Is it desirable and possible to bring intellectual proofs for G-d's existence?
The Rambam believed so [even though as we saw in a recent post, revelation was preferable and beyond proofs].
"I will begin the subject of this chapter with a simile. A king is in his palace, and all his subjects are partly in the country, and partly abroad. Of the former, some have their backs turned towards the king's palace, and their faces in another direction; and some are desirous and zealous to go to the palace, seeking "to inquire in his temple," and to minister before him, but have not yet seen even the face of the wall of the house. Of those that desire to go to the palace, some reach it, and go round about in search of the entrance gate; others have passed through the gate, and walk about in the ante-chamber; and others have succeeded in entering into the inner part of the palace, and being in the same room with the king in the royal palace. But even the latter do not immediately on entering the palace see the king, or speak to him; for, after having entered the inner part of the palace, another effort is required before they can stand before the king--at a distance, or close by--hear his words, or speak to him. I will now explain the simile which I have made. The people who are abroad are all those that have no religion, neither one based on speculation nor one received by tradition. Such are the extreme Turks that wander about in the north, the Kushites who live in the south, and those in our country who are like these. I consider these as irrational beings, and not as human beings; they are below mankind, but above monkeys, since they have the form and shape of man, and a mental faculty above that of the monkey.
Those who are in the country, but have their backs turned towards the king's palace, are those who possess religion, belief, and thought, but happen to hold false doctrines, which they either adopted in consequence of great mistakes made in their own speculations, or received from others who misled them. Because of these doctrines they recede more and more from the royal palace the more they seem to proceed. These are worse than the first class, and under certain circumstances it may become necessary to day them, and to extirpate their doctrines, in order that others should not be misled.
Those who desire to arrive at the palace, and to enter it, but have never yet seen it, are the mass of religious people; the multitude that observe the divine commandments, but are ignorant. Those who arrive at the palace, but go round about it, are those who devote themselves exclusively to the study of the practical law; they believe traditionally in true principles of faith, and learn the practical worship of God, but are not trained in philosophical treatment of the principles of the Law, and do not endeavour to establish the truth of their faith by proof. Those who undertake to investigate the principles of religion, have come into the ante-chamber; and there is no doubt that these can also be divided into different grades. But those who have succeeded in finding a proof for everything that can be proved, who have a true knowledge of God, so far as a true knowledge can be attained, and are near the truth, wherever an approach to the truth is possible, they have reached the goal, and are in the palace in which the king lives.
My son, so long as you are engaged in studying the Mathematical Sciences and Logic, you belong to those who go round about the palace in search of the gate. Thus our Sages figuratively use the phrase: "Ben-zoma is still outside." When you understand Physics, you have entered the hall; and when, after completing the study of Natural Philosophy, you master Metaphysics, you have entered the innermost court, and are with the king in the same palace. You have attained the degree of the wise men, who include men of different grades of perfection. There are some who direct all their mind toward the attainment of perfection in Metaphysics, devote themselves entirely to God, exclude from their thought every other thing, and employ all their intellectual faculties in the study of the Universe, in order to derive therefrom a proof for the existence of God, and to learn in every possible way how God rules all things; they form the class of those who have entered the palace, namely, the class of prophets. One of these has attained so much knowledge, and has concentrated his thoughts to such an extent in the idea of God, that it could be said of him, "And he was with the Lord forty days," etc. [Moreh Nevuchim 3/51]
The Kuzari [1/11-13] differed and felt that the only way to understand and perceive G-d was through His miracles and revelation:
11. The Rabbi replied: I believe in the God of Abraham, Isaac and Israel, who led the children of Israel out of Egypt with signs and miracles; who fed them in the desert and gave them the land, after having made them traverse the sea and the Jordan in a miraculous way; who sent Moshe with His law, and subsequently thousands of prophets, who confirmed His law by promises to the observant, and threats to the disobedient. Our belief is comprised in the Torah--a very large domain.
12. I had not intended to ask any Jew, because I am aware of their reduced condition and narrow-minded views, as their misery left them nothing commendable. Now should you, O Jew, not have said that you believe in the Creator of the world, its Governor and Guide, and in Him who created and keeps you, and such attributes which serve as evidence for every believer, and for the sake of which He pursues justice in order to resemble the Creator in His wisdom and justice?
13. The Rabbi: That which you express is religion based on speculation and system, the research of thought, but open to many doubts. Now ask the philosophers, and you will find that they do not agree on one action or one principle, since some doctrines can be established by arguments, which are only partially satisfactory, and still much less capable of being proved.