Does a god exist somewhere out there? Or is it the name given to the total silence within us?
Yes, God does exist somewhere both in and out "there".
God isn't just a tangible "thing" that someone can go and find as if it was a lost relic of some kind.
The best scientific pursuit of God's actual tangible existence is in examining the web of consciousness that exists between certain human beings. I say certain human beings, but the point of God's existence is to raise ordinary human beings into a higher level of spiritual functioning.
If you examine the claims made by Joseph Smith Jr. you will see that according to him that the being Jews, Christians and Muslims call "Heavenly Father" is a being who has a body of flesh and bone as tangible as man's body of flesh and blood. This was a rather bold statement to make considering the orthodox Christian climate of the day. However, a close examination of the Bible indicates that the relationship between God, angels and humans is extremely interrelated and even mutually dependent.
When humans become "spiritually born of God" they pass from a state of spiritual death to a state of spiritual life. Of course this is within the context of what it means to be spiritually alive or spiritually dead. There are false spiritual births to false spirits that places human beings into a different spiritual state than others would be in. This is the basis upon which there is a lot of strife between the various sects of religion.
Putting aside the varying degrees and varieties of spiritual bodies human beings can become spiritually born within, they all generally come with some kind of terms and conditions of a covenant they are making. As a general rule they agree to function in some manner as a group where the group starts taking on an identity, personality, behavior, etc. of its own. This is the kind of body that Joseph Smith Jr. is referring to when he said that the Father and the Son as resurrected beings have a body of flesh and bone. As individuals grow in their knowledge of the Word and Will of God, they receive a greater endowment of His Spirit within them. They begin to fall inline and function with the others of their group in a manner that makes them a part of a greater body.
All of these individuals participating in these various spiritual bodies, at varying levels or degrees of glory, have received a spiritual birth. This is what transforms an ordinary human being into an angelic being. There are many people here in the flesh who are spiritually alive angelic beings who also have their bodies of flesh and blood. Angels are "hosts of heaven" because they generally participate in the governing aspects of the body within which they belong.
All of these angelic beings who have joined in the pure and valid covenant that the true and living God offers are collectively the members of the body of God. Paul the apostle teaches this quite explicitly in his various treatises. Ephesians 5:30 comes to mind where he challenges the brethren of the priesthood to consider this point: "Know ye not that ye are members of his (Christ's) body, of his flesh and of his bone?" He is telling them the actual literal identity of the resurrected Lord Jesus Christ. Literally the group of those who believe in Jesus and who receive the Spirit of God in them and they are received into the body of Saints by way of valid ordinances literally become a part of the body of God.
It is in this manner that the Mormon religion advocates that individual human beings can become spiritually born of God and progress through varying levels and degrees of spiritual glory/endowment and in due time have a full understanding that they are literally an integral part of the functioning of the tangible flesh and bone body of God.
In the belief of Joseph Smith Jr., so far as I can tell because he never went on record for having said this, it was his understanding that before the new world would be complete there had to be a process by which the new Creation's Godhead would need to be recomposed. There are some people who will become members of the new Creation's Elohim, which is a plurality of elite human beings who have received the highest spiritual endowment that can be bestowed upon a human being. These are the individuals who, collectively as a functioning covenant body, become the new Creation's Elohim.
The overall point here is that God is a co-essential and co-eternal aspect that is an inherent and synergistic component of the human race, or at least a portion of what people consider the human race.