The Greek word Paul uses for "elemental principles" is stoicheia. In ancient Greek, this referred to things like the letters of the alphabet (the ABCs) or the basic physical elements of the universe (earth, air, fire, water). Paul uses it to describe a rudimentary level of religion, a pretense of piety without reality, —basic rules about physical things that have no power to change the spiritual condition of a man. When believers return to these elemental things after knowing Christ, Paul considers it a regression from maturity to infancy, and from freedom to slavery.
Elemental Principles have no power to mature or perfect the individual in regards to righteousness and godliness nor to give life.
I. Colossians 2: The Failure of Asceticism ("Do Not Touch")
In Colossians, the error was the belief that strict self-denial and harsh treatment of the body constituted "perfection." Paul argues that these regulations are merely human commands concerning things that perish with use.
The Warning:
"see to it that no on takes you captive through philosophy and empty deceit, according to the tradition of men, according to the elemental principles of the world [stoicheia], and not according to Christ." (Colossians 2:8)
The "Touch Not" System:
"if you died with Christ from the elemental principles of the world, why, as though living in the world, do you subject yourselves to regulations—'Do not touch, do not taste, do not handle,' which exists all destined to perish with the using—according to the commandments and teachings?" (Colossians 2:20-22)
The Verdict: No Value Against the Flesh:
"such words indeed have an appearance of wisdom in self-imposed religion, false humility, and harsh treatment of the body, but are of no value against the gratification of the flesh." (Colossians 2:23)
This is the critical connection to Hebrews 9 and Hebrews 10. A person can follow a strict diet, observe religious washings, and fast until they are weak (harsh treatment of the body), yet the "gratification of the flesh"—pride, anger, lust—remains untouched. It is an "appearance of wisdom" without the power of transformation.
II. Galatians 4: The "Weak and Beggarly" Calendar
In Galatians, the "elemental principles" manifested as the observance of the Jewish religious calendar. The Galatians were being told that to be truly perfect Christians, they needed to adopt the Mosaic Law's system of holy days. Paul calls these "weak" because they cannot justify, and "beggarly" (poor) because they have no spiritual riches to offer.
Enslaved to Non-Gods:
"Even so we, when we were children, were in bondage under the elements of the world." (Galatians 4:3)
Returning to Poverty:
"But now after you have known God, or rather are known by God, how is it that you turn again to the weak and beggarly elemental principles, to which you desire again to be in bondage? You observe days and months and seasons and years. I am afraid for you, lest I have labored for you in vain." (Galatians 4:9-11)
Paul views the transition from pagan idolatry to Jewish ritualism not as an upgrade, but as a lateral move from one form of slavery to another. This is what he means when he says do not submit again to a yoke of slavery, for Christ has set you free (Galatians 5:1-2). A worldly or religious system that rely on "elemental principles" rather than the Spirit of the Son in the heart, have no power to overcome the flesh (Galatians 4:6; Galatians 5:17).
III. Shadows vs. The Substance
The core problem with "elemental principles" is that they confuse the shadow for the object casting it.
"So let no one judge you in food or in drink, or regarding a festival or a new moon or sabbaths, which are a shadow of things to come, but the substance is of Christ." (Colossians 2:16-17)
The "perfection" spoken of in Hebrews 6 and Hebrews 10 is the Substance (Christ). The "dead works" are the Shadows. When a man embraces the Shadow (the ritual) thinking it will make him perfect, he is boxing at air. The Shadow cannot cleanse the conscience before God; only the Substance can.
Summary: Why This is Not Perfection
Paul classifies these activities as "weak and beggarly" because they are inherently external.
They are Earthly: They deal with food, drink, and days (Colossians 2:21-22).
They are Powerless: They cannot stop sin (Colossians 2:23).
They are Elementary: They are things destined to perish, not of the substance and reality of the heavenly.
True perfection is found in the "new creation" (Galatians 6:15), where the believer is dead to the world’s elementary principles and alive to the law of the Spirit, a new living way of the Spirit.