For the religion that first brought the message of ethical monotheism to the world, Judaism sure does its best to have us lead much of our daily lives as if there were no G-d, strange and heretical as this may sound.
Principles such as "we may not rely on miracles" (Pesachim 64b), "in the ways of G-d why do you concern yourself" (Brachot 10a), or "It [Torah] is not in heaven" (Devarim 30:12), mandate us to minimize risk; to assess our policies based on rational, i.e. geopolitical, factors even if we seem to be bypassing our faith in the Almighty; and to interpret the Torah without recourse to prophecy or heavenly voices. From our human perspective, the world runs according to the natural laws of science, world events are dictated by cause and effect and we must live our lives accordingly. To plan our lives by ignoring these laws of nature, relying instead on faith in G-d, is not only foolish, it is apparently religiously invalid. The knowledge that G-d operates behind the scenes may be comforting but it is irrelevant to our own behaviour.
A minority strain (see Ibn Ezra Shemot 21:19), rejected by the vast majority of our greatest thinkers, allows the truly righteous to ignore the workings of the world as they place their trust in G-d alone. Yet even those rejected views have no applicability today; we lack the necessary level of righteousness to rely on such an approach. Moreover, the Torah demands that we attempt to thwart the will of G-d if need be. The Talmud (Brachot 10a) is critical of Hezkiyahu , one of greatest kings of ancient Israel , who refused to have children, as he knew through prophecy that his son Menashe would turn out to be evil. Even though he was right, he was wrong; such calculations are left to G-d alone and man is mandated to have children, period.
One need not go back to the Talmud to find such an attitude. It is explicit in the Torah itself. "When you build a new home you shall make a fence for your (flat) roof so that you do not bring blood into your home, ki yepol hanofel mimeno, since someone may fall from it" (22:8). The simple explanation of this command is that one must build a fence in order to prevent injury to oneself or others. However, the Rabbis, attentive to the peculiar double wording of yipol hanofel , literally the falling one will fall, add an additional level of interpretation. They note that the verse teaches that the person who falls ( yipol ) was destined to fall ( hanofel ), irrespective of any fence that may have been built.
The reason we build a fence according to this reading of the verse is that "merit comes through the meritorious and calamity through the sinful" (Rashi 22:8). While G-d may have decreed injury or worse for this person you should not be the one to bring it about. From G-d's perspective your efforts are meaningless as, fence or no fence, those who are meant to fall will fall. However, for us it is not the Divine perspective that matters but rather the human one. We must lead our lives without reference to what G-d may or may not do.
Living while "ignoring" G-d is actually a means to moral refinement. Poverty, hunger, disease, environmental disaster, war, even natural disasters are issues that mortal man must resolve. To rely on G-d to solve these issues would be a sacrilege. While we yearn for closeness to G-d, we must live our daily lives as if G-d is distant and remote. Such is the paradoxical notion of religious life. It is we and we alone who must make this world a better place.