An extremely brief sketch of cosmological arguments. Please note first of all that no cosmological argument has the premise "everything has a cause", and also note that the term "cosmological argument" refers not to a single argument but to a family of arguments which are very different from one another. What they have in common is that they start with some general observation of the way the world is, and reason to some kind of self-existing source. NOTE: This is a work in progress and will be updated over time.
Plato is generally regarded as the originator of cosmological arguments. He argues that there are two kinds of motion: self-motion, and transmitted motion. Something either moves itself, or it is pushed by something else. But we see around us all kinds of change and motion: rivers flowing, trees growing, the sun burning, planets turning, and so on. All this motion is being pushed by other things. For example, the river is being pulled by gravity, which is in turn being actualized by mass, which is in turn actualized by (perhaps) the Higgs boson, and so on.
But all this activity cannot be just transmitted motion and nothing else, for then there would be no source of motion. If something is being received, then there must be a source. There must be something capable of self-motion that is the source of all this change and activity, because if everything were a passive transmitter of motion, then there would be no source and hence, no transmitted motion in the first place.
Self-moving things are alive, and so the source of the activity and change in the world must be something that is alive, or what Plato calls "Soul". He concludes that it must be immaterial, because matter is a passive transmitter of motion and hence cannot be the source of all motion.
First, Aristotle proves that motion (and hence the universe) is eternally old. He does this by arguing that if motion began at some point, then something must have moved in order to start the motion, in which case motion existed before it started, so something must have started that motion, and so on ad infinitum. So the past is eternal.
Like Plato, he argues that motion and change can either be transmitted or self-caused. Much of the change we see around us is transmitted. The tree only grows because of the sunlight, the sunlight only burns because (we now know) of nuclear reactions, which only operate because of gravity pulling the Sun in on itself, and so on in a string of dependent causes. There must be a source of change and motion, and this source must be a self-mover. Capable of moving or changing itself and hence the source of all change we see around us, since most change we see is transmitted change, not self-change. And remember that a receiver implies a source. So the source of all change must be a self-mover.
But all self-movers ultimately reduce to unmoved movers. That is, things that can cause change and motion without themselves changing or moving and thus themselves not requiring a cause.
The unmoved mover must be unchangeable, and thus is immaterial. He also argues that it must be a mind, and that it eternally thinks about itself.
The neo-Platonist philosopher argued that the most fundamental principle in the world must be absolutely simple. If it were not, then it would be composed of further elements or principles that would themselves be more fundamental. So the most fundamental principle must be absolutely simple and not composed of further principles.
This means it is unchangeable, because if it were changeable, it would consist of multiple principles. Namely, the principle of actuality (the way it is now) and the principle of potentiality (the way it could be in the future).
To be continued....