It is helpful to distinguish between "a thought' and the "process of thinking".
If you really look at what's going on, there are no things that are thoughts. We can label a passing moment in the process of thinking 'a thought' , but that is really just a sort of freeze-frame way of looking at the activity of the mind. We can apply that method (and most people seem to be in the habit of doing that) but it isn't really accurate. When we say "thoughts arise" this is a convenient way of expressing the activity of the mind. But when you really analyze it, thoughts do not arise, meaning that there are no "things' that are separate, self-existent thoughts, no little thought-clouds in the head.
So, then the question shifts. Instead of asking about stopping thoughts, you have to ask about stopping the activity of the mind, and you really can't do that. Or, if you do it, you won't be aware of it, and you could have no memory of it! So then, what is this whole idea about? It's about calming the activity of the mind, allowing the activity to settle so that the awareness is on mind itself, rather than on objects of internal or external awareness.
Here is a little experiment you can do. Do you hear that? Shhhh! listen very carefully. (as though there might be some strange noise, either very soft, or very far away that you are trying to hear). Put all of your attention on the subtlest thing you can hear.
While you are doing this, while your mind's full attention is on subtle hearing, just take a little bit of notice of your visual awareness---not at what your eyes are looking at (don't shift your attention away from listening), which may be nothing at all, but just barely notice the visual awareness itself. Just sort of feel your eyeballs with your mind. If you practice this a few times, what you will notice, what you can sort of replicate, is awareness without thinking. Of course, this is limited to visual awareness, but it will give you some sense of "no thoughts arising". You can also do this by focusing on listening and then noticing your body's tactile awareness, or taste awareness.
From this exercise, you can see that it is possible to bring your mind to complete rest, with no object of focus and no thought activity occurring...just awareness itself, and awareness of that awareness without imputing a "self' who is being aware.
Profile Picture: "The Foaming Monk"
The Chinese characters are Fo (buddha) and Ming (bright). The image is of a student of Buddhism, who, imagining himself to be a monk, and not understanding the true meaning of the words takes the sound of the words literally. Likewise, People on web forums sometime seem to be foaming at the mouth. Original painting by P.Volker /used by permission.