It has to do with how a system is comprehended by someone. That system, in order for either word to apply, cannot be simple.

As you try to understand something, it is complicated. When you understand something complicated, it is complex. Take for example euler's formula (from my memory): x=e^-i*pi. To me, this is complicated, I've never understood it. It has imaginary numbers, pi radians, and e-- known as euler's number. But I've seen some great explanations, from people who really got it, and I know that it can be complex and beautiful to those who really understand. I think for me to really understand, I would need a reason to need it.

Of course, this applies to startups.

An idea must be complex inside (to the founders), for otherwise it is nothing and weak. It must not be complicated inside, for then the founders do not understand their own creation, and more time must be spent.

An idea must be simply represented on the outside (to the users). In a world rich with information, understanding something complicated is a heavy investment on the part of the user. This can only rise in correspondence with the popular belief of the worthiness of your product.

An idea that is complex on the inside, but complicated on the outside is ahead of it's time. Users need catch up, so that it becomes more simple and obvious.

Lets look at some example:
Twitter: simple outside
Facebook: simple outside
Myspace: equally simple, but less thought on the inside.
Paypal: simple outside (a hell of fraud prevention on the inside)
Amazon: simple outside
Google wave: complicated (outside, I have no idea about inside, perhaps complex, perhaps complicated)
Microsoft word: complicated
Google docs: simple outside. Had it not been for Microsoft word, it would have been too complicated.
Google buzz: simple outside, simple inside.
Square: simple outside

Etc, etc, etc. Nobody wants a more complicated world, folks. Find a problem, and making it simple means solving it. Then take those solutions, combine them, and make them simple. That is evolution of software.

FOOTNOTES:
The phrase "understand something" is a complete misnomer in the opening paragraphs. Certainly from my perspective of machines and moving parts, you don't understand a lone object but a bunch of relationships; for me to know something is to learn all the interactions between the things you know and the things that you are trying to know. Of course, when you realize something new about something old, this is what's happening, much to your surprise. Eureka!

An example of a simple learnable may be someone's name, where a complicated one might be a personality or a complete machine. Nothing is too complicated, it just needs to be broken in to smaller steps (in the words of DHH and many before). A simple idea is neither complex nor complicated, it has few relations between what you know and what you dont. (for the sake of the beauty of the English language, few relations means few relationships to build.)

When I said I would need a reason to understand eulers number, I think that's because there would be so much more in my head to relate things to.

As a final note, I think that the idea of beauty, in an intelligence sense, comes quite close to this discussion. In some cases a complex idea is beautiful from it's very nature. Or for example, the English language's beauty in matching our thoughts well. There is something to be understood about beauty and a well-running mind. I wonder if it is relative to out experiences or our DNA, but that is the topic of another post.

Sent from my iPhone