How is mathematics related to creativity?

[fresh_42]

I found this interesting article about the balance between mathematical creativity and necessary content.

When instructors develop an environment where students are willing to put themselves “out there” and take a risk, interesting moments often happen. Those risks can only build one’s creativity, which is the most sought-after skill in industry according to a 2010 IBM Global Study.

https://blogs.ams.org/matheducation/2019/01/15/reflections-on-teaching-for-mathematical-creativity/

I don't want to make the discussion exclusively a pedagogic one, which is why I posted it here. Personally, I think mathematics is far more creative than commonly thought - at least if I take the many comments about mathematical rigor which I've read over the years into consideration. In my experience, one learns a lot and gains many different views on a topic, if forced to teach it. Every student you want to convince of a mathematical fact has their own way to it, so one needs different approaches. This is part of the mathematical creativity in my eyes, but certainly not the only one.

I'm a regular reader of Terence Tao's blog, which is another great evidence of mathematical creativity. Of course there is plenty of room between Tao's brilliance and the necessity of ideas while teaching. Additionally, other than in physics, a new idea is usually easy to decide: either you can prove it or not. And if there is a proof, it is automatically no longer a personal theory - just more or less of interest.

What are your views?

 

[.Scott]

The study that they are referring to is this one: https://www-03.ibm.com/press/us/en/pressrelease/31670.wss
This is a CEO survey.
It focuses more on management than engineering staff. So, I would say that, in this context, they are not closely related.

But in the context of the American Mathematical Society - which you cited, there is one.
Although AMS can rightfully claim to encourage "creativity", I would consider their association of this with the IBM study to be a bit of puffery.

It's an issue of semantics. You need to be creative in your interpretation of "creativity".

 

[jedishrfu]

As I read about the experiences of these teacher, I got the feeling that its best to foster a Socratic scheme of teaching to encourage the students to collectively direct the teaching of a topic. In this way, more avenues of thought are explored until the "best" one is found. However, with the pressure to teach to the clock then only the "best" track is taught and the students never know about the alternatives that while not good for this topic may come in handy on future topics.

As far as creativity goes, I often think it has something to do with not knowing/remembering something you need to know/remember and so the brain tries to fill in the missing pieces. It may do say differently each time and hence you have a "creative" thought that ma in fact be quite superior to prior thoughts on the topic.

I was once on a caving tour where the tour guide said you can ask me any question and if I know the answer I'll tell you and if I don't I'll make something up. That's the creative spirit both positive and dangerous at the same time.

 

[fresh_42]

Let's phrase it a bit more provocative and less based on the AMS article:

Can you even do mathematics without being creative?

I remember a situation during an oral exam when the student had been asked to explain what a linear transformation is. She could repeat the definition without any mistakes, but when asked what it means and to give an example she was lost. I'm not sure whether this already counts as a creative act, but obviously the facts alone weren't sufficient. I think she got a C in the end, but couldn't understand why. One may say this is imagination and not creativity, but aren't those two sides of the same medal?

 

[jedishrfu]

I remember an exam like that where I couldn’t remember the formula for finding the five complex roots of a real number and then I remembered the angular symmetry of the roots and with that completed the problem.

I think it was deMoivres theorem.

https://www.ck12.org/book/CK-12-Trigonometry-Second-Edition/section/6.7/

 

[opus]

From a student at a much lower level of mathematics, I'd definitely agree! I've been doing trigonometric integration all week and was getting very frustrated until I let myself kind of "play" with things. Not only has it made me much better at them, but made them a ton of fun. It is difficult to get rid of the algorithmic mentality to solve problems that we're brought up with, but it comes to a point (as I've noticed) that there is no cookie cutter method to solving problems. It's a shame to hear people say that they hate math when they haven't had the chance to see what it's really capable of. I had the same mentality when I was younger as the algebraic and geometric algorithmic problem solving processes are so dull!

 

[Auto-Didact]

It is evidently clear that mathematics and physics, while highly technical disciplines, also require genuine creativity to tackle the open problems that lay before us all: it is in precisely this arena - far more than in the rest of everyday conventional mathematics - that creative mathematics usually of a highly intuitive and conceptual nature is at least as, or perhaps even more, important than mere technical, logical strictly rule-based skill.

Big conceptual leaps are 'always' the work of one creative individual who is able to see beyond the foreseeable and pierce into the seeming impenetrable by seriously entertaining or embracing an unconventional, often purely intuitive, point of view; in this manner he is able to see a forest or even the outline of a new fruitful land mass, while thousands of technically skilled workers surrounding him are only able to see trees, or even worse, insist that that the trees are all there is. To paraphrase Smolin, a craftsman cannot do the work of a visionary because he thinks and approaches matters in a fundamentally different manner.

Three relevant works about creativity in mathematics:

Jacques Hadamard penned down in a short book what he learned about creativity and invention in mathematics among its practitioners in "An Essay on the Psychology of Invention in the Mathematical Field".

Henri Poincaré in his masterpiece "The Foundations of Science" carefully explains the main branches of mathematics up to his time and how they relate to differences, both in approach and viewpoints, among practitioners.

Sir Michael Atiyah recalls the life and work of one of the most creative 20th century mathematicians, Hermann Weyl, growing up in Hilbert's world of modern - i.e. non-creative - mathematics, in this biographical memoir.

Edit: this recent obituary for Michael Atiyah, written by one of his friends within the mathematical community, illustrates the necessity and importance of creativity in mathematics. To end, using Atiyah's own words: "In the broad light of day mathematicians check their equations and their proofs, leaving no stone unturned in their search for rigour. But, at night, under the full moon, they dream, they float among the stars and wonder at the mystery of the heavens: they are inspired. Without dreams there is no art, no mathematics, no life."

 

[Tom Kunich]

I found this interesting article about the balance between mathematical creativity and necessary content.

https://blogs.ams.org/matheducation/2019/01/15/reflections-on-teaching-for-mathematical-creativity/

I don't want to make the discussion exclusively a pedagogic one, which is why I posted it here. Personally, I think mathematics is far more creative than commonly thought - at least if I take the many comments about mathematical rigor which I've read over the years into consideration. In my experience, one learns a lot and gains many different views on a topic, if forced to teach it. Every student you want to convince of a mathematical fact has their own way to it, so one needs different approaches. This is part of the mathematical creativity in my eyes, but certainly not the only one.

I'm a regular reader of Terence Tao's blog, which is another great evidence of mathematical creativity. Of course there is plenty of room between Tao's brilliance and the necessity of ideas while teaching. Additionally, other than in physics, a new idea is usually easy to decide: either you can prove it or not. And if there is a proof, it is automatically no longer a personal theory - just more or less of interest.

What are your views?

I had several jobs in which physicists presented me with mathematical problems to build into electronics designs or program into computer code. The problem was that their analysis of the problems was incomplete, or in one case, totally incorrect. I had to analyze and work the entire problems out myself after many hours of having incorrect working of the machinery. And one of them meant the savings of perhaps hundreds of lives. I don't know if I would assume that mathematics increases your ability to be creative or if you have to be creative to understand the problem in enough detail to use mathematics to prove it.