M381 mathematical logic - or Escher, Goedel, Bach - Revisited

Although I believe the academic world is at least as closed as it always has been, the access to ( mathematical knowledge ) is ( almost ) a level playing field for those in and those outside the thick walls of academia. All thanks to the development of Internet. Some people say we live in a short period of a truly free Internet that once the technology stabilizes ( whenever that will be ) more and more sites will become accessible to the elite only. A bit like today's access to scientific journals for example, which give professors a six month to a year lead.

To the point: I still haven't watched the MIT video-series 'Gödel, Escher, Bach: A Mental Space Odyssey' because it is stored in Real format and I don't want to install Real with it's bogus Adware. But these files play well in VLC Media Player which is my favorite media player. Now is the time to watch these videos, I suppose. In the middle of doing M381 mathematical logic. Now that I have two ( real ) books on the subject the Open University booklets seem so much more accessible. In retrospect I think that M208 is an extremely difficult course IF you go by the booklets alone. Enough said over that subject.

I noticed that understanding a mathematical topic is something continuous it's not a discrete 'I get it' versus 'I don't get it'. The Jigsaw pieces are slowly getting on their right place and the image becomes clearer every day, Goedel's Incompleteness Theorems. I remember that when I read Hofstadter's book that I thought that he was an absolute genius ( he might be ) and that it was impossible for me to =ever= understand Goedel at a mathematical level. And now, years later, I am getting closer to that every day. I still have this idea about unreachable highs of mathematical knowledge represented by mountains disappearing in the clouds. I am far from the mountains on low ground. As long as I keep walking I must get there one day.

MIT video, six lectures on GEB.

About M381 mathematical logic or how to study an Open University booklet

How to study an Open University booklet? Well, don't if you have an alternative classic source. Study the source first, then go back to the booklets and do the exercises. Just because the TMA questions are modeled after them and the model answers are written in that style. Remember that your tutor wants to get through your work asap. The more they look like the model answer the better. The rule is 'when in doubt it is wrong.' Tutors are NOT professors, so don't expect them to be.

Studying the source is essential. Because that's what the authors of the booklets used when they wrote them. You should be able to write these booklets yourself and sooner than you think. Besides, the vast body of mathematics is written in source classics, not in the form of OU booklets or lecture notes. - ( An Open University booklet is roughly comparable to a set of lecture notes brick university students get, I suppose. )

Take for example the M381 mathematical logic booklets. What a Layman would understand as Mathematical Logic is called Propositional Logic and is covered early in the track in the MS221 course. The title of the course 'Mathematical Logic' is thus somewhat confusing. And it is not just the title that is confusing. I noticed on the forums that students considered the M381 logic part rather 'difficult'. The principal author of the booklet is Alan Slomson, now a professor in Mathematics at Leeds University. ( A host of other names are listed in the credits including Jeremy Gray: Wise Beard Man, the face of the OU ). Anyway, the source book is: "Computability, an introduction to recursive function theory" by Nigel Cutland.". ( I checked all available books on Mathematical Logic, I have no doubt that this is the book that was used. ) The thing is that the material in the course booklet looks a bit fuzzy, somwehat unclear, rather difficult to be honest. Well, it isn't. Not if you read Cutland's original first. Then you recognize what they ( Slomson ) are -trying- to explain. It's just the presentation in the booklet that makes it foggy. It would have been better if they simply have used Cutland's book ( like Apostol's in the Analytical Number Theory course ) and write a Reader's Guide. In a Reader's Guide you can use informal English to explain things while for the formal math definitions you can refer to the book. - ( Under Slomsons wings another book on Computability has been published by Brian Cooper : Computability Theory. Here too you'll find Cutland's URM and a lot of his original examples. )

Despite my occasional criticism I feel quite at home at the Open University.

The Open University and MathCad

I wonder if there is any university out there in the world where the professors choose MathCad for their students. The answer is of course No, N-O. Only managers or others on the receiving end of the kickback pipe can take a decision like that. - MathCad is a joke. If I had known on beforehand how bad MathCad was I would not have chosen to study mathematics at the Open University. I am very serious about this.

Think twice if you are still deciding. And go elsewhere if you have an option.

What I am complaining about is the following.

I do my TMA's in four rounds. First I solve them with Mathematica. That part is easy and fast. Then I typeset the answers in LaTeX because most ( not all ) tutors don't really understand what they are marking. They insist a presentation a toddler can understand. This round is time-consuming but I like to become proficient in LaTeX so who cares. Then in the third round I descend down to the gates of hell. In that phase I have to redo the answers in MathCad. This is Guantanamo Bay Home Edition. Then round four: the psychiatric ward. Because the OU does not accept electronic delivery I have to PRINT sometimes up to 100 pages of which 80 come from MathCad. I call this insanity.

Thanks MathCad.

Linear Programming ( M373 )

A series of 24 video lectures on Operations Research of which the first 12 are on Linear Programming. These 12 lectures cover more or less block 2 of course M373. The lectures are easy to follow since they go in a slow ( but steady! ) pace with lots and lots of examples. TIP!

More here: NPTEL

As usual Mathematica can be your virtual math coach through the commands:
- Maximize
- Minimize
- LinearProgramming
( See the Mathematica docs for more. )

Social media for mathematicians: Math Overflow

I wouldn't be surprised if ( research ) mathematicians were so ( Asperger / social anxious / Nerd / whatever buzzword you like ) that they stay away from Facebook. - The quality of Facebook is that you meet people that you wouldn't have met otherwise, because they are completely out of all the circles you are in. - Anyway, a place where research Mathematicians ask questions ( in a way programmers do for example ) is the site Math Overflow, it is set up like Stack Overflow, a well known site for programmers. Registration is open to anyone. Only questions on the graduate and research level though. No place to search for help with TMAs but a fascinating place to get a feel for what it is like on the frontline of our science.

Math Overflow ( Research )

Math Stack Exchange ( B.Sc. students )

Equivalence classes in Mathematica

An equivalence relation like 'has the same remainder after division by 5' partitions a set in equivalence classes.

The relevant Mathematica function is GatherBy[ set, equivalence relation] -> {equivalence classes}:

In[5]:= GatherBy[{1, 2, 3, 4, 5}, Mod[#, 5] &]

Out[5]= {{1}, {2}, {3}, {4}, {5}}

In[6]:= GatherBy[{49, 33, 11, 1, 2, 3, 4, 5}, Mod[#, 5] &]

Out[6]= {{49, 4}, {33, 3}, {11, 1}, {2}, {5}}

Or when using the equivalence relation 'Is Odd?' on the same set:

In[7]:= GatherBy[{49, 33, 11, 1, 2, 3, 4, 5},OddQ]

Out[7]= {{49, 33, 11, 1, 3, 5}, {2, 4}}

M381 Exam setup

Today I received pack 2 of M381 materials and read some interesting news about the exam setup. ( Open University ) course M381 has two fairly independent tracks: Number Theory and Mathematical Logic. The TMA's are thus mostly 50/50. The exam however has a different setup.

The exam consists of 16 questions.
Questions 1-8 cover Number Theory. ( My educated guess is 1 questions for each book / topic.
- divisibility
- prime numbers
- congruence
- Fermat / Wilson theorems
- multiplicative functions
- quadratic reciprocity
- continued fractions
- linear Diophantine equations.
Questions 9-16 cover Mathematical Logic. ( My guess is as follows )
- computability
- primitive recursive functions
- Church's thesis
- formal systems
- formal proofs
- formal number theory ( 3 questions )

All questions are marked and ranked. The nine best results are used to count your grade for the exam. There is one restriction however you can't use more than six questions from one subject. So if all number theory results are better than the logic ones than they take the six best nt questions and then the best three from logic.
eaxc
If, and I suppose it is, ( isn't it always? ) an extremely time-constrained exam then there are quite a number of strategies to follow for sequencing the work at the exam. I will have to think about that when the exam date comes closer, I suppose. A lot depends on my self-

Solving linear congruences ( NOT the M381 way )

In M381 Number Theory Unit 3 I read "...The way to become proficient at solving linear congruences is through plenty of practice at applying the above strategy. ..." - What did I just read?!

Well. This is NOT how it should be done. You need as much 'strategies' for solving linear congruences as you would need for solving a linear equality equation in x. There is a simple formula that solves -any- linear congruence equation.

a x ~ b mod m <=> x = Mod[a PowerMod[b, -1, m], m] where a, b can be ( negative ) fractions. ( Understanding the formula only requires understanding Greatest Common Divisors and the Euclidean Algorithm or one of its improved alternatives )

Example.
7 x ~ 41 mod 101 <=> x = Mod[41 PowerMod[7, -1, 101], 101] <=> x ~ 78 mod 101
Since:
7 * ( 78 + 101k ) - 41 = 505 + 707k which is clearly a multiple of 101 for any k.

TMA week ( Pomodoro style )

Studying mathematics at the Open University means that you have to send in a Tutor Marked Assignment ( 10 questions on which I spent as little as 5 hours but there were TMAs that cost me 40 hours ) per 7.5 study-points. So, if you do 60 points in a year you have to do 8 of those ( and one or two exams ). Unfortunately it is not allowed to publish any TMA questions on the Internet.

Today and the next thee days until Saturday are for M381 TMA02 ( Number Theory and Mathematical Logic ). Although I haven't planned to complete the TMA by then I want to see how far I come now that I have almost three weeks of Pomodoro experience. The drop in posts to my blog is definitely a result of my implementation of the PT. - I can say that the Pomodoro Technique is fully compatible with GTD, one could argue that PT fixes GTD so that it effectively cures procrastination. ( That the PT is compatible with Hubbard's Management Methods and Techniques is a no-brainer. )

More later...

Sum of a list in Mathematica

Problem: We want to calculate the sum of the elements of a list of numbers. Suppose this list is named l and has been assigned the value {1,2, ..., 20}

Method 1: ( Use the Mathematica function to total the values in a list. )
Total[l]

Method 2: ( Use the formula for the sum of an arithmetic sequence, with l[[1]] being the first element, l[[Length[l]]] being the last element and Length[l] the number of elements.)
(l[[1]] + l[[Length[l]]])*Length[l]/2

Method 3: ( Go through all the elements from the first to the last and add them to a running total. )
Sum[l[[k]], {k, 1, Length[l]}]

Method 4: ( Recursively define the total of a list. )
s[{}] := 0
s[l_] := First[l] + s[Rest[l]]

Method 5: ( Use some intricate method of the Mathematica language. )
Fold[#1 + #2 &, 0, l]

All five methods give the same result 210.


Set l := Array[#&,20];

Or use the following to test the commands.

Total[Array[#&,20]]

(Array[#&,20][[1]] + Array[#&,20][[Length[Array[#&,20]]]])*Length[Array[#&,20]]/2

Sum[Array[#&,20][[k]], {k, 1, Length[Array[#&,20]]}]

s[{}] := 0
s[l_] := First[l] + s[Rest[l]]
s[Array[#&,20]]

Fold[#1 + #2 &, 0, Array[#&,20]]

Maintain fitness

If solving yet another mathematical problem tires you out. Don't blame the problem. You might need some serious mental fitness. Athletes and their coaches have developed several statistics to determine their fitness for the job. If stats are down a good coach can develop a training program to regain healthy stats. Athletes are centuries ahead of mental workers like us. We should strive for that kind of perfection though. - The saying a "Healthy body, healthy mind" is true but it is not enough. A healthy body is merely a prerequisite for a healthy mind.

What is important for a mental worker? And can it be measured?
- Regular and enough sleep. Yes, hours sleep. Time in / out of bed.
- Trained eye-muscles. Speed-reading exercises. Reading-speed can be measured.
- Focus and concentration. Number of internal ( self induced ) interruptions.
- Quiet and healthy work/study environment. Number of external interruptions.

Some books on the subject.





The best route to mental fitness is of course Yoga but that route is long and hard.

Another 'Done TMA'

I haven't posted here as frequently as I used to the last couple of weeks or so. It's not that I am done with blogging or anything like that. There are two reasons for the sudden drop in posts and I am not entirely sure if they are related.

-1. I have been very busy with and focused on completing an M373 Optimization TMA which took considerably more effort ( 56 pages ) and time than planned. No time for pleasant 'self-chosen' ( beautiful ) problems, or just reading math books not directly related to one of my OU courses. Time to do that is very, very precious which you only realize when it's gone. ( Last time I was in a 'crisis' like this I promised myself to let it never happen again. It still did but only because I underestimated the work. Never underestimate an OU TMA even if it looks simple. There are no simple TMAs. And if they are for you then ask yourself if you aren't fooling yourself by doing a course on a topic which you already own anyway. )

-2. In order to keep my study going, to stay focused, keep it organized during several years of life ( with its ups and downs ) I use all the planning and control tools I can get. Including those to improve reading and concentration. One of the recent methodologies is the Pomodoro Technique. The purpose of the PT is helping you DO things, focused and with concentration. Unlike tools like GTD that are more of an aid in handling the in-tray, filing and deciding WHAT to do. Since I don't write down that I should create a blog post in my planner ( I use the advanced MyLife Organized ToDo list program ) I can't move it to a Pomodoro which explains it all.

Why now? It's late and I am contemplating about the future direction of this blog.

Unfolding

Definition: An unfolding is the cutting along edges and flattening out of a polyhedron to form a net.

Somewhat related to unfolding is the pop-up ( as called by the artist ), but what's in a name?



More paper-art here: http://www.l.van.breemen.scarlet.nl/

SF versus Reality

Science Fiction: Glenn

Reality 2011: Justin

Remark:
Justin's software runs on external computers. In the movie, Glenn is a stand-alone system ( although she is rebooted using a laptop early in the movie ). What if the Glenn's of the future ( million's perhaps ) are all operated from the same computer? They would be like a swarm.