But D, I would not offend you for anything, especially at this time
of the year, but the point I am trying to make is that the cooking
brigade (Bless them! I love good cooking!) seem to insist on treating
2 apples + 2 apples = 4 apples
as the same kind of statement as
2 + 2 = 4
They
are not. The second statement says nothing about anything doing
anything or representing anything but counting --- enumeration, as most
mathematicians like to call it. The moment that you start adding THINGS to the transaction, things that have their own attributes to the
numbers, in such a way that your enumerations are affected by those
things, then we are no longer thinking about the same statement, but
something different. For example, suppose you go to a shop to buy two
sets of underwear, you would not be satisfied to find that each set came
with a man wearing it, would you? Not even if the salesman pointed out
that you had received the exactly the right number of the right kind of
items? Nor would he mollify you by offering to replace them with
underwear on baboons instead of men?
I am sure that mollification would require more sophisticated maths than that! ;-)
Now,
let's see. Eggs. Apples. Nicholas. That sort of thing. That 2+2=4 thing
is maths and purely axiomatic as long as we are speaking of formal maths.
As soon as you start talking about 2eggs+2eggs=4eggs, you change the
game. You are no longer speaking of formal maths, but applied maths, a thing so bastardised that G.H. Hardy refused to regard it as
having anything to do with maths at all. He was wrong of course; he
should have asked me before making such a fool of himself. (Mind you,
since I was not born yet...)
Now the thing about applied maths is
that its axioms must match the behaviour of the system you apply it to.
If I want a certain mass of egg juice, then if the egg masses are not
constant enough, then I cannot trust the calculation of masses from the
munber of eggs. Fair enough. No one said that they did. Certainly no
mathematicians, formal or applied. OTOH if the eggs are pretty well
constant in mass, then the outcome will be close enough. In ordinary
enumeration close enough means exact. No more and no less. Precision is
truth, and Truth is constructed in such a way that it can't be
exaggerated. (Exaggerate it and it no longer is true, right? You can't
improve an over-buttered pudding by adding more eggs, or produce perfect
toast by toasting it till it smokes, then 30 seconds less, even though
subtraction is a valid opeartion on enumerable numbers.)
We can
think of all sorts of examples where Nicholas would use hs nous and put
in 3, 4, or 5 eggs to produce a perfect result in doubling a recipe, but
if he had to produce enough eggs to fill four egg cups, he would no
longer be in any doubt how many to use (if they were hens eggs' and egg
cups for hens' eggs anyway; but I didn't think that we were discussing
silkworm or emu eggs!)
The thing is that now it is enumeration
again. We assume that Nicholas can be trusted to handle eggs without
dropping them (or he would never have stayed the course for 25 years!),
so we could say that:
Because 4 = 4
therefore:
it takes 4 eggs to fit into 4 eggcups.
Now,
my wife would explain to you that if she were silly enough to trust me
into the kitchen, the reasoning woudl go something like this:
Because if Jon is fetching the eggs, all bets are off,
Because 4 = 4
therefore:
it
takes 6 eggs to fit into 4 eggcups with two on the floor or forgotten
on the way, or ready to sit on if you don't check before first look.
But that still won't satisfy the mathematician that 6=4.
He is talking about one thing (enumeration) and you about another (quantification.)
You
would not blink at his enumeration of the eggcups, and he would wonder
(as Nicholas very rightly did) about why one uses enumeration of eggs
when what one cares about is the quantification of egg mass.
And on slightly different bases, one might for example find that enumeration and quantification go slightly wrong if you substitute the same mass of turtle eggs for hens' eggs, or Jersy milk for Friesland milk! These are things that could be very relevant to applied maths but meaningless in formal number theory.
Am I getting warmer?
