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Old 12-03-2004, 06:13 PM   #1 (permalink)
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The Role of Culture in Mathematics and the Natural Sciences

What role does culture play today in Mathematics and Natural Sciences? Has the role of culture changed over time? Does culture SUPRESS the development of knowledge in the Natural Sciences, or does it give the push to attain new acheivements?
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Old 12-03-2004, 06:25 PM   #2 (permalink)
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Culture guides scientific research, the values instilled by it alters the results that scientists choose to find (to an extent), and in that guiding, it also limits the scientific field. A perfect example is stem cell research in the U.S. which I find very odd, as the government bans possibly life-saving research, but has no problem with keeping its enormous cache of nuclear weapons or developing technology to eliminate M.A.D.
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Old 12-03-2004, 06:30 PM   #3 (permalink)
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Hmm I would agree with you on that Suave. But was it not culture that gave the initial push to even look into Stem Cell research? It seems to me that although the scientific method requires the scientist to completely let go of cultural influences, these cultural influences always find a way to get into research. Does Science perhaps neeed a more definite, concrete system of gaining knowledge, such as the one used in Mathamatics?
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Old 12-03-2004, 06:46 PM   #4 (permalink)
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I would argue that culture does not play that much of a role in mathematics anymore. Gone are the days where some mathematical issue could mean a revolution in the way we think (like the addition of zero to the West, or the square root of two to the Ancient Greeks).

The contentions come within the field, or from philosophers. I'm thinking of things like nonstandard analysis, set theory axioms, fractional calculus, and the battles of different philosophical ideas about mathematics, like intuitionism versus formalism. This is all academic, though they can be very important to scientists and engineers in some cases.

As far as greater culture, it plays no sigificant role in mathematics---people in China are doing similar things to people in the USA now---unlike thousands of years ago, where the way you live played a role in what you studied. For example: Chinese focusing on solving various algebraic problems (writing some nice textbooks) instead of trying to prove abstract theorems.

I believe the public sees true mathematics as something of a black box, and they are happy about that. People tend to not care (seeing it as mundane), or tend to see it as some sort of magic or game. See the large pile of books about numerology, puzzle books for enthusiasts, and study guides for algebra. This is not so much the public engaging in mathematics as it is coping with mathematics in various ways.

There is one exception I think, and that is statistics. Though very few people really understand even the basics, everyone reads newspaper polls and likes to hear about the latest pharmaceutical study. The utility of mathematics is something everyone can appreciate now. Mathematics for its own sake is a different story.
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Old 12-03-2004, 07:56 PM   #5 (permalink)
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I agree with you Phukraut. Mathematics seems to have been reduced today to a method which utterly removes any form of cultural influence. The mathematical language and method used today relies on the setting out of a proof in the mathematical language, which is defined and valid only in the world of mathematics, although it has its applications.

But culture does seem to have some influence still on Mathematics..

Like you said Statistics is influenced by culture in some way. Mathematicians today are still looking for better and better mathematical models to model situations that occur in the natural world. The Bell Curve was a result of cultural influence, and we can see that the influence of culture on Mathematics seems to have opened another door in knowledge.

If culture can open doors in Mathematcs, like in Statistics, are we depriving ourselves of information in Mathematiccs because we dont have culture in Maths ?
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Old 12-04-2004, 12:50 AM   #6 (permalink)
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Are we depriving ourselves? Who knows? I would guess though there is sufficient influence from academic subcultures to make up for it: mathematicians talking amongst themselves, scientists requesting math models, etc.

Here is something I missed before. Mathematicians, particularly amateurs or number theorists, will look at things produced by culture for inspiration of research. Examples include describing the mathematical implications of Chess moves, or mathematical patterns in music or visual arts (see Escher and group theory).

Here we have the fruits of culture influencing mathematical discourse just for the hell of it---it's a curiosity or it's for Beauty's sake, or whatever the reason. Sometimes these cultural explorations can lead to deeper results.

You also implied something worth talking about. What is considered mathematics? When one sees a paper in a journal, how can one tell it is a mathematics paper? If something counts as mathematics only when it's discussing proofs, or logical progressions, then there is exactly zero cultural influence, yeah.

On the other hand, if we view mathematics like a jelly donut, where gooey center is the logical discourse, and the outer pastry includes everything else---like speculation and conjecturing, exploration of real world examples, and empirical analysis to guide proofs (very common in number theory)---then culture may indeed have definite influence, such as we both just talked about.

Here's a question for you: who is missing out more from the lack of interaction between culture and mathematics---the general public, or mathematicians? Is that even a fair question?

Last edited by phukraut; 12-04-2004 at 01:04 AM..
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Old 12-04-2004, 09:48 PM   #7 (permalink)
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That answer would be affected by what your perception of the role of Mathematics is. The public would be more concerned about the actual applications of Mathematics, but in Pure mathematics in subjects such as group theory, there are almost no applications whatsoever. In pure mathematics, TODAY i dont think there is any influence of culture, and I dont think that the mathematicians are missing out in any way. Pure mathematics is carried out for the mathematical knowledge that is gained from these subjects, and it is the desire for pursuit of knowledge that drives the Mathematicians, not any cultural influence. I also believe that in pure mathematics, we have exhausted our sources of revelation from culture, and the influence of culture is minimal - just as it should be .

In applied mathematics, the maths that is carried out is usually for some form of application to the general public. Take for example statistics and the bell curve, and its use in the SAts, IQ tests... A lot of what is done in Mathematics is also imported into other areas of knowledge.. for example Physics where the trig ratios and graphs are used in the analysis of AC currents. Culture here plays a pretty significant role, but as you said it's not part of the 'jelly.' it's what CAUSES the maths to be opened up, but does not affect the ACTUAL math. so culture does play some sort of role in Maths, or rather DID, the role of culture now even in applied maths is minimal, everything that is needed for our lives seems to have already been found.

So the answer to your question is... we may never know. We just dont know enough about our situation in this world to answer that question. We may think that we have unlocked most of the doors in Mathematics, but there may still be things out there that we cannot even imagine..something like the other dimensions. But to me, it seems that no one is missing out because of the mininmal influence of culture.
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Old 12-05-2004, 12:04 AM   #8 (permalink)
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Sure, the problems the Ancients struggled over are now taught in high school, but I'm not sure who thinks we have unlocked most of the doors. I remind those people however that Wiler's proof of Fermat's Last Theorem was released only in 1994. Flipping through texts on group/ring theory (which has quite a few interesting applications actually), graph theory, and analysis (Riemann's Hypothesis is the big one here) suggests we still have a very long way to go.

I find your description of applied maths interesting: it's main thrust is for the general public? Given that, I would agree that everything that is needed for our (the general public's) lives have been found---thousands of years ago when our ancestors made the first calendars. I therefore disagree with your description, and say the handmaiden of applied mathematics is physics first and foremost. If we accept that, then there is still a very far way to go in understanding. Also, it would mean the development of fields like astrophysics drives mathematical research.

I was thinking about that jelly donut argument I made before; I think I may have been wrong. If you recall, back when all that revolutionary reworking in the calculus was going on (Newton, Liebniz, and just about everyone else got in on the action), proofs were often put on the backburner. Here we have an excited generation looking at strange possibilities, driven by naivete and even simple formalism to develop what we know today. The jelly must have been leaking out then, because the calculus was definitely a part of the enlightenment culture---it's no surprise that Liebniz' philosophy about 'monads' feels a lot like his work with infinitesimals in calculus.

It may seem unlikely that such a revolution in how mathematics is practiced may happen again, but I would keep it in mind. A lot depends, for example, on whether Riemann's Hypothesis is true; some mathematics today depends on it, and mathematicians have been a little impatient, deciding to go on. I definitely think Hilbert's list of problems is not the last word, and one day we may see another revolution of the same sort, where mathematicians "lose their heads" for a while.

Lastly, let's explore that idea about mathematicians searching for knowledge being its own reward, that, like you say, they <em>do</em> mathematics to pursue knowledge. Where did this drive come from? I'd guess from the Greek cultural tradition. What about now? The debates today seem to be: can a computer do mathematics---true mathematical research and proof? If the answer is yes, then culture in mathematics is either dead or an illusion. If not, though---if something would be missing from the programming, like the aesthetic quality, or the passion of pursuit---then human beings will always be integral in mathematical research. Aesthetic ideals are not immune to cultural influence.

Look at the jelly again, how proofs are structured. Not all proofs are created equal, meaning that some are more elegant than others. Not everyone liked the proof of the Four Color Theorem for example, calling it ugly. Look at intuitionism---if that took complete hold in mathematical practice then proof by contradiction would be disallowed and considered bad form. The jelly would change, being influenced by cultural ideas of what is considered a "good" proof.

Cheers

Last edited by phukraut; 12-05-2004 at 01:50 PM..
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Old 12-06-2004, 01:59 AM   #9 (permalink)
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Wow phukraut I am having trouble digest all that information.. i am young !!

Indeed I think I have learnt a lot from what you have said, it has given me a much clearer , complete picture of the influence of culture on mathematics and the natural sciences. It is amazing how Mathematicians strive to adhere to the mathematical language devoid of the influence of culture, yet culture still influences them in the way they think, in their motivation as well as the selection of MAthematical areas to study.
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Old 12-06-2004, 01:53 PM   #10 (permalink)
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I guess I have a difficult time understanding what is actually being asked here. "Culture" is such an amorphous word that it needs to be better defined before we can really tell what you're talking about. Mathematics and the natural sciences are a part of our culture, and they only exist in service to it.

All math and science comes about to serve a cultural function, if it doesn't it becomes obsolete: anyone taking alchemy 101? Astronomy and telescopes came about in an effort to better understand messages from the gods in constellation form. Anatomy developed from painters' desire to portray the human body better. Computers were developed to break Nazi codes, and also to keep records of their final solution for the Jews.

Phukraut was right to point out that in the past these distinctions between "culture" and "science" as seperate worlds were non-existant. It's is really only in the past 150 years that we see specialization, mostly through the university system, of different fields of knowledge.

Rene Descartes (1596-1650) was the father of both Cartesian geometry and metaphysical philosophy. I'm sure he could have explained his philosophy in mathematical terms if he had to. Likewise if you delve into mathematics deeply, it's all just philosophical belief--the concept of numbers, zero, infinity, etc.

"Pure" knowledge is never pure, it's always framed by a cultural context and purpose. Even the most arcane mathematical theory is judged on the basis of it's applicability. If that application is to one-up your contemporaries or to impress coeds then it will likely be forgotten soon. Conversely if your theorem is useful in creating nuclear fission or predicting the weather, you will be remembered and celebrated.
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Old 12-06-2004, 03:08 PM   #11 (permalink)
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Originally Posted by Locobot
"Pure" knowledge is never pure, it's always framed by a cultural context and purpose. Even the most arcane mathematical theory is judged on the basis of it's applicability. If that application is to one-up your contemporaries or to impress coeds then it will likely be forgotten soon. Conversely if your theorem is useful in creating nuclear fission or predicting the weather, you will be remembered and celebrated.
A miniscule fraction of modern mathematics research is ever applied to 'practical' problems. Mathematicians largely do what they do for their own amusement. However, their work is still viewed as "applicable" by their peers (at least by the small handful who might understand it). A result is considered applicable to them if it is interesting and leads to more interesting problems. "Interesting" here basically means surprising and/or beautiful, not practical in any technological sense.

Applied mathematicians are usually interested in physics, but there is still a big gap between those two fields. Surprisingly enough, physicists hardly talk to the applied math people, which is mostly because they're too hard to understand. The math people don't really make an effort to change this either, as they're usually concerned with problems that physicists don't consider interesting (e.g. existence proofs).

In general, most mathematicians seem to write without much regard for how their work will be received by others. They will often spend a very large amount of time making sure something is perfect, and then write it all up more as a way of preserving the knowledge they have discovered rather than teaching it. Most of the sciences are much more competitive. Perfectionism is rare, and usually leads to failed careers because it takes too long to publish perfect papers. Peer acceptance is also a much bigger priority.

Anyways, I suppose the more practical sciences are affected somewhat by popular culture, but there are branches where I don't think there is any connection at all.

I don't know if you're trying to limit this discussion to popular culture though. Math and science both have their own internal cultures, and these very strongly influence which problems people choose, and how those problems are approached. These cultures are often very specialized. In my own field, you can often tell who someone's Ph.D. advisor was by looking at the style of their papers.
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