Can mathematics be characterized as a universal language?

First look at the cartoon on this link:

http://www.cs.utah.edu/~draperg/cartoons/jungle.html

This author/artist views math as being a usable universal language. I however, strongly disagree that I could communicate using math ONLY with a person who spoke a different language. The mathematical symbols may be universal and the concepts may be universal, but unless one understands the concepts in his or her own respective language FIRST, there is no purpose in attempting to communicate. For example, I could write the problem 2 x 3 = ___ , take it to one of the German exchange students, and she would (hopefully) write the number 6. She knows to write the number six because she has been taught IN GERMAN that the little "x"means multiply just like I have been taugh IN ENGLISH what the "x"represents. Without our own languages, we would have no idea what to write in the little blank after those two parallel lines becuase we would have no grasp of the concept of multiplication. (ex quote: "There are 10 kinds of people in the world, those who understand binary math, and those who don't" -anonymous)
Also, without other languages, the answer 6 is completely devoid of meaning. The 2 and 3 and 6 are all supposed to represent something else, but there is no common language to illustrate what they are representing. If there are three people and each person has two oranges, obviously there are 6 oranges involved...but I don't know how to say orange in German, I can only write arbitrary numbers down on a sheet of paper. Yikes! the bell just rang, more writing later!!

Prime Time

What do you think of this?

Are all mathematical statements either true or false?


I think I have to go to Kurt Godel, the mathematician/logician to answer this question properly. And in agreement with him, I'm going to say no; mathematical statements may be true, false, or undecidable. Godel defined such undecidable statements as those that may be true, but cannot be proven true or false.

Godel published his work Principia Mathematica in which he arrived at the "incompleteness theorem." The incompleteness theorem is often compared to the old philosophers' trick: "This statement is false." Is that statement true or false? Of course, if it were true, then it would be false. If it were false, then it would be true... It's an undecidable paradox. Godel applied this idea to mathematics with the incompleteness theorem that basically said the following:

"Godel essentially constructed a formula that claims that it is unprovable in a given formal system. If provable, it would be false, which contradicts the fact that in a consistent system, provable statements are always true. Thus there will always be at least one true but unprovable statement."
(Don't judge me, but that's Wikipedia's surprisingly concise description of the Incompleteness Theorem.)

As a result, mathematics is not simply statements that are either true or false. It is, as Godel revealed, a "logical mess," like language or anything else.

What does it mean to say that mathematics can be regarded as a formal game devoid of intrinsic meaning? If this is the case, how can there be such a wealth of applications in the real world?

Mathematics is a formal game devoid of intrinsic meaning because it begins and ends with arbitrary human inventions. A mathematical investigation starts with formulas, numbers, variables and the like -- all of which mean absolutely nothing if not placed in a very specific context and subjected to intense scrutiny. This separates mathematics from other areas of study because its results, as well as its questions and processes, are not of any value alone -- they can only be used to explain or relate to other ideas. The "discovery" of a new mathematical theorem simply means that someone has designed a new way to play the game -- a new way to use an artificial system to produce artificial answers from artificial input.

This being said, there are certainly fields relating to and using mathematics which are quite valuable and meaningful. Mathematical formulas are valuable shortcuts in solving problems in various areas of science which do produce intrinsically significant results. Though the specific formulas used in designing jets are arbitrary, the fact that they can help create a flying machine is useful.

These applications exist because mathematics has been designed around them -- we made the rules, so we can play the game wherever we want. We fool ourselves into thinking that we understand the world because we have discovered the mathematical truths at its core when in reality we have simply attempted to mirror what was already there. The examples of natural patterns Lillian mentioned are indeed all mathematical. However, this does not mean that we "found mathematics" after careful study of nature.

The patterns are there no matter how we describe them. Mathematicians cannot claim to have "discovered" mathematics in nature -- the patterns were in nature; they designed the mathematical system around them. I could just as easily (and more directly and specifically) describe the same phenomena in words. Mathematics is necessary because complex situations would render the use of words far too cumbersome -- we streamline and stylize the situation by using our system. The complex problems of the modern world are greatly facilitated by mathematics -- just we might design a game to aid in our understanding of any occurrence. The game has no value in itself -- the points are meaningless, the rules are arbitrary, and the score is not an ultimate answer to any question. However, the game of mathematics can lead to greater understandings just as a simulation involving students orbiting around a room might serve as an analogy for the solar system.

The fact that there are additional patterns in nature which cannot be described by our mathematical system serves as definitive proof of the seperation between natural patterns and human mathematical representations. On some level, everything must be a pattern and occur for a reason -- the cycle and operation of life, the function of the basic forces of physics, the creation of the universe. Our mathematical system cannot mirror these truths because we cannot yet comprehend them. If we could create mathematical representations of these collassal patterns, it would mean that we would have come to understand everything -- but it wouldn't be mathematics that we were understanding.

mathematics questions

We can use mathematics successfully to model real-world processes. Is this because we create mathematics to mirror the world or because the world is intrinsically mathematical?

i don't think math was created by people at all. i think we have created symbols and terms to make it easier to discuss and teach, but it has always been there. the example i have is how the fibonacci series appears in nature...like the nautilus shell is a perfect model for the golden rectangle. (and some flower petals, and tree branch patterns and i think there are others). so if that showed up in nature, then fibonacci or whoever it was who calculated the golden mean was really just discovering what was already there. that had already been produced by nature. i mean i guess that might tell some people that the whole universe is mathematical, or god really exists or something...i think it's really important though, and it shows some order in the design of the world. so yeah, i guess the world is intrinsically mathematical. or at least some of it has always existed. math's so different from other subjects, like english or something...if you write a book it has to be something completely new and different (i mean, generally) and make some breakthrough or present some new sort of style, it can't be something that's already been written. and i guess the mathematicians who really gain fame are the ones who make totally new breakthroughs, BUT the fibonacci series will always be the same as it was when the greeks used the golden rectangle in their architecture. math stuff like that doesn't change or become obsolete. the pythagorean theorem always works.(..which is why math is so boring) so. math wasn't created, it just has been defined more concretely by stuff like "the fibonacci sequence"or "the pythagorean theorem" or "the set of natural numbers" and stuff like that. mathematical order has always existed, i think.

Math Questions

I have put a set of questions from the IB Diploma Progamme Guide for TOK here. Look at them and start some discussions. You have only three guidelines:

1. You may not begin a discussion about a question on which someone else has started a discussion.

2. You may, however, reply to a discussion about any question from the guide.

3. Please cut and paste the particular question about which you are starting a discussion and post it one color, then begin your discussion in a different color. Please see Lillian's excellent example in red and green above.

Have fun!

I believe...

I think I'm sort of agreeing with Molly's post and disagreeing with it at the same time here. I'd like to make my own distinction between "I believe" and "I know." I agree that often times "I believe" is used in place of "I know" simply to avoid offense. But I don't think that is always true, and in fact I know ( ;) ) that "I believe" can be a very appropriate qualifier. For example, I believe that God exists. For me, this means that I in fact do think strongly that the existence of God is real, yet I acknowledge that others dispute the idea, there are plausible arguments against such belief. I qualify my belief as just that--a belief--because though I find it to be true, I recognize that it cannot be proven either way. Thus it is personal knowledge--a belief--rather than completely objective "I know" knowledge. On the other hand, I have no problem saying "I know" in cases where I feel I have arrived at objective knowledge even on a controversial issue. Let's take abortion, since it's already been mentioned. Allow the debates to start flowing, but I will say I know that abortion is wrong in some cases (to emphasize: in some cases). I base this upon what I believe to be objective ("I believe", since thoughts on objectivity cannot be proven truly objectively) , such as generally accepted moral code coupled with science, etc. (I can elaborate if it's really desired, but my point here isn't to argue my knowledge, it's only to make the point that I consider this an "I know" statement as opposed to an "I believe" statement.)
"I know" this post is starting to confuse me but "I believe" it makes a bit of sense.

Although the previous posts make some good points, I personally think that "I believe" is just a politically correct term intended to prevent "offending" others.  (Laurel started to say this in class.)  You either know something or you don't.  Belief should apply only to religion.  Religion is based on (for the most part) blind trust in a supernatural being. so despite arguments, no one will ever KNOW which religious follower is correct.  One religion is correct, though, whether it be the athiests, Catholics, Jews...etc.  Most people would say it should apply to ethics, since most people claim to be ethical relativists.  They say that what you "believe" is right is not necessarily right because other opinions exist.  Therefore, you do not "know" you're right--you "believe" you're right.  As an ethical objectivist, I KNOW that the highest obligation one has is to himself and his own happiness.  "Knowing" is often perceived as arrogance, so people use the word "believe" to appear more modest and "respectful" of others' beliefs.  Ironically, these people usually still think they are absolutely correct.  I don't love the words "I believe."  I hate the words "no offense."  You know what you know.  Admit it.

I Know I Believe I Think?

Going back to a topic we have touched upon in some of our previous classes, I believe that "I believe that..." (irony besides) is a strong statement in that it is simply difficult to refute. When someone prefaces a statement of knowledge with "I know...", it is often subject to immediate criticism and debate. For instance, the statement "I know abortion is wrong" would most probably generate more controversy than the statement "I believe that abortion is wrong." The first generates controversy because it seems to impose upon other people's personal beliefs. For instance if a pro-life advocate and a pro-choice advocate are having a conversation, the statement "I know abortion is wrong" implies, "I know that your belief is wrong." Whereas "I know" is an aggressive proclamation of knowledge, "I believe" is more emotionally based. Think about the things that we typically preface with "I believe": religion, abortion, politics, emotions, and cultural and ethical issues. Most would agree that there is no universal "truth" on many of these issues. Instead, truth and knowledge is based on an internal evaluation of the issue (similar to the article on introspection that we read a few days ago). Using "I believe" in conjunction with these issues indicates that the speaker has introspectively found personal knowledge -- however, "I believe" also implies that this knowledge is indeed personal and is not universal. When one says, "I believe in God", it does not imply that this is the only truth. While it may be the "truth" to that person, that statement also seems to acknowledge other beliefs at the same time. You can say that you "believe" something and also acknowledge other people's beliefs. Therefore, it seems as if "belief" is often used to describe issues that rely on emotion and faith as ways of knowing. Emotion and faith are difficult to refute because they are unique to the individual, and it is difficult to say that a particular emotion or faith is "wrong." When one claims to "know" something, it also implies that this knowledge is universal and supported by irrefutable reasoning.

Thus, "I Believe" derives it's strength because it is difficult to challenge a personal belief. On the other hand, "I know" derives it's strength because it seems to indicate the "knowledge" of a universal truth that can be explained through irrefutable evidence and reasoning. "I Belief" has the power of pathos while "I know" is an appeal of logos.

Belief mistaken for Knowledge

I think it's worth mentioning just how easily a fervent belief can be mistaken for knowledge.

For me at least, when I hear "I believe that..." (as opposed to "I know that..."), it suggests a degree of uncertainty. Although it may be believed just as vividly as any statement of absolute truth, a claim prefaced with "I believe that..." seems to me to be one that lacks sufficient proof or empirical evidence. Statements of knowledge, on the other hand, should obviously be more grounded in fact.

So here's where I see a little bit of a conflict: When someone holds a belief strongly enough, he/she often claims it to be knowledge when it actually is not. If you believe something firmly, it can actually be pretty hard not to do. For instance, I'm sure you could find two kids at our school at any moment with religious beliefs that directly oppose each other's (i.e. they are incompatible and cannot be simultaneously correct) who would each claim to know his/her belief to be true. But if knowledge can only be applied to true belief, then at least one of them should not be saying "I know..." I guess my point is that it's very difficult for the believer to distinguish between knowledge and firm belief.

Belief, Thought, and ANSWER CHUNG!

First of all, Chung raises a great question in the previous post. Get in their and discuss it!

The previous post brought up a discussion from last week regarding the statements, "I think that...," "I believe that...," and "I know that...." One notion I would like to throw out there is that believing, rather than knowing or thinking, seems most tied to action. Knowing that there is poverty, for example, affects many people does not, in and of itself, move most people to action. Thinking that it is important may move some. Believing that it is important seems to move the most to do something about it. Let us assume for a moment that this is the case, that belief is what is most closely tied with action. If so, then is a belief statement stronger than a knowledge statement?

Should the Omission of Ethics be necessary or even right for attaining a position of authority?

i was watching over some of McCain and Obama commercials and speeches and found that some of them are really rude and dismissive of ethical concepts, instead of talking primarily about their goals and purpose they bash the other person with argumentative diction, I was just wondering whether a person attempting to reach a position of authority over a large or even selective group of people should employ non-ethical methods in attaining that position?

Truth and Knowledge

I was thinking about what we talked about last on Tuesday about the difference between the statements "I think that," "I believe that," and "I know that." We stated that "I know that" was a stronger statement than "I think that" and "I believe that." It was also mentioned that perhaps "I think that" and " I believe that" were synonymous. Although I think that we often use those two phrases synonymously, there is a difference or else there would be no reason to put both of those up. 

When somebody states that they "know" something, we assume that they have evidence and facts regarding the situation. 

...Tying into the example between "This is a book" and "Obama is having an affair." 

I think that it is not so much about "Obama" having the affair as much as it is an accusation of someone having an affair. If "Wild Bill" was having an affair, wouldn't it require the same evidence as Obama to prove that it was true. "This is a book" is an assumed truth. We assume that since we've seen books before and the book in front of you matches the identifications that we have from past experience with books, it is a book. Even if someone is often caught having affairs, we still want SOME evidence for the accusation. 

The problem between truth and knowledge is that we often use truth to gain knowledge while we are also using knowledge to gain truth.