"Which statement seems more true: (1) I have a brain. (2) I am a brain."
What it means: Is it true to say "I have a brain"? But the idea of saying that comes from your brain, so maybe it's more true to say "I am a brain".
"The "Strange Loop" phenomenon occurs whenever, by moving upwards (or downwards) through levels of some hierarchial system, we unexpectedly find ourselves right back where we started."
Simple: A "strange loop" happens at these times: when we are moving up or down through levels of a system, and we are surprised to find that we are right back where we started.
What it means: A "strange loop" is something that talks about itself, for example, a sentence that says "This sentence is not true".
"One of the basic tenets of Zen Buddhism is that there is no way to characterize what Zen is. No matter what verbal space you try to enclose Zen in, it resists, and spills over. It might seem, then, that all efforts to explain Zen are complete wastes of time. But that is not the attitude of Zen masters and students. For instance, Zen koans are a central part of Zen study, verbal though they are. Koans are supposed to be 'triggers' which, though they do not contain enough information in themselves to impart enlightenment, may possibly be sufficient to unlock the mechanisms inside one's mind that lead to enlightenment. But in general, the Zen attitude is that words and truth are incompatible, or at least that no words can capture truth."
Simple: One of the important ideas of Zen Buddhism is that there is no way to say what Zen is. Every time you use words to try to close Zen into a small space, it fights back and comes out of that space. You might think, then, that trying to tell what Zen is is a waste of time. But Zen masters and students don't think that. Zen koans (short stories) are an important part of learning about Zen, and the koans are made of words. Koans are supposed to be small things that make big things happen. They don't have enough information to give someone enlightenment, but they may be enough to open the things in someone's mind that can lead to enlightenment. Mostly, Zen students believe that words and truth are two different things, or that words can't catch truth.
"This type of paradox is quite characteristic of Zen. It is an attempt to "break the mind of logic"."
Simple: This kind of paradox happens frequently in Zen. It's someone trying to stop the mind from using logic.
What it means: A paradox is something that can't make sense.
"Perhaps the most concise summary of enlightenment would be: transcending dualism. … Dualism is the conceptual division of the world into categories. … human perception is by nature a dualistic phenomenon— which makes the quest for enlightenment an uphill struggle, to say the least."
Simple: Maybe the best way to say in a few words what enlightenment is is: going further than dualism. Dualism is the way we divide the world into categories in our minds. The way humans think is always by dualism: which makes looking for enlightenment into hard work like walking up a hill.
What it means: Trying to find enlightenment is like trying not to think the way we usually think.
"Relying on words to lead you to the truth is like relying on an incomplete formal system to lead you to the truth. A formal system will give you some truths, but as we shall soon see, a formal system, no matter how powerful—cannot lead to all truths.
Simple: Using words to find the truth is like using an incomplete way of writing mathematics to find the truth. A way of writing mathematics will give you some truths, but there is no way of writing mathematics that will lead to all truths.
"No reference is truly direct—every reference depends on SOME kind of coding scheme. It's just a question of how implicit it is."
Simple: No way of talking about something is completely plain. Every way of talking about something uses a language or code. Some are very obvious, and some need some thinking to understand them.
"Hofstadter's Law: It always takes longer than you expect, even when you take into account Hofstadter's Law."
Simple: Doing something takes more time than you thought it would take, even if you already planned that it would take more time.
Replying to following question by Deborah Solomon in Questions for Douglas Hofstadter: "Your entry in Wikipedia says that your work has inspired many students to begin careers in computing and artificial intelligence." He replied "I have no interest in computers. The entry is filled with inaccuracies, and it kind of depresses me." When asked why he didn't fix it, he replied, "The next day someone will fix it back."
Simple: Deborah Solomon asked Douglas Hofstadter a question. Deborah Solomon said something that meant, "Your page on Wikipedia has made many students think about having jobs with computers and the way computer's think." Douglas Hofstadter said something that meant, "I do not like doing things with computers. The page has many errors, and I am sad at that." Solomon then asked Hofstadter why he did not fix it. Hofstadter said something that meant, "I did not fix it become someone will fix it back the next day – they will put the errors back in."
What it means: Hofstadter is saying that while Wikipedia can have errors sometimes, it can be fixed very fast by one of its many users.