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definitions_in_mathematics [2013/08/20 16:36]
bmwoodruff 		definitions_in_mathematics [2013/08/20 16:46] (current)
bmwoodruff
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 Mathematical definitions are //prescriptive//. The definition must prescribe the exact and correct meaning  of a word. Contrast the OED's descriptive definition of continuous with the the definition of continuous found in a real analysis textbook. Mathematical definitions are //prescriptive//. The definition must prescribe the exact and correct meaning  of a word. Contrast the OED's descriptive definition of continuous with the the definition of continuous found in a real analysis textbook.
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-A function $f:A\to \mathbb{R}$ is //continuous at a point// $c\in A$ if,  for all $\varepsilon>0$, there exists $\delta>0$ such that whenever $|x-c|<\delta$ (and $x\in A$) it follows that $|f(x)-f(c)|<\varepsilon$. If $f$ is continuous at every point in the domain $A$, then we say that $f$ is //continuous on// $A$.((This definition is taken from page 109 of Stephen Abbott's //Understanding Analysis//, but the definition would be essentially the same in any modern real analysis textbook.)) +A function $f:A\to \mathbb{R}$ is //continuous at a point// $c\in A$ if,  for all $\varepsilon>0$, there exists $\delta>0$ such that whenever $|x-c|<\delta$ (and $x\in A$) it follows that $|f(x)-f( c )|<\varepsilon$. If $f$ is continuous at every point in the domain $A$, then we say that $f$ is //continuous on// $A$.((This definition is taken from page 109 of Stephen Abbott's //Understanding Analysis//, but the definition would be essentially the same in any modern real analysis textbook.)) 
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 In mathematics there is very little freedom in definitions. Mathematics is a deductive theory; it is impossible to state and prove theorems without clear definitions of the mathematical terms. The definition of a term must completely, accurately, and unambiguously describe the term. Each word is chosen very carefully and the order of the words is  critical. In the definition of continuity changing "there exists" to "for all," changing the orders of quantifiers, changing $<$ to $\leq$ or $>$, or changing $\mathbb{R}$ to $\mathbb{Z}$ would completely change the meaning of the definition.  In mathematics there is very little freedom in definitions. Mathematics is a deductive theory; it is impossible to state and prove theorems without clear definitions of the mathematical terms. The definition of a term must completely, accurately, and unambiguously describe the term. Each word is chosen very carefully and the order of the words is  critical. In the definition of continuity changing "there exists" to "for all," changing the orders of quantifiers, changing $<$ to $\leq$ or $>$, or changing $\mathbb{R}$ to $\mathbb{Z}$ would completely change the meaning of the definition. 
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 Of course, memorization is not enough; you must have a conceptual understanding of the term, you must see how the formal definition matches up with your conceptual understanding, and you must know how to work with the definition. It is perhaps with the first of these that descriptive definitions are useful. They are useful for building intuition and for painting the "big picture." Only after days (weeks, months, years?) of experience does one get an intuitive feel for the $\varepsilon,\delta$-definition of continuity; most mathematicians have the "picking-up-the-pencil" definitions in their head. This is fine as long as we know that it is imperfect, and that when we prove theorems about continuous functions mathematics we use the mathematical definition.  Of course, memorization is not enough; you must have a conceptual understanding of the term, you must see how the formal definition matches up with your conceptual understanding, and you must know how to work with the definition. It is perhaps with the first of these that descriptive definitions are useful. They are useful for building intuition and for painting the "big picture." Only after days (weeks, months, years?) of experience does one get an intuitive feel for the $\varepsilon,\delta$-definition of continuity; most mathematicians have the "picking-up-the-pencil" definitions in their head. This is fine as long as we know that it is imperfect, and that when we prove theorems about continuous functions mathematics we use the mathematical definition. 
  
-We end this discussion with an amusing real-life example in which a descriptive definition was not sufficient. In 2003 the German version of the game show //Who wants to be a millionaire?// contained the following question: "Every rectangle is: (a) a rhombus, (b) a trapezoid, (c) a square, (d) a parallelogram." +We end this discussion with an amusing real-life example in which a descriptive definition was not sufficient. In 2003 the German version of the game show //Who wants to be a millionaire?// contained the following question: "Every rectangle is: (a) a rhombus, (b) a trapezoid, <nowiki>(c)</nowiki> a square, (d) a parallelogram." 
  
-The confused contestant decided to skip the question and left with \euro 4000. Afterward the show received letters from irate viewers. Why were the contestant and the viewers upset with this problem? Clearly a rectangle is a parallelogram, so (d) is the answer. But what about (b)? Is a rectangle a trapezoid? We would describe a trapezoid as a quadrilateral with a pair of parallel sides. But this leaves open the question: can a trapezoid have //two// pairs of parallel sides or must there only be //one// pair? The viewers said two pairs is allowed, the producers of the television show said it is not. This is a case in which a clear, precise, mathematical definition is required.+The confused contestant decided to skip the question and left with \euro FIXME 4000. Afterward the show received letters from irate viewers. Why were the contestant and the viewers upset with this problem? Clearly a rectangle is a parallelogram, so (d) is the answer. But what about (b)? Is a rectangle a trapezoid? We would describe a trapezoid as a quadrilateral with a pair of parallel sides. But this leaves open the question: can a trapezoid have //two// pairs of parallel sides or must there only be //one// pair? The viewers said two pairs is allowed, the producers of the television show said it is not. This is a case in which a clear, precise, mathematical definition is required.
  
  
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 ====Tags==== ====Tags====
-{{tag>appendix dana needsreview }}+{{tag>appendix dana needsreview rben}} 
 + 
 +====Comments==== 
 +Dana, like the expository approach to explaining why definitions are so crucial. My only concern is that this might be too long.  
 +  *Your 2nd and 3rd paragraphs could probably be combined and halved in length, when you compare and contrast the two definitions.  
 +  *Your 4th and 5th paragraph might be shortened as well.  I'm guessing that if a student knows enough about epsilons and deltas to make the changes you describe there, they probably already know a lot about definitions. 
 + 
 +Ben
definitions_in_mathematics.1377030972.txt.gz · Last modified: 2013/08/20 16:36 by bmwoodruff

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