As you may have surmised from previous posts (see Pegs and Holes and Pascal’s Triangle), I’ve got a thing for the mathematical sciences, and for letting my brain wander. And oh yeah, I’m one of those computer science majors, with a dash of linguist and philosopher thrown into the mix, too. An odd combination, to be sure, but this only means I’m fun to have beers with.I got to thinking (that's trouble) about how one concept is applicable to all of these realms. Let's go one at a time:
Mathematics
Even if you haven’t studied it ad nauseum like I was forced to in college, you’re probably familiar with something called the inverse property, or an inverse function. You would agree that 5 + 1 = 6, and 6 – 1 = 5. It’s a simple example: adding a number is the opposite, or inverse, of subtracting it. One step further, if you take a number, do something to it (like add one), then do the opposite to it with the same value (like subtract one), you end up with what you started with.
This idea can be generalized to all sorts of stuff. Muliplication/division, square/square root, any combination of these, yada yada yada. It’s actually a pretty powerful concept once you start studying it closely, but this idea can be applied outside of the mathematical realm.
Linguistics
Suppose you’re translating a phrase from one language to another. For instance, how do you say “I am tired of reading this crap” in Italian? Assuming Google Translate is correct (I’ll get back to this later, by the way), that’s “sono stanco di leggere queste stronzate.”
Language translation ideally should satisfy the inverse property. If I translate the above Italian phrase back into English using the “inverse operation” (Sorry, I’m getting all math-y on you again.), then I should get my original phrase back. But this is not necessarily the case: languages are not as clear cut as logic and math. As any high school foreign language teacher would tell you, there’s more than one way to translate phrases into English, and vice-versa.
Software
But what if you do attempt to treat the translation of languages as an “inversable” operation? TranslationParty.com is a nice little web application that attempts to do just this. Using Google Translate to do the gruntwork, it will translate any phrase you wish into Japanese, and then back to English. Trouble is, you’ll likely end up with something different from what you started with. So for grins, it’ll translate it again into Japanese and back into English, and will keep doing this until it gets two of the same English phrases back-to-back and, as the website calls it, reaches equilibrium.
Example: shoving “The quick brown fox jumps over the lazy dog” into the translation machine will move back and forth through the language barrier until it reaches this equilibrium, eventually ending up with the English phrase, “The lazy dog jumps to a simple brown fox.” Totally butchered. (For the record, in Japanese, that’s "シンプルな茶色のキツネに怠惰な犬にジャンプします。").
(This write-up on TechCrunch.com has a slightly more hilarious example, but I just couldn’t bring myself to add a Star Wars reference to this article. I’m thinking it’s nerdy enough as it is.)
There's a disconnect here. The laws of logic (and software) don't mesh with the laws of language. If they did, these translations would be seamless and unambiguous. But there's an interpretive side that logic just can't grasp.
I just blew your mind.
6 comments:
Nice post!
But perhaps it's the other way around... imprecise concepts such as natural language are ill-suited for expressing the rules of logic.
And not all languages are created equal - not all (if any) can express every possible idea. The same is true rolled up one level of abstraction to communication mechanisms in general.
I don't think it's possible to communicate an idea without some form of language, so logic itself can't be thought about without aid from some form of language.
And that brings up an interesting question - how can we be sure of logic/mathematics since we are confined to use imprecise language to explore it?
My thoughts on inversion of natural languages:
Suppose I have a translator "f" that can translate an idea "x" from one language "L1" to another language "L2". Also suppose "g" can translate "x" from "L2" to "L1".
In mathematics, both "L1" and "L2" are guaranteed to be capable of expressing "x". It seems reasonable to say, then, that both "L1" and "L2" actually belong to the same language, namely that of mathematics.
And mathematical operations that stay within the boundaries of the language of math represent a different class of idea manipulation than do natural language operations that span multiple languages.
Also, not all mathematical functions have an inverse.
Also, you'd like "I Am a Strange Loop":
http://www.amazon.com/Am-Strange-Loop-Douglas-Hofstadter/dp/0465030785
There's a considerable number of pages devoted to what happens when you translate between the language of formal logic and natural language.
Interesting stuff. Ok, I'll stop rambling in your comment section now :)
GREAT question about our use of imprecise language to explore logic/math. We always try to think "outside the box", but our tiny little brains only have a particular way to approach logic/math, and its so indelibly embossed that there are some things we'll never be able to learn.
As a math minor, it took a while to unravel some of the concepts we had learned so that we could wrap the laws of mathematics around bigger, better, more generalized concepts.
bryan says "Not all mathematical functions have an inverse."
In case you're curious (because I was, and I'm rusty), the square operation (y=x^2) is not invertible. Why? Because given its output, it's impossible to deduce the sign of its input. (If y = 9, does x = 3 or -3?)
Former philosophy major. Just stumbled across your blog at work. Here's my take:
I think what you're saying kind of demonstrates that there are some inconsistencies in the language we use to describe the world. One problem along similar lines comes from our "theories of reference."
For example, the town Dartmouth technically refers to the town at the mouth of the river Dart. But, what if the river were to change its course such that the town is no longer at its mouth? Would it still be Dartmouth? What if a new town were built at the new mouth?
The gist is - in naming the town 'Dartmouth' are we referring to that town which is at the mouth of the river? Or, are we referencing some metaphysical concept of a town which happens, at the time of naming, to be located at the mouth of the river?
Problems like this demonstrate, some would argue, a fundamental flaw in our systems of language. See Kripke, Naming and Necessity.
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