i have a friend who will not accept the fact that there is endless room for creativity in mathematics. how can i convince him otherwise?
Perhaps certain aspects of math, like pure computation, is not very creative, especially if computers can do it too. However, doing proofs is something that no technology (that I know of) can duplicate, and thus stems completely from the human mind. Some proofs can be very, very creative indeed. And they don't need to be long to be creative. When I first read Euclid's proof of the infinitude of primes as a kid, I was very impressed by the creativity of that proof, even though it was very short and easy to understand. Cantor's proof that there is no surjective map from a set A to its power set was another (very short) proof that really wowed me in the creativity department. Even more creative is developing new ideas and concepts in math that turns out to be very useful in other sciences. Einstein had difficulty developing his theory of general relativity in the beginning because he needed the concept of tensors, which he didn't know of at the time. Fortunately, about 20 years prior, tensors were developed by mathematicians even though they were not a necessary tool for physical sciences at the time, but later proved to be an indispensible tool for a subject that apparently it was not initially meant for. Now that is creative! Also, finding counterexamples is a very creative aspect in math. To prove something is false is often more difficult than proving that something is true. One must find a counterexample, and there is no fixed route for finding counterexamples. One must just find it using deep creative thinking. I once tried to find out if a product of quotient maps is a quotient map, but later learned that it wasn't true. But I wasn't convinced until I found a counterexample. It was hard to find one and I could not think of one myself. When I finally read a counterexample, I was baffled at how someone could have thought of that. Also, it requires a lot of creativity to determine the conditions that does make the assertion true. There are many other creative aspects in math, like making generalizations of a specific well-known concept that then leads to results that apply to other concepts, developing abstract ideas that lead to very concrete results, establishing alternate definitions that apparently seem very different but lends to new ways of loooking at the same thing, alternate proofs of the same theorem that are very different from each other, and the list goes on. And there are too many examples of these other points to list them out. In summary, I believe math is very filled with creativity.
creativity is in the eyes of the beholder, as you can't teach someone how to be creative, you can't as well teach him why something is creative. anyway this is philosophy, and it's mambo jambo anyways.
Of course you can teach someone to be creative. Just as you can teach someone to be more intelligent. Through teaching you wil lexperience new things, discover new things which will ultimately make you more creative.
You can teach a painter how to paint, but you can't teach someone to discover new things and that's all creativity is about, to find and invent new theories or new ideas, ofcourse you need some knowledge beforehand in order not redo something that has been done already, but you can't teach someone how to make a breakthrough, because if you could it wouldn't be his breakthrough would it?!
that kind of reminds me of when one of meno's slave boys "recollect" the geometric truths of a triangle in Plato's "Phaedo", only after socrates gives him step by step instructions!
People here are having trouble with clear concepts for words. Maybe Mathematics is creativity, maybe it is not. If Mathematics development is not creative, which may possibly be true or not, then what do we call it? Maybe we need another set of words the characterise the Mathematics which has been developed and which can still be developed. May we choose another nice word, like "inventive"? Now, is inventiveness different than creativity? Do we need creativity before we may invent? A possible viewpoint is that creativity is something which happens when your intuition does something for you for which you could not initially see the logical steps. Later, when you find the logical steps, you have invented. This might relate to Mathematics, maybe like in Calculus, someone developed the Riemann summing method leading to Calculus'es integration operation. This development on paper with diagrams and algebraic reasoning steps was invention. But before it could figured how to reason and write it, creativity may have been needed. Have fun, people; this is turning into a philosophy discussion.
Sorry to play devil's advocate, but if it's discovered does that mean that there is a limit to mathematics?
There are different types of mathematical philosophies that take sides of this argument, for example, Constructivists, who insist in order to prove a mathematical object exists you must find it, believe all maths is constructed/invented. Other types of course, believe that the mathematics is already there, and would exist regardless if human were here to discover it or not. For before humans existed, the Sun was still radiating energy, the galaxy was spinning and the universe was expanding - How could this system has run without mathematical rules? Of course these tendencies to move in certain ways and to expand at a certain rate are determined by physical properties embedded into the universe. So just because these objects are following physical principals that can be explained by mathematics, does it mean mathematics is being done? Does mathematics, inherently, have to be done by intelligent conscious beings? It's sad that even mathematicians still have this debate, and not take the Quantum Mechanic example. First of all, the question can never be physically or mathematically answered, and hence it is outside the realm of our studies. We need not even think of the notion, as long as our mathematics is correct, everything is fine. Interpretations are put forward to perhaps assist one in understanding the nature of what they study, to make them sleep a tiny bit better a night because they think they know a bit more about their field. In the end, it makes absolutely no difference to the mathematics. PS - Constructivist proofs are hell annoying.
Might be, but that's not what I said. I like to think about it as we create the mathematics rather than discover it. Genious minds find new ways to use the mathematics that is already created. It is weird to be on either side of the discussion as Gib points out though. I would say that mathematics is created like a musician creates a piece of music. You could say that there is a limited set of notes that can be used in a limited order. Even if you think like this, you must admit that it wouldn't be 'discovered' unless there was anyone to discover it. So in either way, mathematics is created.
Ahh but many musicians, usually the extremely modest ones, claim they did not invent that piece of music, but merely discovered it =] For if they had not, that certain sequence of notes would have existed, regardless if it were played.
That's the thing. With that way of thinking you can say anything in the world is discovered, and nothing is created. Abstract things that is.