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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.

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.

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Gib Z

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Don't tell your friend that they have computers which can create mathematical proofs now either.

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MathematicalPhysicist

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anyway this is philosophy, and it's mambo jambo anyways.

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disregardthat

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MathematicalPhysicist

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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!but you can't teach someone how to make a breakthrough, because if you could it wouldn't be his breakthrough would it?!

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symbolipoint

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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.

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This reminds me of the never-ending debate: Is mathematics created, or discovered?

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disregardthat

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I wouldn't say mathematics is discovered any more than any invention we make.

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Sorry to play devil's advocate, but if it's discovered does that mean that there is a limit to mathematics?I wouldn't say mathematics is discovered any more than any invention we make.

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Gib Z

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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.

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Sure there's not a definitive answer to a lot of questions but people still debate them.

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disregardthat

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Might be, but that's not what I said. I like to think about it as we create the mathematics rather than discover it.Sorry to play devil's advocate, but if it's discovered does that mean that there is a limit to mathematics?

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.

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Gib Z

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disregardthat

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Gib Z

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Well, that is exactly what those people what that way of thinking say! =]

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Then just be friends. You can't convince anyone of anything.

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