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I am trying to figure out the meaning of prefix "Meta-" in English.

  1. Quoted from Wikipedia

    Meta- (from Greek: μετά = "after", "beyond", "with", "adjacent", "self"), is a prefix used in English (and other Greek-owing languages) to indicate a concept which is an abstraction from another concept, used to complete or add to the latter.

    In epistemology, the prefix meta is used to mean about (its own category). For example, metadata are data about data (who has produced them, when, what format the data are in and so on). Also, metamemory in psychology means an individual's knowledge about whether or not they would remember something if they concentrated on recalling it. Furthermore, metaemotion in psychology means an individual's emotion about his/her own basic emotion, "or somebody else's basic emotion".

    Another, slightly different interpretation of this term is "about" but not "on" (exactly its own category). For example, in linguistics a grammar is considered as being expressed in a metalanguage, or a sort of language for describing another language (and not itself). Any subject can be said to have a meta-theory which is the theoretical consideration of its meta-properties, such as its foundations, methods, form and utility.

    I was wondering what is the slightly difference between "about (its own category)" and "'about' but not 'on' (exactly its own category)"?

  2. My original problem is to understand the meaning of "metalogic" and "logic". Quoted from Wikipedia:

    Metalogic is the study of the metatheory of logic. While logic is the study of the manner in which logical systems can be used to decide the correctness of arguments, metalogic studies the properties of the logical systems themselves.1 According to Geoffrey Hunter, while logic concerns itself with the "truths of logic," metalogic concerns itself with the theory of "sentences used to express truths of logic."2

    The basic objects of study in metalogic are formal languages, formal systems, and their interpretations. The study of interpretation of formal systems is the branch of mathematical logic known as model theory, while the study of deductive apparatus is the branch known as proof theory.

    In this case, is it correct to understand "meta-X" as some study that treat something (here logical system) as a whole without going inside it, whereas "X" study the same thing by going inside it?

  3. How would you explain the relation between "metaphysics" and "physics", and between "meta-stackoverflow" and "stackoverflow"?

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2 Answers

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1. A simple example: I could look at a sequence of numbers and make a new sequence of numbers in which each number is the difference between two consecutive numbers from the first sequence:

  • 1, 4, 7, 2 (sequence 1)

  • 3, 3, 5 (sequence 2: sequential subtraction, with as its object sequence 1)

Then I could make sequence 3, with de differences between the numbers from sequence 2:

  • 0, 2 (sequence 3: sequential subtraction, with as its object sequence 2)

Sequence 2 does with sequence 1 exactly the same thing (sequential subtraction) as what sequence 3 does with sequence 2. Both the operations carried out by 2 and 3 can be defined by the same name, sequential subtraction. We could say that 3 operates on a meta-level in relation to 2. We could say that 3 is meta-2. In logical language, we could define 2 as SeqSub(sequence 1), and 3 as SeqSub(SeqSub(sequence 1)). As you see, 2 is nested within 3.

On the other hand, consider grammar and normal language. Normal language describes the world, ideas, anything. Grammar describes, too, but it only describes language, not the world or anything. Grammar could be called meta-language. In this manner, we could call anything meta-x as long as x is a theory and meta-x is a theory about x, even if x and meta-x operate in different ways. X should be something vaguely similar to a theory, some abstract operation.

(You could even use meta- with things that aren't theories or abstract operations, but that is normally not done, except as a joke — suppose you had a brush to clean the floor, and a rag to clean the brush; then you might jokingly call your rag a meta-cleaner.)

Now what is the difference between grammar and sequence 3? We could say that grammar does not do exactly the same thing with language as what language does with its object, because grammar cannot, for example, refer to a physical thing directly. I think this what your quote means, the difference between identical operation on the one hand and similar-but-not-identical operation on the other. Sequential subtraction = sequential subtraction; grammar is a language, but language is not always grammar. In logical language, we could describe language as Describes(world), and grammar as GrammaticallyAnalyses(Describes(world)). They are nested, but in two slightly different ways.

Of course this distinction between "identical" and "similar" depends on definitions, which may be somewhat arbitrary. So I do not have full confidence in its strength and meaningfulness.


2. I think I can feel what you mean, but I am not sure I'd phrase it like that.

I'd put metalogic in the second category mentioned above — similar to but not exactly the same as — but I am not sure. The reason why it is called "meta-" is that logic studies language and thinking, which makes logic an abstract operation and a theory; and metalogic studies logic, so that it is on a meta-level in relation to logic. Note that meta-x is always relative to its object: metalogic is not "meta-" in relation to, say, pottery.


3. [Edited] In "metaphysics", the prefix "meta-" is used in its original sense in ancient Greek, which is here "after". Aristotle wrote the Physica, which were about the workings of nature: physis/phusis is Greek for nature. And he wrote the Metaphysica, which he called "The [books/bookrolls] about prime philosophy" — physics was the secondary philosophy. Later Greek scholars catalogued this work as "ta meta ta PHusika": "the things after the Physica", because they came after his Physica in their catalogue. Because his Metaphysica were about causality and other principles at work behind the physical world, it seems people later interpreted the "meta-" in Metaphysica as meaning "on a higher level than", and that is where our use of "meta-" came from.


4. Stackoverflow is a website with questions and answers about programming. Meta-Stackoverflow is a website with questions and answers about Stackoverflow. So M-SO operates on SO the same way as SO operates on programming. In logical language, SO is SO(programming), and M-SO is SO(SO(programming)). You see how one is again nested within the other? That is why it is called Meta-Stackoverflow and not Newname.

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Thanks! So if I understand correctly, meta-X seems to be the same as a mathematics concept called Higher-order function en.wikipedia.org/wiki/Higher-order_function? –  Tim Jan 11 '11 at 2:46
    
I mean under the assumption that X can always be interpreted as a function? –  Tim Jan 11 '11 at 3:01
    
Hmm, if I understand higher-order functions correctly, they are not the same. Meta-x is a specific type of higher-order function, with the additional condition that it can only be so called when it is a function that operates on a function similar to itself. So if f(x)=3x, and g(f)=3f, we have a meta-relationship; but if f(x)=3x, and h(f)=f^2, it is just a higher-order relation. How similar f and g need to be (here they are exactly identical) is open to debate, though I'd say "identical" in mathematics. Correct me if I understood higher-order functions incorrectly. –  Cerberus Jan 11 '11 at 3:02
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@Cerberus: to my knowledge too (I’m a logician), meta-logic (and relatedly, meta-mathematics, meta-theory, etc.) are the most widespread usages of meta- in mathematics. But I feel (and I think most mathematicians would feel similarly) that Tim’s suggestion of higher-order functions would be a much more reasonable thing to call “meta-functions” — your use of “meta-f” for the iterate(s) of f doesn’t quite feel right. Maybe that’s just me, though… (Great answer in general, in any case!) –  PLL Jan 11 '11 at 6:42
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@PLL: metamathematics is used much more than metalogic. Metamathematics is a part of mathematical logic and so is logic, metalogic is just a normal part of logic (an axiomatisation of a theory is itself axiomatisable, e.g., using logical frameworks). @Cerberus: there are both recursive and non-recursive instances of both first-order and higher-order functions. –  Charles Stewart Jan 11 '11 at 7:47
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StackOverflow has questions about a topic. Meta-StackOverflow has questions about the questions in StackOverflow - for example, "is this question on topic?". X and Meta-X usually have similar relationships.

Logic is (roughly) about formal reasoning. Meta-logic is reasoning about logic, or why logic works.

Also, the relationship between physics and metaphysics and other similar pairs of words doesn't follow these "rules", and have independent definitions; you should look at their definitions and ignore the common root.

This is my admittedly informal understanding. I don't have a more formal understanding, but I don't have any practical problems with the concept either. Hope this helps!

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