At first glance this particular proverb seems kind of intense. But when you consider what happens when a child is not taught to bring his selfish tendencies under some kind of discipline and control - you are leaving them in a very precarious position in life.
We are to discipline our children early. That is what is being said by saying to discipline a child while there is hope. The hope is that by helping a child to recognize selfishness - and to fight its destructive ways - you are giving that child hope for better things. Consider if you will the start of life on this planet. We are taught that Adam and Eve had two sons. Jealousy led one son to kill his brother. The early chapters of Genesis inform us of a remarkable downward spiral that took place in the human race. By the time God decided to destroy all but Noah's family, the imagination of man was set continuously on evil. It did not take long for the Fall of mankind to manifest itself in every kind of sin and rebellion imaginable. This is why we MUST discipline our sons. Because of the Fall mankind is basically and terminally selfish and self-centered. Left to himself man would destroy himself with this selfish bent. Discipline at an early age helps to combat this natural selfish tendency. We bring this discipline to our children in hope that they will have their basic, natural selfish tendencies held in check until a time when they are convicted of sin by the work of the Holy Spirit. It is not that we think we can overcome the flesh by child-training, but we do see a need to put limitations and boundaries in their lives so that the flesh is not completely unchecked.
To withhold discipline from a child is to "desire his death." There are some who translate this as meaning that a father is not to discipline so severely that he winds up killing a child, but the evidence behind this translation is very weak. The King James version translates it as a call to discipline a child and not be swayed by their crying. When you discipline a child properly, most will cry. Some will cry because they were spanked - others will cry because they are responding to the guilt of being caught - while still others will cry to try to get you to stop the discipline process. Whatever the case, this passage, if translated this way is trying to steel the heart of a parent who may decide against discipline because their heart just can't handle having their child cry or be in any kind of pain. This is actually selfishness on the parent's part because they should be looking at the long term effects of the punishment and discipline, not the short-term reaction of the child.
The other main way this is translated is to warn the parents of the long term effects of not disciplining a child. The natural selfishness of a child will lead them to reject the Lord and embrace their full sinfulness. This, if left to fester and grow to its fullness will bring a child to a point where they will embrace a lifestyle without God. In the end, their willfulness will kill them - if not with actions that are harmful to their lives - then with a rejection of God and His discipline. Hebrews 12:5-9 reminds us that God disciplines us as a godly parent would. If a child is only used to getting their own way, they will reject such a God - and will ultimately reject Christ.
Disciplining a child is serious business. We are to do so with a godly set of standards and choices. These guide a father as he disciplines with a view to godliness - and also as he reigns in his own anger and refrains from having discpline turn into abuse. But the real end in disciplining a child is to turn them away from their own selfish, fleshly tendencies. The process of making a child face correction and conviction for doing wrong is preparation for God's future rebuke of their conduct. This one will come by God's grace as they are convicted of their sin - and brought to repentance and faith in Jesus Christ as the answer for their sinfulness and need of salvation. In light of THAT day, discipline your child and make them face their selfishness and sinfulness - for your preparatory work will TRULY prevent them from death - ultimate death due to sin.