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.”
I would like to take a moment and discuss a topic that is probably not a favourite among the majority of the population: Mathematics. I will now give you a moment to compose yourself and allow the nauseous feelings to subside. You may now be wondering why in a blog about corporate innovation and promoting an innovative environment I choose to talk about such a dreadful topic. My reasoning is rather simple: in reading about innovation and how to get people to be creative, the benefits of Mathematics are never mentioned and yet it is in everything from Art to Music. We focus so much on allowing free thinking and encouraging others to look beyond and outside the box. What about Math?
Unfortunately, many assume that thinking about Math will place a person in an analytical rather than creative frame of mind and may inhibit innovation. Is this true? If you’ve struggled with Math in the past you’re probably hoping that it is! Sorry…I am going to prove otherwise and hopefully convert you to a Math lover. No, I am not intending to turn you into the next famous Mathematician, but rather to develop in you an appreciation of this complex subject and show you how something perceived as “structured” can actually promote abstract and boundless thinking. Weird, huh?
When we think of Math in a general term, it includes complex subtopics like Calculus, Multi-variable Calculus, Geometry, Algebra, Differential Equations, and so on. Yes, these are very complicated in application, but one need only look at a picture or painting or listen to a symphony or rock song to appreciate them in real life. In addition, there are more advanced topics like Logic, Abstract Algebra, and Linear Algebra but these are for enhancing logic, reasoning, and visualization and their benefits will be addressed in another post.
To see the complex but paradoxically simple beauty of Math, you can look at almost any masterpiece of painting or sculpture. What is the best part about this? Knowing how to solve complex differential equations and triple integrals is not even necessary. In fact, most of the beauty we see is more from applying simple geometric patterns over and over in some sequence. When done this way, it gives rise to what Mathematicians term Fractals – a detailed repeating pattern that makes itself obvious when zooming in on the picture. One person in particular, Benoit Mandelbrot, has become known as the father of fractal geometry. As stated in a Wikipedia article about him, Mandelbrot “emphasized the use of fractals as realistic and useful models of many “rough” phenomena in the real world. Natural fractals include the shape of mountains, coastlines and river basins, the structures of plants, blood vessels and lungs, the clustering of galaxies; and Brownian motion. Fractals are found in human pursuits, such as music, painting, architecture, and stock market prices.” I encourage the reader to type into a search engine the term ‘fractal’ and enjoy all the interesting pictures that result.
Besides fractals, another interesting common mathematical wonder (for lack of a better term) is the Fibonacci sequence – a sequence of numbers defined by the linear recurrence equation Fn = Fn+1 + Fn-2. Although it looks really complex, it basically means you get specific numbers: 1,1,2,3,5,8,13,21…(to infinity). So why is this interesting? Well, if you are a fan of Dan Brown’s novel The Da Vinci Code or the TV show NUMB3RS, then you may already have some familiarity with this. Fibonacci numbers are found in the structure of crystals, the spiral of galaxies, and in the design of a nautilus shell. A rather new innovative application is found in the song “Lateralus” performed by the progressive metal band Tool. Maynard James Keenan, the lead singer of the band, not only sings of spirals but the syllables of the lyrics follows the first few Fibonacci numbers. It is worth sharing, as it is rather cool to see:
(2) white are
(3) all I see
(5) in my in-fan-cy
(8) red and yel-low then came to be
(5) reach-ing out to me
(3) lets me see
(2) there is
This is just part of the song, but you can already see him following the Fibonacci sequence 1,1,2,3,5,8..and then reversing back…5,3,2,1,1 etc. Watch the video below to hear mathematics in action:
So what does all this mean to the osmotic innovator? Many times we are faced with what seems like an insurmountable challenge. We attack it head on and think that with enough critical thinking and brainstorming we can somehow solve it. We may even go so far as to call in specialists and technical experts for their point of view. Does this really help us approach the problem differently, or are we just getting lost in the proverbial complex landscape? Instead; take a problem, break it down into its constituent parts and handle each one of those separately. Mathematics teaches us to take a step back – don’t get lost in the extreme complexity, for even in something as complex as the nautilus shell there is a simple structure to be found. We don’t all need to be a mathematical genius to solve something that seems rather complicated. Remember, the lead singer of Tool, James Maynard Keenan, used the Fibonacci sequence in a song. He took something difficult, simplified it, and turned it into music. Now think of the possibilities available to you as osmotic innovators…they’re infinite!