In ancient Hebrew thought man is composed of two physical elements:
The making of man into these two elements is graphically described by Moses:
Gen 2:7 then the LORD God formed the man of dust from the ground and
breathed into his nostrils the breath of life, and the man became a
living creature.
In other words, YHVH scooped up some dirt and molded it into a statue of himself and then animated it with the breath that he himself breathed. For Moses the breath had supernatural powers. It animated the lifeless dirt and imparted self awareness, God awareness and general intelligence.
This anatomy persists throughout the scriptures:
Job 33:4 The Spirit [breath] of God has made me, and the breath of
the Almighty gives me life.
Joh 6:63 It is the Spirit [breath] who [that] gives life; the flesh
is no help at all. The words that I have spoken to you are spirit
[breath] and life.
Rom_8:2 For the law of the Spirit [breath] of life has set you free
in Christ Jesus from the law of sin and death.
Rom_8:6 For to set the mind on the flesh is death, but to set the
mind on the Spirit [breath] is life and peace.
Rom_8:10 But if Christ is in you, although the body is dead because
of sin, the Spirit [breath] is life because of righteousness.
Rom_8:11 If the Spirit [breath] of him who raised Jesus from the dead
dwells in you, he who raised Christ Jesus from the dead will also give
life to your mortal bodies through his Spirit [breath] who dwells in
you.
1Co_15:45 Thus it is written, "The first man Adam became a living
being"; the last Adam became a life-giving spirit [breath].
2Co_3:6 who has made us sufficient to be ministers of a new covenant,
not of the letter but of the Spirit [breath]. For the letter kills,
but the Spirit [breath] gives life.
Gal_6:8 For the one who sows to his own flesh will from the flesh
reap corruption, but the one who sows to the Spirit [breath] will from
the Spirit [breath] reap eternal life.
- the breath gives self-awareness, God awareness and general intelligence:
Job_32:8 But it is the spirit [breath] in man, the breath of the
Almighty, that makes him understand.
Pro_20:27 The spirit [breath] of man is the lamp of the LORD,
searching all his innermost parts.
Rom_8:27 And he that searcheth the hearts knoweth what is the mind of
the Spirit [breath], because he maketh intercession for the saints
according to the will of God.
1Co 2:10 these things God has revealed to us through the Spirit
[breath]. For the Spirit [breath] searches everything, even the depths
of God. 1Co 2:11 For who knows a person's thoughts except the spirit
[breath] of that person, which is in him? So also no one comprehends
the thoughts of God except the Spirit [breath] of God. 1Co 2:12 Now
we have received not the spirit [breath] of the world, but the Spirit
[breath] who is from God, that we might understand the things freely
given us by God. 1Co 2:13 And we impart this in words not taught by
human wisdom but taught by the Spirit [breath], interpreting spiritual
truths [truth from the breath of God] to those who are spiritual
[filled with God's breath]. 1Co 2:14 The natural person does not
accept the things of the Spirit [breath] of God, for they are folly to
him, and he is not able to understand them because they are
spiritually discerned. 1Co 2:15 The spiritual person [full of God's
breath] judges all things, but is himself to be judged by no one. 1Co
2:16 "For who has understood the mind of the Lord so as to instruct
him?" But we have the mind of Christ.
The "mind" then is an abstraction, a manifestation of the operation of the breath interacting with the body and not a "thing" in itself. The modern term to describe this phenomena is "supervening":
https://en.wikipedia.org/wiki/Supervenience
So the LXX translator was using synecdoche and emphasis in his translation. He is using synecdoche by referring to "mind" to refer to the "breath" obliquely and he's using emphasis by specifically pointing to a single function of the breath that is most relevant to the point he is making.