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I want you to read the following statement twice, slowly, before continuing:
How does that statement sound to you? It’s a stand-alone sentence, isn’t it? There is a subject (2+2), a verb (=), and an object (4). Sometimes this realization comes as a surprise: who thinks of a mathematical equation as a sentence? In mathematics, as in every other subject, clear (and pleasant!) communication is crucial. I am aware that mathematicians come off as stuffy and quite disconnected from reality—and sometimes for good reason—but the fact of the matter is that we quibble among ourselves about writing just like poets fret about the final stanza. We want others to read what we’ve written, after all, and enjoy it!
In this sense, mathematics is a creative endeavor not dissimilar to writing fiction or even turning pottery. We are artists who wish to convey a beautiful vision, and how to best convey that vision is a subject not lightly undertaken. In general, our vision includes “proving” abstract statements; thus part of our work is to write a convincing and logically correct “proof”. These mathematical proofs are not courtroom “beyond reasonable doubt” proofs, nor are they scientific “we can do it over and over again” proofs. The idea is probably best conveyed in an example.
Consider the question “Is the sum of two even numbers always itself an even number?” If the answer is “Yes” (and the answer is indeed “Yes”), a proof must tell you precisely why that is always the case and leave no room for doubt of any kind. In many cases, the tools for constructing a proof include applying definitions or previously-proven results to the problem at hand. For this specific question, we need to know the definition of an “even number”, and we need to know the previously-proven fact that multiplication “distributes”.
Certainly you remember the even numbers, stemming back to a time where your parents asked you to “count by twos”; they are 2, 4, 6, 8, 10, etc. Mathematically, the even numbers are numbers that are 2 multiplied by an integer. The integers are those first counting numbers you learned, 1, 2, 3, etc., along with 0 and the negative numbers -1,-2, -3, etc. Thus the following are even numbers, equated to 2 times an integer to illustrate the definition:
The other necessary fact, that multiplication distributes, is illustrated by the following examples:
In the first line, notice that the 2 that is multiplied by (1+3) “distributes” its multiplication to 1 and 3; and similarly for the second line (recall also that writing 2(3) simply means “2 times 3”).
We’re now ready to construct the proof. First, think about the goal: we want to make a statement about the sum of two even numbers…any two even numbers. So, let’s just pick two of them randomly: call them a and b. Now remember that all even numbers look like two multiplied by an integer, so we can rewrite a and b as the products 2x and 2y, respectively. Now add them:
If we apply a sort of “reverse distribution” from what we illustrated above, we can write
Now stop and ask yourselves what (x+y) is. It’s just another integer, right? And 2(x+y) is therefore an integer times 2, which is even. Proof complete! We have shown that no matter what two even numbers we add together, the sum is also an even number.
The preceding exposition, nice as it may be, is not “written mathematically.” A formal mathematical proof would be written like this (read it as a paragraph!):
Let a and b be two even numbers. Then a=2x and b=2y for some integers x and y. Taking their sum, we find
Since a+b is an integer, 2(a+b) is an even number. We conclude that the sum of two even numbers is always an even number.
Given the difference between the formal proof and our initial formulation, it is not surprising that non-mathematicians find reading mathematics hard. Formal mathematics is concise, and often written with shorthand notation. For example, we could rewrite that same proof as follows:
Let a,b∈E. Then ∃x,y∈Z such that a=2x and b=2y. Note,
Believe it or not, a mathematician reading this proof aloud would make is sound just like he was reading a well-written paragraph—just like the statement 2+2=4 can be read as a sentence.
As far as the writing itself is concerned, the appropriateness of a proof depends on the audience. In a number theory course, the “notation” proof would be preferred, while the first proof may be given in a proof-writing class (and yes, such classes exist!). The fact that the audience matters even extends to the computations we turned in as high school assignments: there, “content” was all that mattered and no thought was given to “style”. Beyond calculus, style and audience considerations are imperative for proofs; as in the examples given, there are often computations which need to be embedded into the proof itself.
If you’re still not convinced that writing is taken as seriously in mathematics as it is elsewhere, consider the following comment from a recent assignment for a graduate-level abstract algebra course. I used the word “whence” just before stating the conclusion of my proof, and the instructor responded with:
You may be amused to note that my wife and I ended up having a bit of an argument over your use of 'whence.' The short version of which is that I was concerned that you should more properly be using 'hence' in this context and she being concerned that it is a pretty archaic word at this point and should probably be avoided (in other words, we talked past each other badly).
A search of the web yielded the usual collection of grammar Nazis arguing that you should never say 'from whence' since the 'from' is redundant, however the pairing 'from whence' occurs frequently in written English going back to the 13th century, so while it may be a logically redundant usage, insisting that English usage be logical is probably an act of pedantry. There's also a pretty neat debate about this at [http://english.stackexchange.com/questions/109882/when-to-use-whence-instead-of-hence]... None of which helps any of you, of course, except to say it is probably correct here, but a little old-fashioned.
Certainly this exposition illustrates that mathematicians do more than stare at blackboards while smoking pipes. As I asserted earlier, mathematicians really are artists in disguise—the patterns underlying their work are simply hidden in notation and jargon. Like editors, mathematicians sometimes quibble long into the night…
K is currently a graduate student in mathematics at the University of Alaska Fairbanks. When he fantasizes about free time, K pictures himself reading classic novels, fishing, writing poetry, and visiting his friends and family back home in Wisconsin. If K ever finishes school, he'll probably teach math and science.
I've never thought of mathematics as a creative endeavor!!! As a side note, I once was part of a group of people formulating a job description for the State of TN. It was an OFFICIAL job description. A roomful of IT professionals, many of whom were directors, shot me down when I tried to tell them mathematics had an "e" in it. The description went on to the next phase with it spelled "mathmatics." I don't know if the final version had the e...I left soon after because I had a book deal with a major publisher and a freelance writing business, but I wonder to this day if they ever realized they were ALL wrong. Probably not!ReplyDelete
Looks good! The proofs could be in block quotes, but I think it's easy enough to see where they start and stop. Thanks for the opportunity to post!ReplyDelete