In the first post in this series, I explored how despite the common tendency to categorize mathematics as a subject concerned with rules and complexity, it can actually provide ways to lead one to think creatively. Instead of getting lost in the specific complexity of Calculus and Differential Equations, we took a step laterally and showed how these areas manifest themselves in the creative beauty around us.
In this second part, I would like to introduce another area of study within mathematics that is commonly viewed with dread when one decides to either major or minor in the subject – mathematical logic and proofs. Many (if not all) math textbooks contain a section proving how and why different theorems work. Although ignored by many students due to their high complexity, they are the reason why the theorems work and are rather important to the field of study. So why does a course in something like mathematical logic help enhance creativity? Put simply, it forces you to think differently. In mathematics, the goal is to find truth and proofs are the explanation we use to convince ourselves and others. I am not going to now go into any further discussion on how to write mathematical proofs, but instead focus on some of the simplistic components. In writing a proof, you have a few options:
- Simply find an example of something that works
- Contrapositive – which simply means negating both sides of the statement
- Induction – try using a low number and then if it works, prove that it will work for when that number is increased by 1
There are many other ways, but I don’t want to get too caught up in the details. So knowing this, you may now wonder how it could be applied in your next innovation session. As a starting point, it is important to note that each of these techniques enhances reasoning and enables you think creatively by forcing you to look logically and break things down, analyze them, and build them back up. Therefore, you may want to try a few of the following:
- Break the challenge statement down into its components
- Ask questions assuming the opposite situation is occurring
- Use contradiction – find examples of things that didn’t work and ask why. Then add something incremental to it (e.g. a motor, magnet, sensors, etc) and ask if that works
- Examine a new product idea that really resonates with consumers. Ask why as many times as possible to get to the core as to its success
- Take something from a completely different industry and try to apply it to your challenge
Three prisoners have been sentenced to long terms in prison, but due to overcrowding, one must be released. The warden devises a scheme to determine which prisoner is to be released. He tells the prisoners that he will blindfold them and paint a red or blue line on each forehead. After this is done, he will remove the blindfolds and a prisoner should raise his hand if he sees a red line on at least one of the other two prisoners. The first prisoner to identify the color of the line on his own forehead will be released. Of course the prisoners agree to this. The warden blindfolds them and then proceeds to paint a red line on all three prisoners. He removes the blindfolds and, since each prisoner sees a red line, each prisoner raises his hand. Some time passes when one of the prisoners exclaims: “I know what color my line is! It’s red!” This prisoner is then released. Now, we must ask: How did this prisoner correctly identify the color of the line painted on his forehead?
I will let you think about that and have some fun with it. Hopefully by now in reading the two blogs about mathematics, you have some better appreciation and understanding how such a subject can indeed enhance creativity and exercise the mind.
Mathematical techniques like proofs challenge the practitioner to become adept at understanding the process by which you reach a conclusion. Having all that skill can improve innovation and creativity by allowing a person to inherently examine the truth in a problem and solution – not to just take it for granted. That level of analysis can manifest itself in recognizing new solutions or incorrect assumptions to create better innovations
Creativity, as we all know, comes in many forms and is a huge part of Mathematics. Allow me to end with the quote from well-known writer J.K. Rowling (author of Harry Potter novels).
“Sometimes ideas just come to me. Other times I have to sweat and almost bleed to make ideas come. It’s a mysterious process, but I hope I never find out exactly how it works.”