Biblical Hebrew uses three words to describe three different kinds of vows. (All three appear in Numbers 30.)
The first word is neder, which describes pledging property as an offering to God. Such a pledge is an obligation until fulfilled. In the Hebrew language a person does not "pay a pledge" but will "shalom a neder": making peace by fulfilling the obligation.
The second word is eesar, which means "bound". Normally it means physically bound: either tied up, imprisoned, or both. But in Numbers 30 (only) it is used to discuss a personal promise to "bind your soul", which tradition interprets as temporarily abstaining from food, sex, or other pleasure to better stay focused on God. Someone who "binds their soul" is voluntarily assuming an second kind of obligation (to abstain from pleasure) to reach a spiritual state of heightened innocence.
The third word is shvooah, which is used for an "oath of completion": an interpersonal obligation. A shvooah is when one person swears (to another) that he/she will do something. In most cases a shvooah is not a spiritual issue. But a person may make a personal eesar public by "swearing to bind his/her soul"; in other words, by making that promise of abstention publicly before witnesses the eesar also becoms a shvooah.
There are countless examples of vows in the Tenach. Swearing any of the three types of vows was a common practice.
Note that if the vow was purely personal then the common practice was to use God's name when making the vow. Two examples of this kind of personal (but public) eesar vow are when Abraham vows to keep his hands free from greed (Genesis 14:22-23) and Boaz vows not to give in to lust (Ruth 3:13). Deuteronomy 6:13 agrees this is proper procedure for personal vows.
Oddly, the first example of swearing "As Adonai lives!" is the double occurrence in Numbers 14:21 and 14:28, and it is God himself who invents the idiom!
God's name was also used for interpersonal vows. Jeremiah, with a contrast between verses 4:2 and 5:2, taught that a shvooah vow of interpersonal obligation is only good when it is declared regarding actions of truth, justice, and righteousness. How would someone know that an action will have truth, justice, and righteousness? Chapters 2 answers by instructing us to ask "Where is Adonai?" (2:6-8), to pursue scripture as valuable (2:8), to fear Adonai (2:19), to seek Adonai in times of prosperity as well as times of trouble (2:27), and to be open to receiving correction (2:30). According to Jeremiah, habitually doing those five things will cause a person's actions will be in line with God's desires of truth, justice, and righteousness.
One type of vow deserves comment because it involves both a public proclimation and a private abstention from a certain pleasure. The Nazerite vow of Numbers 6 was called a neder, not at an eesar. The shaving and re-growing of hair was physical and visible to the community, and thus more significant that the personal abstention from grape products.
Now we are ready to understand Numbers 30, which discusses how family relationships affect certain kinds of vows. This chapter is not about the usual shvooah oaths of interpersonal obligations. (With one exception we will describe later, the Torah never claims that people are limited by age or gender in how they may commit to interpersonal obligations.)
A careful consideration of the Hebrew vocabulary we have learned, as it occurs in Numbers 30, teaches the following:
To summarize: (a) Fathers/husbands own most of the household property and other family members cannot give it away without permission; (b) Self-denial should be private, but may be public for married adults; (c) A woman's self-denial requires the father's/husband's approval.
The strange omission is of requirements for young boys: it seems odd to allow a young boy to freely donate family property to the Tabernacle.
Apparently the phrase "As Adonai lives!" was no longer in use in the first century. Also, two details of making vows changed for Yeshua's followers.
First, in Matthew 5:33-37, Yeshua taught that if a vow is not purely personal it is not proper to use God's name. Interpersonal obligation should be sealed by personal integrity without invoking God's name.
Second, First Corinthians 7:4-5 extends gender equality to eesar vows of abstainance. Now neither the husband nor the wife alone should vow to abstain from sexual intimacy: this should be a joint decision.
In other was making vows was unchanged. Note that First Corinthians 7:4-5 does teach that it is still appropriate to vow abstainance from sexual intimacy to improve a time of fasting and prayer.
Acts 5:1-11 shows an example of a neder vow used by Yeshua's early followers. Hananyah and Shappirah voluntarily put themselves under an obligation, and when they violated it God killed them.
Acts 21:22-24 shows that the Nazerite vow was another type of neder vow used by Yeshua's early followers.
Consider Jeremiah 2-5 again. In these chapters, God is chastising the Israelites for no longer looking to him. The people are not bother to acknowledge God lives in their deeds. The example of vows is only one aspect of how they are treating God as dead. Vows can be tools in acknowledging that our God lives!
A neder vow is still an appropriate way to help God's people. For example, most communities of Yeshua's followers practice tithing even though Yeshua never told his followers to tithe aside from the fest-day Temple tithes.
Yeshua taught in Matthew 25:14-30 that non-monetary "talents" we are blessed with may also be appropriate items to dedicate to God in a neder way.
An eesar vow can still aid prayer. In Matthew 17:21 Yeshua mentions that self-denial can help our spiritual warfare. Self-denial can also help us focus our energy to metaphorically "feed" our new nature empowered of God's Spirit while "starving" the old nature that is empowered by the flesh.
As Yesuha's bride-to-be we should realize that Yeshua can annul our imagined ways of acting holy and our rash utterances of promised behaviors.
We must not make deals with our Adversary or pledge away our inheritance.
All of our vows must be for truth, justice, and righteousness, through a lifestyle of habitual awareness of God, studying scripture, fearing God, seeking God at all times, and receiving correction.
The Kol Nidre prayer has created a historical confusion regarding Jewish culture and vows. Historically the prayer was used to anually repent of rashly made vows to God. However, in the eleventh century it was reworded to the future tense, to ask in advance for forgiveness for the future year's rash vows, and the new wording neglects to specifically mention that it is about vows privately made to God. Thus Gentiles who heard the prayer believed that their Jewish neighbors could not be trusted, for apparently they annually prayed that no upcoming vows would be binding.
The practice of occasionally repenting of rashly made vows to God has merit. Here is an example of a modified version of the Kol Nidre prayer that is more explicit about its intent.
All our vows and any other ways we have committed ourselves without seeking God: we regret, and we repent!
All our plans and dreams we have set our hearts towards without considering God: we regret, and we repent!
All our promises we have made to ourselves without thinking of God: we regret, and we repent!
All our dedications we have made to God without following his will: we regret, and we repent!
And may atonement be granted to all the People of God and to the stranger who lives among them, for all have transgressed unwittingly.
Forgive the sins of your people in accordance with your great mercy, as you have continued to forgive them from the days of Egypt until now.