Friday, January 1, 2010

Math - Grammar - Law

Geometry tends to bore people, for example, me. It might be interesting in one way though. There's a connection we feel between Math, Grammar, and Law (within this essay, "MGL") that emerges when we do Geometric proofs. I don't think there's any other subject where there's a convergence like this. What allows them to work together appears to be the rule structures underlying each concept.

Familiarity, intuitive use
Most of us who are not professionals in MGL fields still operate with them enough to have a feel for them. In Math most of us recall that different theories seem to apply to different types of problems, for example, commutativity and additive inverses. Without knowing the names of the rules, we understand that 3 - 2 is not the same as 2 - 3, but that 2 + 3 and 3 + 2 are equal. In Grammar, we know that adverbs describe actions, but even if we've forgotten the name of that rule, it feels wrong if we mistakenly use an adverb to describe a noun. Eg, "An unused heavy weight makes a good doorstop." looks correct, as compared to "An unused heavily weight makes a good doorstop". In Law, most of us understand that, if contracts are broken, there might be a lawsuit, whether or not we happen to know the applicable law. So in each of these areas, most of us operate out of habit without needing to consult the specific MGL rule that applies. In Geometry, when we make proofs, we have to be explicit about only the Math portion, but there are other laws at work.

Grammar is determined by usage and social convention. In the case of US English, media status and academia tend to promote one form of usage into Standard English and Received Pronunciation. These then normatively reinforce that usage over others. Outside of this politicking however, grammar "rules" are descriptive, not normative. they are descriptions of what we observe across languages. One of the potential hiccups is agreeing on the linguistic terms. Linguistic terms are themselves words -- defined partly by usage -- so that they risk a circularity of using themselves to define themselves. We break the circle fairly effectively by first having a conversation about what our linguistic terms refer to, a meta-conversation, attempting to solidify what our discussion will point out in a language before we start examining languages. Outside of Linguistics study, in the world. language operates without bounds. Inside Linguistics, that is, while studying worldly language effects, we want own words to point to agreed-upon concepts.

As an example, let's suppose we're Linguists who agree about the meaning of the word "case", insofar as language is concerned. Using this definition, we're able to observe the number or nature of cases across languages in way we both understand. In German, we could agree there are likely at least 4 cases; the nominative, accusative, genitive, and dative cases. Observing English, we might agree that typical English usage does not split these apart so neatly. The language rules society follows come through usage, the rules we use to describe them must come through academic agreement. In one sense, Grammar is a posteriori, but the study of Grammar requires a priori definitions.

With Math, we believe we establish rules based on logic first, and experience second. So, the trajectory is deductive, we start with principles and build logically from that point to conclusions within that logic, or must expand it. The basis is a priori, but the expansion of axioms to encompass new information is a posteriori. Of course, Kant considered Mathematics "synthetic" for this blend. But what about our meta-conversation in Math, like the one we had in Linguistics? Isn't it true that we must first have a conversation about what Math words mean and agree upon what they point to, before we subsequently do that Math? For the part of Math that is arithmetic, that uses symbols, such as 2+2=4, it's relatively easy to agree because there are quantities these point to, not just concepts. We can place two items in front of anyone in any language and, once it's clear we are discussing a quantity, need only agree on the symbol. So for quantitative Math, there can be agreement about the language. But Math is not all quantities, it's also performing operations on quantities and applying theorems to those quantities, and these operations and concepts may need clearly agreed terms to discuss.

With Law, the statuatory scenario is a priori; the law is true by its definition as declared, not through observation (inductively). In this way it is like the first discussion we had about Linguistics, we had to first agree what we meant by the words we were going to use with the However, with precedents, the law also allows for interpretation which can affect its implementation in an a posteriori sense.

Geometry connection
One thing that seems to combine all three of these is Geometry. Geometric proofs utilize deductive logic, the obviousness of quantities and graphical constructs, and clear definitions. Perhaps Geometry is not so boring after all.

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