I have heard Atheists talking about proving that Gd doesn't exist or that there is no proof of Gd's existence.
I'll explain how to know Gd exists and why prayer is effective.
There are two ways I think about this. One is rooted in science and reality so it should appeal to people who aren't interested in mystery - facts only.
The second is pretty Jewish.
Contemplate the existence of the world. You know that our world exists in space and that space is vast. When I took Astronomy in college we learned that there might be something outside of the three dimensional existence we all know and understand.
But think about everything in existence. Everything. All of it. What we know about and what we don't know about. The complete all of it.
That's hard to think about but let's say that's what Gd is. Gd is everything all put together.
How to make sense of this?
Think about your own self. You might think of yourself in various ways but we all know that we exist in our bodies. That's the limit that we know of our existence. We can't step out of our bodies or lose our bodies. Our body is all of us.
What I'm trying to explain is the idea that we each are an infinitesimally small part of the whole. The whole is Gd.
Why does that matter?
Think about your own body again. You might say that it's different because you have consciousness and your own mind and thoughts. But you can easily come to realize that the various parts of your body have their own minds as well. How do you find out about what the various parts of your body want? They let you know in various ways, don't they?
They absolutely do.
And my belief in Gd is that even though I am tiny, I am a part of Gd and the louder I scream out about what I want or need (through prayer) the better the chances are I'm going to get it.
And of course as a human I'm not much like Gd. I can't understand all of everything. I can't comprehend the complete totality of the sum of all that exists.
But if I can understand that when my stomach growls I know to put something in it and when I'm tired I rest and when something hurts I give it attention then surely Gd, who is total perfection, can perceive and respond to my prayers.
To take this toward Judaism, I believe that Gd is a step beyond all of everything. Gd is the creator of everything. This part is hard to explain but imagine total nothingness.
If you go back in time far enough at some point there was nothing. Absolute nothingness is probably the most perfect thing you can try to imagine. At some point that nothingness created everything and sustains everything to this day.
Again, if you want to understand this think of being a parent and having a child.
You love your child unconditionally. No matter what they do, no matter how many times they screw up. You always love them. You always want to help them. And what do you want from your child when they need your help? You want them to come and ask. You want them to recognize that they need help and ask you for it. You want them to be sorry about what they did wrong but you always want them to be happy. You want to see your kids succeed and shine and be their best selves.
And that's just between a human parent and child. Not this perfection that is everything from which everything springs forth. Imagine how Gd responds when we pray and say that we need help, that we feel sorry for doing something wrong and ask for assistance.
I know there are so many people who think Gd and religion are just garbage. Made up stuff. But I believe that when we ask earnestly in prayer Gd answers us.
I don't mean that we are given the things we ask for, but I do believe that when people pray to Gd in humility they are given an answer. And one that will lead us to happiness (so if you hear Gd tell you to hurt someone - it isn't Gd!!!!)