Context, Language, and False EquationsDate: 02/12/2001 at 18:03:06 From: Rheanna Subject: Algebra Is there such thing as a "false equation?" My math teacher says no, but a friend and I are trying to prove him wrong by taking a survey. So far yes and no are tied with 4 each. I need you to break the tie. Date: 02/13/2001 at 12:55:49 From: Doctor Ian Subject: Re: Algebra Hi Rheanna, Have you ever heard the story of the Emperor's nose? It's a story about a Chinese man who decides to find out how long the Emperor's nose is by asking a lot of different people... none of whom have ever seen it! The moral of the story is that you can't prove anything by taking a survey. (A modern, real-life version of the story occurred during World War II, when a group of about 100 university professors in Nazi Germany signed a letter saying that Einstein's Theory of Relativity was incorrect. Einstein's response was instructive: "If they were correct, one signature would be enough.") Having said that, let's consider your question. Is there such a thing as a false equation? We can certainly write down an equation that isn't true. For example, there is nothing to prevent us from writing 1 + 0 = 0 Leaving aside for the moment the philosophical issues involved in deciding what '+' means, this is clearly an equation. And it's clearly false. So it's clearly a false equation. Ultimately, an equation is just a shorthand version of a sentence. And a sentence can certainly be false, can't it? I can write: "The current President of the United States was born in 1492." This is a sentence, which happens to be false. Using mathematical notation, we might write the same thing this way: year_of_birth( president( united_states, 2001 ) ) = 1492 This is a false equation, corresponding to a false sentence. There is no big problem with having false equations. The only thing you need to remember is that a false equation, like a false sentence, isn't really useful for anything, except possibly for misleading people. And for constructing indirect proofs. Well, okay, there _is_ one other thing that you need to remember, which is that an equation, like a sentence, _always_ assumes a context. And it's really difficult (perhaps impossible) to write an equation - or a sentence - that isn't true in _some_ context! This is easier to see with sentences. Try to think of a sentence that is clearly false - for example: "Frogs have wings." Certainly toy frogs can have wings, and fictional frogs can have wings, and it's not impossible that a mutant frog might have wings.... So depending on the context of your sentence - which is to say, how you define words like 'frogs' and 'wings' and 'have' - it may be false, or it may be true. Similarly, an equation like 1 + 0 = 0 is false in the context of standard arithmetic, but it's easy to set up algebraic systems in which it happens to be true. So you can't really say that an equation is false without _completely_ specifying the context in which the equation is to be evaluated. What this means is that you can assert that '1 + 0 = 0' is false, and I can assert that '1 + 0 = 0' is true, and unless we've established that we're talking about exactly the same context, we can _both_ be correct. Which raises the question: Is it possible to _completely_ specify a context? Read this story by Lewis Carroll before you answer: What the Tortoise Said to Achilles http://www.lewiscarroll.org/achilles.html This is probably more information than you wanted, isn't it? :^D Believe it or not, I'm trying to let you in on a secret about math, which is that these kinds of conversations about basic notions like 'true' and 'false' can be quite a lot of fun... and very valuable in learning to defend yourself against people who want to manipulate you with subtle forms of lies... and essential for understanding what the limits of mathematics are. (A fun game to play is to take turns with a friend trying to say something that is absolutely true or absolutely false - that is, true or false in _any_ possible context - and then trying to find a context that makes it false or true.) It also turns out that the issue of determining the context in which a sentence (or an equation) is to be evaluated is one of the things that makes it so difficult to get computers to be able to understand natural language. This is because most of what people communicate to each other using sentences (including equations) isn't contained in the sentences at all, but in their shared understanding of the world. Think about the last time you had a conversation like this: "Mom, can I go to the mall?" "Have you finished your homework?" On the surface, the question and response have nothing to do with each other. A complete explanation of _why_ this question generates that response could fill a book. If you heard it at a friend's house, you would have no trouble at all understanding it. But how would a computer make sense of it, without already knowing about school, and homework, and why kids are always trying to get out of doing their homework, and why parents consequently bug them about it, and what people do at malls... ? The central problem of getting computers to understand language, then, has much less to do with dictionaries and grammars than with getting computers to experience the world in the same way that humans experience it, so that they will use similar contexts when evaluating the truth or falsity of sentences. This is a tough problem, a problem that will only be solved by people having conversations like the one we're having right now. I hope this has been helpful, and not too overwhelming. Let me know if you'd like to talk about it some more, or if you have any other questions. - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ Date: 02/13/2001 at 19:56:02 From: Rheanna Subject: Thank You! Thank you for the answer on if a false equation is real. I had never thought about it that way. I really appreciated your work in sending that e-mail. It was not too overwhelming, and it was just what I was looking for. Thanks again! |
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