Making Da'wah to people of other religions may have similarities, but there are different ways to do it. For example, there is one way to make Da'wah to Christians that is different from the way you would make Da'wah to Jews; additionally the way one might make Da'wah to a Christian may vary according to which sect they follow. Is there a way to make Da'wah to a non-religious person -- specifically, a person who doesn't believe that there God exists? What are the major points to keep in mind while making Da'wah to an atheist? What major points should be considered aside from the fact that they do not believe in God?
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Someone who does not believe in God cannot be started to be argued based on Qur’an, Ahadeeth, and any quotation from the scholars. They will never accept any such reasonings as these are not authentic to them at all. What we can do is to start from a position that they agree upon as well, whether ourselves approves it or not.
Most of the atheists I have met very much believe in science and accept either witnesses from science (and especially the famous scientists whose theories are widely accepted) or any bare intellectual reasoning. So here is the starting point to discuss things with them. You should be able to rationalize the religion for them and if there is any controversy you should be able to explain why your idea can be the true answer and the science's position be the wrong answer. Some weeks before this I explained a rigorous proof for existence of God which immediately proves many other characteristics about Him as well, that He has existed from -oo and will exist to +00, that He is unique, and etc. . He was trying not to accept the proof (برهان وجوب و امکان) but then I brought to him some criticizes objected to it by some critics like some famous Western philosophers and explained to him why their criticisms are not valid. After some hours he accepted the concept of God as we do believe, but the discussion ended before the other arguments about the prophets, the Hereafter, Qur’an, and etc. being argued. He just refused to accept anything that I was to say. I guess such things are not easily acceptable by those who were once proud of believing differently, like thinking modern (متجدد), all at once, that will need time for them to adapt and get ready for hearing the rest of the reasoning thread. However, if they accepted Qur’an and alike issues as well the rest will be easy.
So far it is addressed where to start, but how to start is also important. The atheists today believe they are so reasonable and science based, we can approach them by showing them how lame is their knowledge about the universe. When their ME collapsed, when they become humble, then they will be ready for learning new things, otherwise you will always find them building a rough guard against you whenever that you approach them for a constructive argument. This I have learned from a great scholar (I don't remember his name but maybe he was Ayatullah Milani), seems correct to me for my personal experiences as well.
Godspeed for the difficult task
PS 1. I know the answer was like being at my soapbox, but I'm not sure if you were also asking for references and citations? My personal experience more or less.
PS 2. Actually this was based on my personal experience, as I had no great success in convincing the atheists using verses of Qur’an. However, this fact must not be ignored that the holy prophet --peace be upon him-- invited the atheists and Mushrik's all around by Qur’an, rightly choosing the verses and rightly arguing with them based on such verses. Such that the person could find the truth understood be his hearth, no matter if he was then to deny or accept it. / Another way is to use some specific Ahadeeth without stressing over the narrators, but instead stressing over the context of Ahadeeth to change his mind about the Ahadeeth. This will bypass their excuses for not accepting the quotations merely because they are quotations!
One could argue that an atheist lacks reasons to believe in God, while tending to approve the scientific method.