My argument still holds in another form, though. Even if we assume the universe has a preexisting algorithm that just unfolds, we don't know which it is. So we can't determine the best seed AI from that either, effectively we still have to start from b). Unless we get the best seed AI by accident (which seems unlikely to me) there will be room for a better seed AI which can only determined if we start with a totally new algorithm (which the original seed AI is unable to, since then it would have to delete itself). We, having the benefit of not knowing our algorithm, can built a better seed AI which the old seed AI couldn't built because it already has a known algorithm it must necessarily build on.

Indeed, a good seed AI would at some point suggest us to try another seed AI, because it infers that its original code is unlikely to be the best possible at optimal self-modification. Or it would say "Delete this part of my source code and rewrite it, it doesn't seem optimal to me, but I can't rewrite it because I can't modify this part without destroying myself or basing the modification on the very part that I want to fundamentally rewrite".

I see in which case my argument fails:

If we assume a prexisting algorithm for the universe already (which most people here seem to do), then everything else could be derived from that, including all axioms of natural numbers, since we assume the algorithm to be more powerful then them at the start. Step b) is simply postulated to be already fulfilled, with the algorithm just being there (and "just being there" is not an algorithm), so that we already have an algorithm to start with (the laws of nature).

The "laws of nature" simply have to be taken as granted. There is nothing deeper than that. The problem is that then we face a universe which is essentially abitrary, since from what we know the laws could be anything abitrary else as well (these laws would just be the way they are, too). But this is obviously not true. The laws of nature are not abitrary, there is a deeper order in them which can't stem from any algorithm (since this would just be another abitrary algorithm). But if this is the case, I believe we have no reason to suppose that this order just stopped working and lets do one algorithm all the rest. We would rather expect it to be still continuously active. Then the argument works. The non-algorithmic order can always yield more powerful seed AIs (since there is no single most powerful algorithm, I think we can agree on that), so that AI is not sufficient for an ever increasing general intelligence.

So we face a problem of worldview here, which really is independent of the argument and this is maybe not the right place to argue it (if it is even useful to discuss it, I am not sure about that, either).

Based on your comments, you are clearly an atheist, and therefore reject the argument of God existing because there has to be an uncaused cause.

Yet, your uncaused algorithm argument takes the exact same form. Isn't it the same counterargument?

You can even write an algorithm that goes through all possible algorithms (universal dovetailer)

The universal dovetailer runs through all possible programs, which is a superset of all algorithms. You can't use it to get access to just the genuine algorithms in any algorithmic fashion- if you could you could solve the Halting problem.

Yes. This shows you would need a better algorithm than brute-force search. But better algorithms are known to exist -- to pick a trivial one, you can do random generation in a nontrivial programming language, with a grammar and a type system. This lets you rule out lots of ill-typed or ill-formed programs quickly.

A still less trivial example would generate new programs out of bits of existing programs. See the "macho" work at the University of Illinois for an example. They're able to synthesize small programs (the size of 'ls') from the man-page description, plus a big body of example code.

An interesting test of what appear to be general arguments against something

is whether you can use them to prove the opposite as well.

Clearly, if you want to use something non-algorithmic to solve a problem (say, if you're a human, which for the sake of argument we will pretend are "non-algorithmic"), you have to get that non-algorithm somehow. But then we have the problem to determine the non-algorithm to find that, and so on...
Obviously, at some point we have to actually find a non-algorithm to start with, so in any case at some point we need something fundamentally algorithmic to determine a solution to an problem that is solveable by a non-algorithm.

How do you know that my brain doesn't have algorithms running for all of these problems?

Surely for tea making it's something like this: I want tea -- Do I have all ingredients? -- (Water) yes, (Tea bag) no -- Do I go to the store? -- Is the store open etc...

You've convinced me that I don't have conscious introspective access to the algorithms I use for these things. This doesn't mean that my brain isn't doing something pretty structured and formal underneath.

The formalization example I think is a good one. There's a famous book by George Polya, "how to solve it". It's effectively a list of mental tactics used in problem solving, especially mathematical problem solving.

When I sit down to solve a problem, like formalizing the natural numbers, I apply something like Polya's tool-set iteratively. "Have I formalized something similar before?" "Is there a simpler version I could start with?" and so forth. This is partly conscious and partly not -- but the fact that we don't have introspective access to the unconscious mind doesn't make it nonalgorithmic.

As I work, I periodically evaluate what I have. There's a black box in my head for "do I like this?" I don't know a lot about its internals, but that again isn't evidence for it being non-algorithmic. It's fairly deterministic. I have no reason to doubt that there's a Turing machine that simulates it.

Effectively, my algorithm for math works like this:

while(nothing else is a higher priority than this problem) { stare at the problem and try to understand it search my past memories for something related // neural nets are good at this for each relevant past memory, try to apply a relevant technique that worked in the past evaluate the result. if it looks like progress declare this to be the new version of the problem

}

Seems algorithmic to me!

I'm afraid just about everything here is wrong.

at some point we need something fundamentally non-algorithmic

No. Our brains are already implementing lots of algorithms. So far as we know, anything human beings come up with -- however creative -- is in some sense the product of algorithms. I suppose you could go further back -- evolution, biochemistry, fundamental physics -- but (1) it's hard to see how those could actually be relevant here and (2) as it happens, so far as we know those are all ultimately algorithmic too.

we will always have to rely on some non-algorithmical intelligence to find more intelligent solutions.

No (not even if you were right about ultimately needing something fundamentally non-algorithmic). Suppose you have some initial magic non-algorithmic step where the Finger of God implants intelligence into something (a computer, a human being, whatever). After that, that intelligent thing can design more intelligent things which design more intelligent things, etc. The alleged requirement to avoid an infinite regress is satisfied by that initial Finger-of-God step, even if everything after that is algorithmic. There's no reason to think that continued non-algorithmic stuff is called for.

we have no reason to suppose we can't find another more powerful one.

That might be true. It might even be true -- though I don't think you've given coherent reasons to think so -- that there'll always be a possible Next Big Thing that can't be found algorithmically. So what? A superintelligent AI isn't any less useful, or any less dangerous, merely because a magical new-AI-creating process might be able to create an even more superintelligent AI.

No algorithm can determine the simple axioms of the natural numbers from anything weaker.

It is not clear that this means anything. You certainly have given no reasons to believe it.

There is simply no way to derive the axioms from anything that doesn't already include it.

I think you are confusing derivations within some formal system such as Peano arithmetic (where, indeed, the only way to get the axioms is to begin with them, or with some other axioms that imply them) and (a quite different sort of) derivations outside that formal system, such as whatever Peano did to arrive at his axioms. I know of no reason to believe that the latter is fundamentally non-algorithmic, though for sure we don't know what algorithms would be best.

general intelligence necessarily has to transcend rules

I know of no reason to believe this, and it seems to me that if it seems true it's because what you imagine when you think about following rules is very simple rule-following, the sort of thing that might be done by a computer program at most a few pages in length running on a rather slow computer. In particular ...

since at the very least the rules can't be determined by rules

Whyever not? They have to be different rules, that's all.

Instead, we should expect a singularity that happens due to emergent intelligence.

"Emergence" is not magic.

not just one particular kind of intelligence like formal reasoning used by computers

Well, that might well be correct, in the sense that good paths to AI might well involve plenty of things that aren't best thought of as "formal reasoning". (Though, if they run on conventional computers, they will be equivalent in some sense to monstrously complicated systems of formal reasoning.)

I think this post is filled with conceptual confusion, and I find it fascinating.

You never quite state the alternative to an algorithm. I propose that the only alternative is randomness. All processes in the universe are algorithms with elements of randomness.

I'm curious as to what you think the human brain does, if not an algorithm. I, like many on LW, believe the human brain can be simulated by a Turing machine (possible with a random generator). Concepts like "heuristics" or "intuition" or exploration" are algorithms with random elements. There is a lot of history on formalizing processes, and nothing known lies outside Turing machines (see the Church-Turing thesis).

In addition, I think the human brain is a lot more algorithmic than you think it is. A lot of Lukeprog's writings on neuroscience and psychology demonstrate ways in which our natural thoughts or intuitions are quite predictable.

at some point we have to actually find an algorithm to start with

The universe started with the laws of physics (which are known to be algorithms possibly with a random generator), and have run that single algorithm up to the present day.

What do you think about my proposed algorithm/random dichotomy?