number in Hen's rational.

Â Namely all the perfect squares. Okay, that's only still out.

Â You don't really need to say anymore. That's it,

Â now here's that it's true and it tells you why it's true.

Â And that's the proof.

Â because you won't ask really to sort of, say whether it's true or false.

Â But it was this that I was interested in.

Â Can you construct a proof of its negation? Of it, or its negation.

Â That that was really what it's about, it's all about proving things here, okay?

Â 10:09

Well, finally question seven is one of these fallacious proofs.

Â There's there's a whole range of these things.

Â quite amusing.

Â usually they depend upon dividing by zero in some disguise form.

Â this one's different, okay?

Â So let's follow it through, because normally they,

Â the point is just to find the mistake.

Â We're clear that there's going to be a mistake because one doesn't equal two.

Â and indeed one of the things

Â that I asked you to do was to find a mistake.

Â But, the focus here isn't so much on finding the mistake.

Â It's how would you grade this as a proof?

Â And remember when we're grading proofs logical correctness is certainly part

Â of it, so there's going to be some losses of grade here.

Â But also it's about communication, okay.

Â So let's go through this one and just see how we would grade the thing.

Â Okay.

Â Well for example, is it, is it clear? Well it's

Â absolutely clear, so we're going to have to give

Â four mark for, for clarity, this is very clear.

Â There's a good strong opening, we start with the identity, 1 minus

Â 3 equals 4 minus 6, okay, both sides are equal to 2.

Â So four marks for giving the, the opening.

Â 11:17

the conclusion is stated.

Â So it's 4 marks for stating the conclusion.

Â And reasons given, yes absolutely.

Â Adding 9 plus 4, 9 over 4 to both sides is

Â just completing the square to, to, to, give you perfect squares.

Â Then we're saying that it factors, which it does.

Â That's why we added 9 over 4 to both sides.

Â Now I'm taking square root of both sides.

Â Absolutely!

Â Reasons, reasons, reasons. So in terms

Â of the structure, this is wonderful.

Â above all I'm going to have to give it a 0,

Â right, and that's clear, because the thing is plain false.

Â the question is, you know, what do I give for logical correctness.

Â Well, first of all we need to identify

Â the the evidence where the errors occurring, right?

Â and the errors here. it's in this line.

Â Because when you check square roots,

Â then of course there are positive negative square roots that we had.

Â And in fact, the correct solution is to see that, then,

Â then, minus on the left hand side the negative root on

Â the left hand side, minus 1 minus 3 over 2 and

Â it's a positive root you take on the right hand side

Â 12:27

when you take square roots. So that's what you should have.

Â In other words, 3 over 2 minus 1 equals 2 minus 3 over

Â 2, in other words, a half. It equals a half.

Â Okay, so now we dig f, I mean not so. That's nice, the world still exists.

Â a half is equal to a half, even though one doesn't

Â equal two.

Â So the issue here was when your taking

Â square roots there are two, two possible signs.

Â And and the one that makes it valid is is is

Â the negative on the left and the positive on the right.

Â Okay.

Â So, this is where the mistake is.

Â The only question is, do we give partial credit

Â for the the fact that there's logical correctness everywhere else?

Â This is a judgement call. I would say that these steps are so simple

Â it's just ele, very elementary algebra arithmetic.

Â That's this level I'm not going to give

Â particular credit for getting these bits right.

Â And I'm going to say that this is a big mistake.

Â This is huge.

Â Okay, when you take square roots you have to take a plus or minus.

Â That's some mathematical mistake.

Â Realizing that, that, that not recognizing that you need

Â to worry about the signs when you check square roots.

Â So that's a big error for mathematical

Â points of view, so this grading that I've given it.

Â Which, which means I'm actually giving 16.

Â Means I'm giving no points at all, on the mathematical correctness.

Â 13:56

I'm giving lots of credit on everything else.

Â Now if this was a mathematics course, I wouldn't be using this rubric this way.

Â I would take account of these, but they would have less weight.

Â I mean, these have all got the same weight, a maximum

Â of, of four.

Â and that's because of this focus of this course is on, on proofs and reasoning.

Â And, and common communication. And these are important parts of that.

Â you know the assumption in this course is that you already can do some mathematics.

Â and so I'm not really grading you on that.

Â I'm grading you on mathematical thinking and mathematical communication.

Â so the zero max here on the mathematics.

Â Build this formax and everything else. now this is the reason why I did this one.

Â This is where everything is welded out, but the thing is planned wrong.

Â So it distinguishes between mathematical correctness and the mathematical thinking,

Â and the communication, and then writing a proof out correctly.

Â Having said that, in this case,

Â it's blatantly obvious that something's gone wrong.

Â But very often in

Â mathematics, mathematicians, professional mathematicians

Â make, make mistakes, buried in proofs.

Â And because they're professionals, they can usually write things correctly.

Â We know how, we learn this part of becoming mathematicians.

Â We learn how to write things out, we give reasons.

Â So in fact, this, although it looks absurd

Â here you often find that because mathematicians know how

Â to lay things out well, results get published

Â even though they're absolutely wrong, whenever one gets published.

Â Basically what that means is that the

Â referee of the journal where it's published,

Â has graded it and said this is, this is perf this is actually correct.

Â 15:32

So, you know, this is not an unrealistic situation in principal.

Â When false results get proved.

Â When false results get proved, when false results

Â get published, that is, because they've never been proved.

Â The false results get published when, what's going on then is that the referee

Â has gone through it and and think, and thought, everything's okay.

Â Okay, here it's dramatic because the answer's absurd.

Â And these peyton, they force.

Â But this is actually a not unrealistic scenario.

Â And then so, giving a high grade is, isn't, you know, it's not about.

Â I mean it's, it's, it's, I'm doing this to emphasize

Â the distinction between the various things we're looking at here.

Â And this one I think makes it clear.

Â That there are other issues involved other than, other than logical correctness.

Â But as I said, if this was a regular mathematics

Â course, I wouldn't be giving 16 out of out of 24.

Â In fact, I'm not even sure I'd give any

Â marks for this, if a student came up for this.

Â Because it really this is really a big mistake.

Â Okay?

Â But within this context, within the context of this course this

Â is the kind of thing we're looking for in

Â looking at mathematical thinking and mathematical proofs and communication.

Â Okay.

Â There we go!

Â Well that's the end of the, the problem set questions.

Â But let me leave you with one more tantalizing little puzzle.

Â We are probably familiar with the story of Archimedes, lived in Greece about 250 B.C

Â [UNKNOWN]

Â who was asked by the king to determine whether a crown

Â had been given was actually made of pure gold or not and

Â that involved calculating the density and he knew how to calculate to

Â find out the, the, the, the massively, the weights of the crown.

Â But the question is how do you calculate its volume?

Â now, now Archimedes knew lots of mathematics for calculus

Â in volumes, indeed he'd invented a lot of that mathematics.

Â He was able to calculate areas of circle

Â and volumes of spheres and various other shapes like

Â boxes and rectangles and pyramids and so forth.

Â He knew all of that stuff.

Â So he had a lot of mathematics at his disposal that he could have applied.

Â But it didn't seem to work for something

Â irregular, like a crown, or at least not easily.

Â But then one day when he's taking a bath, this is sort of

Â the story goes, when he is taking a bath, he has this amazing insight.

Â He says to himself, if I immerse the crown in water, it will displace some water.

Â In fact, the amount of water it will displace

Â is exactly equal to the volume of the crown.

Â So if I collect the water when it spills out of the bath,

Â and if I, when I put a crown inside it, into, into the

Â water, then I'll be able to just measure the volume of the water

Â in, in a standard way and I'll know the volume of the crown.

Â And as that story goes, he was so

Â impressed and tickled about his solution that he

Â jumped out of the bath and ran stark

Â naked through the streets of Athens crying out eureka,

Â eureka! Which is Greek for I found it!

Â I found it! now, I've known lots of mathematicians.

Â I certainly haven't known Archimedes, but I doubt if even a mathematician deep

Â in the throes of solving a problem would run naked through the streets.

Â However, I can imagine him being extremely pleased with himself

Â and having an adrenaline rush when he had that insight.

Â Because that's a great example of thinking outside the box.

Â He knew lots of techniques for calculating

Â volumes, he invented many of them, but on this occasion he thought

Â outside the box and found a really elegant solution that was different.

Â And the puzzle I'm going to give you is very much along those lines and it

Â actually is about taking a bath. And here it is if

Â it takes half an hour for the cold water faucet to fill your bathtub and an hour

Â for the hot water faucet to fill it, how long will

Â it take to fill the tub if you run both faucets together?

Â Now this looks like one of those, those frustrating little

Â word problems you get in, in, in, in high school.

Â Okay, where you, you end up, you sort of say that

Â the, that the rate of flow of the cold water be f.

Â And, and then the t.

Â And you, you, you write down some equations

Â and you, you you figure something out, okay.

Â I'm sure you know

Â how to do that.

Â You can apply standard technique for doing this kind of

Â thing involving rates of change, and you'll get the answer.

Â But you don't need to know any of that, you don't need

Â to do any calculations at all in order to solve this one.

Â If you think outside the box you can answer it without doing any

Â of those calculations based on the rates of flow or anything like that.

Â You simply have to think of the problem a different way.

Â