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Friday, November 14, 2008

Understanding Through Mathematical Concepts

My great aunt is a wonderful lady. A worldly intellectual in her own right, she can speak knowledgeably about Thucydides (which she read in the original Greek, not that wimpy translation stuff I read) and other literature of which I could not hope to compile an exhaustive list, as well as hold her own in a discussion of history and politics, especially if it involves Korea (where she was born and raised through much of her childhood before returning home to Canada). I am also a big fan of my great uncle, but since it was my aunt that made the comment I am going to discuss, I will have to wait for another day to sing his praises. I bring up my esteem for my aunt to put in context a comment she made one night when my girlfriend and I were at my aunt and uncle's for dinner, in which she stated something to the effect that she didn't understand how mathematics could hold any draw as a subject since it was such a dry and abstract thing. I think it was somewhat unfortunate for her that she made such a comment at a table with her husband (a retired aeronautical engineer), my girlfriend (who studies physics and mathematics), a Russian fellow who sails with my uncle and his wife (both who studied mathematics and computer science before moving to Canada), and me (a former aerospace student and now student of computational neuroscience), so she may have been a little unfairly outnumbered by those who had ties to mathematics. A great cry went up around the table and everyone tried to explain all at once that mathematics was, in fact, a wonderful thing. I don't think my aunt (a former graduate of the humanities) was trying to be confrontational at all, but I think she really was baffled (and, unfortunately, I don't think any of our answers really cleared anything up at the time, since the best we came up with was simply that it helps you to see the world differently without really giving any examples). I also don't think my aunt is alone. For many people, mathematics remains a dry and stuffy subject, handy for balancing the books and maybe work in research and design (but even then, there are a fair share of engineers who forsook mathematics upon achieving their degree and getting a job), but beyond that they don't have a concept of it.

While I am no mathematician, I still enjoy mathematics and dabble in it in my studies. I will therefore endeavour to give an example of how mathematical concepts can help explain aspects of the world using a personal insight about another subject that also commonly baffles people: speciation in evolutionary biology. Among critics of evolution, one of the commonly fallacious argument given is, "if evolution is true, why doesn't a dog give birth to a cat?" (or some other ridiculous combination). While that is probably the most ridiculous formulation of the argument, the basic idea that trips people up is understanding how one species can evolve into another. This lack of understanding often leads to the lamentable "middle of the road" half-cocked compromise in which a person accepts "microevolution" while claiming that he still doesn't believe in "macroevolution". To give some insight into how speciation works, at least from a conceptual standpoint, I turn to probability and calculus.

Take a circle with a spinning dial mounted in the middle. If you mark a spot on the circle (say the spot corresponding with '12' on a clock face) as the 0 mark, then you can spin the dial and it will land with some anticlockwise angle from 0 to 360 degrees. Since there are an infinite number of points on the circle, however, if you take your measurement to an arbitrary level of exactness (landing at 10.0000000000001 is different from landing at 10 exactly), the probability of landing at any distinct spot is essentially 0. The only way to obtain a non-zero probability is to talk about a range of possible angles. The probability is then simply the length of that range divided by 360 (thus, having the dial land within the first 90 degrees has a probability of 90/360 = 1/4). Thus, the circle can be divided into regions, each one representing a range of possible angles and thus having a non-zero probability. However, at the borders we see that which region we are in becomes a harsh cut-off over a seemingly negligeable difference. For example, if we divide our circle into four regions of equal size (each representing 90 degree increments), 89.99999999... would fall into region 1 while 90.0000000...001 would fall into region 2, despite an arbitrarily small difference between the two of them. Take the idea of that circle and now morph it in your head to represent an evolutionary lineage. The population of organisms at each moment in time represents one single location on the circle. A region represents a species, and thus a species is said to evolve into another if its region precedes the other. But remember that the demarcation line of our regions was essentially an arbitrary cutoff, a boundary imposed to provide meaning to the system. There is no drastic change in the dial's position when we go from region 1 to region 2, but rather the change can be as infinitesimally small as we want. Likewise, the change from species A to species B is not some drastic, single moment of monumental change such as a dog giving birth to a cat, but is rather a collection of tiny bumps in the dial position as it gradually creeps along the circle going from region 1 to region 2. However, when one compares the dial position from somewhere near the middle of each region, it looks to be very far apart.

This is no lofty or profoundly insightful thing I have come up with. I also recognize that I may have taken some liberties with the specific terms and workings of both mathematics and evolutionary biology, so for anyone who is actually in those fields and upset with me, I apologize (and you have full permission to admonish me in the comments). However, it is something that I have discovered a surprising number of people never really put together on their own. The concept of infinitesimal steps from calculus is a profound thing, and with it many other concepts in the world can be illuminated more fully. That, to me, is how mathematics is not dry or dull. Its concepts are wide reaching, elegant, and profound. If a person understands mathematics, there is a huge variety of subjects that suddenly become easily grasped.