Saturday, December 11, 2010

Symbol and Meaning

I made an abbreviated effort to express my current epistemology of mathematics, in which symbols take on a primary role, to one of my fellow grad students. His response was, "But they're more just than symbols."

Just symbols? I can see where he's coming from. Anyone who's tried to explain mathematics to someone else knows the frustration of trying to express concepts as more than mere symbolic manipulation. As grad students we're expected to teach mathematics to students who are often pretty uninterested in math, and who have often been taught their whole lives that math is nothing but symbols which have no connection to the real world. Trying to convince these students that there is real meaning behind these symbols can be exceedingly frustrating, especially if they care about their grades but not about the material. Students who try merely to memorize symbolic manipulations without understanding any of the concepts will do poorly in a college calculus class, and the worst part is they might not ever know why.

But on the other hand, mathematics cannot exist without symbols. Even a blind mathematician must find some representation in order to embody his conceptual framework--symbols don't have to be purely visual. This "deeper understanding" which supposedly exists beyond the symbolic manipulation is, ironically enough, only expressible through symbols, that is, through words--all words are symbols.

What is the difference between "merely a symbol" and a symbol with content behind it? It all has to do with relationships. Someone can memorize certain symbolic manipulations and do pretty well on a certain set of calculus problems. However, this person will inevitably come out of a college level exam complaining that his teacher never showed him those kinds of problems before. This is because the only connections he has ever made are between the mathematical symbols themselves--put these symbols in a new context, and he is totally lost. Sometimes even just changing problems he has already seen before will confuse him so much that he won't know where to start. What he must learn to do is relate mathematical symbols to other symbols. The symbols "d/dx" stands for derivative, and students can quickly memorize symbolic relationships like "d/dx x^n = nx^{n-1}." What is more important, however, is connecting the symbol "d/dx" with things like the picture of a tangent line to a graph, or ratios like speed (miles per hour) or marginal cost. This ability to relate different symbols to one another is not merely an aid to learning; it is learning.

There is no meaning apart from symbols. Although we tend to insist that there is meaning beyond the bare symbol, this often presupposes that this meaning could exist even without the symbol. Yet if the symbol doesn't exist, does the concept really exist? If an idea is expressed by no one, is it really believed by anyone? Ideas seem to form gradually through a series of experiences leading the mind to make many connections which it could not make a priori. But an idea never truly comes to life until it is expressed symbolically through language; and once that expression has been made, the idea continues to evolve as this symbol inspires new experiences and connections.

There is no mere one-to-one correspondence between a symbol and its meaning. We understand only by developing our own matrix of relationships between symbols. Relationships are the key to understanding. Our understanding is really the sum total of the interaction between all the various symbols we use to coordinate our response to the world around us. The stronger the connections we make between symbols that ought to be connected, the better our ability will be to deal with complex issues.

To me, this is all Christological. God can't but be tied to his Incarnate self, Jesus of Nazareth. And Jesus is tied to the symbols through which we come to know him--the church, the scriptures, the sacraments. We don't really know God directly in the way that certain philosophers would hope. God reveals himself in symbols, and the knowledge of God comes in dealing with these symbols properly.

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