[ndsipa_pomu]

> There is no contradiction because there is no limit of knowledge.

I don't think we know enough to be able to state that definitively. It's feasible that the universe behaves mathematically (it seems to so far) and thus possible to gain a thorough understanding of the underlying principles, if not the specific facts (c.f. with understanding how to produce integers yet not "knowing" all the integers).

Even if the universe doesn't have underlying rules to be discovered, there's still a limit to number of configurations available to particles etc. within our visible universe. Although that number might appear to be infinite to us, it's actually drastically closer to zero than to infinity.

So, if there is indeed some finite limit, then using y = e^x would be the wrong function as that doesn't approach a finite value.

[fnovd]

This leads to a more fundamental question: What is the universe?

Is the optimal move in an a given chess board considered knowledge? If so, can't we create entirely new sets of knowledge from the emergent properties of an arbitrary set of rules called a "game"? If we can create an infinite set of arbitrary combinations of rules and states (games), then knowledge should be infinite. Maybe not all knowledge is scientifically applicable, but we have learned a great deal about science and engineering from studying chess. In fact, we are starting to learn more about learning as a process and not as some magical thing that human beings can do, just from studying the best way to make decisions in this totally-contrived and scientifically-useless game.

Taking this a step further, let's look at the animal kingdom. If learning about the intricacies of the mating habits of birds can help an arbitrary bird increase its impact on the future gene pool, is that knowledge not worth something to the bird? To bird society? Are the things we learn about ourselves knowledge? They certainly have utility. Is there any limit to what we can learn about ourselves, about the stochastic process of life? Is life not part of the universe?

Is computer science even knowledge? It seems if we're more directly concerned with the physical nature of the universe, we ought not to care about what the system of a computer actually does; we only need to care about what it is, about its physical structure. Except, that's not actually how we pursue knowledge or science at all.

In my view, Asimov's sentiment can be reduced to a complete tautology: we're at the point where we know almost everything there is to know about the things we think we can know.

[ndsipa_pomu]

There aren't an infinite number of chess positions, moves or even games, so that's not a good example. It's possible to come up with a number game that could have infinite possibilities, but that doesn't mean that the universe could even contain some of the options within our visibility. Our current state of knowledge about the universe strongly suggests that there's a finite limit to the available knowledge (I.e. between the Planck scale and the visible horizon due to the speed of light).

A googolplex looks to be the first number we've found that is too big to be contained in our universe.

[fnovd]

You're right--chess is a decidedly finite game. Even so, we have not "solved" this simple, finite game--not even close! If we're not close to solving such a trivial game, how can we be close to the limit of the knowledge of the universe?

A googolplex is "too big to be contained" in our universe yet here we are talking about it. We can perform operations on this number, compare it to other numbers, and even come up with mathematical proofs showing that it's too big to exist. There are an infinite amount of numbers larger than a googolplex and we could have an infinite amount of conversations about them. The material limit of the universe does not limit our ability to create information, to learn things.

There isn't enough space in the universe for an infinite series, either, yet we can (and do) still use them, we reason about them, we learn from them. We can even reduce some infinite series to a finite number. The material bounds of the universe are not a limit of knowledge.

[ndsipa_pomu]

I think you're mistaking the map for the territory. A googolplex is a representation of a number, but not the number itself, although it's simple enough that we can get away with using the representation as it's obvious what the form of the number would be. However a number such as tree(3) is unimaginably bigger, but more crucially, we don't know anything about the form of the number beyond its size and we can't sensibly use it in calculations.

Now both of those numbers are finite and we could try to figure out how many numbers we could "describe" such as tree(3), but that would be limited by the number of symbols (i.e numbers, operators, letters and words) that could be used (i.e we would have less than a googolplex different numbers that could be represented using maths, language and thought). That's still going to be a finite number.

[fnovd]

If the Universe is the territory than Knowledge is the map. I'm not at all mistaking the map for the territory: I'm pointing out that the set of maps that can describe a given territory are virtually infinite. Asimov is saying the map is almost complete and I'm saying there are an infinite number of maps left to go.

Cartographers in the 18th centuries were "basically done" mapping out the Earth. In the 20th century we were able to use satellite imagery to get the "full picture". Does that mean we have perfect knowledge of the Earth? Absolutely not. There is never a final frontier of knowledge.