infinity vs GODELS theorem

i was wondering if Gödel's incompleteness theorems are related to the infinte numbers in mathematics....... we can see that there are several unanswered quuestions because of the limitation mathematics........ besides, Can we think of any other ways to describe the existance and actions of our daily life? is mathematics based on numbers or describtion of things by the numbers?

QUESTION OF THE WEEK - 27th MARCH 2009

This is first of all a reminder that your contributions to the blog will be taken into account when semester grades are awarded. I notice that only a minority of you have responded to last week’s question. Those who have responded have made some great points, so why not take the chance to join them by making your contribution now?

In the mean time, we need to post the next question of the week. This time we are going to take a slightly different approach – please read on carefully.

Here is a Theory of Knowledge essay title:

Compare the roles played by reason and imagination in at least two Areas of Knowledge.

The question is – how would you go about responding to this title?

You would have a word-limit of 1,600. What form would your essay take and why? What would each paragraph be about? What areas of knowledge would you consider and why? How many areas of knowledge should be included? How would you set up comparisons between reason and imagination? How would you explore what these two terms mean? What examples would help? What claims and counterclaims could be made? Let’s explore in a practical way how a title like this could be the springboard for an outstanding essay.

I think this could be a very useful discussion in the build-up to the full-length essays you will be writing very soon…

aesthetics of mathematics

Following on from today's session, here is a version of the proof of the infinity of primes:

1.Each number is either prime or not prime (ie composite)
2.All composite numbers can be broken down into prime factors (eg 154 = 2 × 7 ×11) – this is the fundamental theorem of arithmetic
3.Let’s suppose that there is only a finite number of primes
4.Let’s call the members of this set: a, b, c, d, and so on, up to z, which would be the largest prime
5.Multiply all the members of the set of all primes together: a × b × c × d × …. × z – let’s call this number P, the product of all primes
6.Add 1 to make P + 1
7.P + 1 must be composite since it is larger than the supposedly largest prime
8.So, according to statement 2, there must be a prime number – let’s call it p – that divides P + 1
9.But for p to be a member of the set of primes a, b, c, d… z it would have to be also a divisor of P, and no number larger than 1 can divide two consecutive natural numbers
10.Hence for any set of prime numbers there will always be at least one more prime outside the set; therefore the number of primes must be infinite – QED

Do you see how some people can sense elegance or beauty in such a construction? More broadly, does mathematics have anything in common with the arts? Historically, mathematics has sometimes been regarded as an art...

Please also note that the powerpoint presentation that I showed you today can be found on the local network in the folder at K:\Staff_To_Students\IB Subjects\Core\TOK.

Necessity begets invention

Is mathematics a discovery or an invention?

I believe that the hunger humans naturally have for certainty cannot be overlooked. Did our need for certainty result in the invention of the least ambiguous area of knowledge; mathematics? Or was mathematics discovered when Adam and Eve realised that they each had 5 fingers on each hand, and had even more when they put both their hands together?

QUESTION OF THE WEEK 20TH MARCH 2009: IS THE HISTORIAN MORE LIKE A PHOTOGRAPHER OR A PAINTER?

IS THE HISTORIAN MORE LIKE A PHOTOGRAPHER OR A PAINTER?

(NB: Thanks Chris for anticipating our question. This happens to be the question of the week)

ABADI BABY!

Kamau COMMANDED me to write about the TOK trip. This is long overdue and I'm sorry but i WILL blame it on the absurdly hefty workload that IB decided is necessary to get my 45. However, i guess this will serve as reminder of a time when it wasn’t so bad to be an IB student and bring back more pleasant memories...

So we left the school like a minute or two after 3pm (so technically we were OFF SCHEDULE, YES I was looking forward to that) and got to Abadi at roughly 4:30pm, I believe. Sadly we left the charming females by the lake and looked for encampment further up in the dark dangerous hills. Much to the dismay of certain people who thought themselves a trio of wise men, they got to sleep in the village huts on HAAAARD beds, the lucky ones got the ever-so comfy chalets and the hardest bunch of 22 guys got to sleep in the bar surrounded by a myriad of booze bottles, [shout outs to Mr. Togbor and the female bartender (was her name Esther or something?)] a TV with DsTV which was unavailable and an AC which decided to breakdown when it was most needed.

OK, so we settled in and then the charming females came up to join us on our side where we had supper and then went on to do an activity. Firstly, we were shown a jar of candy and told to estimate how many pieces were in it -at this point I thought this was pretty dull. Later we did another activity. Surprisingly (for me at least), i found this quite interesting for many reasons- Firstly because the task was unusual and secondly because our initial idea underwent mind-boggling modification before our final answer. Also, it was quite amusing and intriguing as to how other peoples ideas made so much more sense(and vice-versa;) that it made you wonder what kept you from thinking further. Oh almost forgot the task: We were split into our different class groups (for some reason green was supposed to disperse so I joined Blue) and each group was given a black box with something in it. We were supposed to describe the structure of the box and what could be in it.

After this we were advised to break up into our groups to prepare for the presentation on Saturday morning. Needless to say some did, others didn’t- C'est la vie. I met with my group by the bar (strange location...?) and we were done in like 10mins. Everybody else was done at about then and somehow the barroom turned into some mini concert hall with me and Ibi on the vocals and Elom on the strings and Zahra and Nah D.....well i guess they could pass for the video girls. Oh yes, all this while Mrs. Hutchful hung out with ‘The Boys’ trying hard to not look thoroughly bored at being surrounded by an excess of testosterone

OK, ffwd, the girls eventually leave *sniff* and it’s literally Boys Night Out. Between ordering countless plates of chicken and drink (nothing naughty, pinky swear), Isaac telling Mr. Togbor about IB HGIC and his whole life story, James browning Isaac, Albert and Jerry thinking they were too deep to let go of their earphones and Leonard just causing havoc among other things we somehow got through the night up until Kamau decided that people needed to sleep and so the lights had to go but the voices could remain below hearing threshold. Consequently, Albert got into deep philosophical discussion with Kimi, Jerry stole my sleeping bag even though I was lending it to him, I tried to steal Caleb’s pillow and at about 3am I DID steal James’s cover cloth- sorry blad but I was cold. Oh yes, I did see Mr. Togbor boozing and then picking a mattress of me to sleep on (this resulted in my pilfering of James’s cloth- his expression was priceless in the morning!)

The next day we were supposed to get ready at about 6: 30, before the girls arrived. The chalets’ water wasn’t running (its called karma) and hence all bathing was done in some basement-like bathroom and a hut serving as a bathroom further down in the Village. The charming members of the opposite sex arrived and we had breakfast. Do I need to state that at this point I was a bunch of nerves because the presentations started immediately after breakfast?

Anyway the presentations started with the Red, Blue and Purple groups beginning the day. I cannot remember which group I joined but it had presentations from Paapa’s group and Harriet’s group and was evaluated by Kamau and Mwalimu. I was very entertained, informed but mostly intimidated especially by Harriet’s group which was especially impressive with an in depth dissection of the subject of Knowing Beauty or The Perception of Beauty (they flagrantly exceeded the time limit though. Ha!). Sources say the other groups were also very good but I can only base this knowledge on language…After this session we had snack.

Drum roll, trumpets blow, my group is the first group for the second session (At this point we had been split into two groups-Green, Yellow). Contrary to my expectation it went better than expected. We talked about language as a way of knowing and it’s role in limiting or redefining knowledge with the Biafra War as the case study, David dropping a couple of jokes(?) here and there. Sadly, Mr.Kitching felt that the topic wasn’t completely dealt with*sniff*. Zahra’s group talked about knowing when killing is justified and Dwayne’s group talked about Mathematics and the mathematician lack of the ability to express himself – Gartaa, trust him, would not take this and this made for some comic relief. I hear Yasmine’s presentation and Ibi’s presentation, both on History, were engaging and insightful (respect!) We were then asked to evaluate our own presentations.

Lunch was next- a choice between banku and jollof. Those of us who considered ourselves European/ Asian/ American/ (any other appropriate description) opted for jollof which by the way someone described as ‘horrid/cardboard-tasting’…. Me, I was hungry and had a third helping as did many others of the male gender (and some females too…). Next thing I knew I was sleeping as were many others until Mr. Kidane came to force us out of Dreamland and all the way to the female encampment.

Here we sat by the lake feeling all exotic and in touch with nature till another activity came along. We broke into our larger groups again and were given profiles of a tribe which we were supposed to act like. My group, Green considered itself divine descendants of a king-deity, Gbalazu who was given divine status and grace along with all his descendants by the river god, Abugri. We were then supposed to interact with the other groups bearing our identity in mind. Later we were given another scenario where our island was sinking and we had to share another island with people from other lands. We were shuffled such that there were 4 people from each group in every group. Bearing, our identity in mind we were supposed to come out with a constitution for the island. After arguing our divinity out ¼ hour the meeting somehow disintegrated.

Some guys played soccer while others hung out in the girls’ rooms…well, until Mrs. Hutchful seemed to think that this was risky (lol! with a large number of both sexes in an uneven ratio). Eventually the guys left to shower and get ready for supper and the looooong -awaited jam.

Supper was Fried rice and Chicken which could definitely have done with some spice. Then somewhere between trying to fix the obdurate AC and Jerry and Edwin doing things with the music the jam, started… it was insanely hot (the weather, that is) and the speakers bungled twice. Eventually, it was moved to Chalet 1(or was it 3?), where I can only say it was a success, thanks to HAMJ. All this time, Paapa, Anna and Edwin (Buddha himself) were making the kebabs- thanks guys, Bra Albert is proud of you.

The girls left after exchanging jets of water from water sachets with the guys outside. The night ended with more chicken and a discussion on dwarves (yes, its true) among other things by which time I had seized Leonard’s sofa.

Sunday, we had breakfast and got back our estimates of the quantity of sweets and were supposed to come up with consensual estimates (experiential ones) for our groups (the Black Box ones). Sefa then won the jar with an estimate of 214 sweets, which was actually 210, I think (London juju) and all hands were on deck for its consumption. Next, we were supposed to make 2min evaluation-presentation of the blog. Reviews were generally good but EVERY group wanted facebook unblocked (hint, hint). Some more chicken and then we hit the road. En route, Isaac organized a turkey-backside (chofi/kyofi) feast at the back of bus. I’m not sure when we got back but expressions were ambivalent.

This trip taught me that the idea of 22 boys sleeping in a bar is not as absurd as it sounds, opening one’s mind to other opinions does wonders and that I have to get out of IB!

Pretty long; guess I got carried away by the memories

*Did I mention that the bus left 2 people and that the dwarves had their revenge on Isaac? -Only in the graduating class of 2010!*

Mathematical Knowledge

Dear All,

Next week we will be starting to examine the famous discipline of Mathematics. What is mathematics? Why do we all have to study it? Why do some people love it and others hate it? What kind of knowledge does mathematics give us? What methods are used in mathematics to generate this knowledge? What do words such as axiom, theorem, conjecture and proof actually mean? Why does mathematics seem to be so useful in other subjects? Why does it appear to us that the entire universe obeys relatively simple laws that can be expressed mathematically? Big questions.

A couple of things you could do while waiting impatiently for these lessons to come around? OK, why not ask your mathematics teacher about the subject? Why did s/he decide to learn and teach this subject? What would s/he say is the best thing about doing mathematics? And so on...

Also, try answering the following questions yourself. On Tuesday, we will all meet in the Auditorium in order to discuss them and move on into more exciting territory...

1. Mathematics is a subject about
A. logical thinking
B. illogical thinking
C. things that exist in nature
D. things that do not really exist at all
E. things that are certain
F. things that are not certain

2. Problems in mathematics can best be solved by using:
A. clever tricks
B. experiment
C. computers
D. graphic calculators
E. trial and error
F. investigations
G. discussion
H. the answers in the back of the textbook

3. Mathematics is a subject that should be studied by:
I. people who are interested in it
J. engineers and other people who want to apply it
K. people who are challenged by it
L. people who want to become better thinkers
M. people who are intrigued by aesthetic qualities
N. people who want to become better artists
O. people who want to improve their overall academic performance
P. people who want to improve their college entrance exams
Q. all Diploma Programme candidates
R. people who are poor at it

4. Which of the following best describes mathematics?
A. a body of knowledge
B. a practical tool
C. a cornerstone of philosophy
D. the perfection of the logical method
E. the key to understanding nature
F. an intellectual game
G. an aesthetic experience

5. Which of the following quotations best captures the essence of mathematics?
A. A mathematician is a blind man in a dark room looking for a black cat which isn’t there. (Charles Darwin)
B. Pure math is a game. It’s fun to play it for its own sake. It’s more fun than applying it. Most of the math that I teach is never used by anyone. Ever. (Ted Williams, Prep School Mathematics Teacher)
C. If you ask your mother for one fried egg for breakfast and she gives you two fried eggs and you eat both of them who is better in arithmetic, you or your mother? (Carl Sandburg)
D. Mathematics is a game played according to certain simple rules with meaningless marks on paper. (David Hilbert)

social science

I was wondering how Economists tried to measure the satisfaction that one can gain from using certain goods or service.to what extent we can justify the calculation made by measuring the satisfaction one can get?As originally conceived, utility was taken to be a subjective measure of strength of feeling. An item that might be described as worth “40 utils” was to be interpreted to yield “twice as much pleasure” as one valued at 20 utils.1
As we can obseve from our class lesson measuring and equating variables in social science is challenging.In fact a thery which is backed by mathematical equastion is more relaible.so are we tring to approach where natural science reach through the help of mathematics?can mathemacis be a best tool to analize theories in social science.
source
1.http://www.britannica.com/EBchecked/topic/620713/utility/34299/Utility-measurement-and-ordinal-utility

How far back do we go in search of causes of historical events?

There’s the old tale of a country that was lost because a war was lost

The war was lost because the battle was lost

The battle was lost because a general didn’t receive orders

The general didn’t receive orders because the message was delayed

The message was delayed because the messenger stopped en route

The messenger stopped en route because his horse’s shoe came off

The shoe came off because of a sharp rock in the road.

How far back do we go in search of causes of historical events?

Seeing things?

Okwase and I noticed something really cool this week...You know the posters put up for science and technology day? well you may have noticed that the face has been blacked out and they say "guess who this is".And yes, like many of you, I guessed it was Mr. Kitching. Well now that the poster with the true face has been put up next to the blacked out one, you can see a face in the blacked out one. Take a look at it the next time you go by one. Sense perception limitations? Or reasoning playing with our eyes?

QUESTION OF THE WEEK – 13TH MARCH 2009

To explain is to be able to predict. Is this true?

seek knowledge

to get a knowledge should we seek for it or knowledge
develops whith in a period of timeunknowingly?

TOK essay assignment - March 2009

Dear Students,

Here is the title for the next TOK written assignment:

If distinctions can be made between the natural sciences and human sciences as different areas of knowledge, could one be described as better than the other?

The word limit for this assignment is 800 words.

You must use this title exactly as given - do not alter it in any way. This assignment is intended to be a stepping stone from the 400-word assignment from last semester towards the full length essays that you will soon produce. The criteria for assessing this assignment are given below.

Your response is to be submitted to your class teacher at the start of the lesson on TUESDAY 24th MARCH - ie in two weeks.

A: Knowledge Issues (maximum = 5)
Skill objective: to produce a piece of work that recognizes knowledge issues that are related to the title and shows understanding of them through actively illustrating them with reference to different aspects of the TOK course
Marking Guidelines: Reward the work if knowledge issues relevant to the title are explicitly recognized, and if Ways of Knowing and/or Areas of Knowledge have been linked and compared in the service of understanding these knowledge issues. A low score is appropriate if the knowledge issues are not relevant or only implicit and/or left unconnected to Ways of Knowing or Areas of Knowledge, thus showing a poor understanding of them.

B: Independent Thinking (maximum = 5)
Skill objective: to produce a piece of work that explores the title imaginatively through both the student’s own perspective and that of others, using real and understood examples from the student’s own knowledge
Marking Guidelines: Reward creative, imaginative responses to the title which show an awareness of different perspectives on knowledge (including an awareness of the student’s own), supported by examples drawn from academic and personal experience. A low score is appropriate for dogmatic, unreflective responses, or for the over-use of hypothetical or commonplace examples.

C: Analysis (maximum = 5)
Skill objective: to produce a piece of work in which logical rigour is evident, including an awareness and analysis of things often taken for granted.
Marking Guidelines: Reward responses to the title in which the main points are justified and arguments are coherent, and in which counter-claims, implications and assumptions are recognized and explored. A low grade is appropriate for sloppy reasoning as evidenced by a failure to achieve some or all of the attributes listed above.

D: Organization (maximum = 5)
Skill objective: to produce a piece of work that shows respect for knowledge by setting it out with attention to format and factual accuracy, and acknowledges where appropriate the achievements of others who have helped to discover or create it
Marking Guidelines: Reward well-structured work that is clear to read and understand, in which factual claims are correct, and sources have been properly referenced so that they can be traced. A low score is appropriate for careless formatting, language that needlessly interferes with understanding, factual inaccuracy or missing/inadequate references.

PENALTIES
· No double-spacing of text: 2 points subtracted
· Missing name: 1 point subtracted
· Revision of (or failure to write) the prescribed title: 1 or 2 points subtracted
· Missing references that should have been included: 1 or 2 points subtracted
· Incorrectly formatted references: 1 point subtracted
· Failing to quote an accurate word-count at the end: 1 point subtracted.
· Exceeding the maximum permitted word limit: 3 points subtracted (and the assessor will not read beyond the permitted word-count)

Atheism

Each time I search google for definitions or notes on important TOK terms such as 'sense perception' and 'reason', the first website on my results list is http://www.strongatheism.net/. What could this possibly imply? was TOK begotten by atheism or simply by the natural inquistiveness of the human mind?

USEFULNESS OF THE BLOG

1.It provides us with information and reminders.
2.It is very efficient when it comes to sending information to the whole IB 1 class concerning TOK.

IMPROVEMENT OF THE BLOG

1.Instead of discussing everything in class, vital information should be extended to the blog in order for other students to part in the discussion.

SUSTAINING AND MAINTAINING THE BLOG

1.Students must be encouraged to clearly express their views and ideas so that other students can appreciate them.
2.Most importantly, students and teachers who acquire their information from other sources should manage to state their sources.
3.People should manage to give their views and responses on knowledge issues raised on the blog.
4.Controversial issues must be raised.

Isn’t agreement very crucial for absolute truth?

Here I have an interesting question. If there wasn’t any agreement, then we would not have the same answer for math. Even in any other subjects.
My question is some how related to agreement and reasoning.

If there are three houses, house A, house B and house C being arranged as A, B, C.
Then assuming that these houses have their own house numbers from 1 to 3 on the other side of each house and if house B was built first, then house A and at last house C was built. Which house could most probably be house number 1, house number 2 and house number 3?

Group Blue's views on the blog

EVALUATION OF BLOG
-Blogs posted seem unconnected.
-The blog is visually unappealing as colors are dull and there are not enough pictures or diagrams to attract us.

IMPROVEMENTS
-Layout of web pages should be enhanced.
-There should be a general topic each week for effective discussion so that the blogs posted are connected to the topics.
-If possible, notifications of posted blogs should be sent to the members of the blog.
-There could be a portion where interesting news headlines are posted to derive knowledge issues from them.

STUDENT PARTICIPATION
-Blogs should be connected to student life.
-Interesting and mind boggling blogs should be posted to attract us and to encourage us to comment on them and post our own blogs.
-Fun facts on TOK could be posted to make students participate.

DOES THE BLOG MEET THE TOK AIMS?
-There is hardly any connection across knowledge as there are separate pages for different subjects.
-The blog has posts with diverse perspectives on a topic.

Postcard from Kuwait

Hello everyone,

Well, I have struggled to get to this point on the blog because all the text at the top comes up here in Arabic. Now that is a new challenge for me! Harder than Google in Kiswahili! Sometimes it is good to have your taken-for-granted props taken from under you - anyone got experience of this? Of course you have - we have a rich linguistic diversity in this group - many of you must have found yourself linguisitically disabled in certain situations - when this happens, is it merely a hindrance to knowledge acquisition, or can we learn something from it? Emirates tries to sooth any discomfort by announcing even before takeoff the list of languages that its cabin staff can cope with. Is this good for us? What do you think?

I would like to add that I thought the last weekend was very successful in many ways, although there are quite a number of loose ends that need to be cleared up - deliberately so, because such an event needs class time (or blog time) to tie up the issues we touched on. In the mean time, keep the contributions coming - particularly the other groups who haven't posted on the Sunday morning activity.

What's that I hear on the west breeze? "What am I doing here?" More soon...

TOK WEEKEND

YELLOW GROUP 1

TOK BLOG
An extension of our class where opinions and ideas are shared and expressed via the net (www.)

EVALUATION
So far, the blog is serving its purpose. Hopefully, through continuous use and input to the blog communication between teachers and students will improve. After all, learning DOES extend outside the 4 walls of the classroom.

In order to do so………………

• Blog posters should also try and post into that we as student scan relate to.
  
• Access to internet (to enable us to access the blog) throughout class.
  
• UNBLOCK facebook (.com). This will give us the opportunity to share knowledge obtained from TOK with a larger audience.
  
• Possibly post notes on TOK and important details on the blog (e.g. the schedule for the TOK weekend). This will encourage students to check the blog more frequently.
  
  AIMS
• A means of obtaining knowledge on a diverse scale. Through this blog, ideas, opinions and perspectives of knowledge and knowledge issues are effectively communicated. This in return broadens and improves OUR perspective and understanding of issues at hand and knowledge in this course.
  
  

hi guys.....
i was just wondering if it was possible for 'intuition' or 'having a knack for something' could serve as a way of knowing

wat is lying?

lying is not just saying something that is clearly made up or false, it is stating a degree of emotion which is simply untrue

NOTE: What is "wat"? Please let's uphold an acceptable standard of English here - this is not a "chatroom" (JHK).

TOK WEEKEND PRESENTATION

GROUP PRESENTATION FOR PURPLE A

1. How do we evaluate the blog?

RESPONSES:

a. Strengths

Opportunity to ask and discuss anybody’s question or knowledge.

It’s educative and stimulates intellectual thought.

b. Weaknesses

Important TOK information shouldn’t be posted only on the blog this is because we don’t always get time to check.

Not enough time to visit the blog and contribute.

2. How should the blog be improved?

Multimedia facilities should be enabled so that things like videos, pictures can be posted.

Links to materials discussed in class should be posted there.

Answers to a “poster’s” question or knowledge issue should be directed to their inbox.

Post well specified and important information.

3. What can we do to sustain and maintain the blog?

Make references to interesting questions posted on the blog during class.

Invite other teachers and students from other IB schools to the blog.

Follow up suggested improvements.

4. Can the blog meet the TOK aims?

Providing a platform and springboard for discussion of knowledge and its issues.

Provision of tools for knowledge evaluation- research on knowledge issues can easily be done since we would already be online.

AUTHOR: JACOB YAMBASU