Wednesday, 10 December 2014

Proofs and problem solving, part 1

The human visual system alters and reshapes our perception of reality. Highlights appear where there are none, shadows are created that do not exist, and identical colours somehow end up looking dramatically different. We are so accustomed to this that it passes without notice. 

The effects can be very strong. I remember being astonished in the mid 1970’s when I first learned about the highlight and shading effects in the Cornsweet illusion and Mach band effect. There are also amazing difficulties in accurately perceiving colours. See for example the examples here (scroll down a screen or so to the "blue and green spirals"). I urge you to visit these illusions before reading the rest of this page. 

There was a time, well before the invention of photography, when artists strived to portray things in a realistic fashion. The perceptual distortions of light and colour mentioned above must have been a challenge: if an artist were to paint what he “saw”, he would include those false visual effects, and a person viewing the painting would be looking at something that differed significantly in light, shade, and colour from the actual scene. As a result, the viewer's perception would likely differ quite a bit from the painter's perception. Sort of like the increased blurriness that you get when you save a jpeg copy of a jpeg, or the degradation that occurs when you make a Xerox copy of a Xerox copy. Passing an already filtered version back through the same filter may have a detrimental effect.

Yet, despite these obstacles, some artists were able to achieve remarkable realism. It makes me think that they were fully aware of the artifacts that their vision introduced, and that they found ways to expunge the distorted light, shade, and colour from their paintings. Exactly how they did this is not clear.

One solution is offered in the film Tim’s Vermeer. It recounts Tim Jenison’s attempt to describe how, in the 17th century, Johannes Vermeer could have used relatively simple optical devices to help him create the light and colour in the painting The Music Lesson. Although not explicitly stated as such, the film is all about how Vermeer could have circumvented the problems inherent in his own visual system and thereby realistically duplicate in his painting the physical highlights, shadings, and colours that were present in the scenes that he observed.  

Viewing the film through my filter as a mathematician, I think of it as somewhat of a sustained metaphor about the process of problem solving and its relationship to proof. In a university lecture, the presentation of a solution frequently follows a path like this: Here’s a problem, here’s my solution, and here’s proof that my solution works. But this is what happens after the solution is obtained. What most likely happened before the solution was completed is that the solving and the proving were thoroughly interweaved.

In the film we see Tim Jenison’s solution to the problem of  accurately reproducing physical light, shade, and colour, along with his proof that his solution works. His proof that Vermeer could have used optical devices is to use those devices himself to recreate Vermeer's Music Lesson.  As the film proceeds, Jenison sticks to his basic premise, but continually adjusts and adapts the process, much like what happens when one is solving a mathematical problem. 

By the time I have solved a problem, my first thoughts about how to solve it have been mostly forgotten. It may be a personal peculiarity, but I find it very difficult to recapture my state of mind as it was when I first considered the problem. When I present my solution, I can polish and burnish it, and can point to logical insights that, once you notice them, inexorably lead to a solution. However, I usually garner these insights as I work my way through the solution; they are not the initial thoughts that led me to the solution.  In other words, the manner in which I present my solution is not always a true reflection of how I solved it. 

This is not a happy state of affairs when teaching. If you show your ultimate solution to a class—no matter how polished and clear it is—if your students do not experience some of  the missteps, stumbles, and side avenues that you took along the way, are you really doing anything different than reading from a textbook? 

George Polya has written books about problem solving, and he gives suggestions on how to get past the initial stages [see the wikipedia article about How to Solve It]. I am not going to dwell too much on that part of problem solving. But what I do want to examine is what happens after you are pretty sure that you are going in the correct direction, how you can continue in that direction, and how during that process you are actually developing an argument that your solution is valid. In other words, I will try to show how proof and problem solving are intimately intertwined just as they are in the the Tim’s Vermeer film.

This will be the content of my next two posts. and I will use two well known simple puzzles to try and illustrate the process. 

Here is the first puzzle:
There are three boxes that contain red and white balls. One box contains 10 red balls, one contains 10 white balls, and the third contains 5 red and 5 white balls. The boxes have been labelled “red”, “white”, and “mixed”, but each label is on a wrong box.   
All of the boxes have lids so you cannot see what is in them, but you are allowed to reach inside the boxes without peeking and take one or more balls out and look at them. Taking out as few balls as possible, figure out what the correct labels should be.
So go ahead and solve it, and convince yourself that you have a solution. Come back in a few days and see if you followed a path similar to mine. 

If you are interested in the Tim's Vermeer film, there is more information about it here and here.  The film is controversial, especially among art historians who view it as an attack upon the artistic integrity of Vermeer. Moreover, in The Music Lesson, Vermeer did not depict everything accurately, but deliberately added and removed light and shade to enhance the painting (see here). This, however, does not disprove that Vermeer used physical devices to help him understand light, shade and colour. And, unlike some of the art historians, I do not view the film as belittling artists: to me it demonstrates the intelligence and ingenuity of visual artists like Vermeer.

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