Daily RC Article 80

Is mathematics a kind of language?


Paragraph 1

What it means to "explain" something in science often comes down to the application of mathematics. Some thinkers hold that mathematics is a kind of language--a systematic contrivance of signs, the criteria for the authority of which are internal coherence, elegance, and depth. The application of such a highly artificial system to the physical world, they claim, results in the creation of a kind of statement about the world. Accordingly, what matters in the sciences is finding a mathematical concept that attempts, as other language does, to accurately describe the functioning of some aspect of the world.

Paragraph 2

At the center of the issue of scientific knowledge can thus be found questions about the relationship between language and what it refers to. A discussion about the role played by language in the pursuit of knowledge has been going on among linguists for several decades. The debate centers around whether language corresponds in some essential way to objects and behaviors, making knowledge a solid and reliable commodity; or, on the other hand, whether the relationship between language and things is purely a matter of agreed-upon conventions, making knowledge tenuous, relative, and inexact.

Paragraph 3

Lately the latter theory has been gaining wider acceptance. According to linguists who support this theory, the way language is used varies depending upon changes in accepted practices and theories among those who work in particular discipline. These linguists argue that, in the pursuit of knowledge, a statement is true only when there are no promising alternatives that might lead one to question it. Certainly this characterization would seem to be applicable to the sciences. In science, a mathematical statement may be taken to account for every aspect of a phenomenon it is applied to, but, some would argue, there is nothing inherent in mathematical language that guarantees such a correspondence. Under this view, acceptance of a mathematical statement by the scientific community--by virtue of the statement's predictive power or methodological efficiency--transforms what is basically an analogy or metaphor into an explanation of the physical process in question, to be held as true until another, more compelling analogy takes its place.

Paragraph 4

In pursuing the implications of this theory, linguists have reached the point at which they must ask: If words or sentences do not correspond in an essential way to life or to our ideas about life, then just what are they capable of telling us about the world? In science and mathematics, then, it would seem equally necessary to ask: If models of electrolytes or E=mc? say, do not correspond essentially to the physical world, then just what functions do they perform in the acquisition of scientific knowledge? But this question has yet to be significantly addressed in the sciences.

Topic and Scope:

Math as language; specifically, whether language is fixed or changing, and the implications of that for using math to convey scientific knowledge.

Purpose and Main Idea:

Well, this is about as dense a passage as has ever come in CAT—the kind of passage for which the concept of gist was invented. The gist of the author’s purpose is to describe the debate over the nature of language, and then to touch upon the impact of that debate on how mathematics (the basis of scientific explanation) can be used and understood.

Paragraph Structure:

The passage gets dense right away, but let’s keep our heads and transform it into something manageable. Paragraph 1 is all about how mathematics is used as the “language” in which science is explained. Math meets the common definition of language, after all, and thus “tells” scientists things when applied to “some aspect of the world”. The phrase “Some thinkers hold...” is the problem here, because those with CAT experience are used to anticipating “But other thinkers believe otherwise,” a contrast that never comes; lines 3-6 turn out to be a sentiment that the author flatly agrees with and uses as evidence. But that’s a trap that should cause only momentary aggravation at worst. By the end of Paragraph 1 we simply need to have gleaned the idea that “math is the language of science.”

Paragraph 2 begins with a helpful sentence, helpful in that it cements for us the topic (“the issue of scientific knowledge”) and scope (“At the center of the issue...[are] questions about the relationship between language and what it refers to.”) Then we do get a “Some argue X/Other argue Y” construct, and it hinges on the issue of whether the meaning of language is fixed, solid, and essential, or whether it’s fluid and dependent upon common agreement. (Even on a superficial level one can begin to see the implications of this debate in terms of Paragraph 1. If language is fluid, and if math is a language, then how can math be used with precision?)

Paragraph 3 also begins helpfully, in that the author announces himself squarely in the second, “fluid” camp. Language in most disciplines, we’re told, depends on what’s going on in the disciplines themselves and will vary over time. Why should scientific “language” be any different? asks sentence 4, and the rest of the Paragraph goes on to reply “It shouldn’t and it isn’t.” The sense of the rest—and it is difficult, no doubt about it—is that math isn’t so much a precise statement, as an imprecise metaphor or analogy that will work until a better one comes along. Unsure about exactly what he’s talking about? Doesn’t matter; we have the gist of it. Let’s plug on.

Paragraph 4 announces the change of scope right away: “the implications of this theory” (line 45). The author implies that in non-science fields, such as literature and history perhaps, a dilemma exists: If words have no fixed meaning, then what can we truly learn from spoken or written language? And by the same token, in science—where math, lest we forget, IS the “language”—what can we really learn from E = mc2, a seemingly precise “equation” which, according to the author’s theory, isn’t precise at all, but a bunch of vague metaphors that may change over time? (This is more or less what we speculated at the end of Paragraph 2. See above.) Anyhow, the author concludes by intoning that science hasn’t begun to explore those questions. Fortunately, we don’t have to explore any of it any further. We just want the payoff (in points) of having mastered this thing.

The Big Picture:

  • “Transformation.” That’s what we need to do with every passage, especially the ones that employ difficult concepts in difficult language. Boil it down to simple terms— transform it into something you can mull over—and don’t worry if you can’t make sense out of every single point. Trust that the passage wouldn’t be on the CAT if it weren’t, on some level, do-able. Hang in there, THINK about the basic ideas, and the rewards will be waiting for you.
  • Always be on the lookout for phrases like “Some thinkers hold...” (line 3). Recognize that they usually indicate that a contrary view is coming up, but also keep in mind that (as in this passage) the author can surprise you.
  • Often when a passage is of greater than average reading difficulty, the editors will see to it that the opening of each Paragraph gives special help. So it is here. Keep that in mind when you encounter a tough passage; expect the 1st sentence of each Paragraph to provide real assistance. Let that be an anchor for your technique and your confidence.
  • Check out, in advance, how many questions a passage carries with it, and let that guide you in your strategic attack on the section. This one only comes with 5 points attached, and certainly one or two of those have to be low-to mid-difficulty, answerable on a pretty superficial level. So attack such a passage with a sense of breezy fun. It’s just not worth getting all bent out of shape over. Even if you absorbed it all—unlikely, given the time constraints—there’d be no real payoff for all that exhausting work. So chill. Concentrate on topic, scope, and purpose, and on paraphrasing the sentences that most communicate the author’s views. And that will be enough for the lion’s share of points.

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