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.

Xenophobia and Alien Fascination

I am going to indulge myself in some fanciful psychology now. While I often rail against the credulity and imprecision of psychology as a general field, that doesn't mean it isn't a fun way to pass the time. My main problem is its lack of rigour and self-correction, not the fun that can be had in utilizing the field to explore human nature. It is like literature and art - while some may make the argument that those should be taken into the fold of science (I haven't actually read the book and I enjoy Jonah Lehrer's blog, but I disagree with the premise that the book is touted to have. Perhaps reading it will change my mind, and I wouldn't mind giving it a shot at some point), I would rather say that much of psychology should be relegated to the social science and humanities status that encompasses art and literature. It is fascinating and fun to explore our humanity with words and conjectures, but it is rarely done rigorously and powerfully like the work done in most of the other sciences.

Anyway, with that preamble out of the way, I am going to launch into my exploration of one of the aspects of the human psyche. There were two events that got me thinking about the subject today of non-human animals. The first was an experiment on mice that we were performing in my lab course this morning and the enjoyment I got out of watching the adorableness that was the mouse's whisker twitches as he explored the box he was in (I certainly would not want to be a behavioural neuroscientist or psychologist. Giving injections or any of the other things done to animals in a laboratory I would find too difficult. While I recognize the benefit, I am squeamish and soft-hearted and enjoy my cognitive dissonance, thank you very much). The second event was a slide containing information about the variety of different somatosensory representations in different species, notably raccoons (and how they had a very similar proportion of cortical space devoted to finger control as that found in primates). That prompted me to think about how neat I find raccoons and their cleverness and aptitude for mischief, and then a brief internet search into raccoons as pets (apparently they do not make good pets, as they tend to have an intense ornery streak as adults including indignant defecation on owner's belongings and a tendency to bite). From my brief sadness at finding out that, like many of the other creatures I have often wistfully thought of as having as a pet, I was never going to have a raccoon as a pet, I got to thinking about the odd propensity of people for domestication of other species. While much of domestication can be explained by the utility of it (and we are not the only ones who engage in such practices, with fascinating ecological tales of ants and aphids or butterflies (the guy's voice in the video I actually found kind of annoying, but you get the idea) and other such relationships found in nature textbooks), there are clearly indirect psychological benefits as well (not for everyone. Some people genuinely have no enjoyment in interaction with nonhumans. While I find that baffling, it clearly exists).

The thing is, domestication simply facilitates the process. It makes the other organism more prone to like us and accept interaction with humans, but there are many examples of non-domestic animal friendships. Some of my favourites include Jessica the Hippo and Owen and Mzee (odd that they both include hippos). Clearly, there is at least a small propensity for interspecies bonding even without domestication. What makes this such an appealing concept? Before I address that concept, though, I want to bring up another one: xenophobia. Basically, fear of those foreign and different is a powerful psychological motivator at the heart of ingroup-outgroup conflicts. It doesn't have to include humans and humans, but is also at the heart of the general trend to treat non-human animals as less deserving than humans (after all, if the colour of one's skin is enough to inspire contempt and hatred, not even sharing the same form is an even more blatant marker of difference. The main reason it is rarely labelled as xenophobia, though, is people take it as defacto that other animals are so different from us that it doesn't need to be acknowledged). So how can interspecies relationships work and be such a powerfully uplifting thing (for at least some people) when there is an inborn tendency for mistrust of those who are different that sometimes fails to even reach past boundaries within a species, nevermind to another species?

At this point, I began to ramble on in a long-winded analysis of the intricacies of interspecies relationships. However, this post has already gotten longer than I intended, so perhaps I will leave such an analysis to a later time and jump ahead to the conclusion I was intending to reach. It seems to me like there is a competing psychological need for empathic understanding instilled by our social nature and the instinctive mistrust of strangers needed for simple survival. While the mistrust of strangers is necessary without being able to guarantee the trustworthiness of others, it isn't pleasant. Conflict is messy and uncomfortable, and though xenophobia can often be gratifying and exhilerhating, that is only when one is surrounded by likeminded individuals railing belligerently against the out-group that is not present. When one is forced to come face to face with the brutality and hate that was so previously euphoric (especially if one is forced to deal with the outgroup on relatively equal terms, like in a war, rather than in unequal terms like a lynch mob), the rush of happiness inspired by belonging with the ingroup is tempered somewhat by the unpleasantness of conflict. I think most people who are not insanely bigoted understand this and frown upon that aspect of their psyche. Forming a bond with a member of an outgroup, therefore, is a way of throwing off the shackles of xenophobia and searching for a linking trust rather than dwelling on the separating differences. Instead of saying "You walk on four legs and I walk on two, so I look down upon you", it is saying "I like to scratch you behind the ear and you like it when I do, so let's be friends." We can think that if we can set aside our mistrust and fear of the other party in this situation, maybe we can do so again in the future and have a more peaceful life. Or maybe I'm just a dork wrapped up in his own fanciful psychology. Whatever the case may be, I know some day I'd like to rub a tiger's belly, scratch a hippo behind the ear, or play a game with a raccoon. I also know none of those are going to happen, and even if given the opportunity I'm not sure I'd have the courage to go through with it. In a way, that thought makes me very sad.

I knew it....

I knew you couldn't trust that Putin...

Remembrance Day

This isn't exactly a holiday to be happy about, so rather I'll just point out that it is happening here in Canada. It is Remembrance Day today, so spare a few thoughts for the fallen.

How to Think About Science

I was listing to the opening segment of an interesting set of broadcasts collectively entitled "How to Think About Science" and I thought it had some very interesting points. The one I thought that was most interesting was when Simon Schaffer pointed out that science, while normally celebrated as promoting skepticism and a reliance upon personal evidence and observation, was in reality a systematic organization of trust. You will have to listen to the broadcast for his full argument, but it is essentially that no one can practically witness evidence for everything one accepts as true, but where science excels is in giving a powerful framework for deciding who and what should be given credence.

I thought that was a very interesting and thought-provoking observation. It is quite simple and seems obvious after hearing it, but in many ways those are the best thoughts to have. I found myself thinking about it this morning as I read the news. So many of our world's problems, especially in the political sphere, are based on issues of trust. It is one of the exceptionally messy aspects of politics that makes me want to practically avoid the field. It is also why pseudoscientific things like creationism/intelligent design and alternative medicine continue to flourish outside of the scientific world (in the realm of the popular and political) where there is not that system of rigorous evaluation to keep them in check.

Some Words of Wisdom to Start the Week

"Women are really much nicer than men:
No wonder we like them."
"No pleasure is worth giving up for the sake of two more years in a geriatric home in Weston-super-Mare."
- Kingsley Amis, English novelist and poet, 1922-95

"Children's talent to endure stems from their ignorance of alternatives." - Maya Angelou, American writer, 1928-

"God is on everyone's side... And, in the last analysis, he is on the side of those with plenty of money and large armies." - Jean Anouilh, French dramatist, 1910-87

Top-down Processing in Visual Perception Part I: Introduction and Some Examples

One of the subjects I have written about before is machine vision and the incredible difficulty of developing a robust visual processing system that can equal the robustness of our own visual system. It shouldn't be entirely surprising, though, that our visual system is as incredibly powerful as it is, since a huge proportion of our brain is utilized primarily for visual processing. One of the interesting debates in perception psychology and neuroscience is whether the brain performs bottom-up or top-down processing. As with most things (especially in psychology), neither one is entirely correct and your brain utilizes a combination of the two. Optical illusions and trick images are one relatively simple way to explore the way our brain processes visual information, and they are also fairly fun to look at.

Bottom-up processing basically means your brain reads in the raw visual information captured by the retina and gradually figures out what it means as one moves farther along the processing chain that is your cerebral cortex. Top-down processing means you start with an idea of what you ought to be seeing (most likely determined by recent sensory information, other sensory clues, and your past experience). Your brain clearly does some bottom-up processing, since you react to raw changes in the visual stimuli even if there was no reason to expect that change. What is fairly surprising, though, is top-down processing is also clearly involved in visual processing. Effectively introducing top-down processing into artificial visual systems, however, is quite difficult, and it would seem that the top-down algorithms instituted by our brains (and their handy parallel architecture) are what keep us currently so far ahead of computers.

One example of top-down processing that is fairly easy to demonstrate is the blind spot. In your retina you have a small area devoid of receptors where nerves and blood vessels enter and leave your eye. This is normally not a problem since the blindspot of each eye falls on a different area of your visual field, so the sensory perceptions of one eye can compensate for the other. Also, your eyes are almost constantly performing saccades (small jumps around to focus on different regions of the visual field). However, if you close one eye and keep your other eye locked on a specific target, your blind spot becomes anchored in place. You do not realise this, though, because your brian manages to fill in that area of your visual field with its best guess as to what is there. A quick way to demonstrate this is to take a piece of scrap paper and put two X's on it about eight centimeters apart. Then close one of your eyes and stare at the opposite mark with your open eye (for example, if you closed your left eye, look at the left X with your right eye). Hold the paper about half an arm's length in front of you and gradually move it closer. At a certain point, the X on the periphery of your vision should disappear. When it does, it is sitting in your blind spot, and your brain fills in that area with it's best guess (in this case, blank white paper).

Another example that occurs slightly higher up in your visual processing is the Necker Cube, shown here.

This simple drawing forms a three dimensional clear cube. It is ambiguous, though, whether it is intended to be in one of two possible orientations: are you looking slightly down onto the cube, or slightly up at it (in other words, are the bottom two corners corners on the front or back face of the cube)? For most people, there is a default orientation when they first see it. However, after staring at the cube for a few moments, they can cause it to 'flip' into the other orientation. At no point, though, can both orientations be held in one's head at once (at least, I cannot manage to do that). It would seem that your brain takes the visual information provided about the cube's edges and then tries to fit an interpretation on it. Since more than one interpretation is possible, your brain alternates between them. However, whenever one particular interpretation is selected, the others are suppressed to avoid conflicting interpretations of a visual scene.

Continue reading in Part II: Faces.