3:49

I've made the observation that they're both one more than a multiple of eight,

and then I sign off by stating that this is, this in fact proves the theorem.

So I've got a beginning, a middle, and an end, so it succeeds communicatively.

I've given reasons for each things. Notice I didn't just square these things.

I didn't simply square them, I said I was squaring them, so that the reader's not

left wondering what I'm doing. Now, to red it might be a little bit

[INAUDIBLE] why I started in this way and that's basically experienced

mathematicians can fairly regularly moderately come up with this kind of

thing. If you're a beginner it can take you

longer. Okay, and then when I've dome that

calculation I make a specific conclusion, and this is a conclusion.

A conclusion to the reasoning, not to the whole proof, because the conclusion of

the whole proof is the observation that this proves the theorem.

This is the drum roll, this is where the cymbals crash, this is where we take a

bow. We've done it, we're out of there.

Okay, so the answer is yeah, it's valid. And we just checked that it's valid by

going through all of these arguments, okay?

Let's move on to number three. Well, this proof is in fact valid, okay?

Let's just make sure that it works on all levels.

we're arguing by contradiction, okay? So, this is the beginning and this is the

beginning step, suppose the conclusion's false.

Then there'll be a natural number n for which a is not true, so there'll be a

least one. Now about the first condition, in our

case, at A one, we know that the m can't be equal to one, because A does at one,

so m is bigger than one. So m equals n plus one for some n.

Since n is less than m, we know that n holds because n is at least one of which

it doesn't. Then by the second condition, A n plus

one holds, i.e., A m. That's a contradiction, and that proves

the result. the only thing to notice is that here the

n has been used in, in a quantified form, so this is a variable that's been

quantified, right? Okay, in this case, what we're saying is

if the, if the, if it, if this fails, then there'll be one of these n's for

which it fails. And at this stage, the n is sort of a

variable. But at these points, when we put the m

in, m is a specific number. We don't know what its values is.

In fact, we're going to show that no such m exists.

But on the basis of a false assumption, there will be a, a place where it fails.

And m is going to be a specific number, so the m here is specific within the

proof. Okay, when I get down to here, the n is

also specific within the proof. Because it's equal to m minus one.

So in these cases, once I'm down here, I've got a specific m and a specific n.

Up here, it's just a general n, okay? So everything here is specific.

7:19

Now, in fact, there never was such an m, and hence there never was such an n.

Because the conclusion as a result is valid.

But within the text of this argument, these guys are specific.

Here, there's a, there's a, this is a variable that's quantified, alright?

In any case this is proof this is true. On number four, we have to use the course

rubric again. first of all, let's see what, what the

theorem says. It says that if we take the Fibonacci

numbers. And we square them and we take the first

n of them and square them another. Then the result is equal to the nth

Fibbionaci number, the last one in this sequence here, multiplied by the one

after that. now this is actually, this is true, by

the way. This is true, it's a valid theorem.

It's one of many identities about Fibonacci numbers that show that they're

connected in In what at first, a very surprising ways.

they're very typical in, that they're almost always proved by induction.

And when you look at the induction proof, you realize that it's really that the

identity, which at first, seems surprising.

Is actually just a disguised version of the definition of the Fibonacci numbers.

What's behind the, all of these interesting identities.

Is the fact that the the n plus second Fibonacci number is the sum of the nth

Fibonacci number and the n plus first Fibonacci number.

So they all go back to the fact that it's defined in this iteratively additive way,

okay? well, let's just see if this one, see,

see how this one goes, okay? first of all, we'll look at logical

correctness. worrying about things like reasons and so

forth later. Let's check that the, the first case is

true. F1 is equal to one, so F1 squared, and in

the case, n equals one, there is no sum it's just F1 squared.

So the left hand side is just one, on the right hand side is F1, which is one.

And of course the second Fibonacci number is also one.

Because the Fibonacci sequence begins with a pair of ones.

So we have two ones on the right. So, this is logically correct.

then there's the, the induction step. Let's just check the algebra here.

here it's just the taking taking a sum up to n plus one, and pulling out the second

one, the last one, the last term in the series.

So we've got the sum of n plus 1 is the sum of the first n together with the last

one. we'll look at the issue of, of reasons in

a minute. this will be the induction hypothesis,

and nicely stated as, as a reason. that this sum equals Fn plus one, this is

just Fn plus one squared I carried through.

take out Fn plus one as a common factor okay.

let's see, this is definition of Fn plus two, is the sum of that one plus that

one. And then we've, we've got the other

identity n plus 1. Okay, so, so all of the logical steps are

correct. This is a valid induction proof.

So I'm going to give four marks for that. is it clear?

Yes, I think this is clear. again, there's going to be some issues of

reason as to explain things, but everything is clear.

It's well laid out, it's easy to follow steps, even when I had to figure out what

the author meant. I'm going to, we'll talk in a minute as

to whether, whether I should have had to. I, it, it was, it was easy to follow in

that sense. So it, so it was clear.

there's an opening, it's well opened. It's a proof by induction, there's a

standard method. And, and they, they, the good way to

start. The correct, well, almost to start to

prove using a standard method is to state what the method is.

So I want to get full marks for that. conclusion when the conclusion is stated

I'll, I'll look at whether it's properly stated in a minute.

But it certainly it, it, stated when the proof is complete And it's been laid out

that it's going to be an induction proof. So, I think we're going to get four for

that reasons, a couple of quibbles. I think the order should have said,

separate out fine term. this is good, stating the, the use of the

induction hypothesis and an induction proof is, is always, is critical.

I, I, I think that's this is such an important step.

by algebra, yeah, you could say something like check out a common factor but the,

this is, this level of mathematics. When we're doing proofs in number theory

as I mentioned with the last problem set. we, we can assume that people can, can

spot things like taking out common factors.

this I think is important. and this, this this is critical, the fact

that we're using the, the definition of the, of the Fibonacci numbers.

the, these kinds of identities, as I mentioned a moment ago, these kind of

identities actually only hold because of the way the Fibonacci sequence's identity

is, is defined. which establishes the identity for n plus

one. That's good that's a local conclusion,

which is good. the proof is complete, here the person

should have said, by induction, or by the principle of induction.

let's put out in four principle of induction, because there is a powerful

fact about what the, the natural numbers that's been used here.

Okay, so, what I'm going to do for reasons, I'm going to give two, I think.

Because I can't give four. you know, I, if, if this one arguably I,

I, I would tend to lean on having this in especially since it's an opening step in

the proof. But if that was the only thing that was

missing in this context, I might not have even deducted any marks at all.

this one however, I, I have to deduct at least one, and I think, I think I've

really deducted, I think really I've deducted the two for this one.

It was, it was a bit of a judgement call. you know, I have to allow for the fact

that the author of a proof maybe made a slightly different judgement call.

you, you have to try, judge how well you think a persons putting down a proof.

and, and you can't really say, I always do it this way, therefore you always

should do it this way. Because people have different, they come

on different sides of these issues. we're really looking to grade this as an

overall thing. You know, one of the problems with using

a rubric is we're trying to take something that's holistic.

And is basically an overall judgement call and reduce it to number of

parameters. This is not how professionals go ,they

look overall and say ,this is a good proof and then assign a number ,but now

space for years of expertise. Pulling us apart this way is a good for

beginners because it allows you to focus on one only individual things, but a

professional looks at all these things in, in one.

And you have to sort of balance things out.

and that's why we're going to all these videos, to try and give you some

indication of how a professional, and in this case the professional is me.

How we go about it and how we, tacitly, and when we're doing this, this part of

our every day work, when we're grading work.

whether, whether we're grading student's work or we're evaluating proofs of other

mathematicians. This is all part of the tacit process of

grading. and, and in writing out a proof, I've

just, tried to isolate the things that I implicitly and automatically look at in

grading proofs. as indeed does, does any professor,

professional mathematician when they're grading a proof, okay?

and so, and what I'm really saying is, this is.

Not really perfect in terms of giving reasons.

arguably this one is more important than that one, for example.

You know, it's overall, however, I'm going to get four.

because these are sort of niggling. and the reason they're niggling is the

person has laid out the fact that this induction, has made it clear that they're

assuming it for n, have proved it for n plus 1.

they've definitely stated the use of the induction hypothesis.

as I said, this one would have been nice, I would have liked to see it, but this

author. Presumably decided it really was, was,

was patently obvious what's going on. this I think is an important one, because

that's critical to the proof. this is not something that's typical.

The farther you can pull this in depends on the, where the Fibonacci sequence is

defined, and this is the only part in the proof where we make use of that fact.

So you really should Should mention this one.

So this ones important, and this one's important.

And the reason is until we've got to the last line, all we've really done is we've

shown that the thing is true for an equals one, two.

Actually, further equals one, use the further and F2 was what it was.

So if there were few, they were put through for the first case, or observed

in the first case. And then we've shown that if it holds

stage n, it follows, and it holds stage n plus one.

So we've proved two simple facts. One, the fact about the first one, well,

it's the first two for that matter. And secondly, we've proved an implication

from n to n plus one. The conclusion is that this holds for

infinitely many number, for all of the actual numbers.

So somehow proving, two statements, one simple observation and one implication,

has proved that something is true for the infinitude of all natural numbers.

Now admittedly, induction has a natural it's always an obvious thing, this sort

of a, self evident truth to the principle of induction, you know.

You can think of it in terms of, of knocking rows of dominoes over or

something. So But the fact that the, that the, that

the principal of introduction, or the part, the method of induction has got a

sort of intuitive obviousness to it. Shouldn't obscure the fact that this is

actually a deep result. Making a conclusion about an infinite set

is non trivial, you know? The, the Hilbert Hotel tells us that

infinity's a very, paradoxical domain. We've got to be very careful.

So, this actually holds. We can make the conclusion that it's true

for all n. Because of the principle of mathematical

induction. This, in other words, this is a big deal.

This is a big, big deal. And when big deals are involved, you

should mention them. You know, if there's a big guy in the

room. it's polite, if not, [LAUGH] a matter of

self preservation to observe that fact and make it clear.

So you really do need to state the principle of induction here to state that

it's been used, or at least to say, by induction.

I, I, you know, if I was feeling, if it wasn't for the fact that the rest of the

proof was laid out so nicely. I might well have just deducted more

here. But as it is given everything else was

laid out so well and given that the proof is nice and elegant.

And this is a slick proof, there's almost no superfluous lines in.

And I think overall simply deducting two marks is about right.

So I've got 22 out of 24 for this one. and I feel reasonably good about this.

you know, the, couple of a small points, [COUGH] well, it's one small point, one

moderate big points, one really huge point, I think.

I think, this is being generous. But I think this is proof deserves

generosity. overall, I'm happy with that.

And, and this is really how it, it splits up.

Okay, let's go ahead and look at number five.

And number five is another of these Fibonacci sequence results.

This one says that if we take the first n Fibonacci numbers, add them together, the

result is the the next but one. Fibonacci number, we skip over Fn plus

one, we go to Fn plus two. Okay, so, okay, so as is typical for

these results it is proved by induction. So see how the proof goes.

For n equals one, the left-hand side is F1, that's why you get if n equals one,

there's no sum, it's just F1 itself. And the right-hand side is uh-oh, oh

dear. If n equals one, the right-hand side

would be F3 and F3 equals two, and one does not equal two.

So this isn't even true, it's not valid for n equals one, which means the

theorem's not valid. Oh, good grief, this is such an obvious

mistake. It's the kind of mistake that anyone

could make. Doesn't really reflect on their ability

as a mathematician. It's just a human error.

So common this kind of thing, and yet this is mathematics.

Ultimately in mathematics things are right or wrong.

I mean, you know, if, if this mathematics being used by an engineer to build a

bridge, and the bridge falls down and people get killed.

You know, that, that engineer could be held liable.

i am just defining or so you know at the end of the day we call that thing goes to

false ,i am have to 0 for logical correctness before, we are go further ,i

mean it is just a plan to evolve false result you could make the result correct

,you could make first result correct by subtracting one And it fact it turns out

as, and I'll come back to this, that if you put a minus 1 in here, then the

identity is true for all n. So there is a theorem here, and the get,

we get at the theorem by noticing what went wrong with this proof.

Incidentally, this is very typical in mathematics.

often in mathematics, the statement of the theorem when it's proved isn't the

one that the author originally tried to, to do.

Very often in mathematics we, we make a conjecture.

We try to prove it. the proof has gone wrong and so by

analyzing the proof we've thought of go back and change the statement.

So it's often the case that statements of theorems actually come after the proof.

21:57

Not many proofs in mathematics began as proofs of something else that failed, but

then the, the statement has changed to what's been proved.

So it's not always the case that the mathematicians sort of formulated

theorems and then proved them. They often formulate a theorem, develop a

proof, find out the proof is wrong, go back and restate the theorem so the proof

works for that restatement. Okay, that's just the way mathematics

advances, it's, it's part of the process of, of getting you knowledge.

Okay, [COUGH], well we're going to have to come back and, and, and sort of look

at how the thing works as a proof but, but it, it isn't a proof as it stands.

Okay, well let's go through the mechanics of it and see if, if all of the other

steps are okay. So, assume the identity holds for n, then

well what's going on here? This is a case of separate, I'll, I'll,

I'll give grades for, for these ones later, but let's just mention that what's

going on here is separate out the last term.

Okay, let's see, that's the reason here. Sum up to n plus one is the sum up to n

together with the final term. this is induction hypothesis.

That guy, equals that. Incidentally and I'm jumping ahead to

when I corrected this, at this point there should be a negative one in here,

and if we do that it carries through. So this, this is the proof as we're going

through it will work validly. Providing that we stare at the theorem

correctly. Okay, so I'm not going to give double

jeopardy deduction for this. We've already knocked a ton off for that.

okay, so the rest of the thing is, is actually, the logic is correct, in of

itself. Expect for this glaring mistake at step

one. But that, that topples the whole

interphase. So I'm going to give four for clarity

because it's absolutely clear, well laid out.

it's proof by induction, it states it, stated the method at the beginning.

So four for the opening, the conclusion is certainly stated, I'm going to give

four marks for that. What's the problem I have with this is

that, it's not mentioned that this is an induction proof.

And as I elaborated on a great length with the previous question, was question

four. the fact that something is, you know the

fact of an infinite three/g, and infinite three statement.

The statement about something which we've all end.

The fact that that follows from a couple of little facts like an observation and a

simple implication that's a big deal. and that takes us into the realm of the

infinite. And what takes us into the realm of the

infinite is the fact that we have this thing called The Principle of

Mathematical Induction. Which is difficult to prove if you try to

prove it, or you have to assume it is, it is an axiom or some kind of principle.

So this is not a trivial thing. It's intuitively clear, I know.

But it's not it's, by no means easy to prove it.

So you know, their, we're pulling on something powerful here.

We should state that we're pulling on something powerful.

Okay, no reasons, I mean I was addressing those as I went through.

this ones missing, that's in, that's good.

You know, the, the, the, the fact that the, the definition of the, I mean it,

thi, this res-, this result holds, I mean, when I modify it to make it true.

It holds by virtue of the way that Fibonacci sequence is defined.

So you should stipulate the fact that you're using that.

this is good, using the induction hypothesis.

I've got something good here, I've got something good here.

that's a bit of a problem. That should've been in.

this one, I, you know, again, this is this is a judgement call, I would like to

see that there, but that's just me. All of the things being equal, I would of

ignored that, but all of the things I'm seeing, of course, this person's already

missed this out. So, got a missing reason here, missing

reason here, but a couple of good ones here.

so I'm going to give two, okay. I think it's about right, overall

valuation? The thing is false, I mean, I can't

possibly give four overall for a theorem that's plain false, you know.

Much as I, I'm sympathetic for the fact that it was a simple slip, the most I can

give is two. And I think I'm being generous there,

quite frankly because it is a false result.

On the other hand the, the, the course is focusing on mathematical thinking and

mathematical communication, the ability to formulate and present a proof, and

there's lots of aspects to that. It's not just about whether things are

right or wrong. You know high school mathematics is, is

largely focused on, K through 12, mathematics is largely focused on things

being right or wrong. we've, we've moved beyond that now.

Right and wrong is still an important factor which is why I give zero to this

part. But there's other things we're looking

at. So I will be giving 16 for this one,

okay? Well, that's said eight max lost out of

24, so this pair lost a third of the max for this.

So one could say that this is generous for false proof for false result.

On the other hand there's a lot of good stuff here.

It was simply a silly mistake right here at the beginning.

Okay, that was very unfortunate. Okay, life's like that at times, let's go

on and look at number six. Well, this is another one about the

Fibonacci sequence. what does it say?

It says that the n Fibonacci number is at least equal to the number three over two,

to the power n minus two. Okay, let's see how this person does

this. we have, oh, okay.

Interesting way to start, making a statement about F1.

which is equal to one, true, I mean, I, I wouldn't look at logical correctness.

I'll just follow it through, and see what this person's doing.

and two over three is equal to three over two to the negative one.

Okay, well I guess that's showing that it's true for F equals F1.

so maybe this is an induction proof, although there's no way of knowing just

by looking at it, right? well this is interesting.

The person now goes on and, you know, you usual with an induction proof you prove

it for n equals 1 and then you stop. This person now proving it for F 2.

well, let's see what they're doing. So they're saying F2 equals one, which is

does. And one is indeed three over two to the

power zero, which is true, because anything to the power zero equals one.

So the inequality is valid for n equals one, two, alright, absolutely correct.

Now assume the inequality holds for n, where n greater or equal to two, so, we

assume, but we're not told, that this is going to be an induction proof.

This person just jumped in, so this is, this is already not a, not a proof.

This isn't telling a story. It's, it's, it's, it's presenting us with

a, with a who done it or a what done it. Or what are they doing?

so this is going to be a mystery, where n's greater than or equal to two.

Okay, then, let's see. Fn plus one equals Fn plus Fn minus one.

True, that's a definition of the Fibonacci sequence.

Although there's no explanation of that fact.

that's greater than or equal to, well, ha.

Where's this coming from? well looking ahead to the fact that this

is the, the person's is almost certainly doing an induction proof, but hasn't

written it, written it down. What should of been said here, was for

wholes the in, inequality wholes up to and including it that, I think, is what's

is what's meant here. Because we're assuming, we're using it

for two cases here. We're using it for the two previous

cases, and we're saying that is greater than or equal to that one, that's greater

than or equal to that one. and then then we're taking out a common

factor. this is interesting, there's almost no

reasons given for anything else. And certainly the person makes the

obvious statement that this is bialgebra. Which indeed it is, taking out 3 over 2

to the nc is a common factor. And one of them now is a digit of one

there. then 3 over 2 plus one equals 5 over 2.

Now we're spelling everything out in goal we have mathematical detail.

5 over 2 is 10 over 4, why are you doing that?

Well, because because then you can put a 9 over 4 here and make it smaller.

And then nine over four is just 3 over 2 squared.

Which gets you everything back to 3 over 2.

Which establishes the inequality for n plus one.

Okay well, this person can certainly manipulate fractions and also really

impressive. Because you know a large percentage of

the world's population has trouble dealing with fractions and inverting

fractions and things. So, so this person has a lot of

procedural skill with fractions, but is it a proof?

Heavens no, I mean this, this so much missing here.

I'm going to give four for logical correctness, because the manipulations

and the logic and everything was, was, was fine.

as a professional mathematician, able to figure out what's going on.

I could recon I am, and I could recognize fairly early, fairly early on that this

was an induction proof. But no thanks to the person writing it

down, and just because I've got a lot of experience.

never the less I'm going to give four for the, for the logical correctness.

I'm just going to get four for the clarity, because it was sufficiently well

lit out. That one side realized that there's an

induction proof going on here. I was able to follow the steps.

so I didn't have to solve, bury myself in a, in a marage of details.

To find out what was going on it was clear.

Opening, well there wasn't one, there absolutely wasn't one.

This person just straight in, and did so very obscurely, because even if looking

at the first case indicates that we're doing induction.

Why look at the second case when we almost never do that in induction.

And then we had to make a, a modification to, to even make sense of what was going

on. So, no opening, what about the

conclusion? Well no, there isn't a conclusion.

what the person presumably should have said, or maybe meant to but didn't.

Hence, by induction, we could simply say the theorem is proved, that's fine.

That's absolutely fine. It bears fruit, this is alright.

but it's missing. There's no conclusion stated, and

there'll be a zero here. Reasons, there aren't any.

There aren't any reasons. simply no reasons.

I mean, this one doesn't count. I'm not going to give credit for that.

That's just a, here, we should have said something like well, by, by the

induction, well actually, there's two things.

By the definition, of the Fibonacci sequence is the first equality, okay?

And the second one, is the induction hypothesis.

33:49

This is by the definition of the Fibonacci sequence.

you know, it should be said. I mean, this is so self evident.

and either way it's written, because this is literally the definition.

You know, you could maybe, let this person have have some, some leeway, if

everything else was good. But nothing else is good, so I'm not

going to, I'm, I'm not inclined to give credit generously.

When the so lack of, there's so much lack of reasons give.

This is the induction hypothesis applied twice, okay?

Once for that one and once for that one, and then the rest is algebra.

Okay overall, the only you know I do, you know, if, if I simply leave it at that

And say that this is just terrible. Because it is terrible as a proof.

The person gets eight out 24, and yet, this is some, this is quite sophisticated

by most people's standards. I'm going to give two for the overall.

And I'm going to give some sort of compensatory credits in here.

For the fact that this person's done some really intricate, manipulations.

there's some intricacy in here, this is not trivial but it still just gives 10

out of 24, I mean this is a, a low mark. Now in terms of a grade for mathematical

dexterity that would be low, this person clearly has considerable mathematical

dexterity. But proving things and communicating

things in mathematics is much more than dexterity.

it's about telling a story, it's giving reasons.

It's establishing why something is true, you know?

I'm prepared to believe that this person convinced themselves that this is true,

and they understood why it was true. But what they haven't done is express

that fact properly. this doesn't even come close to being a

proof, as a proof, this one sucks. I mean, this, this is not a proof at all,

okay? no mention of reasons.

we've still got this mystery of why did the person.

Why did this person do the first two cases?

Well that was actually an obser, a good observation.

mostly with induction proofs, you have to prove.

Just, you, you still have to prove the first case.

But this one, you have to prove two of them.

Because if you look at the induction step, it uses it twice.

You have to use it as the two previous cases.

So in this case, the induction step depends on having it twice.

You've got Fn greater than or equal to that.

And you've got Fn minus one greater than or equal to that.

So we've got two instances of the induction hy-, the induction hypothesis

here. Well, it would, the induction hypothesis

has to be pulled in twice. We use two inequalities.

So you have to prove the first two cases. And you also have to, when you, when

you're making the assumption. You're not just assuming it for n you're

assuming it n and n minus one. Okay, you're assuming it for n and n

minus one. and, and the simple way to say that is

you assuming it up to and including n. So there's a lot in terms of missing

reasons here. the fact that the proof, the fact that

this person does what he or she does Indicates to me that they almost

certainly understand these issues. You know, simply observing you have to do

it for two cases, is significant. So I've got a lot of sympathy with this

person. They've obviously thought deeply.

I'm convinced, actually, that they know what they're doing.

And they know why they're doing it, otherwise, they wouldn't have done this.