Sunday, 20 April 2014

WFFs, or P versus NP

I was surfing the BBC website the other day when I came across this interesting link: http://aeon.co/magazine/world-views/what-is-left-for-mathematics-to-be-about/


In and of itself the author says nothing new in the article - in fact, it's about one of the oldest arguments about mathematics since the Greeks took it out of everyday, particular world into something abstract and general: what is mathematics? The main premise of James Franklin, the author, says there is a link between mathematics and the real world, and that this link is often overlooked in the discourse of metaphysics of mathematics.


Don't get me wrong. I'm not dismissive of Franklin's article. I appreciate his arguments, and he does say more than is apparent.


But it got me thinking about the similarities and differences between the language of everyday and the language of mathematics. And there are similarities between the two, to be sure. But the are also profound differences: the notion of "closed rings" for instance.


A closed ring, in mathematics, is an important concept that allows orderedness and constraint (ie, goes here but no further) that are so important to why we can trust results and derivations in mathematical operations, or, more precisely, arithmetical operations: for addition and multiplication, the positive, whole numbers are sufficient (a closed ring); for subtraction and division, we need to expand the notion of number to negative and ratio numbers (the positive and the other two "new" notions of numbers, in turn, comprise a larger closed ring for the four fundamental operations of arithmetic); geometry requires more than just that though, because some geometric constructs/relations require the notion of irrational numbers for some of its results to make sense (ie, the square root of two, the ratio between the circumference of a circle to its diameter - to cite just two famous irrational numbers). There are more closed rings beyond these numbers, even unto the "imaginary" numbers which are so useful for engineers and physicists...


But I digress.


To those initiators of our modern notions of mathematics (ie, the ancient Greeks), mathematics, ethics and metaphysics comprised only parts of a larger, more encompassing discourse called, philosophy. Mathematical notions were used to "prove" philosophical arguments, and vice versa. All serious, all the time. The abhorrence of beans as much as the abhorrence of irrational numbers - all fit into the worldviews of some cults that gave birth to august academies.


One of the unfortunate artifacts of this is the contamination of mathematics by unfortunate choice of terminologies - namely, the notion of "truth". In the canon of humanistic corpus, the notion of "truth" is, ironically, the first word of its satanic verse - the father of sophists (ie, postmodern criticism), as far as I'm concerned. The insidious nature of "truth", in this sense, is that it is an undefined assumption and people just never bothered to ask critically. Yeah, eh!


There is another mathematical paper that I tried to read and didn't give up on - just put aside for now if only to contemplate on it and compare it to other works. The paper was written by Max Tegmark and called, Is "the theory of everything" merely the ultimate ensemble theory?


In it Tegmark says a lot of interesting things. But the most interesting one (which makes part of his premise) is a mathematical notion called, a "well-formed formula" (the WFF in the title of this entry) and pronounced "woof" by logicians - I kid you not.


A well-formed formula is an important concept in mathematics in much the same way as a syntactic tree is to linguistics. A syntactic tree - in terms of structural principles - allows you to analyse elements of a phrase to determine whether it is grammatical or not. It makes no claims to the truth of a phrase, and that is its beauty, its strength. It has allowed linguistics to figure out that a well-formed phrase does not necessarily mean that it'll make any sense at all.


For instance, "little, green ideas dream furiously" is a well-formed phrase but makes absolutely no sense at all. It is what is called, a linguist accident. "Linguistic accidents" are very useful things: At the phonological level, it is exploited especially well by marketing agencies to come up with brand names.


Another phrasal accident that has stymied logicians since the ancient Greeks is "this sentence is a lie" (or, "all Cretins are liars"). Linguistics has developed a rather more sophisticated regard of such WFFs than mathematics, and, in fact, has developed a component of linguistic analysis called, Pragmatics.


This analytical component in the linguistics toolbox allows the formal study of how social, cultural and even temporal/geographical context and semiotics contribute to enciphering and deciphering meaning:


In this respect, pragmatics explains how language users are able to overcome apparent ambiguity, since meaning relies on the manner, place, time etc. of an utterance. (http://en.wikipedia.org/wiki/Pragmatics)


Unlike the apparent "timeless profundity" of results/insights in mathematics, everyday language is dynamic, mutable, chaotic, and utterly supple and living-breathingly alive. And, thus, linguistics so.


Surely, conceptual cognates of this nature must exist in mathematics. One would think that this conceptual notion has some bearing upon the P vs NP question, whether a given problem is a waste of polynomial time...an analytical pons asinorum perhaps.


Jay

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