@Algemon: Yet, it would be good if we understand the time we started getting used to the thing. By the way, I think that wise fellow was John von Newmann.

@Todd Trimble: Dear Todd, the discussion wasn't in English. But I assure you the peoples involved are now quite aware of this post. I think you do agree I cannot give more information until they want to react personally. There will be another meeting soon in which I will report on what I've learned here at MO. I will try to convince them to contribute personally here.

@fedja: Sorry, I deleted the part of question that you put your answer for. Indeed, as my above comments show, I found part of your answer quite related to my main concern (reflected in the main question). However, It wasn't that easy to find that part of your answer! So to keep, the future possible answers more focused I deleted the question of your answer! Please for more details have a look at the meta related to this post. With all due respects.

@Todd Trimble: It was a two hours hot discussion. I did my best to report the core of the discussion as accurate as possible. And as you can see, I didn't mention the name of Milnor before checking with him. Let's see it other way. After all, what would be the difference between "somebody has said that" with "something could be said". If you think that something is worthy of attention you may decide to ponder the issue. However, I understand that the simple sentence "life is beautiful" would have a different meaning for most people (including me) if they know Albert Camus has said so!

If there is only one potentially brilliant mathematician who let the two compartments merge into each other, it would be a curious attempt to see how it works. But, If there is an education system that tries to do so, it won't be a good idea to sit back and see how it works. Moreover, it won't be enough to say "see, it is not the way today's professional mathematicians work in general", or "consider them a bunch of flies" (it could be very harmful!) You should try to understand carefully and clearly the opponent to find a way to communicate with it. Now, you see what makes my nights sleepless.

But, let us think of religion in a more narrower sense. Indeed, I am much worried about "reverting the craft" to its pre-renaissance level than pre-Greek level. I should be worried since I live in a society with a conscious attempt to do so. I may say to myself, no worry, it would not be likely that the mathematics community as a whole goes in that direction again. But, I think it is better as you said "look at what real results [if any] they achieve[d] (or they think they may achieve) with their approach instead." (continued)

"If religion plays a role in their [today's professional mathematicians] lives, it is kept in a separate compartment and seems not to be a source of professional inspiration." What if a mathematician does not want to keep it separate? What if he lets it sneak into its other compartment (i.e., mathematics)? It brings me to the first part of the first sentence of your fifth comment! Somehow, it also gives me a religious feeling in a very broad sense of religion.No worry, since "Most mathematicians are platonists on Sundays but formalists on weekdays". (continued)

@fedja: What an answer. Surely, I agree with some parts and disagree with others. However, I just go with the parts I agree to keep things less "argumentative". Indeed, your comments encouraged me to tell part of the story I avoided to tell all the way from start. Somehow it is frightening (at least for me). I try to be as cautious as possible since I've just started to understand the situation. Let me start by quoting something I have recently read in my "sleepless nights". It is from Philip J. Davis' paper,"A Brief Look at Mathematics and Theology": (continued)

I am pretty sure that we should give the credit of that "almost-complete nonsense" you have mentioned to Euler :)