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This question is but a specific case of the problem of evil, a matter which Christian apologists have responded to many times. I won't offer a lengthy discussion of the problem of evil here, but I'll offer a brief logical & theological refutation of this specific argument.
Logical argument
The argument is at its core attempting a reductio ad absurdum - endeavoring to show that, given xyz evidence, it is absurd to believe that God exists. However, what is presented here is not a valid reductive argument--the flaw is subtle, but crucial.
To construct a reductio ad absurdum argument against God one must begin by accepting, for sake of argument, that God exists. This is the way the first premise is formulated for any reductio ad absurdum--one must start by accepting the truth of the claim in question, in order to show via the absurdity that follows that the original premise should be rejected.
This reductive argument tacitly assumes--as most atheistic arguments do--that there is no afterlife, there is no eternity, and so any injustice not cured in this life will never be cured. This assumption destroys the reductio ad absurdum--to state the premises formally, this argument approaches the problem as follows:
P1: God exists
P2: There is no afterlife
The best the reductio ad absurdum can do is show that these premises cannot both be true; since no Bible-believing Christian would accept premise 2, the reductive argument is entirely ineffective.
The moment we open the door to an afterlife--especially an afterlife that is far greater in scope & duration than mortal life--theological explanations for human suffering in mortality abound. Atheists who do not wish to engage with theological arguments should not attempt reductio ad absurdum arguments against the existence of God, because they'll have to start by assuming (for sake of argument) that God exists. Once that premise is on the table, theological arguments are fair game, and the reductive argument will progress no further without responding to theology.
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Theological argument
God is certainly capable of healing limbs that have suffered permanent damage (examples here and here), but most of the time He chooses to wait before doing so. The maladies of this life are healed in the resurrection (see 1 Cor. 15:54); they are not always healed prior to that time.
The question then becomes why would God allow His children to experience temporary pain? (and mortal life starts looking extremely brief when stacked up next to eternity)
Any parent knows it would not be loving--in the long run--to shield their children from everything painful or difficult. Mortal parents certainly try to protect their children from injuries that will affect them through all of mortality, and our Eternal Parent offers protection from injuries that will affect us through all eternity, but a loving Father who wants us to grow will not withhold from us the very refiner's fire that will allow us to develop. I discuss suffering more extensively in this video on my channel: The Parable of Ana and the Volcano.
To borrow an idea from Michelle Craig, challenges are part of the reason we are here.
Spencer W. Kimball effectively illustrated why a loving God should not remove every obstacle:
If all the sick for whom we pray were healed, if all the righteous were protected and the wicked destroyed, the whole program of the Father would be annulled and the basic principle of the gospel, free agency, would be ended. No man would have to live by faith.
If joy and peace and rewards were instantaneously given the doer of good, there could be no evil—all would do good but not because of the rightness of doing good. There would be no test of strength, no development of character, no growth of powers, no free agency, only satanic controls.
Should all prayers be immediately answered according to our selfish desires and our limited understanding, then there would be little or no suffering, sorrow, disappointment, or even death, and if these were not, there would also be no joy, success, resurrection, nor eternal life
...
Being human, we would expel from our lives physical pain and mental anguish and assure ourselves of continual ease and comfort, but if we were to close the doors upon sorrow and distress, we might be excluding our greatest friends and benefactors. Suffering can make saints of people as they learn patience, long-suffering, and self-mastery.
...
I would likely have protected Paul against his woes if my power were boundless. I would surely have healed his “thorn in the flesh.” And in doing so I might have foiled the Lord’s program...Paul many times could have lost himself if he had been eloquent, well, handsome, and free from the things that made him humble.
...
With such uncontrolled power, I surely would have felt to protect Christ from the agony in Gethsemane, the insults, the thorny crown, the indignities in the court, the physical injuries. I would have administered to his wounds and healed them, giving him cooling water instead of vinegar. I might have saved him from suffering and death, and lost to the world his atoning sacrifice. (source)
My own experience gives me the conviction that God does hear & answer prayers, but He does not always give the answer I want. God could give us what we want today, but He loves us enough to offer something eternally better.
Addendum--response to questions
Why would God heal other ailments but specifically exclude healing (in this life) for those who had a part of their body removed?
I provided an example (linked) above that God has performed such healing and has not unilaterally excluded such injuries. That said, there are so many ways a human being can be injured it would be unrealistic (and contrary to the Biblical principal of faith) to believe in God if and only if He demonstrates His ability to heal at least one instance of every possible form of injury. I'm unaware of God ever raising from death a person who died while skydiving, but that does not lead me to conclude that He doesn't love these people or lacks the ability to heal this precise form of injury.
If God can raise the daughter of Jairus, the son of the widow of Nain, Lazarus, Eutychus, and Dorcas from death, I see no reason why He couldn't raise a skydiving victim from death if He saw fit to do so.
Does the reductive argument really require premise 2 above?
Yes, consider what the argument would have to say otherwise. If we grant the possibility of an afterlife--for sake of math let's say the afterlife has a duration of 10^500,000 years (of course if we accept an eternal afterlife it's much longer than that)--the question becomes, how is it fair that:
The force of the original argument is lost.
Why doesn't God just make people already in a state of eternal life?
A more complete answer to this question would require a discussion of what is meant by eternal life--and I personally have a very high view of the term--but the short answer is:
Upvote:1
The question assumes that there is no positive value to be mined in being an amputee, but that would privilege it as suffering, which I am sure is not the intent of the person asking. What amazes me in this is that someone would think this is a harder situation than the resurrections of dead people like Lazarus. Now, that does not happen often so it mainly establishes that God can do it and if HE does not do it before your general resurrection there is a reason. I think the motive is to say, I will only accept someone else's miracle as a prod to Faith and nothing else. No wonder you have no faith then, right? You can be open to God giving you Faith but you can't put God to the test out of curiosity.
Upvote:3
Similar arguments could/have been made about other groups in other bad situations, so this really boils down to Why doesn't God answer prayers (in this case amputees, but there are plenty of other groups in worse conditions that similar questions could be asked).
This answer to 'What is the purpose of prayer?' has some good points:
Prayer is more than just asking for things. Tom Duckering's answer does a good job outlining this, but to gloss it again, prayer is communication with God which involves both talking to God and listening to him. As our primary form of communication with God, it reinforces in us our relationship with him in all its forms.
Prayer does include asking for specific outcomes. There are far too many examples to ignore: Abraham conversing with God over Sodom and Gomorrah, David pleading for Bathsheba's son, Hannah praying for a child, even Jesus praying that the cup be taken from him. God wants us to ask for things, even though he doesn't always give us what we ask for.
From OP:
God chooses to ignore the prayers of amputees
'Unanswered' is not ignored. God knows what is best for us.
God created us and knows infinitely more than we know. He knows what is best for us, and what would not be good for us. If you have children, when they were very small, sometimes they asked for things that would not be good for them, or would harm them. For good reasons sometimes parents do not always give their children what they ask for, when they ask for it. Parents give them what is best for them.
It is the same way in our prayers to God. God gives us what is best for us. We are God’s children and He gives us what is best for us, and at a time when it is best for us. ...
God's timetable is not the same as ours. He knows better than we do when is the best time for our prayers to be answered. (See Hebrews 6 :13-15). God is eternal and does not measure time as we do.1
CS Lewis wrote in Problem of Pain
“If God were good, He would wish to make His creatures perfectly happy, and if God were almighty He would be able to do what He wished. But the creatures are not happy. Therefore God lacks either goodness, or power, or both.” This is the problem of pain, in its simplest form. ...
On the one hand, if God is wiser than we His judgement must differ from ours on many things, and not least on good and evil. What seems to us good may therefore not be good in His eyes, and what seems to us evil may not be evil....
When we want to be something other than the thing God wants us to be, we must be wanting what, in fact, will not make us happy. ...
Whether we like it or not, God intends to give us what we need, not what we now think we want
This post also assumes all amputees pray for the same outcome. There are amputees who pray in gratitude that it was just a limb/partial limb, not their life that was lost (or some other variation of gratitude).
Upvote:4
One Christian apologetics site I frequently use has published an article that refutes the arguments made by the anti-Christian website articles you refer to. The articles suggest that because God does not heal amputees this is proof that God does not exist, that prayer is useless, that so-called healings are coincidence, and that religion is a myth.
Got Questions counters by suggesting that the argument against the existence of God is based on seven false assumptions. Here is a brief summary with only a few partial quotes:
Assumption 1: God has never healed an amputee: To say, "I have no empirical evidence that limbs can regenerate; therefore, no amputee has ever been healed in the history of the world" is to deny that Jesus healed lepers, many of whom would have lost facial features and hands and feet.
In each case, the lepers were restored whole (Mark 1:40-42; Luke 17:12-14). Also, there is the case of the man with the shriveled hand (Matthew 12:9-13), and the restoration of Malchus’s severed ear (Luke 22:50-51), not to mention the fact that Jesus raised the dead (Matthew 11:5; John 11), which would undeniably be even more difficult than healing an amputee.
Assumption 2: God’s goodness and love require Him to heal everyone: What a load of rubbish! If that were true then there would be no sickness, no disease and no death. Facts show that God is not duty-bound to save humans from the consequences of sin.
Joni Eareckson Tada suffered a diving accident that left her a quadriplegic. In her book Joni, she relates how she visited faith healers many times and prayed desperately for the healing which never came. Finally, she accepted her condition as God’s will, and she writes, "The more I think about it, the more I’m convinced that God doesn’t want everyone well. He uses our problems for His glory and our good" (p. 190).
Assumption 3: God still performs miracles today just as He did in the past: In the Bible there are only four short periods of time in which miracles were widely performed. Since the death of the last apostle miracles are rare.
Any ministry which claims to be led by a new breed of apostle or claims to possess the ability to heal is deceiving people. "Faith healers" play upon emotion and use the power of suggestion to produce unverifiable "healings." This is not to say that God does not heal people today—we believe He does—but not in the numbers or in the way that some people claim.
Assumption 4: God is bound to say "yes" to any prayer offered in faith: After Jesus’ ascension, the apostles were given power to perform miracles as they spread the gospel (Acts 5:12). God is not bound to answer all prayers and sometimes the answer to a prayer made in faith is “No”.
Jesus twice uses the phrase "in My name." This indicates the basis for the apostles’ prayers, but it also implies that whatever they prayed for should be consonant with Jesus’ will. A selfish prayer, for example, or one motivated by greed, cannot be said to be prayed in Jesus’ name.
Assumption 5: God’s future healing (at the resurrection) cannot compensate for earthly suffering: Typical human reasoning, based on a total lack of spiritual vision and looking only to the short term, i.e., the here-and-now and not the here-and-after. Believers have God’s promise of future physical wholeness.
Jesus said, "It is better for you to enter life maimed or crippled than to have two hands or two feet and be thrown into eternal fire" (Matthew 18:8). His words confirm the relative unimportance of our physical condition in this world, as compared to our eternal state. To enter life maimed (and then to be made whole) is infinitely better than to enter hell whole (to suffer for eternity).
Assumption 6: God’s plan is subject to man’s approval: Really? Since when is the created in a position to make demands of his creator and tell Him what should, or should not be done?
God is perfectly just (Psalm 11:7; 2 Thessalonians 1:5-6) and in His sovereignty answers to no one (Romans 9:20-21). A believer has faith in God’s goodness, even when circumstances make it difficult and reason seems to falter.
Assumption 7: God does not exist: The whole point of the question asking why God does not heal amputees is based on the premise that God does not exist and that religion is a myth. This foregone conclusion is foundational to the argument. Here is the conclusion of the matter:
God can heal amputees and will heal every one of them who trusts Christ as Saviour. The healing will come, not as the result of our demanding it now, but in God’s own time, possibly in this life, but definitely in heaven. Until that time, we walk by faith, trusting the God who redeems us in Christ and promises the resurrection of the body.
The personal testimony at the end of this Got Questions article is worth reading: https://www.gotquestions.org/God-heal-amputees.html
Upvote:5
The question given wasn't why won't God heal amputees, but "Have any Christian apologists published refutations to the objections posited by the site "Why Won't God Heal Amputees?"?"
The answer is yes. Here are a few.
These are just the first few I saw by a simple web search on the phrase "Why won't God heal amputees?" I haven't evaluated the arguments, or verified whether they are comprehensive. But, anyway, the answer is clearly yes.