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The nature of our Universe imposes certain limitations on human mathematical
abilities:

1. one cannot perform infinitely many consecutive actions in finite time;
2. one cannot perform infinitely many independent actions in parallel.

These two limitations put together, imply that whatever can be done,
can equivalently be done by a Turing machine. Here lies the main cause of
various incompleteness and undecidability results in mathematics.

There are further superficial limitations; like finiteness of human life,
and the necessity for a human to understand the meaning of what one's doing.
These secondary limitations are gradually being overcome by extensive
use of computers; and by conditioning humans to be able to take interest
in whatever needs to be done, be it even writing suchlike texts; the latter
conditioning being achieved by higher level social structures.

No matter how severe and unsurmountable these secondary limitations be; and
how spectacular be our attempts to overcome them; the basic restrictions (1)
and (2) stand unchallenged, albeit in our Universe only.

So, let us speculate on other Universes, free from Restrictions (1) and/or (2),
and on the shapes mathematics would take in those Universes. It goes without
saying that the absolute world of mathematical objects is one and unique, and
does not depend on the physical nature of various Universes, or the abilies and
disabilities of mathematicians therein.

To clarify this point, an example is in order.

Imagine a species, call it A, whose mathemathicians have no idea of natural
numbers greater than 20. For them, the question of whether 13 plus 11 is
odd or even would be unsoluble; all answers like "odd", "even", "we'll never
find out", "this question is meaningless", - would be equally acceptable.

Imagine another species, say B, who can grasp the idea of natural numbers,
but whose short attention span prevents them from counting beyond 20. For them
the same question is meaningful: they know that the answer is either "odd"
or "even"; and they might hope of eventually producing a genius who would
find a definite answer.

Proud of being superior to both these imaginary races, let us turn around and
look in the opposite direction.

Recall two famous mathematical problems: the (3n+1)-problem, and he Continuum
Hypothesis.

The first one deals with a seemingly simple function on natural numbers:
f(n)=3n+1 when n is odd, and f(n)=n/2 when n is even. The question is: what
happens to a natural number when you iteratively apply to it the function f?
The conjectured answer is that, no matter where you start, eventually you
reach the value 1 (and then the sequence repeats itself as 1,4,2,1,4,2,...).
This was confirmed for quite a few starting values but remains not proved
in general.

The Continuum Hypothesis (CH) states that the next largest infinite set
after the set of of natural numbers is the set of all real numbers. Its
situation is quite different: it is proved that in different models of set
theory CH can be either true or false, and it is also proved that it's
impossible for human mathematics to decide whether CH is true in "reality",
or even to decide whether this notion of "reality" has any definite meaning.

Now, mathematicians of a species not chained by Restriction (1) would not
consider (3n+1)-problem as challenging; for them it would be a simple exercise.
To these hypothetical creatures, we would look as pathetic as the species B
looks to us.

Those restricted by neither (1) nor (2) might find Continuum Hypothesis
just as easily verifiable (or refutable, as the case may be). For them, our
feeble attempts would be as pitiful as the pointless speculations of the
race A for ourselves.

The point of this exercise in imagination is this: as 13+11 is meaningful, and
is even, despite any beliefs and ignorances of races A and B, so we must
assume that neither our shortcomings imply anything about the absolute world
of mathematical objects.


Prodolzhenie, mozhet byt', posleduet.