Goedel, Escher, Bach - Lecture 6

In the first six minutes or so Curry gives a fairly good summary of Goedel's Theorem. Unfortunately this is the summary of the previous lecture which was not recorded. It seems nothing is free, not even free video lectures because it turns out the best ( not implying the rest is good ) is missing.

After rushing through formal stuff he wastes five minutes about a three-layer stupid joke about a book he had not read.

I quick-scanned through the rest of the video. Not worth watching, really. Too bad. I looked forward to this.

The take home message of the course. All provable things are true but not necessarily al true things are provable.
Justin Curry

Now that I am mostly through all M381 stuff I am glad it included mathematical logic. I would -not- have done it as a stand-alone course. Logic is hard in the beginning, like most new subjects. It needs time to work on you. I will get back to this in the next M381 post.

Previous posts on the series:
- Lecture 1
- Lecture 2
- Lecture 3
- Lecture 4
- Lecture 5

About M381 (1)

Regular readers of this blog know that I have a sort-of rage-button like the Hulk: it is called MathCad. Thank God, I have the anti-dote almost always open and ready: Mathematica. I am not going to repeat why MathCad is a danger to your mental health, but I have to press the MathCad button at least once.

In almost all mathematics courses you can do at the Open University there is software involved. They either deliver a standard package, or ship custom software especially developed for the course ( i.e. MT365 ). The house-package of the Mathematics Department of the Open University is MathCad, version 2001. I have argued that MathCad alone is a reason -not- to choose for the Open University. What a disgrace...

( Calming down. )

They do however recognize that software, computers, tools are relevant in mathematics. Especially in Number Theory computers are used in active research. Another area where they use software in active research is: mathematical logic. Stronger: research in Number Theory is impossible without computers.

These facts are not even mentioned in M381. There are many open source tools available for Number Theory, even more for Mathematical Logic. Not a word about it in M381. One, if not -the- reason is the fact that course development in the Open University is done in a project organization. A project is created with the objective to create course X which will then be used for the next 10 or so years. It is exactly the opposite of what one would expect of a university education. It is not reasonable to expect the Open University to be at the forefront of mathematical research. Simply because other universities in the UK have that role. But it is reasonable to expect more than a static expose of 19th century Gauss number theory and early 20th century logic from Church, Turing and Goedel. In fact, the field is presented as abstract and of theoretical importance only. But Number Theory and Mathematical Logic are extremely relevant and applicable in many industries! But I did not learn that from the course and that is sad.

It took me a lot of work but I found some relevant learning tools in the fields of number theory and mathematical logic. More about those later in this blog.

(*) - I may have misunderstood the concept of 'University' in the UK. I think many universities in the UK are what we call in the Netherlands 'schools'. They deliver professionals with a degree in all fields through excellent education but they don't do research and so on. They don't add to the body of knowledge. They process and transfer knowledge. That description fits the Open University as well. - A marketing issue is that students like to have a 'university' education. And marketing people love empty heads boxes, they have a fancy word for it too: the 'packaging'. Does that make sense?

Goedel, Escher, Bach - Lecture 5

A few months ago I started to watch the MIT video lecture series on Goedel, Escher, Bach. Due to time constraints I wasn't able to complete watching the entire series. Today I continued with watching lecture 5. I have learned quite a lot on the subject through M381 and I am about to really 'get it' as far as the Goedel Incompleteness Theorems are concerned. My first reading of GEB took months and now parts of the book begin to look simple. If you don't know what I mean browse through a mathematics book you thought was hard, a few years ago. It often seems if there is 'nothing in the book'. The odd thing with Goedel ( and with all mathematics, I suppose ) is that in your mind you think you can explain it to a laymen in one or two sentences. ( It is -that- simple, I am afraid. ) The power of mathematics is that it can capture an entire knowledge tree in a single word. That word remains meaningless without understanding of all the words in the knowledge tree.

A bit about lecture 5.

Dress shows arrogance

I wonder if Justin Curry would go to a job interview in that Club Med outfit. Students are paying customers ( and a pool of cheap labor for lucrative research deals the university makes ) deserving respect from teaching staff.

Formal number theory ( as in M381 )

Justin talks about Typographic Number Theory, ( formal number theory in M381 ). For example $$\forall x ( \neg x = \mathbf{0} ( \exists y x = y') )$$ can be interpreted as

"Every x that is not equal to 0 is the successor of some y."

Leibniz was the first to propose a formal language for number theory. He asked whether it was true that an algorithm could decide if a statement in number theory was true. - Although in M381 this question is answered negatively that does not mean computers can not play a role in proving mathematical propositions. There is an abundance of ( open source ) software for proving theorems.

( Not in video: ) Isabelle a formal proof theory assistant has been used in testing an operating system kernel written in C and assembler. It not only verified that the spec was implemented correctly but it also discovered hundreds (...) of programming and design (...) errors which were not found by traditional testing methods.

Previous posts on the series:
- Lecture 1
- Lecture 2
- Lecture 3
- Lecture 4

Neutrino's supposedly faster than light.

Off-topic:

Neutrinos Travel Faster Than Light, According to One Experiment

I got really excited when I read that "... particles had been found that travel faster than light ...". ( local teletext ). Wouldn't that be amazing? I mean, all the major scientific discoveries seem to have been made in the distant past. ( Physics and mathematics, I mean ). That would change overnight if Einstein's law of relativity got broken.

My guess? A measurement error. They are talking about a particle that traveled 60 nanoseconds faster than expected. One nanosecond is 1/10^9 part of a second. I am sure they have found some ingenuous way to measure speeds involving figures like that, but NOT when the figures depend on GPS! Come on, GPS.

I am afraid that this could turn into something very, very embarrassing for CERN. I hope their P/R people are as brilliant as the scientists they employ. ( Except the guy who was -?- responsible for measuring the speed of neutrino's, of course. )

Exercise ( algebra )

Given that $$x^n-y^n = (x-y) (\sum_{k=1}^n x^{n-k}y^{k-1} )$$ with for example: $$x^4-y^4 = (x-y)(x^3 +x^2y +xy^2 + y^3)$$.
a) How would you factorize $x^5 + y^5$?
b) Generalize.
c) Prove the identity above for $x^n-y^n$ using mathematical induction.

Exercise ( logic ).

Knowledge of mathematics is not required to solve the following exercise, but it will sure help ;-)

Dr. Who asked you for a ride in the Tardis. Naturally, you couldn't decline, it might be a matter of national, if not global, importance. The Tardis landed on the Planet of Truth which is is inhabited by people who always tell the truth. A minority however decided to lie, always. It is Doctor Who's mission to seek and destroy all liars. First you must get to the Capital of the Planet. Two roads fork out. Should you go left, or right? An inhabitant approaches, greets you and gives you the privilege of asking him one Yes/No question.

Ask him where the capital is, left or right on the fork, but beware he might be a liar!

Credit follows with the answer.

Take the challenge, test your ability to think logically.

Study Tip - 3 ( mathematics does not have to be difficult )

Being a programmer who studies math (, a math studying programmer, whatever ) gives me access to the literature of the foundations of computer science. Now and then I browse the libraries. I thought to have a found a book that fits my level, so I read the preface to check if I was right. The author Alan Parkes made the following interesting statement.

For Students : I wrote this book partly because when I studied this material as part of my own Computing degree, I had to work really hard to understand the material, a situation which arose not because the material is too difficult, but because it was not well presented ...

from Preface of A Concise Introduction to Languages and Machines by Alan P. Parkes, Springer 2008.

I have seen video of a lecture on MSRI where the researcher / professor held 'a talk' ( there is a subtle difference between 'a talk', a presentation and a lecture, 'a talk' is a euphemism for a not well prepared presentation ) about a subject from scribbled notes. Most of the lecture he stood with his back to the audience mumbling while scribbling math. End of lecture. Goodbye. It was expected of the audience to write up notes from his mumblings and scribbles. - Folk like that, probably based on a brilliant thought they had decades ago, get the opportunity to write books too. Once in the hands of ( then ) students like Alan Parkes they make the learning experience troublesome to say the least.

Well written, illustrated mathematics followed by examples, more examples, examples of exercises, exercises with written solutions and so forth make studying mathematics easy. ( Think of an Open University booklet, a good one ). Open University booklets are written by course teams for students and not by professors who write to impress... who actually?

Tip 4: Change books

If you have difficulty with a ( mathematics ) subject then find another book about the subject. Check how the other author explains the topic. Try a third, fourth if necessary. This method is guaranteed to work. When you are close to revision for an exam it stimulates to read through the material written by someone else. Different exercises, and so on.

Finally: authors who write for students are read, respected and remembered.

Off-topic: 9/11 Revisited (4-4)

What a totally insane day this is. Everybody knows that the official 9/11 story is a lie and the mainstream media are repeating that same lie. The US government is criminal, period.

People that have been very important to 9-11 truth are:
- Alex Jones,
- Dr. Stephen Jones,
- David Ray Griffin,
- Prof. James H. Fetzer,
- Dr. Judy Wood
- and many others
I deeply respect their courage to come forward and tell the truth.

I say no more.

Visit
- Scholars for 9/11 Truth
- Architects and Engineers for 9/11 Truth


Previous posts:
- Off-topic: 9/11 Revisited (1)
- Off-topic: 9/11 Revisited (2)
- Off-topic: 9/11 Revisited (3)

Reading fiction about mathematics

This is a lighthearted, enjoyable novel about a bumbling but likable mathematics professor at an unnamed university who believes that he has discovered a solution to a famous mathematical puzzle known as "Beauregard's Wild Number Problem."

A novel set in the world of contemporary mathematics: The Wild Numbers by Philibert Schogt. Doesn't that sound great? I mean actually having the time to leisurely read novels, stroll through the city, drink coffee somewhere. Haven't read it yet, but I will, someday, I must really. :-)

Is M381 True or False ?

In facebook I have a photo album with mathematics related pictures. Sometime ago I added the picture below. I thought it was a joke...

... but I could not have been further from the truth. The picture could have been taken at any Open University M381 tutorial. This student is not clueless, on the contrary: he contemplates about some deep mathematics.

Anyway, if you have always wanted to give an answer to the layman's question: "But can you -prove- that 1+1 = 2?" then M381 is your course.

Prime?

A prime p is a number with two positive divisors: 1 and p. Note how this definition nicely excludes 1 which has only one positive divisor. Then the primes are: $$2,3,5,7,11,13,17,19,23,29, \cdots$$
Wait! What about:
$$ 5 = -i \cdot (1 + 2i) \cdot (2 + i)$$
This is an example of a factorization in the quadratic field of Gaussian Integers. It is therefore not enough to say that a number is prime. Primality is relative in relation to the number field.

The mathematician Lamé thought to have cracked Fermat's Last Theorem in 1847. Needless to say that his proof contained an error. He overlooked the fact that prime factorization was not unique in a number system he used in his proof.

In Mathematica factorization is done with:

FactorInteger[n],
factorization using Gaussian integers is done with:

FactorInteger[5, GaussianIntegers -> True].

Off-topic: 9/11 Revisited (3)

What have -you- seen on 9/11? Are you sure that what you saw on TV really happened?

Ten years, what does that mean? A first year math student of 18 was only 8 in 2001. The story must be told and re-told because 9/11 is an important part of the history of -the world-. - Anyone older than, say 28, probably remembers 9/11 as if it was yesterday. I do, definitely, I can't forget 9/11. Especially in periods like this, with the 10th anniversary and so forth, it keeps going through my mind. Why?! I have asked that question many, many times. Because I have seen things on TV that are scientifically impossible but everybody accepted it as the truth. That brought me in a condition of doubt. Therefore the following video came as a relief to me. Things that I thought to have seen were finally confirmed.

This is the video that confirmed what I thought to have seen. ( Three parts, part 2 and part 3 on YouTube ).



Previous posts:
- Off-topic: 9/11 Revisited (1)
- Off-topic: 9/11 Revisited (2)

Perrin numbers

Let $P(0)=3, P(1)=0, P(2)=2$ and $$P(k) = P(k-2) + P(k-3.)$$ These numbers are called the Perrin numbers. They have the interesting property that $\mod{[P(k), k]} = 0$ in almost all cases when k is prime. ( Otherwise we would have found a true prime generator! ). In any case the property holds until $k=271441.$ Interesting, isn't it? See the table below for the first 40 Perrin numbers.

k	P[k]	Mod[P[k],k]	PrimeQ
2	2	0	True
3	3	0	True
4	2	2	False
5	5	0	True
6	5	5	False
7	7	0	True
8	10	2	False
9	12	3	False
10	17	7	False
11	22	0	True
12	29	5	False
13	39	0	True
14	51	9	False
15	68	8	False
16	90	10	False
17	119	0	True
18	158	14	False
19	209	0	True
20	277	17	False
21	367	10	False
22	486	2	False
23	644	0	True
24	853	13	False
25	1130	5	False
26	1497	15	False
27	1983	12	False
28	2627	23	False
29	3480	0	True
30	4610	20	False
31	6107	0	True
32	8090	26	False
33	10717	25	False
34	14197	19	False
35	18807	12	False
36	24914	2	False
37	33004	0	True
38	43721	21	False
39	57918	3	False
40	76725	5	False

Off-topic: 9/11 Revisited (2)

I am a big fan of the BBC because of its documentaries, Sherlock, Doctor Who, Torchwood and even EastEnders ( although I am 'clean' these days ). For many years I looked at the BBC as -the- independent source of world news, but 9/11 changed that, unfortunately. Reading the following below may shatter your trust in the BBC.

Links:
- The BBC’s Instrument of 9/11 Misinformation ( from: VetaransToday by prof. Jim Fetzer )
- Off-topic: 9/11 Revisited (1)

Fibonacci Sequence - Revisited

Some thoughts about my progress in maths. - I browsed through some old blog posts to get a feeling for how much I progressed with my maths. That was the original purpose anyway. Six years ago I wrote about the Fibonacci sequence as I am nowadays. It often feels if I make no real progress. I see mountains ahead of me, when I look back I see a tiny road. I don't believe in working hard, working smart should do the trick. On the plus side I can say that today I can prove Binet's formula and give at least three alternative formulas for the series. I don't know I even knew the formula in 2006. If progress is thinking about the Fibonacci series as something trivial then that is not what I expected. When I was in secondary school the mathematics teachers seemed to have acquired infinite knowledge about mathematics. I am sure that I know at least as much about math as the average high school teacher. That should give me a sense of satisfaction, achievement. It does not. Link: - Max Cohen draws the golden spiral

Off-topic: 9/11 Revisited (1)

My blog is about mathematics and in particular studying mathematics. I have kept the blog ad-free and I have avoided posts about politics and/or religion ( with one exception, a one-time post about 9/11 ). In little over a week it will be ten years ago since the 9/11 attacks. These attacks defined the major events in the last decade and shaped the world as we know it today. Counting down to the 9/11 memorial I'll make a few posts about 9/11. I hope you don't mind.

The first 9/11 post is a link to VT Veterans Today, the Military & Foreign Affairs Journal. The VT article is called 9/11: Video of Missile Hitting Pentagon Leaked.

It concerns the following video.

Exercise

For $x,y \in \mathbf{N}$ the equation: $$x^2 - 11 y^2 = 1.$$ has infinite many solutions. Find three solutions other than $(10,3)$.

Hint: Use continued fractions.