December 15, 2003 2:11 AM   Subscribe

how many levels of meta are possible, in practice? [more inside]

"going meta" means going to a higher level of abstraction - a formal way of defining "thinking outside the box".

you can go to an unlimited number of levels trivially with tricks like repeatedly asking "why?" (like a child, questioning each answer), or with "i know that you know that i know that...". but non-trivial examples only seem to go a few levels before bottoming-out. does anyone have any examples of non-trivial abstraction to more than a couple of levels?

in functional programming it's common to have higher order functions. functions that return functions are common. functions that manipulate functions that manipulate functions sometimes exist. but i don't think i've ever gone beyond that.

in philosophy, the ability to "go meta" is closely linked to consciousness (or perhaps self-consciousness) - but that only seems to require one level. dennett, in elbow room, mentions this and makes the observation that we seem to be limited in our ability to go to higher levels. he cites nietzsche, but his example (valuing values) isn't exactly mindblowing (to little ol' me, at least).

in askmeta a few threads below i posted an answer discussing how limits in maths are one level of abstraction above the normal way of thinking. taking that another level gets us to conversations like this, which discuss going meta. but what's the level above this?

if there are no good examples of deeply nested meta-abstraction then what's the source of the limit? are we simply too stupid? or is abstraction so powerful that after a few applications it includes "everything", making further use pointless?

has anyone got any good references to discussions of this problem in the literature?
posted by andrew cooke to Computers & Internet (15 answers total) 3 users marked this as a favorite
Godel Escher Bach is of course always suggested reading material, and gives a nice example of slightly less trivial infinite meta with Escher's picture of two hands drawing eachother. Cheating perhaps,
but even with two levels there's an infinite number of meta-ing steps.

Also have a look at functional programming and lambda calculus,
a lot of deep nesting there.
posted by fvw at 2:43 AM on December 15, 2003

Gödel's Proof by Nagel and Newman is recommended reading, and if you've got the math (or want the math) Cohen's forcing method in set theory is an interesting abstraction of the process of collapsing or telescoping abstractions. I recommend Kunen's Set Theory: An Introduction to Independence Proofs if you can find or borrow (it's part of a series that a good University math library should have) it.

Also maybe read some literature from the intuitionist school of math, Heyting and Brouwer have written many good short articles . Also read Putnam's "Mathematics Without Foundations" (short piece). I saw him when he came by my city to talk about it and it was excellent.
posted by j.edwards at 2:54 AM on December 15, 2003

I point a video camera at a display of the camera feed, or stand between two parallel mirrors.

I get the feeling that it could go on infinitely.
posted by armoured-ant at 5:02 AM on December 15, 2003

Godel Escher & Bach, to repeat, is readable and contains approachable and not-entirely-abstruse discussion of your question. Some math and logic required, but not to get the gist of what is said.

The 'strange loops' and 'tangled hierarchies' which emerge from his discussion of recursive meta-questioning (and many other topics) form, he suggests, the crux of consciousness and the explanation for the emergent behaviour of the mind, and he offers the possibility the consequences of Godel's theorem could indicate that 'there could be some high-level way of viewing the mind/brain, involving concepts which do not appear on lower levels, and that this level might have explanatory power that does not exist - even in principle - on lower levels.'

Fun stuff. I reread GEB every couple of years.
posted by stavrosthewonderchicken at 5:26 AM on December 15, 2003

I vote for G.E.B. as well. That book makes my head explode about every six months. I've never seen a book that explores so much of reality in such simple words.
posted by oissubke at 5:54 AM on December 15, 2003

Hear, hear for G.E.B.! It almost seems like andrew cooke was trolling for us to mention it.

Every six months!?!?! It took me two years to finally get through it!
posted by notsnot at 7:43 AM on December 15, 2003

If meta is a form of explanation, that tries to explain by placing things in perspective, maybe it depends on whether your model of knowledge is concentric (as in a core of 'truth/reality,' surrounded by onion-skin layers of meta that each contain all the previous layers), or relativist, in which case meta can become an ironic stance that is outside of but not necessarily encompassing that which is being explained. In either case I guess three or four layers/stances of meta is probably enough for anyone.

As I'm speaking from a social science/humanities point of view, a useful model of learning here for me has been Gregory Bateson's "Logical Categories of Learning," in his book "Steps Towards an Ecology of Mind."
posted by carter at 7:49 AM on December 15, 2003

"going meta" means going to a higher level of abstraction - a formal way of defining "thinking outside the box".

I'm sorry, but it simply has nothing to do with abstraction. Meta typically means something like, concerning or describing but not falling within the bounds of. (Though of course it is currently inordinately popular to abuse the meaning and apply it to anything that is about something else. MetaTalk is, you could say, meta-MetaFilter, if you see it as being outside of MetaFilter; while MetaFilter itself isn't actually meta anything; it's about the web, but it's also on the web.) But abstraction is something else entirely; MetaTalk, for instance, is in no way any more abstract than MetaFilter itself.

And I won't even touch the "outside the box" thing, except to say that it's only one place in your comment above where I honestly have no idea what you're actually talking about, or asking, exactly. Could you give some examples of what it is that "bottoms out"? Are you talking about applications of mathematics, or thinking about certain kinds of concepts, or what?
posted by mattpfeff at 7:56 AM on December 15, 2003

A sidenote from a non-mathematician: you needn't have higher level study in mathematics or logic to understand Godel Escher Bach. A significant portion of what seems confusing is, when contemplated thoughtfully, somewhat common sense. It's a truly-one-of-a-kind read, and as a non-mathematician whose made it through it, I wanted to make sure some of the warnings above didn't dissuade andrew cooke from reading it.
posted by JollyWanker at 7:59 AM on December 15, 2003

Andrew, I think the question you are really asking is more philosophical than mathematical. (Otherwise, we could talk about the use of powersets and russell's paradox and Z-F set theory)

That is, we can describe a class of things, like objects in the world. To go Meta in this case is to then create a new language for how we describe, not the original things, but the "class of things" themselves.

Usually we invoke Meta in order to avoid recursion. For example:

This sentence is false

We can clear up the apparent paradox that that sentence creates by creating two classes of sentences: 1) sentences that talk about things 2) sentences that talk about sentences that talk about things. So, then the construct above is not allowed but we can have constructs like:

1) The House is Red.
2) Sentence #1 is False.

And we can maintain consistency that way. Now, what if we want to talk about meta-sentences? We create a third class that can only address sentences of the 2nd class. And so on.

Now, your question as I understand it, is: Why in our thought processes and in our everyday thinking about the world, do we never need to go beyond 2 or 3 levels? Is it something about the world or about how our minds work?

I dont have an answer myself. I am mostly re-stating your question and hoping I am right.
posted by vacapinta at 1:21 PM on December 15, 2003

ok, sorry for being so vague. vacapinta's re-stating above is more-or-less what i was asking.

i've read geb and, as far as i remember, it takes a very long time to explain that you get in a mess when you don't have a clear separation between language and meta-language.

my phrasing was more complex because (1) vacapinta thinks more clearly than me and (2) i think i was (and possibly still am) confused about whether there's a qualitative difference between using a meta-language once and using it many times.

it seems to me that there's an intellectual leap in using meta language once that's not repeated when you use it again - repeated applications become trivial (you're just doing the same trick). if that's the case, then how do you repeat that leap. what's the thing beyond meta-language?

i've heard it said that most things can be characterised by "none, one, many". maybe i'm doing nothing more than restating that.

so anyway, i think vacapinta nailed my question, in case anyone has a simple answer. thanks for the various references. if anyone has any comments on the qualitative leap ramblings above i'd like to hear them, and apologies again for being unclear. i didn't realise how vague i was being until i read the replies (and i think i tried to make the question too "simple").
posted by andrew cooke at 3:44 PM on December 15, 2003

i'll need to think more about the difference between abstraction and meta-language. to me they seem to be very closely related.
posted by andrew cooke at 3:47 PM on December 15, 2003

Here's my further take:

Higher levels of Meta are just generalizations on generalizations. I think your instinct was right when you first said "is abstraction so powerful that after a few applications it includes "everything", making further use pointless?"

I think thats right. Underneath it all there is a fundamental class of ideas. Layers of generalization impose a cognitive hierarchy which is limited by the fact that the underlying or base class is usually finite.

As an example, we have ideas, then we have ideas about ideas (memetics?), then we may have ideas about ideas about ideas (general properties of idea theories?) and so on...

We cant get much farther because we are the top of the pyramid - underlying objects are getting scarcer (there are not many ideas about ideas so its harder to generalize)


The second point you make is more philosophically interesting - is there a SuperMeta that is to Meta as Meta is to just things? Is that right?

Thats a profound question. It reminds me of when I was a kid and looked at ants and wondered about how they lived in the same world as us, with the same mathematics and the same quantum physics, but how incomprehensible (and you can see how the word 'incomprehensible' is so inadequate) all this was to their world.

Likewise, were there higher forms of looking at the world which made our capacity for abstraction look like the prescribed actions of ants?

For now, the highest level of abstraction I know about is mathematics and, within that, I'd say category theory.

Email me if you find further information on this.
posted by vacapinta at 4:08 PM on December 15, 2003

not sure it's a profound question. it might be no more than "what's another good idea?" have to dash to catch an (overnight) bus. will think some more. cheers.
posted by andrew cooke at 4:52 PM on December 15, 2003

is there a SuperMeta that is to Meta as Meta is to just things?

I think in essence that that's precisely what Hofstadter was talking about in the bit I quoted above from GEB, and his answer, as a philosophical implication of Godel's work, is that yes, there is...
posted by stavrosthewonderchicken at 6:35 PM on December 15, 2003

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