"Even while considering the universe to be finite, one can do mathematics
symbolically as a game with a system of rules. If the game doesn't have
enough pieces we just add new pieces, with new properties or allowed
moves as required. All that matters is that the enlarged system is
compatible with the old system; that the smaller game is a subgame of
the large one; that the smaller system can be embedded in the larger
one.
When does something exist ? Well if there isn't something
from amongst the objects under consideration that has the properties we
want then we just create new symbols and define how they relate to the
old ones.
If we were a pythagorean and the only numbers that
exist are rational numbers, then we wouldn't call 2√ a number, but if we
were also finitist symbolicists then we could embed any collection of
numbers into a collection that contains not only numbers but also
"splodges" which is what we're going to call 2√. It's important to
always keep in mind that 2√ isn't a number - it's a splodge. In this new
system of arithmetic we've invented we can add numbers to splodges to
get new splodges like 1+2√. What a fun game. Let's add some more
splodges. We're bored with algebraic splodges so let's add some
non-algebraic splodges like the one in your question. Of course that
expression is a bit cumbersome so we'll give it the shorthand symbol π
instead.
Given a splodge x, it would make calculus easier if
there were a splodge x+o that was nearer to x than any other splodge,
however that isn't possible so we embed the splodges in a larger system
called the hypersplodges that contains not only splodges but also
vapors, and contains not only the concept nearer but also the concept
"nearer". Vapors like x+o are "nearer" to x than any splodge could ever
be, and when you're finished using them they evaporate leaving just a
splodge.
We want a splodge that satisfies x2+1=0 however there
isn't one, so we embed the splodges in a larger system called weirdums
in which we've added a piece called i with the rule that i2=−1, and
under the new system we can "add" splodges to weirdums to get new
weirdums like 1+i.
In solving differential equations we'd like a
function which is zero everywhere except at a single point but which has
a non-zero area under the curve. There is no function that behaves like
this so we'll go to a larger system that contains not only functions
but also spikes which do have the desired property because the larger
system contains a rule about spikes which says they do. Conveniently
certain calculations involving spikes cancel out leaving just functions.
A
finitist or ultrafinitist shouldn't recognise the concept of infinite
sets therefore the only sets are finite-sets and since all sets are
finite the adjective finite is superfluous therefore from this point
onwards we just use the term "set". Some people want to consider sets
that contain things they haven't put in there themselves - which of
course can't be done because a set only contains the items we've put
there. So we embed the system of sets in a larger system that includes
not only sets but also dafties. In this daft system the rules are that a
daftie can have an "affinity" for things whether those things have been
previously mentioned or not. Dafties have an affinity for things in the
same way that sets contain things. A compatible embedding of a system
of sets in the daft system means that a set has an affinity for the
items it contains when the set is considered as part of the daft system,
therefore by daft reasoning one can say things not only about dafties
but also about sets. To each set one can attach a number. You can't do
this with dafties so we embed the numbers in a larger system containing
sinners and attach a sinner to each daftie. Sometimes there is a need
for something that looks like a daftie but has no sinner - such things
are called messes. A mess can have an affinity for collections of
dafties that no daftie could have an affinity for. This could go on, but
you need gobbledegook theory. The set system has zero gobbledegook. The
daft system is level-1 gobbledegook. The messy system is level-2
gobbledegook. Finitists like to maintain a zero level of gobbledegook.
Analysts are usually happy with one-level of gobbledegook and
category-theorists are comfortable with any amount of gobbledegook.
There are two ways to compatibally extend a system:
1) a conservative extension adds new items but doesn't say anything new about the old items that couldn't be said before;
2) a progressive extension does say new things about the old items but only about things that were previously undecidable
P.S.
We can combine the vapors, splodges and linedups in a system called the
messysplodges but they haven't been studied much because they're a bit
messy."
(http://mathoverflow.net/questions/102237/what-is-the-status-of-irrational-numbers-within-finitism-ultrafinitism
by Steve L. Cowan)
