Thanks to everyone who entered, or who just emailed me with their thoughts about math. If I didn't acknowledge your communication, I apologize. Things got sort of crazy there for a couple of weeks.
Kids Winner:
Dear Mr. Sartwell,
Hi my name is Jenna and I am writing to you to explain the number seven. Seven is the name for my finger. I named my fingers one, two, three, four, five, six, seven, eight, nine, and ten. When I add, subtract, divide, and multiply I use my fingers and their names. This makes math a lot more easier. I think math is very helpful in everyday life and should not be taken out of schools. It may seem confusing at some times but is very useful in the future. I am very sorry that you do not understand the number seven, but as I told you before, seven is the name of my finger. I hope you understand seven a little bit better now and you live a successful life and excel in your math skills.
Sincerely,
Jenna
jenna, hi. you are the kids' winner in my contest to define '7'. i really like the idea that 7 is the name of one your fingers. my only question is: where was the number 7 before you were born? if you want to send me your address i will send you some (smallish) prize! look at my website to see the adult winners and your own little essay, with my comments. thanks!
crispin sartwell
Dear Mr. Sartwell,
Hi, I would like to say thanks for choosing my entry for the winner of your contest. Before I was born, the number 7 was the name of my mom's finger. The name 7 is passes down from generation to generation in my family. When I am grown and have a child of my own, then I will pass down the name 7. Thanks again for choosing my entry.
Sincerely,
Jenna
Comment: "seven" does in many ways seem to operate in the language like a name. If it is a name, however, the big problem is to figure out what the heck it's the name of. I like the idea that it's the name of Jenna's finger, because at least that's a real thing. I personally feel that there is going to have to be an empirical account of the meaning of mathematical notions, though the barriers to such a thing seem at first glance to be immense.
Adult winners:
entry 5
integers can be defined recursively. Zero is the cardinality of the empty set. One is the
cardinality of the set containing zero. Two is the cardinality of the set containing zero and one.
Symbolically, 0={}, 1={0}, 2={0,1}={0,{0}}, et cetera. Each integer is defined as the
cardinality of the set containing all its predecessors. Thus, 7={0,1,2,3,4,5,6}.
david ross
entry 22
As one of your previous contestants has pointed out, and as you were presumably aware before
announcing your contest, definition is a hopeless goal without an agreed-upon list of undefined
terms from which to begin. Likewise, truth is not demonstratable except in relation to certain
unchallenged axioms. These facts are not a curse to mathematicians, but a double blessing. First,
we have made a standard list of undefined terms and axioms: it is astonishingly short and simple,
and the conclusions drawn from them are boundless. I suspect that mathematics is the only field
of human endeavor in which careful thought has been given to these questions. Second, we gain
generality: any concrete interpretation of the undefined terms, shown to satisfy our axioms, is a
valid model, and all the conclusions of mathematics apply to that model (whether or not they are
"interesting" in that context). So your mental construct of mathematics is as good as mine!
Mathematics, the eternal "if-then," acts like a steel-jawed trap: if you step into it with a model,
and trip the switch by verifying the axioms, the jaws snap shut on you and force you to accept
conclusion after conclusion. Enough philosophy -- on to seven. Seven is a basic construct, on the
level of set theory, as many writers have suggested. Indeed, seven is so basic that its definition
can be typed in one (non-recursive) line, without using any but the standard undefined terms of
mathematics: 7 = {{{{{{{{}},{}},{}},{}},{}},{}},{}} The brackets enclose sets. Some of these
are empty, and others contain sets. The active principle is the distinction between the empty set
{} and the set {{}} containing the empty set as its single element. There are 7 objects in a set if
and only if there is a 1:1 correspondence between that set and the set {1,2,3,4,5,6,7}, or, in the
present spirit,
{{{}},{{{}},{}},{{{{}},{}},{}},{{{{{}},{}},{}},{}},{{{{{{}},{}},{}},{}},{}},{{{{{{{}},{}},{}},{}},{}},{}},{{{{{{{{}},{}},{}},{}},{}},{}},{}}}. It is therefore (by functional
composition) totally logically sound to count on one's fingers, as long as one uses the right
number of them. Finally, I would think that a professor of philosophy would be embarassed to
conclude from a lack of contestants that everyone "had no idea how to say what `seven' meant."
In PHIL 120, we learned that arguing from ignorance is a fallacy. It is a wonderful thing to be a
mathematician -- one needs only a pencil, paper, and a wastebasket. But it is far better to be a
philosopher -- one needs only a pencil and paper.
James Swenson
Comment: Contrary to Swenson, these are both (I think) recursive definitions, at least it seems
his forest of brackets arises through a recursive accretion. Now I have the feeling that both of
these definitions are circular. But though in some cases the definitions have been obviously
circular, in these cases the circularity is at least elusive. Those who know something about this
assure me recursive definition is notr circular, but I'm not the only one who thinks they have a
funny smell. Ross's definition also makes use of the concept of "cardinality," and it does seem
that defining seven as the cardinality of a set with seven elements is circular. That's also not
exactly what he said. So at a minimum I can't put my finger precisely on the problem. To the
extent that Swenson's def depends on a 1:1 correspondence between his sets and the first 7
integers, it is baldly circular. But it is not clear to me that it does rely on that. Even if it does, the
answer could be filled out by recursion.
The recursive definitions suggest an ontology of mathematical notions: that they are stipulations created as a system of relations.
Yipes! So I am not convinced that mathematics is not a bizarre pagan cult, worshiping entities of which it has no conception, bizarre gods who will finally be overthrown. But I'm slightly more confused than before, anyway.
