How does the New Testament define and describe a disciple?
In the New Testament a "disciple" is more than just the basic definition of the word (mathetes) as a "learner" or "follower."
Jesus's call to be a disciple went like this:
34 Then he called the crowd to him along with his disciples and said: “Whoever wants to be my disciple must deny themselves and take up their cross and follow me. 35 For whoever wants to save their life will lose it, but whoever loses their life for me and for the gospel will save it" (Mark 8:34-35).
The Jews knew all to well what a "cross" was. It was a brutal method of execution the Roman Empire had used extensively to keep them, and other conquered nations, in line. At times, hundreds of people would be publicly crucified along a major road just to make a point.
Jesus is saying you cannot follow Him and follow your old self-centered will. You must nail it to a cross if you want to be a disciple.
Of course, we spend our lives learning to do this, one day at a time, dying to self so we can follow Christ. In Luke 9:23 Jesus said,
“Whoever wants to be my disciple must deny themselves and take up their cross daily and follow me."
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How can we "lose our life to save it?"
Paul said of himself, "I die daily" (1 Corinthians 15:31), meaning that not only had he set aside his former life as a Pharisee, but also as an apostle he actually put his life at risk often for the cause of Christ.
Why would he do this? Because of his hope in the resurrection (1 Cor. 15:32) and spending eternity with Christ.
In other words, he was willing to lose his life here for Christ because of the life to come, and his desire for others to share in that life.
There may be times in following Christ that we can't see any earthly upside to it. We must trust that God will make it up, and far more, in heaven (2 Cor. 4:17).
Someone said, "Until you are heavenly minded, you are no earthly good."
Jesus has commanded us not just to be disciples, but to also "make disciples" and teach them to obey all He commanded (Matthew 28:19-20).