It is well recognised even by scientists that evidence of senses is no evidence. There are plenty of things that escape the senses due to their inherent limitations. The senses do not report accurately even what they receive as information from the outside world. They are only the first means for receiving the impacts from the outside physical world. Even so there are many other forces, energies, beings that our present organisation of senses cannot register. They have to be perceived within through our subjective senses, for example, love, fear, happiness, sorrow and many other psychological movements that cannot be seen but can be felt. They are not false simply because we cannot see or measure them.
So too God can be felt, initially through a hidden faculty of intuition that is higher than the mind though not yet fully developed. Later as we engage in sadhana which is a specialised process for inner self development, certain latent faculties develop in us through which we can have glimpses of God and in rare instances see Him and hear Him. Plenty of yogic experience is proof enough. It can be verified if we are willing to fulfill the conditions of the sadhana which are not easy though. But that is true of any highly specialised process. Until we can develop these inner subtle senses we can either take it with faith that God exists because there are plenty of yogis who have experienced Him as well as shown us the path for us too to experience. Or else we can deny the existence of God and choose to stay within the prison of our ignorance and the limitations of our present framework of senses and the mind.
Our beliefs and non beliefs make no difference to God but certainly makes a difference to us. Belief in God or better still faith in God opens our doors to receive His Grace while doubts shut the door and we fail to perceive the Divine Grace even when it is there. To each his own choice.
Affectionately,
Alok Da