[Swizec]

> We have absolutely no contract with the universe or nature that guarantees us that there's "always more to invent", even less so that "the possibilities keep increasing".

Think of human knowledge as a circle. The edge of that circle is the edge of human knowledge. Inventing a thing makes a little bump in the circle that pushes the edge outwards. The bigger the circle, the bigger the circumference, the more possibilities to invent something.

http://matt.might.net/articles/phd-school-in-pictures/

[Joe-Z]

That's just a metaphor, and exactly of the type which GP argued against.

To illustrate GP's point metaphorically: Think of human knowledge as discovered areas on a map. The dark areas are what is still unknown to us. Inventing a thing makes a little spot on the map visible. The more you have discovered, the less you still have to discover.

(And to extend it a bit: Of course you can make the already discovered areas more detailed, even to levels unthought of when initially discovered. But as discoveries pile up, there probably won't be many 'woah, there's a whole continent here!' moments anymore.)

I don't have a strong opinion in this debate. I just wanted to provide a counter-point to your metaphor.

[radioactivist]

As an isolated example of this kind of thing, consider the shape of the Earth as discussed in Asimov's "Relativity of Wrong" [http://chem.tufts.edu/answersinscience/relativityofwrong.htm]

[munificent]

You are both right. Animal reproduction is exponential as long as resources are freely available -- the more animals you have, the more they can mate. Like the parent post, the greater the perimeter of your knowledge, the greater the boundary you can now explore from.

Once animals hit the resource limit of the environment, the exponential curve flattens out. Like you note, the more of the map you cover, the less there is left to explore.

The interesting question is how big is the map and how close are we to reaching it's boundaries?

Personally, I think it is effectively infinitely large. If you consider "discovery" to include new combinations of existing things, then you're talking about permutations. If our universe consisted only of 52 playing cards, there would still be 80,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 different ways for us to discover that they can be shuffled together.

[marcosdumay]

Ok, now postulate that we can only invent things on the border of that map (because of technological limitations). Then you'll get a very complex dynamics, of the invention possibilities increasing and decreasing, some times fast, and changing almost unpredictably.

[dwaltrip]

If the illuminated areas are a very small portion of the entire map, then the circle metaphor actually may be more appropriate. But I see your point.

Hmm... Perhaps better still would be an immense valley of very uneven steepness and roughness, that we are climbing out of.

[anonytrary]

Does the set of everything exist? Why or why not?