NO

I note specifically that you invoke big-G God in your question. I.e. the commonly used titular name of the Abrahamic deity. That is, your question is equivalent to "Is it correct to say “any effort using scientific way to prove or disprove Yahweh is in vain”."

This is different from the concept of there being some deity; the idea that there may be a little-g god. The latter is a position that cannot be disproved, any more than the concept of there being fairies can be disproved (they might be tending the toadstools at the bottom of your garden. But they're also very good at hiding and you haven't caught them at it yet. Those toadstools do look really well maintained, though!)

However, once a property is explained, or a prediction is made, a hypothesis becomes a part of science, can be tested and, potentially, disproved. The Bible is a book which puts forward a hypothesis of how this specific deity works, and how it affects the world and universe about it. It puts forward testable predictions. Therefore, this specific concept of a deity is not outside the realm of science.

Just to be inclusive, it's worth saying that the very same argument goes for any other deity whose properties are explained one way or another: Vishnu, Odin, Quetzalcoatl, The Triple Goddess, Ma'at, Hephaestus, Minerva, Mithras, Wadd, and so on.

This is a proposition about the ultimate nature of reality

Some persons think that God cannot be evidenced through natural philosophy, because the propositions about God are such that no observation of their validity or lack thereof could possibly exist.

Some people think that one can evidence for/against the existence of God with natural philosophy. These people are divided over whether the world evidences for or against the existence of God.

What sort of God there is or isn't, and what sorts of things are epistemologically valid as evidence thereof, are significant factors in which camp one's beliefs lead one to. Easy consensus about the limits of science does not exist here.

It is incorrect to say this. Science is most certainly capable of proving the existence of God, and not just any God, but the Christian God, and it is profitable to pursue this approach of understanding God.

Why is this so? If a person of finite ability and means were to construct a universe of moderate subtlety, or a random act of a dead universe, or if God were to create such a limited universe that falls far short of what He is capable, then you could not distinguish the results. You could not tell one such universe from another. But the Bible does not declare that we live in such a universe.

The heavens declare the glory of God;
    the skies proclaim the work of his hands. (Psalm 19:1)

If God is infinite and glorious, then His infinite subtlety, majesty, attributes of love and justice and all the other characteristics that make God God must be present in His creation. Of course, being finite beings, we cannot discover the full pattern of what God created. However, we should find there is no end to the mysteries. Our science should never reach the end of the matter.

In chapter 28 of Job, the suffering man laments that though he search every part of the creation, he can never find wisdom; only God knows the way to it. After speaking of scientific attempts to search out wisdom though human effort, such as digging in the earth for precious gems, Job concludes with this:

Where then does wisdom come from?
    Where does understanding dwell?
21 It is hidden from the eyes of every living thing,
    concealed even from the birds in the sky.
22 Destruction and Death say,
    “Only a rumor of it has reached our ears.”
23 God understands the way to it
    and he alone knows where it dwells,
24 for he views the ends of the earth
    and sees everything under the heavens. (Job 28:20-24)

This would seem to back up your statement that man cannot find this ultimate wisdom (which is true), except for one observation. In God's speech to Job from the whirlwind (Job chapters 38-41), He argues that every facet of the world, from the stars in the heavens, to the birds in the sky, to the earth and its creatures, to the grave and the forces of evil behind it all testify about His wisdom and reveal it. We cannot ascend to Heaven on our own effort, but in Christ, God has descended to earth to give us the Holy Spirit, who shows us the path to wisdom. The Spirit-filled woman or man can, by observing and reasoning about nature (science), retrace the path to God's wisdom which formerly was impossible to follow.

The inability to find wisdom in nature was Job's despair (and humility) speaking. The ability to find wisdom in nature was God's gracious gift, but it requires an infinitely capable interpreter - the Holy Spirit - in order to deliver accurate truths.

9 However, as it is written:
“What no eye has seen,
    what no ear has heard,
and what no human mind has conceived”—
    the things God has prepared for those who love him—
10 these are the things God has revealed to us by his Spirit.
The Spirit searches all things, even the deep things of God. 
11 For who knows a person’s thoughts except their own spirit within them? 
In the same way no one knows the thoughts of God except the Spirit of God. 
(1 Corinthians 2:9-11)

One writer used an analogy that I like: ink on paper. The ink and paper represent nature. They are material and cannot truly show the wisdom of God. However, if God takes that ink into His pen and writes with that ink upon the paper that is the world, then his existence and intelligence and purpose can be clearly manifest. The ink has not changed. The paper has not changed. The only difference is that that ink is now formed into words. Those words carry information that was not there before. That information is not a material property of the ink and the paper, it is an immaterial property. God's wisdom is such an immaterial property and it is written upon our world. And as God told Job, it can be understood.

It is the glory of God to conceal a matter; 
to search out a matter is the glory of kings. (Proverbs 25:2)

God would not urge us to search out what He has concealed if it were futile.

The first thing I would say here is: read about Carl Sagan's dragon, which discusses this theme.

I'm not sure your 7-step conversion program is going to be very workable...

1. Science is correct as long as it is telling the truth;

   The word Science there could be replaced with anything, without changing the meaning; the more interesting question, then, is how we evaluate "truth". That is what science brings to the table: a mechanism for discussing verification and testability of claims.

2. Science have never and will never disprove the possibility of God

   Indeed, the definition of God claimed by most Christians is very non-testable, so is not very applicable to regular science. What science can discuss is the things claimed in scripture, but of course even when something is scientifically impossible, "miracle" and "omnipotent" make that hard to apply. It can also discuss the claims of observable effects of God in this world (for example, prayer efficacy) - and every time that is done in proper conditions, the answer is "no observable effect".

3. You will always have an ever-enlarging unknown zones if you are trusting science

   (also noting your meaning in comments); the unknown zone has not changed just because we are aware of it.

4. It is fully logical and possible (50%) and make sense if a supreme God is in control;

   That is a huge claim; I'd go back here to Carl Sagan's dragon: is that "logical"? is it "possible?"; is it necessary?

5. So you are disbelieving, is not because science shows God is not there, but because you choose to interpret scientific results with a no-God conclusion, you are shutting down the other 50~100% percent of possibilities.

   Actually, since it is not testable, the scientific conclusion is not "no-God", but "the claim of God has not been proven"; not proving something is different to disproving something. But: what makes that "50-100%" of possibilities? Do you "shut down" fairies in your garden? Thor? Where is the line?

6. To know God, actually one has other ways (then I will go special revelations, like Bible, sin, testimony, even though most of them are not supernatural)

   Again, this comes back to how we evaluate truth; all of the things mentioned have many simpler, non-religious answers; since you say science shuts down supernatural possibilities, is it not true that you are ignoring possibilities too? simpler possibilities?

7. Ask him/her to think about her personal life, the problems, the families, the relationships. Which means I will go in this part ask him/her to think many un-scientific data.

   Not sure how this relates to the context.

There's a reason "modern scientific achievement" is respected: it gets demonstrable results that are too numerous to list. Many people can find a way for these two things (religion and science) to co-exist, but there comes a point when to many, Christianity is just another dragon in the garage, alongside many others.

I guess I'm just saying: don't be too upset if some people don't spontaneously agree with you and convert to Christianity - they have reasons for their world view too.

I was impressed by your question because I had a similar experience along the last 5 years.

I'm also an academic (I work in scientific institution),and before changing institution and job, I was also used to be 'cut into pieces' during the lunch I had with my colleagues because they are also "science-is-the-ultimate-answer" people, and criticized me a lot.

Well, I think there are two different kind of approach to talk about God and faith, when talking with non - believers:

1. the philosophical/academic approach
2. the personal experience approach

In the Philosophical or Academic approach you talk about doctrine, faith, philosophy, view of the world, historical facts. In this sense, you can find very valuable resources in this forum, like the good answer of @aceinthehole. I can contribute just suggesting you to read this Enciclical letter, Fides et Ratio, from the former Catholic pope John Paul II, which in its foreword explains the spirit of the book saying "Faith and reason are like two wings on which the human spirit rises to the contemplation of truth".

Science is not in contrast with faith at all, because God created us, created also our brains and gave us the responsibility to use them. Using science for proving or disproving the existance of God is something that many tried to do in the past, but unfortunately it cannot be studied by science because He is not a natural phenomena, He has 'his toughts', 'his freedom'... He's an existing and living being. I differ a bit from what @aceinthehole was saying because I do not agree that God or 'what God does' is not empirically observable. On the contrary it is (think about the miracles), the great problem is that it is not repeatable. So, we have observability but not repeatability --> it is out the scope of what science can study.

In general, this kind of approach is good (in the sense that it can bring to a REAL dialgue) if you have the resources (I mean if you studied a bit philosphy and history) and if the people you talk with are really willing to discover something about faith (this is more or less my view of the episode of Nathanale in the Gospel of John 1, 43).

The personal experience approach. After years of being murdered by the number of criticism by my colleagues, I understood the following things which are valid in the most of the cases (NOT ALL):

1. The most of times, non-believers in science field have many prejudices about believers and faith. They think it is something silly, for kids, for which you don't have to think and use your intelligence. This is for a number of reasons:

   1.1 their knowledge about faith is - in many cases, NOT ALL cases - the knowledge of a 14 years old guy, because they stopped to follow the church because it was "not cool" during their adolescence. Hence, there is a huge problem of understanding each other. Maybe that if you talk about participating to a cult (or mass) you think about relationship with God, while a non believer thinks about loosing 1 hour of his time in doing something meaningless. These 'onthology' and semantic problems have to be taken into account.

   1.2 some non-believers think that believing means do not use only your brain as an approach to reality and it scares many. you have to trust

   1.3 Non - believers in science field, are often angry with churches for what they did in the past. Which can be reasonable, even if in my experience this criticism is exasperated and exagerate

2. Very very often at least in the case of my working place, when non-believers talk about faith, God and all those matters, they are actually talking about something else. You can even arrive to demonstrate that God exists. Well, do you think something would change? I can say that a few people would be interested in this. Mostly, they had bad experiences with the church, priests. They received some uncorrect informations about faith by the medias (newspapers, tv...). So, often they are talking about their personal experience, some blessure they had in the past from church-people.

3. Some non believers think that reason and science are against faith. Many thinks that faith has to do with feelings or abstract things. This conception is false in my opinion. As an exaple I can cite a colleague of mine becaming a believer because, during his PhD in biophysics, because he was amazed about the physics laws which where behind the behaviour of the cells in contact with a metal target.

In conclusion of this long (sorry!) answer, I hence come back to your question and your 7 points.

Concerning your question, I would say that science can be used as a path which lead to the the experience of God. It cannot be the only instrument which lead to him because the experience of God covers us entirely (mind, spirit, body).

Concerning your 7 points, my comment is that I would concentrate only on the last three, which I agree with. The first thing I would do is to listen, rathern than trying to demonstrate that God exists. Secondly, I would try to understand if they really refer to the subject they are talking about or it's just a consequence of some other hidden issue. Then I would talk about this "hidden issue". There is an high probability that this hidden issue do not only implies 'science' but also more personal issues.

When I succeded in doing this, I discovered that my colleagues were much more open to a dialogue and that it was very fruitful both form them and for me.

I would really trash away any theoretic discussion because they do not lead to nothing if they are not linked to our lives. Faith is indeed, in last analysis, something that deals with our lives, not an academic matter of discussion.

I apologize for the long and personal answer, but this is the fruit of my experience and thinking.

The short answer to the high-level question: Yes. Science, nor any line of reasoning, can ultimately prove or disprove God. God is trans-rational.

I think you might find some interest in a lecture given by Fr. George Coyne, former Director of the Vatican observatory in Tucson. He's a very bright scientist (and priest, of course). And I'd argue you won't find a man more zealous about keeping his science pure.

In the course of his talks, he makes a few notable points. Firstly,

Faith is a gift.

It's not something you reason yourself into. It is a gift. And while you can present good reason for believing in God, you cannot present absolute proof of God.

The climax of his lectures generally occurs when he poses a question, referring to the natural processes that brought the universe and life to their current states, "Did God do it?"

He answers like so.

I don't know. There's no way I can know. I'm doing science. I'm seeking natural explanations for natural events. So, if I haul God in, I'm betraying science -- and God by the way. Or if I throw God out, I'm betraying science -- and God by the way. So, we have to separate the two. We have to be very careful that we cannot conclude from this¹ to the existence or the nature or the fact that God created the universe.

And an abridged quote from his QA section of the lecture at another university (watch a few minutes starting at this point).

Faith is not irrational. ... But faith is arational. It goes beyond reason. It transcends reason. ... But look at our human lives. I am not a walking intellect, I hope. You know I have emotions. I have stomach aches. I have headaches. I like being with this person better than that person. You know, all of these things in our lives that are not based on reason. ... Life is not just thinking.

He goes on to make it clear that God is not an explanation for things. He can't be. He shouldn't be. Rather, God is love. (Which, by a Catholic understanding, is not an emotion, but a "seeking the good of another as other.")

I'd also encourage you to take a glance at a few videos by Fr. Robert Barron, who stresses the notion that God, properly understood, is not in competition with science.

http://www.youtube.com/watch?v=3ZkHv8iTJPo

And in a few other videos, including this one, Fr. Barron makes a relevant claim:

One of the signs that you're not dealing with authentic faith is that you've had to sacrifice your mind.

More directly to your question

You cannot lead disbelievers of any sort down a scripted path to belief. Faith is a gift. If anything, people are convinced one way or the other by the poetry of their lives, not by reason. Reason only reinforces what we've already intuited.

If I had a few suggestions, they might be ...

• Understand God properly. Understand that God is a mystery, and that God is not a being among beings; but that God is the shear act of being. Refer to the "name" of God revealed to Moses, I am who am / I am who I am (Yahweh). And make sure your atheist friends are clear on this -- there's no point in discussing anything if they perceive God as one being or thing in the world.
• Cede that God, properly understood, is beyond provability or disprovability -- particularly by scientific methods, which necessarily measure phenomena in nature.
• Note that science is a limited study. It has little, if anything, to say about the qualitative realities that compose the bulk of our lives.
• Avoid the "God of the gaps." God is not an explanation for things we don't understand. It's almost always only a matter of time before another suitable explanation shows up.
• Remember that people are not convinced primarily by reason, but by "poetry." That is, people believe what they do, whatever they believe, as intuited by their experience. Science and reason only serve supporting roles.
• Religion should never demand that you sacrifice your intellect. No truth can disprove the real God. And I'd personally posit that all truths have the potential reveal something about God's personality.
• Don't evangelize because you think you're smart or right, or because someone else is wrong. Evangelize because your relationship with God makes you joyful enough to want to share that joy. If you're not thrilled with God and your relationship to Him, you're not preaching a God I'm interested in.

--

¹ The contents of the lecture, in which he briefly explains many of the natural processes that brought the universe to its current state.

I was trained as an electrical engineer and am by nature a scientist. That is, I discard any idea or concept that does not work or make sense regardless of who says otherwise. However, I take a more drastic approach to the division of science and faith which may or may not be of any benefit to you.

To me, simply put, science is nothing. I see science as an elaborate study of rocks. The gospel is everything. I mean this in terms of evangelism which has do do with the soul and its standing before God.

The Bible never tries to argue for the existence of God, and neither do we need to. Every man who is honest with himself knows there is a God and that he has sinned before him. (Rom Ch 1-3) I do not pretend with anyone that they can escape the moral obligation that they sense in their own conscience in believing the gospel. Yes, many deny it but I simply do not accept the denial and simply state I do not think that is honest. Its is sin not to believe the truth.

In other words if somebody says 'prove there is a God', I say 'no need to.' This may seem arrogant but it the approach the Bible takes and is more caring to the person who foolishly is thinking their is no God.

This basically seems to describe your colleges:

The fool says in his heart, “There is no God.” They are corrupt, they do abominable deeds, there is none who does good. The Lord looks down from heaven on the children of man, to see if there are any who understand, who seek after God. They have all turned aside; together they have become corrupt; there is none who does good, not even one. Have they no knowledge, all the evildoers who eat up my people as they eat bread and do not call upon the Lord? (Psalms 14:1-4, ESV)

It is loving to point out to them that they actually know themselves that there is a God because in the end they will face him in judgment, so why pretend now.

I am not saying that there is no value in apologetics and having friendly intellectual debates, but when fun and games are over, only sharing the gospel can save a soul.

Nobody can be saved through arguments about science. Also nobody who is saved has ever lost their faith by arguments about science. Its all irrelevant. Rocks and the God who made them are totally different subjects. Science in the end is nothing. The gospel is everything in terms of true life.

However in terms of evangelism I do not share my 'narrow views' unless asked of me. I will rather remain silent and listen to people's foolish ideas and be a friend. By confidently listening to a child a child will know you are a parent, then when the time comes, and it may be rare, give them Gods word and you may save a man from the fires of hell.

Evangelism is mostly about love not about intellectual observations or tricky arguments. If you believe in Christ you are the parent, whether you realize it or not. You therefore have the responsibility to rest in confidence while the rest chatter about whatever foolish idea is the most entertaining for the day. As it is all vanity there is no need to get that personally involved.