Jesus begins these comments in Matthew 5:33 by referencing Leviticus 19:12 and Deuteronomy 23:21-23. In this passage, to "swear" is to utter an oath; it is to appeal to a higher being or thing in order to attest to the truthfulness of one's words. Under the Old Law, making oaths was acceptable, though swearing falsely was never viewed favorably.
Over the centuries, the Jewish people had developed the idea that an oath wasn't binding unless the name of God appeared in it. They would therefore use various oaths to suit their purpose of adding weight to their statements or promises, but they wouldn't feel obligated to keep such an oath if they had not explicitly sworn to Jehovah. The Jews had other strange rules regarding which oaths were binding and which ones weren't. For example, consider a portion of Jesus' fiery rebuke to the scribes and Pharisees in Matthew 23:16-22 - "Woe to you, blind guides, who say, 'Whoever swears by the temple, it is nothing; but whoever swears by the gold of the temple, he is obliged to perform it.' Fools and blind! For which is greater, the gold or the temple that sanctifies the gold? And, 'Whoever swears by the altar, it is nothing; but whoever swears by the gift that is on it, he is obliged to perform it.' Fools and blind! For which is greater, the gift or the altar that sanctifies the gift? Therefore he who swears by the altar swears by it and by all things on it. He who swears by the temple, swears by it and by Him who dwells in it. And he who swears by heaven, swears by the throne of God and by Him who sits on it."
When Jesus said in Matthew 5:34, "Do not swear at all" , His meaning was this: Do not swear at all in the manner I am about to illustrate. Do not make oaths that you don't intend to keep. Do not think that you can word a promise in a special way and in so doing make it void or meaningless. A false oath is always false, whether it is explicitly made in the name of Deity or not. To swear by "heaven" or "the earth" or by "Jerusalem", in an effort to make an oath that did not have to be kept was wrong since all of the statements ultimately refer back to God anyway. Heaven is "God's throne", the earth is "His footstool", and Jerusalem is "the city of the great King" (Matt. 5:34,35). It mattered not if God's name was explicitly mentioned! One's word was to be his bond! Those who thought they could speak oaths in a certain way and not be required to fulfill them were only fooling themselves--much like some today who make an agreement but later deny the validity of such because they had their "fingers crossed" (or some other such childish nonsense).
Jesus' example in verse 36 is distinct from the previous illustrations He has offered. "Nor shall you swear by your head, because you cannot make one hair white or black." It is true that one can change his or her hair color today, but this is not our Lord's point. He seems to be declaring that it is not wise to swear by one's own life since man has no inherent control over such.
We will continue our thoughts tomorrow on this important theme of being people of our word.