Generalization and Fulfillment

Generalized Vector Space

Matthew 5:17-20 NIV

"Do not think that I have come to abolish the Law or the Prophets; I have not come to abolish them but to fulfill them. For truly I tell you, until heaven and earth disappear, not the smallest letter, not the least stroke of a pen, will by any means disappear from the Law until everything is accomplished.  Therefore anyone who sets aside one of the least of these commands and teaches others accordingly will be called least in the kingdom of heaven, but whoever practices and teaches these commands will be called great in the kingdom of heaven. For I tell you that unless your righteousness surpasses that of the Pharisees and the teachers of the law, you will certainly not enter the kingdom of heaven.”

Many concepts in mathematics are studied in a spiral. We study a concept in a conceptual setting, then generalize to a more complex setting by increasing dimension, and then generalize again to an even more complex setting by selecting characteristics that appear useful and then removing all other aspects of the conceptual setting. One of the first times the second type of generalization occurs in the undergraduate mathematics curriculum is in the study of general vector spaces.

[Note:  Instances of this type of generalization also occur in non-Euclidean geometry, in group theory, etc.] Students find the lack of a conceptual setting troubling; in fact, instructors of linear algebra often refer to students "hitting the wall" when they first encounter general vector spaces.

Christ also had trouble with his students "hitting the wall" during his ministry. One important aspect of Jesus incarnation was to teach people the meaning behind the law of Moses. His death and resurrection are described as the fulfillment of the Old Testament law. This generalization of the law is beautifully described by Jesus during the sermon on the mount in Matthew 5-7.  A common phrase Jesus used during this sermon is "You have heard that it was said.." Each of these phrases is followed by a generalization of an Old Testament law to include the intent behind the law: murder is generalized to include hatred, adultery is generalized to incorporate lust, and love for neighbors is generalized to love for all. The students of Jesus who "hit the wall" were the ones who thought they understood the law the best--the leaders of the Jewish faith.