A blog about belief

Posts Tagged ‘language’

The Language of Mathematics (1)

In Computer Science, Literature, Mathematics on June 18, 2012 at 7:37 pm

I’m working on a software project (more soon) that involves a notation that is interpreted by computers.  As a way of specifying the language formally, I’m trying out parsing expression grammars, a relatively new alternative to the methods that have been traditionally used to define the syntax of programming languages, like context-free grammars.  I’ve been reading the original paper in which Bryan Ford introduces PEGs, and something struck me about the way in which it builds up to the mathematical definition of the idea.  The paper begins with an “informal” explanation that starts with an example of a PEG written in ASCII text, like you would use as the input to a program:

# Hierarchical syntax
Grammar <- Spacing Definition+ EndOfFile
Definition <- Identifier LEFTARROW Expression
Expression <- Sequence (SLASH Sequence)*
Sequence <- Prefix*
Prefix <- (AND / NOT)? Suffix
Suffix <- Primary (QUESTION / STAR / PLUS)?
Primary <- Identifier !LEFTARROW
         / OPEN Expression CLOSE
         / Literal / Class / DOT


Although the paper explains what it means in a very prosaic way, placing it in historical context and comparing PEG’s practical implications with those of other types of grammar, this bit of ASCII text seems intuitively like the most formal thing in the paper.  The mathematical definition of the construct is set off much less from the text of the article than the ASCII example, which is in a fixed-width font and embedded as “Figure 1.”  The definition begins:

Definition: A parsing expression grammar (PEG) is a 4-tuple G=(VN, VT, R, eS), where VN is a finite set of nonterminal symbols, VT is a finite set of terminal symbols, R is a finite set of rules, eS is a parsing expression termed the start expression, and VN ∩ VT = ∅. Each rule r ∈ R is a pair (A, e), which we write A ← e, where A ∈ VN and e is a parsing expression. For any nonterminal A, there is exactly one e such that A ← e ∈ R. R is therefore a function from nonterminals to expressions, and we write R(A) to denote the unique expression e such that A ← e ∈ R.

One reason why the informal explanation is informal in comparison with this is that it describes the syntax of PEGs using a PEG, making it a circular definition.  But two things jump out at me.

  1. The mathematical notations in the formal definition are interpolated into a paragraph of written English, while the informal definition describes the syntax of the system in a way that a computer could understand.
  2. It would be much harder to see what the formal definition is doing without reading the informal one first.  If the paper had started talking about 4-tuples right off the bat, it would be unclear in what sense the objects it defines could be considered “rules” and “parsing expressions.”  There is something, a sort of mathematical anamnesis, that the reader takes away from the circular definition at the beginning of the paper that makes it possible to see the meaning of the more rigorous math that follows, in a sense of the word “meaning” that is not yet clear to me.

Speaking objectively

In Philosophy, Science on November 11, 2011 at 1:24 pm

The “plain style” of speech that has just about completely supplanted the “high” rhetoric of the past is in part a remnant of an early version of the scientific method that claimed to root knowledge in objective observation.  The ideal of dispassionate language has persisted even as this interpretation of scientific practice has given way to more sophisticated ones, and I’m not sure of the extent to which we ought to hold on to it.  One thing that I’ve learned from Twentieth-century philosophers of science like Karl Popper and W.V. Quine, who have more in common with post-structuralism than anyone wants to admit, is that science works because observation is subjective, because there is no way of reporting what we see that cannot be disputed by someone who sees it differently.  Every act of perception is an act of interpretation in terms of a particular doctrine, and this is a good thing, because discovery occurs at the points at which our doctrines come into tension with what we perceive, at which, that is, there is something that we just can’t make fit with our present beliefs.  Objectivity is the process in which we actively seek this sort of tension.  It’s not the opposite of subjectivity, it’s what happens when subjectivity runs up against the world.

The thing is, this process only works if the doctrine in terms of which one interprets things is in a certain sense rigid.  If the associations that constitute one’s beliefs are flexible enough, then one can finagle any sensory data one comes across to fit them; I have called this, in another post, paranoia.  The purpose of the sort of formalized language that Quine takes such pains to develop is not to be transparent, but to be sufficiently rigid for science to work.  It is to make the relationships between different statements clear enough that when contradictions arise – when things don’t line up properly – the tension is manifest.  A language made rigid in this way looks quite similar to the “plain” modes of speech advocated by the early Royal Society empiricists, but the theoretical basis for it is very different from what led, for instance, Thomas Sprat to decry “figural” language as a cause of pointless contention.  There is a legitimate place for this formal sort of language, although not for the reasons that Sprat gives.  It facilitates both experimentation and productive debate.

But rigidity is only one of the prerequisites of objective discussion as I have defined it.  The other, the contingency and changeability of the systems of doctrine by which we make claims, would seem to be best served by a type of language that looks quite different from the traditional scientific plainness, and I’m not convinced that the particular formalization that Quine comes up with doesn’t falter in this regard.  Despite the theoretical motive to the contrary, Quinean language still looks like it’s meant to be transparent, if due to historical association alone.  An ideal sort of language for an objective discourse would be right up front about the fact that each individual statement in that discourse represents a particular, contingent, subjective viewpoint, while still providing as little give as possible when the discussion runs up against one of these viewpoints’ limitations.  This is doubly important in literary criticism, where it’s always import to keep aware of one’s own cultural position, and how that might impact one’s understanding of a text.

Quine and “Quine”

In Philosophy on August 27, 2011 at 9:09 pm

I’m working my way through Word and Object. I’m finding it frustrating in some ways. What drew me to Quine is his idea of the “web of belief,” a brilliant image of how a conceptual scheme for the world could work without an a priori foundation, but, however much it’s been discussed in commentary on Quine, it seems to be little more than a footnote in his actual work. There’s barely any explicit discussion of this metaphor in The Web of Belief, despite the title, and when it comes up in Word and Object Quine finds it sufficient to quote a paraphrase by Virgil Aldrich rather than bother with the exposition himself (12). Instead of backing up the idea with evidence and working out its implications for cognition, Quine spends a good portion of the book speculating about language acquisition (in the comments section of this New APPS post Jon Cogburn talks about Quine’s “weird a priori stories of language acquisition”) in a way that is, for someone so expressly concerned with science as Quine, strangely lacking in concrete evidence.

Someone recently quipped (I can’t find where) that the significance of Quine’s ideas comes through better in “cover versions” of his philosophy by others than in his own work. I certainly think this is the case with Word and Object. Buried in Quine’s regimentation and speculative psychology are some ideas that are highly relevant today, but they are relevant mostly in areas that Quine had little interest in exploring.

There are, though, some moments in Word and Object that can be quite readily construed in ethical/political terms. The way Quine writes about “native languages” in his section about radical translation has dated poorly, but the section does touch briefly on the ethics of anthropological research. Quine opposes the “doctrine of ‘prelogical mentality'”, in which the anthropologist carefully avoids the assumption that “natives” follow any particular rules of logic, because, Quine argues, we can’t be sure we’re understanding them correctly unless we assume they’re being consistent. “Better translation,” he writes, “imposes our logic upon them” (58). A postcolonialist might bristle at this, but the lack of determinate meaning in Quine’s theory of language places a limit on that imposition:

The speaker can be advised in his paraphrasing, and on occasion he can even be enjoined to accept a proposed paraphrase or substitute another or hold his peace; but his choice is the only one that binds him. A foggy appreciation of this point is expressed in saying that there is no dictating another’s meaning; but the notion of there being a fixed, explicable, and as yet unexplained meaning in the speaker’s mind is gratuitous. The real point is simply that the speaker is the one to judge whether the substitution of S’ for S in the present context will forward his present or evolving program of activity to his satisfaction. (160)

To assume that someone is being logically consistent is to impose on them a set of rules for how logical connectives in a language must work. But if we don’t take speech as the transmission of a “meaning” that exists apart from language, we can infer nothing about what they actually believe with these rules. All we can do is infer, on some hypothesis about their purpose in speaking to us, to what other sentences they might give assent, and in this we can assume neither that they are being honest with us nor that their purpose is fixed. It would take more than just a reading of Quine to develop this line of thought fully, but in his work we have the seed of a rational choice assumption that doesn’t actually assume much about what it means for another person to be rational.


In Philosophy on July 14, 2011 at 11:01 pm

Philip K. Dick wrote that “the paranoid is totally rigid”, but there is an element of flexibility to the systems that paranoid people construct that is critical to the systems’ ability to endure.  The more that a belief is rigidly universalizing – “all sheep are white” – the stronger the possibility that we encounter evidence that would force us to revise it.  If we see a sheep that is not white, then the belief that all sheep are white will wither.  The types of beliefs that are central to paranoia, by contrast, are vague in the sense that there is no clear way of falsifying them, and this gives them a flexibility that enables one to adduce more-or-less any stimulation as positive evidence of their truth.

Things get more complicated, of course, once we start thinking about language.  We do not simply encounter a sheep that is not white; we encounter something that meets our definition of sheep that is not white.  One might say that we have two options when faced with the evidence – we can either revise our belief that all sheep are white, or else we can revise our definition of sheep.  If we accept, as I do, Quine’s thesis that there is no “fact of the matter” about analyticity of sentences, then there is actually no clear difference between these two responses.  Is our belief that all sheep are white a part of our mental definition of “sheep,” or is it information we have gleaned from experience about a concept that is defined by other characteristics?  The answer is that this is not a well-formed question, at least empirically speaking.  Every belief we have about sheep contributes, if only in some small way, to what the word means to us; it is only that some of these beliefs are closer to the surface, where the empirical force can more easily affect them, than others.

Thought in these terms, paranoia is a state in which the deep mental connections to a proposition at the center of one’s worldview are too few, or too weak, to avoid being easily altered by new evidence.  The difference between believing that “all sheep are white” and believing that “the toaster is out to get me” is that there is not adequate mental rigging fixing the meaning of “out to get me” in that context, while presumably there is for the word “sheep.”  Because of this lack of fixity, observational evidence no longer serves to improve the paranoid’s beliefs by forcing revision of them.  Instead, it pulls their existing beliefs this way and that.

Always say never

In Philosophy on July 8, 2011 at 2:28 pm

Years ago, I went to a writers’ group in St. Louis, and I commented that the way a story depicted a real-life volunteer organization seemed too much like an advertisement for it. Someone responded, “Let me see if I’ve got this right.  So you’re saying that people should never use the names of real organizations in their writing?” I got annoyed at this because I never said anything like “never.” This has happened to me a number of other times: I’ve made an argument that I meant to apply in a specific context, and to be just one factor among many to consider, and the other person has responded to it as if it were meant as an absolute.

Call this a failure on my part, and I wouldn’t object. But I think my problem is not that I express myself unclearly. I think it’s that I’ve failed to recognize a bias in how people process the statements made in debate. I’m not sure if this bias is universal or just Midwestern stubbornness (I can’t think of any major instances since I left there), but in either case it’s better we try to understand and account for it than complain. What seems to happen is that a person takes statements made with a fixed variable (Pa → Qa), and adds a universal quantifier to them (∀a (Pa → Qa)); then, rather than interrogating the logic by which these statements were reached, they go looking for counterexamples.

Attack the argument, not the conclusion, philosophers say. But the type of discussion to which this (at this point, only conjectured) bias seems keyed is not a terrible one; it’s just more conducive to inductive than deductive reasoning. If we see the positive statements put forth as hypotheses, rather than conclusions, a debate based on the search for counterexamples to universally-quantified claims could be quite effective at reaching the truth.

What this implies is that the way people tend to debate is not optimized for a situation in which multiple perspectives co-exist. An attempt to respect other points of view by speaking only in contingencies would have quite the opposite effect if the listener winds up universalizing the claims one makes; but that universalization would also allow people to hold those claims to stronger, and more objective, epistemic tests. People can’t be expected to be objective in everything they say, but when a conversation is a debate it can come to a gridlock or worse if the participants deal in nothing but particulars.  The truly productive tension is between the particular and the universal.