Robert Hansen has put up (at various Math-teach threads) some theory/theories about 'creativity in math' vs. 'knowledge in/of math', which don't seem to make much sense to me.
I am therefore directly quoting Robert Hansen on these matters and shall leave it there. (In my comments on the links I have here provided on interesting work on "creativity in math" [and other disciplines], I have commented on Robert Hansen's views).
Here is Robert Hansen's story in his own words (drawn mainly from various posts of his at the multi-part thread on "The Philosophy of Creativity" - he has spoken elsewhere as well about his take on 'the art of math and the knowledge of math' and so on, but I can't hunt those posts out for you. Sorry about that):
Robert Hansen: "Even though knowledge is there on the surface, the creative mathematician is studying the art behind it".
"Real mathematicians are in it for the art of mathematics, not the knowledge of mathematics".
"A mathematician seeks truth in logic and the real numbers and in doing so, increases our understanding of both."
"Yes, (Newton, in regard to his famous quote about 'standing on the shoulders of giants') was crediting the artists that created the knowledge, not the knowledge itself. The product of the art is knowledge.
"If an artist writes a song and gives it to you, do you credit the song or the artist? Did the artist give you 'art' or just the 'product' of art? That depends on you. If you play the song and enjoy it then you are enjoying the product. If instead, the song, in part inspires creativity in you to create another song, then you are enjoying the art.
"Newton enjoyed the art and gave credit to the prior artists".
(Robert Hansen to Joe N.): "Yes, he (Newton) was crediting the artists that created the knowledge, not the knowledge itself".
At which Joe N., apparently giving up on his brave attempt to explain other views about 'creativity and knowledge in math' to Robert Hansen, said:
"Ha ha! You are a funny guy. Its obvious you are just making it up at this point".
If readers are confused (as I was) by Robert Hansen's theory of 'creativity Vs. knowledge in math', they may attempt to disentangle it for themselves at the multi-part thread titled " Scientific American - Excerpt from "The Philosophy of Creativity" starting at http://mathforum.org/kb/thread.jspa?threadID=2632746.
Anyway, on reading Robert Hansen's 'expert' views, I felt the idea of 'creativity and/Vs. knowledge in math' deserved some looking into somewhat more deeply than Robert Hansen does, and here are links to some of what I found.
Admittedly, this is entirely inadequate as a 'study' on the issue.
In order to provide an adequate idea regarding 'creativity and knowledge in math' (and/or other disciplines), we'd have to create 'structural models' *interactively* on these issues; unfortunately, we do not have the facilities here to do that effectively.
So here are a few of the many links I found (alongside some overview commentary from me):
- -- I had earlier mentioned a useful table that Bertie Kingore had put up at her page titled "High Achiever, Gifted Learner, Creative Thinker" - http://www.bertiekingore.com/high-gt-create.htm. (This is more to the issue of how teachers may deal with their students who show the characteristics mentioned in the title, but it's useful as a start, nonetheless - particularly in line with my claim that there is much creativity in "learning math").
In my opinion, the book is well worth reading, though it is NOT, as earler suggested, 'definitive' by any means. It certainly provides some corrective to Robert Hansen's vague ideas about 'knowledge' and 'art' in. 'creative math', and the like.
Chapter 2 of the book explores "The Nature of Mathematics" quite interestingly, going into useful ideas of "Patterns and Relationships" (somewhat cursorily); "Mathematics, Science and Technology"; "Mathematical Inquiry".
This is strictly a commercial venture and I haven't purchased any of their offerings. However, overall, it certainly does seem to be more sound than Robert Hansen's vagueries on 'knowledge' and 'art' in 'creative math'.
Poincare is "(not a) mathematician. Certainly not in the sense I spoke, nor in any sense really. A mathematician seeks truth in logic and the real numbers and in doing so, increases our understanding of both". (Ideas to this effect, as he has done before).
Well, I guess Robert Hansen will pronounce: "Tao is not a mathematician. Certainly not in the sense I spoke, nor in any sense really. A mathematician seeks truth in logic and the real numbers and in doing so, increases our understanding of both". [Ideas to this effect, as he had done before, about others to whom I had drawn attention].
But Professor Tao somehow seems to regard himself as a mathematician (notwithstanding Robert Hansen's 'expert' opinion). Here is what Tao says about himself:
Tao: "I am a Professor at the Department of Mathematics, UCLA. I work in a number of mathematical areas, but primarily in harmonic analysis, PDE, geometric combinatorics, arithmetic combinatorics, analytic number theory, compressed sensing, and algebraic combinatorics. I am part of the Analysis Group here at UCLA, and also an editor or associate editor at several mathematical journals. Here are my papers and preprints, my books, my research blog, and the group blog on mathematics in Australia that I administrate."
Of course, Robert Hansen would doubtless claim that Tao is a mere 'knowledge worker' (or ideas to this effect) and that Tao knows nothing about 'the art of being a creative mathematician like Robert Hansen'.
Well, it is my view that Robert Hansen does not know what he's talking about when he talks about mathematics.
I observe that Terence Tao seems to participate in The Polymath Project. (I have just begun exploring this; have only glanced at it and know very little about it. However, it does appear to be entirely contradictory - in terms of its 'philosophy', theories underlying, and general approach - to Robert Hansen's astonishing views on 'creative art of mathematics' (and stuff like that).
- -- I found a truly fascinating site called "Stack Exchange", which seeks to develop "a network of individual communities, each dedicated to serving experts in a specific field. We build libraries of high-quality questions and answers, focused on each community's area of expertise" - http://stackexchange.com/sites#, and http://stackexchange.com/about.
I know very little about Stack Exchange thus far, but am trying to find out more.
In particular Stack Exchange contains a fascinating section on 'Math problems' - what they call "Mathematics Stack Exchange" - http://math.stackexchange.com/. I do plan to look further at this.
- -- There are literally hundreds (perhaps even thousands) of sites discussing "creativity in math" (and in other disciplines. I have looked at just a few of them that appeared to me to be of some interest. Not one appears to confirm Robert Hansen's views on 'creativity in math' [or in anything else]).