As one of the oldest areas of knowledge, Mathematics is the paragon of certain and unchanging knowledge. For instance: 2 x 2, will always be 4, because someone has made it so, and society has decided to accept it as that. I am not saying that mathematics is entirely comprised of fixed certainties, but the majority of concepts in this field of knowledge have already been predetermined, agreed on, and accepted. It’s amazing how signs and symbols that have been arbitrarily selected by men and women long forgotten are still applicable today. Truly, have you ever considered what exactly “=” means. How do you “+” things? Even in combinations such as “6+1=7”, we are not referring to anything in our world, only simply to things conctructed in one human mind! What is baffling is that math shows flexibility as it is still applicable to our ever-changing world. Through common agreement, we all know we are talking about realities when we say: “Six books plus one book would give us seven books”. Thus what mathematics has helped us do is to merely simplify the task of describing things in our real world. Mathematical knowledge has therefore been inspired by the real world, and must not be mistaken to have really existed on its own. In many ways, mathematics is like an art. People picture mathematics as solving immensely complicated problems by means of massive number crunching, but few things could be further from the truth. Although much of math relies on nature, it is still an artificial way of knowing in that, it is constructed by the human brain. The good thing about it though, is that it deals with absoluteness, and as such, it is considered the foundation of all other sciences, because it makes no hypotheses, or assumptions – in math, it is either right or wrong!
Mr Kitching you make the task sound so easy by saying, "All you have to do..."
To me, this proves the complex nature of math especially since it has the tendency of becoming increasingly difficult as you progress in the subject. One would think that by mastering +,-,* and / you would at least more easily understand as you grow in math. And yet then comes adding numbers to letters, differentiation, the root of a negative number and why is there an 'e'??? Mathematical knowledge in my opinion follows its own 'rules.' I say this because it appears as though math has its own language (although surprisingly when I had the 'wonderful' opportunity of having a maths lesson "en français" it made absolutely little or no sense to me and I wondered if math truly was a universal language)
I do not want to write a 'looong' blog that I know hardly anyone would read if they saw such a length but basically I think mathematically knowledge is definitely abstract and conventional. Of course it is very complex but quite enjoyable to those who enjoy a good challenge
To me, although I have not always been the best at it, math has always been a safe haven when it came to knowledge. In my mind it was meant to be easy because basically any problem you could ever come across has already been solved you just have to have all the knowledge. Mathematical knowledge is set with all its formulas and rules. Like the previous comment 2+2 will always be 4. However, if this was Literature we'd be told there is no right or wrong answer. That to me is more confusing than anything I have ever encountered in mathematics.
In regards to it being a universal language I'm not so sure. Yeah numbers are the same in any language but then with mathematical terms they have to be translated. Trigonometry is not the same in French or Spanish. Basically all I'm trying to say is mathematical knowledge is more accessible than others.
Interesting comments, Tori. You have alluded to the irony that on the one hand mathematics seems entirely divorced from the world we live in (eg nobody walks to school and finds a "2" hiding under a rock), yet at the same time it seems to describe some aspects of that world incredibly accurately (eg look at the simple mathematical relations that lie at the root of so many scientific laws). You wrote:
"...2 x 2, will always be 4, because someone has made it so, and society has decided to accept it as that..."
If you are talking about the symbols (numerals)that are used in mathematics then this makes sense (the conventional nature of numerals is borne out by the experience I once had of trying to work out prices in Egypt). But surely the abstract quantities that these symbols represent NECESSARILY obey this relationship.
Yasmin - I like your anecdote about learning mathematics in French. Perhaps this tells us something about how mathematics is taught and spread - we do not talk "in" mathematics all the time we are doing it.
Chelsea - you wrote:
"...basically any problem you could ever come across has already been solved..."
I know what you mean, but what about the seven "millenium problems" referred to in the link? What does the fact that they have NOT been solved tell us about mathematical knowledge?
hi guys! :) i would like to disagree with yasmin about having mathematics in french. we both had the same experience at the same time. personally, i think mathematics its a language on its own with its own syntax. this is because mathematics itself does not change, whether in french or spanish or english...for instance, using the same example, you may be told to differentiate something but in french you would be told to "................". however, it is the same process. this brings me to mr. kitching's point that we do not always talk in mathematics. we use numbers. 2x2=4. BUT there are different words for "2" and "4" in the different languages. thats what makes the difference...NOT mathematics itself.
and about the questions which have not been solved, i am confused. how were the questions made? and is it that they have never been solved before or that no one has got the answer correct? i think this would mean that our mathematical knowledge has not reached the level of the question and so we cannot solve it......im nt sure..... Mz.Appiah-Korang
I think I understand your confusion. I think it is the use of the word "problem" (ha, language - there we go again!).
You need to distinguish between the kind of "problems" that your mathematics teacher often gives you, which are questions or exercises for you to get through in order to reach the correct answer (which is already known in advance, at least by the person who wrote the question, and is maybe in the back of the book!), and the problems that some professional mathematicians deal with, such as those identified as the millenium problems. And when we come to discuss mathematics in detail, we will look at one of these millenium problems more carefully, namely...
The Riemann hypothesis is a conjecture about the distribution of prime numbers. Most mathematicians seem to think that it is true, but no-one has ever been able to prove it.
[Reminder - TOK is about knowledge claims, the justifications offered for these knowledge claims, and the knowledge issues that arise from the attempt to provide acceptable justifications.]
So think of the Riemann hypothesis as a knowledge claim in mathematics which needs to be justified if it is to be accepted as true. But in mathematics, the standards for justification are very high (what kind of justification will suffice in mathematics, and why the standard has to be so high in mathematics - these are examples of related knowledge issues) - nothing less that a formal proof will do. And no-one has been able to provide one. Yet.
So what have we learned so far about mathematical knowledge?
I agree with Mz. Appiah-Korang on the notion that maths does not have different languages. It is just explained or taught in different languages, the processes used in problem solving are however, the same.
Tori said:
"...2 x 2, will always be 4, because someone has made it so, and society has decided to accept it as that...this field of knowledge have already been predetermined, agreed on, and accepted...we are not referring to anything in our world, only simply to things conctructed in one human mind!"
I don't agree with your point that this field of knowledge is "predetermined" and that "someone has made it so" simply because multiplication (addition, subtraction or division for that matter) are concepts that can easily be proved and demonstrated using the simplest using things around. From a simple transaction with a hawker on the street, to determining the rate of flow of a waterfall, math is simply representing some of the things around us in a manner that is more comprehensible and is therefore not "simply...things constructed in one human mind." People just keep on finding simpler or more accurate ways of representing situations mathematically. I will not say the millennium questions are unsolvable, I will say a (simpler) way of representing the problem mathematically (and thus proving it) with already gained knowledge has not yet been discovered.
HI! I also totally agree with Shawn and Mz.Appiah Korang!! Mathematics is the same all over the world!! I also had the wonderful experience of studying Mathematics in French (with Shawn, Yasmin and Maame) and initially I thought I wouldn’t understand anything that was going to be thought but then I did understand it after all. It was just symbols and numbers that I had to interpret!!! All over the world there are set conventions and symbols in mathematics that cut across and somewhat make Maths a universal language. Whether in English or French or Russian or Spanish, maths is still maths. If it wasn’t so, how else would we all write the same international papers and be assessed with the same criteria?!! Mathematics has its basic rules upon which people try to develop and find out new ways of representing everyday life occurrences. An actuary one once told me that in calculating considered every possible situation that might occur when they are calculating insurance premiums, risks, dividends, and annuity rates. He even sent me an amazingly long “mathematical equation” he formed to calculate the cash flows of a proposed football field and it included the attendance on a rainy day and even the attendance when the team had lost its previous match. This example is just to show how maths is inherent in and applied to our everyday lives. This is also to support Shawn’s statement: Maths is simply representing some of the things around us in a manner that is more comprehensible and is therefore not "simply...things constructed in one human mind." I believe the millennium questions are definitely solvable even though it seems impossible now. Just because we do not know something or know how to express it does not mean it is not there. By extension, in Mathematics just because we cannot solve the questions doesn’t mean the solution doesn’t exist. This is just a question I have: Can Mathematics be classified as a language because it has its own syntax and symbols that mean the same thing all the time and everywhere?! I remember in IG when we were using the scientific calculators, you could get the message “Syntax error” if you didn’t input the argument right!! It seems to have it’s own structure, words and semantics!!
ha ha ha!!! Yeah Yasmin, math from this question is really abstract. how do we even know what we solve every day is correct? Because if it is, any math student with a degree or something higher than that should be able to solve this question. Unfortunately i am certain that less than twenty percent of these intellectuals will barely make it to the correct answer. In my view, this math question is symbolic of the distorted ideas, reasoning and perception of things in the world which makes the world rather complex and questionable. This proves that idealistic humans in the world don’t always find solutions to problems in the world though they try their best this math question proves its difficulty. Funny how you are taught from grade 1-6 that math is only numbers but as you climb higher, you realise math involves alphabets and even English. From all these abstract things you have to reason like your teachers expect you to reason and think the way the world requires you to think like a math student. How are we even certain that such a question like this can be solved and how are we able to add letters to numbers to give us numbers. Well, based on inferred knowledge i don’t remember my grade one teacher giving me a name for the mixture of alphabets and numbers but in grade seven my teacher named it algebra. Algebra? I wonder how i was able to solve algebraic problems. Whether it’s logical or not math will forever be math unlike the humanities where what is taught involves you and requires you to reason and empathise to arrive at a solution, math requires you to accept its facts and apply. Wow, what a world!!!!!!!!! Timah
Mathematical knowledge is based mainly on language.i.e. Mathematical language, reasoning and reason. To me mathematics helps us to represent certain situations for example 2 oranges plus 2 oranges will give you four oranges. But the real question is what is 2? In that I am Mathematics although many laws and theories can be proven through most of the time, induction is very very abstract? For example, if you think about it many of us have seen the functions cos, sin, tan,and many others but thinking about what are they really? For example most of us know cos(90) is 0 but the question is what does the cos do the 90 to make it 0.I would therefore say mathematical knowledge is mainly based on facts discovered or made by somebody who was able to prove it and we as human have accepted it to be so.
Mathematical knowledge or language,in my opinion follows its own rules and format in terms of the various ways to get a right answer. There is a complete difference between mathematical knowledge and lets say... Literature. Mathematical knowledge helps us to apply simple things in daily experience, for example if we go to a supermarket and buy something for GhC8 and we hand over GhC10 mathematical knowledge should tell us that we should recieve GhC2. Mathematics has definate answers according to the different hypotheses and laws created by our forefathers. There is no question about that. The only way we can even begin to question mathematical knowledge, is if previous theories have come out with the wrong or unexpected answers(after many trials of course). In literature on the other hand, we cannot say that something can only have one answer, it depends upon the analysis of the reader. Knowledge in terms of literature does not follow any specific format, it is a subjective subject (lol). It allows the interpretation of many individuals. Math on the other hand does not have this subjectivity. From the mind boggling tasks presented to us to win $1 million it does show that mathematics on its own is a language that needs to be understood and justified before it can be accepted. I for one, think that that theory cannot be done because it just doesn't make sense to me(i am not a math person) but if someone does manage to solve it i will have to be forced to accept it.
I agree with Chelsea up to a certain point. Math is a safe haven when it comes to knowledge. With math an answer is either right or wrong. it is simple and very straight forward. however, literature is not so. an answer is seldom wrong when expressed by a person in such a way that it is their opinion that is being expressed as a result it (to me) is a poor way of acquiring knowledge
HOWEVER, i disagree with her when she says 'In regards to it being a universal language I'm not so sure. Yeah numbers are the same in any language but then with mathematical terms they have to be translated. Trigonometry is not the same in French or Spanish. Basically all I'm trying to say is mathematical knowledge is more accessible than others.' this is because mathematical concepts will remain the same in every language and will never change. Even though 'trigonometry' might have a different spelling or be expressed differently it does not necessarily mean that their concepts would be different. 1+1 would be 2 in any language no matter how the term 'plus' or 'addition' is expressed
It is a remarkable phenomenon that children can learn to speak without ever being consciously aware of the sophisticated grammar they are using. Indeed, adults too can live a perfectly satisfactory life without ever thinking about ideas such as parts of speech, subjects, predicates or subordinate clauses. Both children and adults can easily recognize ungrammatical sentences, at least if the mistake is not too subtle, and to do this it is not necessary to be able to explain the rules that have been violated. Nevertheless, there is no doubt that one’s understanding of language is hugely enhanced by a knowledge of basic grammar - it is almost tautologous to say so - and this understanding is essential for anybody who wants to do more with language than use it unreflecting as a means to a non-linguistic end. The same is true of mathematical language. Up to a point, one can do and speak mathematics without knowing how to classify the different sorts of words one is using, but many of the sentences of advanced mathematics have a complicated structure that is much easier to understand if one knows a few basic terms of mathematical grammar. The main reason for the importance of mathematical grammar is that the statements of mathematics are supposed to be precise, and it is not possible to achieve a high level of precision unless the language one uses is free of many of the vagueness and ambiguities of ordinary speech. Mathematical sentences can also be highly complex: if the parts that made them up were not clear and simple, then the obscurities would rapidly propagate and multiply and render the sentences unintelligible.
For me the fact that there exists certain mathematical concepts that are yet to be proven just shows the complexity of the reality (our world) that mathematics as an area of knowledge is trying to make simpler or understand. Mathematics is just like finding existing patterns that exist in our world and using them as a basis for postulations, where in this case we see the whole logical aspect of it. The fact still remains that these patterns may not hold for all cases in our world hence the need for justification of mathematical knowledge. First of all mathematics, as a way of knowing is universal (public knowledge) because it has its own conventionally accepted language hence is not at the mercy of linguistic misinterpretation or any other problem of that sort. (If everybody developed his/ her own language for mathematics we would hardly ever agree on any knowledge gained through mathematics and the purpose of mathematics would be defeated). Then again the fact that these almost-impossible-to-solve concepts have not been solved/proven does not mean they would never be solved I think it’s just a matter of time, for as humans develop, our areas of knowledge become better and better and increase the accuracy our interpretations of our world. The usual absolute nature of mathematics also brings up the question as to whether there is anything like mathematics provides us with any absolute truth. Is 17+ (-6) always equal to 11. Has mathematics restricted our thinking in anyway? Has the fact that we have all come to accept certain established rules in anyway removed the possibility of 1+1 not being 2. Asantewaa Agyare: "but if someone does manage to solve it i will have to be forced to accept it." Hi guys!
To me, mathematics is the least enjoyable subject and I sometimes wonder what a world without mathematics would be, perhaps we would have no sense of measurement. Mathematics is believed to exist in nature already with concepts like the natural log of e, etc. The one which interests me most is the “GOLDEN RATIO” which in summary states that all things in nature are equally divided. Can you imagine a concept which can be used to determine if a person is actually beautiful?
Mathematics is indeed very important in our world as we look back at Egyptian sculptures and buildings, art works, scientific discovery even in language. When you are given a TOK essay to write you are given a word limit and this only possible because of the existence of numbers. You all make it seem as if mathematics is all about 1+1=2 but you have to know what operation to use at what time because you cannot definitely simply use this to solve a question that requires Pythagoras theorem. Have you ever wondered what one drop of water + one drop of water is equal to?
How sure we that all these laws and formulas are are correct? How many of us have tried to prove Pythagoras theorem or even question its validity? Is it even possible that new laws can be formulated in our age?? With regards to mathematics being a universal language I would refer to it as public knowledge as we as a society have come to appreciate and accept its concepts without any question. To Yasmine if mathematics is really a language then you should understand me when I say 4489!!!! (an exception to this is 143 which means I LOVE YOU.) I agree with Mr. Kitching when he says we cannot find the number 2 under a rock but are we really sure that there are people who do not see numbers lying around?? Take synaesthesia for example many of us thought it weird that some people smell colour but this is real.
Katakyie how sure are you that mathematics is static when we are faced with newer challenges to what existed in the past when the mathematical concepts were being created. This may be the reason why the questions have not been answered. Maybe mathematics needs a bit of updating just like scientific discoveries have been updated over the centuries.
Ntiriwa, when you say 4489 you are quantifying something,however what is being quantified has not been stated. (ambiguity of language?) Mathematics is therefore a language :). I however do not understand how synesthesia is linked with your point?
Well, Math is honestly a bit of confusion to me.Like what Mama said its the least enjoyable subject, for me too.Looking at the fact that we can not live without Math whether we like it or not is something which comes to me like a BANG!! I got the following http://www.trap17.com/index.php/world-math_t49606.html and it says 'I mean we don't really know what we are missing until it's missing. But there wouldn't be equations or anything to figure out problems. Proper dimensions when building something wouldn't really be available. Systematic systems in factories based on precision wouldn't really be available. It's sort of hard to imagine or even for me to explain. There wouldn't really be a currency, just trading one thing for another maybe. Just when you finally think that Math is quite useless just wait to see what would happen if there was no Math in the world.
I like Lloyd's comment and agree with him. From our ToK classes we have come to realise that we do not know everything that is around us or in our world. Mathematics is therefore one way by which we try to understand the various things around us. However, we all interpret thing and because of this I have a few questions. What makes someone's solution to a problem correct and mine wrong? Why should I be "forced" to accept it if I have a different view on it?! Is mathematics limiting our thinking capabilities in a way?! Just like Lloyd said: "Has the fact that we have all come to accept certain established rules in anyway removed the possibility of 1+1 not being 2?"
And Mama, I think Mathematics "updates" itself naturally. Why else would there be new ways of proving certain concepts or solving solutions?! In all situations the fundamental concepts of mathematics is present but then as to how the concepts are manipulated by each individual bring about the "updating" of Mathematics. This is just by the way, I spoke to Okwase once and he said that when he is solving maths, he sees the figures dancing around in his head!! The ability of you to interact and associate with mathematical concepts improves your understanding. For example, we travelled to Lome to interact with French speaking people so as to better our understanding and control of the language. The same applies to Mathematics as it is also a language.
From a TOK point of view mathematics is a rather special area of knowledge. It seems to supply a certain certainty often missing in other disciplines.What is unarguable is the ability of mathematics to yield important knowledge about the world, often in conjunction with other subject areas.By this, I mean a subject like biology would require some math in carrying out some tasks. Also, I very much agree with shawn on his point that mathematics is a language because the fact that mathematics is universal is incontrovertible and i can also say to some extent that there is great beauty to be found in mathematics, although i know some people who would strongly disagree with me.
This is something i just thought of when reading an article. when you use the Pythagoras formula to find the length of a hypotenuse, you use the formula a(squared)+(b(squared)=c(squared) and this gives the length. but actually when you solve it, you actually get 2 values for your length of the hypotenuse which is the positive value and the negative value. what i think about this is that mathematics also has its faults and that we humans have evaluated mathematics in general, toyed with it(obviously since we created mathematics)till it applies in almost every situation and used it to our benefits, which is true because we have applied mathematics to almost every aspect of our lives from finding the distance between the sun and earth to simple equal sharing of cakes(i remember this was the example my teacher used to teach me fractions!!). so now back to what i really wanted to talk about. maybe these 'millennium' questions may be addressing a part in mathematics that has not been evaluated or understood yet and this is why there is so much controversy concerning these 'millennium' questions. maybe mathematics must be toyed with again??
Mathematics is a particularly interesting area of knowledge. I have always found it surprising that a subject that works so much by logical progression is difficult for many people. I would like to hear other people views on this. Why do so many people feel afraid of mathematics?
What do unsolved math questions tell us about math? I believe that they reminds us of the untapped potential of mathematics, of the vast areas of knowledge which we are yet to realise.
Again, should they remain unsloved, they will be important in marking out the limitations of mathematical method.
I think too that these questions remind us of the importance of other "less justifed" ways of knowing. It shows that despite maths reliance on reasoning, reasoning will not solve all our problems. Sometimes, a leap of faith, intuition is required to make breakthroughs in mathematics.
"Mathematics is like love: simple idea but it can get complicated"
From my experience with Mathematics, I have come to believe that Mathematics is a discipline which requires equal parts logic and creativity. The knowledge derived from Mathematics is more often than not abstract, with little or no real use to the "real" world; as demonstrated by some of the ridiculously complex questions on the claymath website.The essence of Mathematics, therefore, may boil down to its intellectual challenge, and the problem-solving skills it bestows on individuals who pursue it. This could explain how "never-ending" the realm of knowledge within Mathematics is - as more problems are solved, more problems are created (usually far more abstract), to further challenge the intellectual capacity of humans.
Hi Shawn! When i spoke about synaethesia, I was referring to exceptional situations in human perception. A person suffering from synaesthesia for your information can smell colour and see music which is different from the normal. So all I was saying is that there may be people who can actually see numbers. And as Sanaa said Okwasw can actually see the numbers dancing in his head when solving questions. Sanaa, can you confidently give examples of mathematical concepts that have updated themselves over the centuries if you claim that they update automatically? Ampeezy productions, when you say there is beauty to be found in mathematics can you elaborate more and defend your opinion? I really like Kojo’s analogy of mathematics to love. It really shows his emotions towards the subject!!
Amazing comment Eleazer...you said you "have always found it surprising that a subject that works so much by logical progression is difficult for many people." Well! It does not surprise me at all...the problem with math is that there is ALWAYS a right answer somewhere..in the air..and you know that it's there! and even though you have solved it and have gotten AN answer..as Maths is a fixed language with rules and all...you know you are wrong. The logicality (is that a word?) of math is not as simple as you make it Eleazer. Because when I solve a math question and i have got it wrong..not because I made a little mistake but simply because my method does not apply to the rules,my logical functions, or reasoning is not necessarily wrong..there have been so many cases where is it actually right! But because of the RULES..it does not apply..
Unsolved maths questions tell us that math is not an automated subject...It is made up...it is formulated..
I doubt that there are any limitations to maths..I believe that humans will continously search for more ways of understanding measurements and distances and money..etc
I agree that, math does require a "leap of faith"...I think this makes it all the more fascinating..that it is created by the human mind in a sense..
I like maths but i think that it should not look as rigorous as it does...for something which "requires a leap of faith"..for something created..
I'd like to leave with a quote of my own btw..:)
"Mathematics is like mathematics...giving you sevens today and five tommorow"..:)
Looking at the site and what the contest entails, I see that mathematical knowledge is structured so that everything fits together. This tight-knit body of knowledge, built on undisputed theorems and laws is an objective area of knowledge that people from different cultures with different perspectives can relate to in the same way. The objective nature of mathematics lends its problems definite solutions that all can agree upon. Now in response to some of the comments already made, the reason why mathematical theorems and laws that are thought up by people are accepted universally to be correct are because they have not been disproven and this is where mathematics has a link with the natural sciences and the scientific method of obtaining knowledge. There are no disproofs because all these theorems and laws fit together so well and like Tori rightly pointed out perfectly relate real life situations and events. Coming to math being its own language, there is no doubt. I'm sure my fellow math HL mates and even the SL students will agree with me on that. We use notations such as 'x є R', which means 'x is a real number' and 'V є Z+', which also means 'for all positive integers'. Clearly, Mathematics has its own language. Mathematical terms are only used to help learners understand the language. It's like learning any other language like French or German. The teacher has to teach from the known tongue. Some may argue that if it's a language, why is it that we can't express everyday speech in math? Language is conceived out of thought and we realize that all the thoughts that need expression have mathematical representations. Also, the structured nature of mathematics helps us to think through problems and realize which problems should have solutions, which ones should, and which ones can't be decided due to unknown variables. For example, we know that one plus one is two and we know that mathematically and logically, no number can be divided by zero. It just does not make sense. But then x plus y cannot be evaluated because there are not enough known variables. Mathematics has been built on these core principles and further thought and analyses have gone into building this completely logical and for me beautiful area of knowledge.
reading this article at first, i could not believe it. Questions that have not been solved is somehow strange to me because i wonder where those who set these questions are that their questions are still there. if math is supposed to be as easy as the 2+2=4 concept we all know, then i think we are all not using our mathematical knowledge to the fullest. Most of the time we are told to apply what we know to solve questions we do not know so then is it that the application cannot be used in this case?
Mathematical knowledge is quite universal. Even though many people are saying that the change in language does not make it universal, the same thought process you would go through is the same. For instance, if you were asked to multiply 3 by five, or “trois par cinq”, you would still reason out in the same way, because you’ll always add 3 fives times. The symbols that are universally accepted mathematically also help to make math universal. I agree with Nana Kwame that math is on its own language and one way to understand it is if you understand the symbols and terminologies. For instance, it is generally accepted that the ‘s-like’ symbol represents ‘integral’ of something and thus, if a question concerning integration was put before an American, a Chinese , a Greek or a Jew, from their own previous knowledge they would know what to do without putting any words there. Secondly, I believe mathematical knowledge teaches one how to reason logically. For example, through Pythagoras’ theorem, we know that the hypotenuse is shorter than the sum of the other sides. It would therefore not be very reasonable, practically, to travel two perpendicular roads when you can cross it diagonally. I love mathematics because of the interesting ways that it can be used to solve problems in our world. When machines are being made, even computers and other electronic devices, it is mathematics that is needed to manufacture and program them as well. This shows how universal the language is. Lastly mathematics, in my opinion, is very objective (unlike literature) and the results to questions remain the same forever. Even though it seems like a ‘difficult’ subject with its complexities, once one is able to apply the socially accepted math language to life’s processes, it becomes simply interesting.
Mama mentioned something earlier on - "To Yasmine if mathematics is really a language then you should understand me when I say 4489!!!! (an exception to this is 143 which means I LOVE YOU.)" If that is your argument, then Mama YOU should understand me when I say "Pao de Acucar" (lol...by the way, it means "sugar bread" in Portuguese. :] ). Math is different, just like French, or Luganda, or Cantonese. The figures in Math may have meanings in our spoken languages, but it is a language in the sense that (as Sanaa rightly said) it has its own syntax, structure and symbols with their universally accepted meanings. It may not be the usual language, but it is one all the same.
I find Math amazing because you can take a statement or idea, and use Math's own rules to prove it and arrive at the answer, sometimes in ways you would never expect or think of. The fact that we can create (or do we discover? - another question to consider) complex problems that many people cannot solve is also very interesting.
I also think that in general we can rely on our mathematical knowledge as long as its right. This is because Math is conventional, and as far as I know it has strict truths that can be proved and which we must obey. Thus, it's either the right answer, or the wrong answer (thankfully)!
Finally, we can, of course, never underestimate the usefulness of Math in everyday life though the average individual may never find use for those complex problems worth 7 million dollars and all. No need to remind you the countless times you were probably reminded by several schoolteachers that Math is everywhere. And the fact that we can even find a problem that someone can pay milions of dollars to have solved shows just how much value we put into this area of knowledge.
Mathematics as an area of knowing is used universally for instance as Ruth said a french saying trios par cinq will do the same thing a Spaniard will do when told tres par cinco and an english when told three times five.This makes math a universal language.Math can also be said to a way a reasoning. When we studied reasoning we learnt that reasoning can either be deductive or inductive. looking at math , when given 2x+3=1 and asked to solve, you will simplify the expression further and further till you arrive at your final answer making math a form a deductive reasoning.This means using math is a form of reasoning. Also, the fact that those mathematical problems haven't been soled is probably due to the reason that no one has been able to reason that deep to be able to solve it. also, mathematical knowledge is obtained through observation and reasoning.
math is another language on its own. a language of only right or wrong. it just involves being able to see through things, predicting, reasoning and understanding. in math we are just accept what was set no addition nor subtraction is needed. its a language that requires deep reasoning in some cases.
Really what is mathematics made up of it is basically symbols which are further complicated to language and so then that is when language plays an important part because 1 is one (in English) moja in swahili and en in french so this shows the complex nature of these symbols so how is it that we all understand it all i think the way we say these mathematical figures also shapes the way we see life for example in English 6 oclock is the sixth hour or 18th hour of the day but if we are to take swahili 6 oclock is the first hour of the day and the first hour of the night and so mathematics plays a part in building our cultures as a collection and it involves reasoning because there is a reason why in English its the 6th and in swahili is the first and somathematics just helps in most of the applications of the world.
Mathematics is useful in our day to life but I honestly don’t understand how some concepts can be applied to our lives, concepts like the logarithms. But it has proved to be very helpful in our day to day lives. Math sharpens the mind.
Maths is a concept that each individual applies to their daily life whether they know it or not really depends on the individual however maths is a universally accepted language no matter how many languages you teach it in every question will have the same answer e.g. 4 plus 4 equals 8 however it is the method that is undergone that there happens to be a variation in skill and language. As we all know math has many equations however there are different messages that can give you the same answer .It is usually through or instinct that determines what method we use. However with languages e.g. French Spanish and English the same thing will be presented in a completely different form e.g. cat, chat and gato.A non English speaker would have difficulty understanding the word cat however if it is in their mother tongue the situation is completely different. The only thing, English and maths have in common is they can both be defined as a language.
Maths is a concept that each individual applies to their daily life whether they know it or not really depends on the individual however maths is a universally accepted language no matter how many languages you teach it in every question will have the same answer e.g. 4 plus 4 equals 8 however it is the method that is undergone that there happens to be a variation in skill and language. As we all know math has many equations however there are different messages that can give you the same answer .It is usually through or instinct that determines what method we use. However with languages e.g. French Spanish and English the same thing will be presented in a completely different form e.g. cat, chat and gato.A non English speaker would have difficulty understanding the word cat however if it is in their mother tongue the situation is completely different. The only thing, English and maths have in common is they can both be defined as a language.
As one of the oldest areas of knowledge, Mathematics is the paragon of certain and unchanging knowledge. For instance: 2 x 2, will always be 4, because someone has made it so, and society has decided to accept it as that. I am not saying that mathematics is entirely comprised of fixed certainties, but the majority of concepts in this field of knowledge have already been predetermined, agreed on, and accepted. It’s amazing how signs and symbols that have been arbitrarily selected by men and women long forgotten are still applicable today. Truly, have you ever considered what exactly “=” means. How do you “+” things? Even in combinations such as “6+1=7”, we are not referring to anything in our world, only simply to things conctructed in one human mind! What is baffling is that math shows flexibility as it is still applicable to our ever-changing world. Through common agreement, we all know we are talking about realities when we say: “Six books plus one book would give us seven books”. Thus what mathematics has helped us do is to merely simplify the task of describing things in our real world. Mathematical knowledge has therefore been inspired by the real world, and must not be mistaken to have really existed on its own. In many ways, mathematics is like an art. People picture mathematics as solving immensely complicated problems by means of massive number crunching, but few things could be further from the truth. Although much of math relies on nature, it is still an artificial way of knowing in that, it is constructed by the human brain. The good thing about it though, is that it deals with absoluteness, and as such, it is considered the foundation of all other sciences, because it makes no hypotheses, or assumptions – in math, it is either right or wrong!
ReplyDeleteMr Kitching you make the task sound so easy by saying, "All you have to do..."
ReplyDeleteTo me, this proves the complex nature of math especially since it has the tendency of becoming increasingly difficult as you progress in the subject. One would think that by mastering +,-,* and / you would at least more easily understand as you grow in math. And yet then comes adding numbers to letters, differentiation, the root of a negative number and why is there an 'e'???
Mathematical knowledge in my opinion follows its own 'rules.' I say this because it appears as though math has its own language (although surprisingly when I had the 'wonderful' opportunity of having a maths lesson "en français" it made absolutely little or no sense to me and I wondered if math truly was a universal language)
I do not want to write a 'looong' blog that I know hardly anyone would read if they saw such a length but basically I think mathematically knowledge is definitely abstract and conventional. Of course it is very complex but quite enjoyable to those who enjoy a good challenge
To me, although I have not always been the best at it, math has always been a safe haven when it came to knowledge. In my mind it was meant to be easy because basically any problem you could ever come across has already been solved you just have to have all the knowledge. Mathematical knowledge is set with all its formulas and rules. Like the previous comment 2+2 will always be 4. However, if this was Literature we'd be told there is no right or wrong answer. That to me is more confusing than anything I have ever encountered in mathematics.
ReplyDeleteIn regards to it being a universal language I'm not so sure. Yeah numbers are the same in any language but then with mathematical terms they have to be translated. Trigonometry is not the same in French or Spanish. Basically all I'm trying to say is mathematical knowledge is more accessible than others.
Interesting comments, Tori. You have alluded to the irony that on the one hand mathematics seems entirely divorced from the world we live in (eg nobody walks to school and finds a "2" hiding under a rock), yet at the same time it seems to describe some aspects of that world incredibly accurately (eg look at the simple mathematical relations that lie at the root of so many scientific laws). You wrote:
ReplyDelete"...2 x 2, will always be 4, because someone has made it so, and society has decided to accept it as that..."
If you are talking about the symbols (numerals)that are used in mathematics then this makes sense (the conventional nature of numerals is borne out by the experience I once had of trying to work out prices in Egypt). But surely the abstract quantities that these symbols represent NECESSARILY obey this relationship.
Yasmin - I like your anecdote about learning mathematics in French. Perhaps this tells us something about how mathematics is taught and spread - we do not talk "in" mathematics all the time we are doing it.
Chelsea - you wrote:
"...basically any problem you could ever come across has already been solved..."
I know what you mean, but what about the seven "millenium problems" referred to in the link? What does the fact that they have NOT been solved tell us about mathematical knowledge?
hi guys! :)
ReplyDeletei would like to disagree with yasmin about having mathematics in french. we both had the same experience at the same time. personally, i think mathematics its a language on its own with its own syntax. this is because mathematics itself does not change, whether in french or spanish or english...for instance, using the same example, you may be told to differentiate something but in french you would be told to "................". however, it is the same process. this brings me to mr. kitching's point that we do not always talk in mathematics. we use numbers.
2x2=4. BUT there are different words for "2" and "4" in the different languages. thats what makes the difference...NOT mathematics itself.
and about the questions which have not been solved, i am confused. how were the questions made? and is it that they have never been solved before or that no one has got the answer correct? i think this would mean that our mathematical knowledge has not reached the level of the question and so we cannot solve it......im nt sure.....
Mz.Appiah-Korang
Hi Mz.Appiah-Korang,
ReplyDeleteI think I understand your confusion. I think it is the use of the word "problem" (ha, language - there we go again!).
You need to distinguish between the kind of "problems" that your mathematics teacher often gives you, which are questions or exercises for you to get through in order to reach the correct answer (which is already known in advance, at least by the person who wrote the question, and is maybe in the back of the book!), and the problems that some professional mathematicians deal with, such as those identified as the millenium problems. And when we come to discuss mathematics in detail, we will look at one of these millenium problems more carefully, namely...
The Riemann hypothesis is a conjecture about the distribution of prime numbers. Most mathematicians seem to think that it is true, but no-one has ever been able to prove it.
[Reminder - TOK is about knowledge claims, the justifications offered for these knowledge claims, and the knowledge issues that arise from the attempt to provide acceptable justifications.]
So think of the Riemann hypothesis as a knowledge claim in mathematics which needs to be justified if it is to be accepted as true. But in mathematics, the standards for justification are very high (what kind of justification will suffice in mathematics, and why the standard has to be so high in mathematics - these are examples of related knowledge issues) - nothing less that a formal proof will do. And no-one has been able to provide one. Yet.
So what have we learned so far about mathematical knowledge?
Hey,
ReplyDeleteI agree with Mz. Appiah-Korang on the notion that maths does not have different languages. It is just explained or taught in different languages, the processes used in problem solving are however, the same.
Tori said:
"...2 x 2, will always be 4, because someone has made it so, and society has decided to accept it as that...this field of knowledge have already been predetermined, agreed on, and accepted...we are not referring to anything in our world, only simply to things conctructed in one human mind!"
I don't agree with your point that this field of knowledge is "predetermined" and that "someone has made it so" simply because multiplication (addition, subtraction or division for that matter) are concepts that can easily be proved and demonstrated using the simplest using things around. From a simple transaction with a hawker on the street, to determining the rate of flow of a waterfall, math is simply representing some of the things around us in a manner that is more comprehensible and is therefore not "simply...things constructed in one human mind."
People just keep on finding simpler or more accurate ways of representing situations mathematically.
I will not say the millennium questions are unsolvable, I will say a (simpler) way of representing the problem mathematically (and thus proving it) with already gained knowledge has not yet been discovered.
HI!
ReplyDeleteI also totally agree with Shawn and Mz.Appiah Korang!! Mathematics is the same all over the world!! I also had the wonderful experience of studying Mathematics in French (with Shawn, Yasmin and Maame) and initially I thought I wouldn’t understand anything that was going to be thought but then I did understand it after all. It was just symbols and numbers that I had to interpret!!! All over the world there are set conventions and symbols in mathematics that cut across and somewhat make Maths a universal language. Whether in English or French or Russian or Spanish, maths is still maths. If it wasn’t so, how else would we all write the same international papers and be assessed with the same criteria?!!
Mathematics has its basic rules upon which people try to develop and find out new ways of representing everyday life occurrences. An actuary one once told me that in calculating considered every possible situation that might occur when they are calculating insurance premiums, risks, dividends, and annuity rates. He even sent me an amazingly long “mathematical equation” he formed to calculate the cash flows of a proposed football field and it included the attendance on a rainy day and even the attendance when the team had lost its previous match. This example is just to show how maths is inherent in and applied to our everyday lives. This is also to support Shawn’s statement:
Maths is simply representing some of the things around us in a manner that is more comprehensible and is therefore not "simply...things constructed in one human mind."
I believe the millennium questions are definitely solvable even though it seems impossible now. Just because we do not know something or know how to express it does not mean it is not there. By extension, in Mathematics just because we cannot solve the questions doesn’t mean the solution doesn’t exist.
This is just a question I have: Can Mathematics be classified as a language because it has its own syntax and symbols that mean the same thing all the time and everywhere?! I remember in IG when we were using the scientific calculators, you could get the message “Syntax error” if you didn’t input the argument right!! It seems to have it’s own structure, words and semantics!!
ha ha ha!!!
ReplyDeleteYeah Yasmin, math from this question is really abstract. how do we even know what we solve every day is correct? Because if it is, any math student with a degree or something higher than that should be able to solve this question. Unfortunately i am certain that less than twenty percent of these intellectuals will barely make it to the correct answer.
In my view, this math question is symbolic of the distorted ideas, reasoning and perception of things in the world which makes the world rather complex and questionable. This proves that idealistic humans in the world don’t always find solutions to problems in the world though they try their best this math question proves its difficulty.
Funny how you are taught from grade 1-6 that math is only numbers but as you climb higher, you realise math involves alphabets and even English. From all these abstract things you have to reason like your teachers expect you to reason and think the way the world requires you to think like a math student. How are we even certain that such a question like this can be solved and how are we able to add letters to numbers to give us numbers. Well, based on inferred knowledge i don’t remember my grade one teacher giving me a name for the mixture of alphabets and numbers but in grade seven my teacher named it algebra. Algebra? I wonder how i was able to solve algebraic problems. Whether it’s logical or not math will forever be math unlike the humanities where what is taught involves you and requires you to reason and empathise to arrive at a solution, math requires you to accept its facts and apply.
Wow, what a world!!!!!!!!!
Timah
Mathematical knowledge is based mainly on language.i.e. Mathematical language, reasoning and reason. To me mathematics helps us to represent certain situations for example 2 oranges plus 2 oranges will give you four oranges. But the real question is what is 2? In that I am Mathematics although many laws and theories can be proven through most of the time, induction is very very abstract? For example, if you think about it many of us have seen the functions cos, sin, tan,and many others but thinking about what are they really? For example most of us know cos(90) is 0 but the question is what does the cos do the 90 to make it 0.I would therefore say mathematical knowledge is mainly based on facts discovered or made by somebody who was able to prove it and we as human have accepted it to be so.
ReplyDeleteMathematical knowledge or language,in my opinion follows its own rules and format in terms of the various ways to get a right answer. There is a complete difference between mathematical knowledge and lets say... Literature. Mathematical knowledge helps us to apply simple things in daily experience, for example if we go to a supermarket and buy something for GhC8 and we hand over GhC10 mathematical knowledge should tell us that we should recieve GhC2. Mathematics has definate answers according to the different hypotheses and laws created by our forefathers. There is no question about that. The only way we can even begin to question mathematical knowledge, is if previous theories have come out with the wrong or unexpected answers(after many trials of course). In literature on the other hand, we cannot say that something can only have one answer, it depends upon the analysis of the reader. Knowledge in terms of literature does not follow any specific format, it is a subjective subject (lol). It allows the interpretation of many individuals. Math on the other hand does not have this subjectivity.
ReplyDeleteFrom the mind boggling tasks presented to us to win $1 million it does show that mathematics on its own is a language that needs to be understood and justified before it can be accepted. I for one, think that that theory cannot be done because it just doesn't make sense to me(i am not a math person) but if someone does manage to solve it i will have to be forced to accept it.
I agree with Chelsea up to a certain point. Math is a safe haven when it comes to knowledge. With math an answer is either right or wrong. it is simple and very straight forward. however, literature is not so. an answer is seldom wrong when expressed by a person in such a way that it is their opinion that is being expressed as a result it (to me) is a poor way of acquiring knowledge
ReplyDeleteHOWEVER, i disagree with her when she says 'In regards to it being a universal language I'm not so sure. Yeah numbers are the same in any language but then with mathematical terms they have to be translated. Trigonometry is not the same in French or Spanish. Basically all I'm trying to say is mathematical knowledge is more accessible than others.' this is because mathematical concepts will remain the same in every language and will never change. Even though 'trigonometry' might have a different spelling or be expressed differently it does not necessarily mean that their concepts would be different. 1+1 would be 2 in any language no matter how the term 'plus' or 'addition' is expressed
It is a remarkable phenomenon that children can learn to speak without ever being consciously aware of the sophisticated grammar they are using. Indeed, adults too can live a perfectly satisfactory life without ever thinking about ideas such as parts of speech, subjects, predicates or subordinate clauses. Both children and adults can easily recognize ungrammatical sentences, at least if the mistake is not too subtle, and to do this it is not necessary to be able to explain the rules that have been violated. Nevertheless, there is no doubt that one’s understanding of language is hugely enhanced by a knowledge of basic grammar - it is almost tautologous to say so - and this understanding is essential for anybody who wants to do more with language than use it unreflecting as a means to a non-linguistic end.
ReplyDeleteThe same is true of mathematical language. Up to a point, one can do and speak mathematics without knowing how to classify the different sorts of words one is using, but many of the sentences of advanced mathematics have a complicated structure that is much easier to understand if one knows a few basic terms of mathematical grammar. The main reason for the importance of mathematical grammar is that the statements of mathematics are supposed to be precise, and it is not possible to achieve a high level of precision unless the language one uses is free of many of the vagueness and ambiguities of ordinary speech. Mathematical sentences can also be highly complex: if the parts that made them up were not clear and simple, then the obscurities would rapidly propagate and multiply and render the sentences unintelligible.
For me the fact that there exists certain mathematical concepts that are yet to be proven just shows the complexity of the reality (our world) that mathematics as an area of knowledge is trying to make simpler or understand. Mathematics is just like finding existing patterns that exist in our world and using them as a basis for postulations, where in this case we see the whole logical aspect of it. The fact still remains that these patterns may not hold for all cases in our world hence the need for justification of mathematical knowledge.
ReplyDeleteFirst of all mathematics, as a way of knowing is universal (public knowledge) because it has its own conventionally accepted language hence is not at the mercy of linguistic misinterpretation or any other problem of that sort. (If everybody developed his/ her own language for mathematics we would hardly ever agree on any knowledge gained through mathematics and the purpose of mathematics would be defeated).
Then again the fact that these almost-impossible-to-solve concepts have not been solved/proven does not mean they would never be solved I think it’s just a matter of time, for as humans develop, our areas of knowledge become better and better and increase the accuracy our interpretations of our world.
The usual absolute nature of mathematics also brings up the question as to whether there is anything like mathematics provides us with any absolute truth. Is 17+ (-6) always equal to 11. Has mathematics restricted our thinking in anyway? Has the fact that we have all come to accept certain established rules in anyway removed the possibility of 1+1 not being 2.
Asantewaa Agyare: "but if someone does manage to solve it i will have to be forced to accept it."
Hi guys!
To me, mathematics is the least enjoyable subject and I sometimes wonder what a world without mathematics would be, perhaps we would have no sense of measurement. Mathematics is believed to exist in nature already with concepts like the natural log of e, etc. The one which interests me most is the “GOLDEN RATIO” which in summary states that all things in nature are equally divided. Can you imagine a concept which can be used to determine if a person is actually beautiful?
ReplyDeleteMathematics is indeed very important in our world as we look back at Egyptian sculptures and buildings, art works, scientific discovery even in language. When you are given a TOK essay to write you are given a word limit and this only possible because of the existence of numbers.
You all make it seem as if mathematics is all about 1+1=2 but you have to know what operation to use at what time because you cannot definitely simply use this to solve a question that requires Pythagoras theorem. Have you ever wondered what one drop of water + one drop of water is equal to?
How sure we that all these laws and formulas are are correct? How many of us have tried to prove Pythagoras theorem or even question its validity? Is it even possible that new laws can be formulated in our age??
With regards to mathematics being a universal language I would refer to it as public knowledge as we as a society have come to appreciate and accept its concepts without any question. To Yasmine if mathematics is really a language then you should understand me when I say 4489!!!! (an exception to this is 143 which means I LOVE YOU.)
I agree with Mr. Kitching when he says we cannot find the number 2 under a rock but are we really sure that there are people who do not see numbers lying around?? Take synaesthesia for example many of us thought it weird that some people smell colour but this is real.
Katakyie how sure are you that mathematics is static when we are faced with newer challenges to what existed in the past when the mathematical concepts were being created. This may be the reason why the questions have not been answered. Maybe mathematics needs a bit of updating just like scientific discoveries have been updated over the centuries.
Ntiriwa, when you say 4489 you are quantifying something,however what is being quantified has not been stated. (ambiguity of language?) Mathematics is therefore a language :). I however do not understand how synesthesia is linked with your point?
ReplyDeleteWell, Math is honestly a bit of confusion to me.Like what Mama said its the least enjoyable subject, for me too.Looking at the fact that we can not live without Math whether we like it or not is something which comes to me like a BANG!! I got the following http://www.trap17.com/index.php/world-math_t49606.html and it says
ReplyDelete'I mean we don't really know what we are missing until it's missing. But there wouldn't be equations or anything to figure out problems. Proper dimensions when building something wouldn't really be available. Systematic systems in factories based on precision wouldn't really be available. It's sort of hard to imagine or even for me to explain. There wouldn't really be a currency, just trading one thing for another maybe. Just when you finally think that Math is quite useless just wait to see what would happen if there was no Math in the world.
Hi everyone!
ReplyDeleteI like Lloyd's comment and agree with him. From our ToK classes we have come to realise that we do not know everything that is around us or in our world. Mathematics is therefore one way by which we try to understand the various things around us. However, we all interpret thing and because of this I have a few questions.
What makes someone's solution to a problem correct and mine wrong?
Why should I be "forced" to accept it if I have a different view on it?!
Is mathematics limiting our thinking capabilities in a way?!
Just like Lloyd said: "Has the fact that we have all come to accept certain established rules in anyway removed the possibility of 1+1 not being 2?"
And Mama, I think Mathematics "updates" itself naturally. Why else would there be new ways of proving certain concepts or solving solutions?! In all situations the fundamental concepts of mathematics is present but then as to how the concepts are manipulated by each individual bring about the "updating" of Mathematics.
This is just by the way, I spoke to Okwase once and he said that when he is solving maths, he sees the figures dancing around in his head!! The ability of you to interact and associate with mathematical concepts improves your understanding. For example, we travelled to Lome to interact with French speaking people so as to better our understanding and control of the language. The same applies to Mathematics as it is also a language.
Bye guys!!
From a TOK point of view mathematics is a rather special area of knowledge. It seems to supply a certain certainty often missing in other disciplines.What is unarguable is the ability of mathematics to yield important knowledge about the world, often in conjunction with other subject areas.By this, I mean a subject like biology would require some math in carrying out some tasks. Also, I very much agree with shawn on his point that mathematics is a language because the fact that mathematics is universal is incontrovertible and i can also say to some extent that there is great beauty to be found in mathematics, although i know some people who would strongly disagree with me.
ReplyDeleteaMPEEZy pRODUCTIONs
This is something i just thought of when reading an article. when you use the Pythagoras formula to find the length of a hypotenuse, you use the formula a(squared)+(b(squared)=c(squared) and this gives the length. but actually when you solve it, you actually get 2 values for your length of the hypotenuse which is the positive value and the negative value. what i think about this is that mathematics also has its faults and that we humans have evaluated mathematics in general, toyed with it(obviously since we created mathematics)till it applies in almost every situation and used it to our benefits, which is true because we have applied mathematics to almost every aspect of our lives from finding the distance between the sun and earth to simple equal sharing of cakes(i remember this was the example my teacher used to teach me fractions!!). so now back to what i really wanted to talk about. maybe these 'millennium' questions may be addressing a part in mathematics that has not been evaluated or understood yet and this is why there is so much controversy concerning these 'millennium' questions. maybe mathematics must be toyed with again??
ReplyDeleteMathematics is a particularly interesting area of knowledge. I have always found it surprising that a subject that works so much by logical progression is difficult for many people. I would like to hear other people views on this.
ReplyDeleteWhy do so many people feel afraid of mathematics?
What do unsolved math questions tell us about math? I believe that they reminds us of the untapped potential of mathematics, of the vast areas of knowledge which we are yet to realise.
Again, should they remain unsloved, they will be important in marking out the limitations of mathematical method.
I think too that these questions remind us of the importance of other "less justifed" ways of knowing. It shows that despite maths reliance on reasoning, reasoning will not solve all our problems. Sometimes, a leap of faith, intuition is required to make breakthroughs in mathematics.
"Mathematics is like love: simple idea but it can get complicated"
Very touching Eleazor...
ReplyDeleteFrom my experience with Mathematics, I have come to believe that Mathematics is a discipline which requires equal parts logic and creativity. The knowledge derived from Mathematics is more often than not abstract, with little or no real use to the "real" world; as demonstrated by some of the ridiculously complex questions on the claymath website.The essence of Mathematics, therefore, may boil down to its intellectual challenge, and the problem-solving skills it bestows on individuals who pursue it. This could explain how "never-ending" the realm of knowledge within Mathematics is - as more problems are solved, more problems are created (usually far more abstract), to further challenge the intellectual capacity of humans.
ReplyDeleteHi Shawn!
ReplyDeleteWhen i spoke about synaethesia, I was referring to exceptional situations in human perception. A person suffering from synaesthesia for your information can smell colour and see music which is different from the normal. So all I was saying is that there may be people who can actually see numbers. And as Sanaa said Okwasw can actually see the numbers dancing in his head when solving questions.
Sanaa, can you confidently give examples of mathematical concepts that have updated themselves over the centuries if you claim that they update automatically?
Ampeezy productions, when you say there is beauty to be found in mathematics can you elaborate more and defend your opinion?
I really like Kojo’s analogy of mathematics to love. It really shows his emotions towards the subject!!
Amazing comment Eleazer...you said you "have always found it surprising that a subject that works so much by logical progression is difficult for many people." Well! It does not surprise me at all...the problem with math is that there is ALWAYS a right answer somewhere..in the air..and you know that it's there! and even though you have solved it and have gotten AN answer..as Maths is a fixed language with rules and all...you know you are wrong.
ReplyDeleteThe logicality (is that a word?) of math is not as simple as you make it Eleazer. Because when I solve a math question and i have got it wrong..not because I made a little mistake but simply because my method does not apply to the rules,my logical functions, or reasoning is not necessarily wrong..there have been so many cases where is it actually right! But because of the RULES..it does not apply..
Unsolved maths questions tell us that math is not an automated subject...It is made up...it is formulated..
I doubt that there are any limitations to maths..I believe that humans will continously search for more ways of understanding measurements and distances and money..etc
I agree that, math does require a "leap of faith"...I think this makes it all the more fascinating..that it is created by the human mind in a sense..
I like maths but i think that it should not look as rigorous as it does...for something which "requires a leap of faith"..for something created..
I'd like to leave with a quote of my own btw..:)
"Mathematics is like mathematics...giving you sevens today and five tommorow"..:)
Looking at the site and what the contest entails, I see that mathematical knowledge is structured so that everything fits together. This tight-knit body of knowledge, built on undisputed theorems and laws is an objective area of knowledge that people from different cultures with different perspectives can relate to in the same way. The objective nature of mathematics lends its problems definite solutions that all can agree upon.
ReplyDeleteNow in response to some of the comments already made, the reason why mathematical theorems and laws that are thought up by people are accepted universally to be correct are because they have not been disproven and this is where mathematics has a link with the natural sciences and the scientific method of obtaining knowledge. There are no disproofs because all these theorems and laws fit together so well and like Tori rightly pointed out perfectly relate real life situations and events.
Coming to math being its own language, there is no doubt. I'm sure my fellow math HL mates and even the SL students will agree with me on that. We use notations such as 'x є R', which means 'x is a real number' and 'V є Z+', which also means 'for all positive integers'. Clearly, Mathematics has its own language. Mathematical terms are only used to help learners understand the language. It's like learning any other language like French or German. The teacher has to teach from the known tongue. Some may argue that if it's a language, why is it that we can't express everyday speech in math? Language is conceived out of thought and we realize that all the thoughts that need expression have mathematical representations.
Also, the structured nature of mathematics helps us to think through problems and realize which problems should have solutions, which ones should, and which ones can't be decided due to unknown variables. For example, we know that one plus one is two and we know that mathematically and logically, no number can be divided by zero. It just does not make sense. But then x plus y cannot be evaluated because there are not enough known variables. Mathematics has been built on these core principles and further thought and analyses have gone into building this completely logical and for me beautiful area of knowledge.
reading this article at first, i could not believe it. Questions that have not been solved is somehow strange to me because i wonder where those who set these questions are that their questions are still there. if math is supposed to be as easy as the 2+2=4 concept we all know, then i think we are all not using our mathematical knowledge to the fullest. Most of the time we are told to apply what we know to solve questions we do not know so then is it that the application cannot be used in this case?
ReplyDeleteEsther
Mathematical knowledge is quite universal. Even though many people are saying that the change in language does not make it universal, the same thought process you would go through is the same. For instance, if you were asked to multiply 3 by five, or “trois par cinq”, you would still reason out in the same way, because you’ll always add 3 fives times. The symbols that are universally accepted mathematically also help to make math universal. I agree with Nana Kwame that math is on its own language and one way to understand it is if you understand the symbols and terminologies. For instance, it is generally accepted that the ‘s-like’ symbol represents ‘integral’ of something and thus, if a question concerning integration was put before an American, a Chinese , a Greek or a Jew, from their own previous knowledge they would know what to do without putting any words there.
ReplyDeleteSecondly, I believe mathematical knowledge teaches one how to reason logically. For example, through Pythagoras’ theorem, we know that the hypotenuse is shorter than the sum of the other sides. It would therefore not be very reasonable, practically, to travel two perpendicular roads when you can cross it diagonally.
I love mathematics because of the interesting ways that it can be used to solve problems in our world. When machines are being made, even computers and other electronic devices, it is mathematics that is needed to manufacture and program them as well. This shows how universal the language is.
Lastly mathematics, in my opinion, is very objective (unlike literature) and the results to questions remain the same forever. Even though it seems like a ‘difficult’ subject with its complexities, once one is able to apply the socially accepted math language to life’s processes, it becomes simply interesting.
Mama mentioned something earlier on - "To Yasmine if mathematics is really a language then you should understand me when I say 4489!!!! (an exception to this is 143 which means I LOVE YOU.)" If that is your argument, then Mama YOU should understand me when I say "Pao de Acucar" (lol...by the way, it means "sugar bread" in Portuguese. :] ). Math is different, just like French, or Luganda, or Cantonese. The figures in Math may have meanings in our spoken languages, but it is a language in the sense that (as Sanaa rightly said) it has its own syntax, structure and symbols with their universally accepted meanings. It may not be the usual language, but it is one all the same.
ReplyDeleteI find Math amazing because you can take a statement or idea, and use Math's own rules to prove it and arrive at the answer, sometimes in ways you would never expect or think of. The fact that we can create (or do we discover? - another question to consider) complex problems that many people cannot solve is also very interesting.
I also think that in general we can rely on our mathematical knowledge as long as its right. This is because Math is conventional, and as far as I know it has strict truths that can be proved and which we must obey. Thus, it's either the right answer, or the wrong answer (thankfully)!
Finally, we can, of course, never underestimate the usefulness of Math in everyday life though the average individual may never find use for those complex problems worth 7 million dollars and all. No need to remind you the countless times you were probably reminded by several schoolteachers that Math is everywhere. And the fact that we can even find a problem that someone can pay milions of dollars to have solved shows just how much value we put into this area of knowledge.
Mathematics as an area of knowing is used universally for instance as Ruth said a french saying trios par cinq will do the same thing a Spaniard will do when told tres par cinco and an english when told three times five.This makes math a universal language.Math can also be said to a way a reasoning. When we studied reasoning we learnt that reasoning can either be deductive or inductive. looking at math , when given 2x+3=1 and asked to solve, you will simplify the expression further and further till you arrive at your final answer making math a form a deductive reasoning.This means using math is a form of reasoning. Also, the fact that those mathematical problems haven't been soled is probably due to the reason that no one has been able to reason that deep to be able to solve it. also, mathematical knowledge is obtained through observation and reasoning.
ReplyDeletemath is another language on its own. a language of only right or wrong. it just involves being able to see through things, predicting, reasoning and understanding. in math we are just accept what was set no addition nor subtraction is needed. its a language that requires deep reasoning in some cases.
ReplyDeleteReally what is mathematics made up of it is basically symbols which are further complicated to language and so then that is when language plays an important part because 1 is one (in English) moja in swahili and en in french so this shows the complex nature of these symbols so how is it that we all understand it all i think the way we say these mathematical figures also shapes the way we see life for example in English 6 oclock is the sixth hour or 18th hour of the day but if we are to take swahili 6 oclock is the first hour of the day and the first hour of the night and so mathematics plays a part in building our cultures as a collection and it involves reasoning because there is a reason why in English its the 6th and in swahili is the first and somathematics just helps in most of the applications of the world.
ReplyDeleteMathematics is useful in our day to life but I honestly don’t understand how some concepts can be applied to our lives, concepts like the logarithms. But it has proved to be very helpful in our day to day lives. Math sharpens the mind.
ReplyDeleteMaths is a concept that each individual applies to their daily life whether they know it or not really depends on the individual however maths is a universally accepted language no matter how many languages you teach it in every question will have the same answer e.g. 4 plus 4 equals 8 however it is the method that is undergone that there happens to be a variation in skill and language. As we all know math has many equations however there are different messages that can give you the same answer .It is usually through or instinct that determines what method we use. However with languages e.g. French Spanish and English the same thing will be presented in a completely different form e.g. cat, chat and gato.A non English speaker would have difficulty understanding the word cat however if it is in their mother tongue the situation is completely different. The only thing, English and maths have in common is they can both be defined as a language.
ReplyDeleteMaths is a concept that each individual applies to their daily life whether they know it or not really depends on the individual however maths is a universally accepted language no matter how many languages you teach it in every question will have the same answer e.g. 4 plus 4 equals 8 however it is the method that is undergone that there happens to be a variation in skill and language. As we all know math has many equations however there are different messages that can give you the same answer .It is usually through or instinct that determines what method we use. However with languages e.g. French Spanish and English the same thing will be presented in a completely different form e.g. cat, chat and gato.A non English speaker would have difficulty understanding the word cat however if it is in their mother tongue the situation is completely different. The only thing, English and maths have in common is they can both be defined as a language.
ReplyDelete