How I feel about mathematical discovery:
You're wandering up and down, it's like wandering in a strange town with beautiful things. You turn around this corner and you don't know whether to go left or right. You do something or other and then, suddenly, you happen to go the right way, and now you are on the Palace steps. You see a beautiful building ahead of you, and you didn't know that the Palace was even there. There's a certain wonderful pleasure you get on discovering a mathematical structure.
I'm perennially fascinated by mathematics, by how we can apprehend this amazing world that appears to be there, this mathematical world. How it comes about is not really physical anyway, it's not like these concrete buildings or the trees. No mathematician believes that the mathematical world is invented. We all believe it's discovered. That implies a certain Platonism, implies a feeling that there is an ideal world. I don't really believe that I don't understand anything. It's a perennial problem to understand what it can be, this mathematical world we're studying. We're studying it for years and years and years and I have no idea. But it's an amazing fact that I can sit here without any expensive equipment and find a world. It's rich, it's got unexpected properties, you don't know what you're going to find.
See also pictures of Conway.