Maths use its own languaje, in witch missundertandings are not possible in the sense that, what you say is always clear, but a thought that drive you to say it is true can always have a "false steep" unseen by specialist. Also, maths don't base its finding in observations, so no missleading comes from here, as use to happend in phisics (unaccurated or missundertanded observations leads to bat theories in phisics every now and them).
But, as in any other science, some times the logic that have to be followed to prove a new theory can look perfect one day, and after admited as true the new theory, some one can pinpoint a weak steep on the demostration making it all to have to be proven again, avoiding or explainig correctly this steep.
It doesn't mean languaje is bad, is just that is not perfect in the sense that a bad prove of a theorem can look true to a the reader that don't have the appropiate background, and as new theries still lack of people that fully understand it, it is possible that latest demostrations finally is proven grown, like it happened with the last Fermat theorem proven some years ago: first prove was accepted for some moths, but it had a weak point, that was uncovered and part of the demostration had to be rewritting, with the risk of beeing even more complex to fix the problem than the theorem it self!
So languaje is not really perfect in maths, and it was proven than a perfect languaje for maths can NOT be found, due to de "incomplettness theorem" from Goedel, that comes to say that maths can never form a complete system in witch all possible theorems that the languaje can express have to be proven worng or right with the time... there always be theorems that can be expressed with maths but not be proveen (right or worng) using this languaje, so maths stop been called "exact science" from that point!
>Mathematics is not absolute, because it is defined with language, and language is difinitive. It is just an interpretation. Can you prove me wrong?
You're wrong.
No matter what language you use, the basics don't change: there is the same set of prime numbers, same integers, rationals. The empty set has zero members, there are as many points on any length of line, The square rot of two is irrational and roughly 1.414... in Greek, Malay, and Uzbekistani.
grg99, i don't agree with you, languaje have to be interpreted as not only equations, numbers, etc., but also the languaje in witch you demostrate me that a new theory or theorem or similar you claim that is true, is really true, and in that demostration, you have to write down your logic into some kind of "dirty" languaje, more or less similar to natural languaje, but not as clear to all people as you imagine.
Imagine that you claim that there is no solution for the equation "X^m+Y^m = Z^m" if m is greater than 5.... this was stated many years ago by Fermat, but no one was able to prove it right or wrong. I agree with you that it si clearly stated in a perfect mathematical languaje, no problem here, but now you try to read and follow the text of the demostration that ensures you that this is true... it is FULL of natural languaje, like "if that is true, and that is also true, then this can not be true" and all this, plus other parts where the logic (not math formulas or numbers) of the prove have to be stated, can be not very clear, or may be even wrong as the author didn't keep in mind some properties of the equations and other objects osed in the prove, just by mistake... no one did notice it, and agreed in the assert that the theorem was true... but the prove was wrong!!!!
Where is perfection in this? The "languaje of maths" include also the languaje of complex logic, and here is where the problems come: Logic is quite more fuzzy than mathematical formulas.
Another example: The "set theory" is a very simple and basic math theory, it was supposed to be 100% true and free of problems, but after many years of using it, some paradox where found in the theory invalidating it completely... if math languaje is perfect, then why was it posible to write down in that languaje a faulty theory, prove it true for many years, and no one noticed it?
Take this other one:
I say that it is true that "1+1=10" and that "2+2=1", then you will say that it is not true, and you are sure because it was written in the sacred languaje of maths, so no doubt about the meaning of the text... but they both are TRUE!!!! 1+1=10 in binary notation (base 2), and 2+2=1 if you are woking in Z/Z3 (numbers mod 3, only 0,1,2 allowed and other stuff).
As you see here, even using the simpliest math languaje, many things have to be explicity stated to be sure you and I speak of the same things... this is a languaje that is NOT perfect at all.
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Mathematics is not absolute, because it is defined with language, and language is difinitive.
The question cannot be answered to everybodies satisfaction. First you have to define "absolute",
Then "definitive". What do you mean by language is definitive?
In any case Godel showed that no axiomatic system can be complete.
Actually its not math but it is our defining language which is so limited that we can not find the solutions to several problems. e.g
any thing divided by 0 has no solution and we show it as infinity because the basics on which human thinking is based are limited. So , no matter what language you use , you may never think about the solution of this problem which I have mentioned, whether you know any language or way of thinking.
It is not possible "TO TELL" without a "LANGUAGE".
"TO TELL" implies the use of a "LANGUAGE".
>I say that it is true that "1+1=10" and that "2+2=1", then you will say that it is not true, and you are >sure because it was written in the sacred languaje of maths, so no doubt about the meaning of the >text... but they both are TRUE!!!! 1+1=10 in binary notation (base 2), and 2+2=1 if you are woking in >Z/Z3 (numbers mod 3, only 0,1,2 allowed and other stuff).
In my humble opinion, in this example you have just NOT used a valid math language. You have willingly hidden some information. A valid math demonstration contains all the informations necessary to avoid any misunderstanding ...just because language is not perfect.
Hi lombard, you are right, any decent demostration must contain ALL the information neccesary, but as it use a languaje in witch things can be "supposed" or "fully stated", the level of detail needed will ruin your demostration.
For instance, if you use "absurd reduction" in your prove, and you don't fully explain why this technique is applicable in this case, some mathematicians will denny the completness of your prove, while other will not bother about absurd being used here... it all depends on what you think is "trivial", because you will stop adding more "small details" to your prove once you reach the "trivial" level.
I wanted to ilustrate this by the two "extreme" examples, but in real world, the problem is VERY serious, and lead to some historic mistakes as the one with Fermat last theorem and others.
For instance, you can make a demostration supposing a previous theory is 100% right, so only mentioning it entitles you to use all this theory, but may be the thoery used prove finally wrong, so your demostration, that where true by all means previously, now result to be wrong... why? because you stop the level of your demostration when you reached a "trivial" treshold that included a wrong theory... it also hapends, and makes math languaje to be very unstable as any assert relay on many many suppositions, and suppositions are the mother of dissaster.
>For instance, if you use "absurd reduction" in your prove, and you don't fully explain why this
>technique is applicable in this case, some mathematicians will denny the completness of your prove,
>while other will not bother about absurd being used here... it all depends on what you think is "trivial",
>because you will stop adding more "small details" to your prove once you reach the "trivial" level.
I agree. And this sentence demonstrate that the "absoluteness" of math is very difficult to be represented with the "incompletness" of language. Math concepts are "absolute" by definition, it is the description that could be incomplete.
Yes lombard, but it is even worst: If you wish to "polish" your prove by adding the "details" about why absourd is applicable in your case, you will find that YOU CAN'T, because absurd reduction is such a simple technique that you just have to agree with it or disagree, but you can not prove it wrong or right! "Absurd reduction" usefullness is an axiomatic true for "believers", and nothing usable for non believers. Here is where Goedel smile at us as a devil!
Math is metaphorical so it would seem that it is based on language. However, it is a metaphor you can demonstrate without any language at all. Math is all based upon the metaphor of containment. Therefore math is based upon visual and spacial dimensions. We just use language to describe mathematics.
>> Mathematics is not absolute, because it is defined with language, and language is difinitive. It is just an interpretation. Can you prove me wrong?
Math is not defined with verbal language, nor is it verbal language. In such, conclusion for language cannot lead to the conclusion for math.
For example in color hex, colors can be defined like #FF0000 or #555500. Each monitor displays colors slightly differently. So because the monitors are not absolute, does not mean that the values are not absolute.
Language is limited because it is only symbolic to the idea/concept. The purpose of language is to simplify the absolutes so that a person can convey the message to another person. It is up to the destination/receiver to interpret what you have spoken into the idea. (Similar to... encoding bmp to jpg/gif then converting back to bmp, take the integration of the derivative of the equation) But unlike machines, we are capable of using all of our experiences to convert back as close to the original idea as possible.
Like I said previously, it is impossible to convey that "math is absolute" accurately. Math is absolute because that is what it is. The word math is defined by language, but the idea math is defined by idea. I can only give you enough information so that you can come up with such an idea. It is kind of like teaching where you cannot teach understanding directly.
Of course, if you want to discuss if an idea is absolute... that's a whole other discussion.
Math is absolute, language is merely a tool by which one can try to describe math. A fair example might be to use the phrase "prime numbers" or "Abelian Group". These phrases should conjure very spcific ideas and concepts which cross all language groups.
Heck, even a mere graphical representation would sufficiently describe prime numbers and this doesn't use a language at all, just a picture.
xx xxx xxxxx xxxxxxx xxxxxxxxxxx xxxxxxxxxxxxx xxxxxxxxxxxxxxxxx ...
Please don't talk much about this because we will waste time. We can do more things which have real benefit for our life !!!
We are living in a "non-absolute" life. You like green colour, I like the red colour... It is hard to say that a thing is True or False...as people have different ideas about things in the world.
However, math is based on a set of definitions and concepts, and we work in this set. The proof of the theories are mostly based on things which are considered absolutely TRUE or FALSE by everybody. So the logic of math is absolute. But depending on the set of pre-defined definitions and concepts, we may have difference "maths" and therefore there is not a unique way to describe our world. Everything is not "absolute"... but don't tell that to your girl friend :-) your love for her must be absolute.... :-) otherwise... :-)
The language of the mathematics tries to make human beings able to communicate its absolute.
If it fails to, the language is to blame, not the mathematics.
In some areas of the mathematics like topology there is a concept of "almost" ; while at first glance it appears as not absolute, it is clearly defined, and, finally, express something "usable" on the absolute side.
a contradiction there, as you state "it is clearly defined", thats exactly my point. It is defined, ( too be a tool of absolutness), which makes it difinitive.
a picture needs to be interpreted. You cannot interpret it without having skills of reasoning. Reasoning requires a form of language. As everyone has agreed so far-language is difinitive.
RE
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"You're wrong.
No matter what language you use, the basics don't change: there is the same set of prime numbers, same integers, rationals. The empty set has zero members, there are as many points on any length of line, The square rot of two is irrational and roughly 1.414... in Greek, Malay, and Uzbekistani.
Language has nothing to do with it.""
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It doesnt matter what language you use to explain a concept. You could intrepret the tools of math in sign language. I think, like most people so far, you miss the point.
Everyone, so far is treating math, as if it were just always there. It wasnt. It is a set fixed of rules used to explain the physics and interactions of things. It in itself is a language. When it doesnt work properly, it is tweeked by inventing a new "word" also known as an Algorithm. 1-10 is the mathematic equivelent of A-Z.
>a picture needs to be interpreted.
usually true, but not always. Pictures can simply be astetically pleasing to look at. But for the sake of argument, I will accept this as true.
>You cannot interpret it without having skills of reasoning.
This phrase doesn't quite "feel" right (to absolute perhaps), but since I cannot think of any counter examples, I again will accept this as true.
>Reasoning requires a form of language.
This comment I disagree with. Reasoning is born through history and past experience. The past experience does not always need to be directly related to the subject matter, but one *ALWAYS* must use past references to reason out any problem. Riddles and Logic Puzzles are perfect examples. People become quite good at logic puzlles if they do them often. They may have never seen a particular puzzle before but becuase they have done puzzles before thay may have had similar components they will do well at that puzzle. In this case, they are using language. But how would you explain Pavlov's Dogs? Or Skinner's Pigeons? Those were just learned/conditioned responses by dumb animals with no language what so ever. learned behaviors and conditioned responses do NOT require language and reasoning is just another form of a learned behavior. Perhas not as dramaic as slobbering dogs, but just as accurate.
>As everyone has agreed so far-language is difinitive.
No, no and no. Language is anything EXCEPT definitive. "Definitive" accordig to Webster is precise or expressly defined. Language is NOT expressly defined. Certain words might be, but many words have several meanings and connotations. For examples of this look in any thesaurus. Just look up the work "Big". You'll find many words that mean the same as big, but they don't really have the same connotation. Large, Grand, Jumbo, Huge... Go to a fast-food resturant and ask for a Large soda, then ask for a Jumbo sodo. You'll get one 32 oz drink and one 44oz drink. Roget's Thesaurus said the two words are synonomus, but they have very differnt connotations.
Moreover, If I were to tell you that Mathematics *IS* absolute could you disprove it? I mean really prove that MATH is not absolute... using very concrete numbers returned from a mathematical process and telling me the results are ambiguous does not mean that math is not absolute. it merely means that your logic is flawed or you do not have eough data to portray an accurate model.
Wouldn't it be that 0-9 are the mathematical equivalent to A-Z? But then, we'd only be speaking of base10 and (primarily) English... What about base2, base16, Russian, Chinese?
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But, as in any other science, some times the logic that have to be followed to prove a new theory can look perfect one day, and after admited as true the new theory, some one can pinpoint a weak steep on the demostration making it all to have to be proven again, avoiding or explainig correctly this steep.
It doesn't mean languaje is bad, is just that is not perfect in the sense that a bad prove of a theorem can look true to a the reader that don't have the appropiate background, and as new theries still lack of people that fully understand it, it is possible that latest demostrations finally is proven grown, like it happened with the last Fermat theorem proven some years ago: first prove was accepted for some moths, but it had a weak point, that was uncovered and part of the demostration had to be rewritting, with the risk of beeing even more complex to fix the problem than the theorem it self!
So languaje is not really perfect in maths, and it was proven than a perfect languaje for maths can NOT be found, due to de "incomplettness theorem" from Goedel, that comes to say that maths can never form a complete system in witch all possible theorems that the languaje can express have to be proven worng or right with the time... there always be theorems that can be expressed with maths but not be proveen (right or worng) using this languaje, so maths stop been called "exact science" from that point!