Number
&
Signs

As mathematics grew from its infancy,

and spread in all manner of strange and unfamiliar directions,

a whole range of different kinds of numbers came to light,

different sets or families of numbers

could be distinguished in an infinite variety of ways.

From the natural numbers with which we count,

first the number line was projected back through zero

to include all of the integers.

But then there was the distinction between the rational and irrational numbers

between the natural numbers,

which along with all the integers are part of the real number family,

along with algebraic numbers

and the transcendentals like pi,

which are real but quite impossible to enumerate completely.

And of course, all real numbers can be distinguished

from the imaginary numbers, situated in a different dimensional plane to the number line,

and from there we can reach the complex numbers,

such as display themselves in the Mandelbrot set.

The world of number seemed to have no constraints,

to stretch to infinity in every direction,

a language in which everything could be defined

extrapolated by the use of a series of logical steps

from a small number of axioms.

But as maths developed,

this Euclidean approach seemed to be growing unfit for purpose,

and when Georg Cantor came up with a theory of sets

to give maths a foundation that could be trusted,

even that came up against Russell's paradox.

Bertrand Russell pointed out

there is always a self-inclusive set

that gives a logical paradox,

as with the tale of the barber,

brought to a village and told to shave every man that doesn't shave himself.

The paradox

assuming that the barber isn't a woman

is Who shaves the barber?

A question that intrigued Kurt Godel

who then developed the Incompleteness Theorem

which bears his name

and that really put the cat among the pigeons.

In it's first part the Incompleteness Theorem states that

In any (appropriate) axiomatic theory,

there exist statements which make sense within the theory

but which cannot be proved true or false within that theory

so no matter how much we would like to completely define any theory by its axioms,

it isn't possible to do this,

and we can always add another axiom.

Then the second part concludes that

It is only possible to prove

that an (appropriate) set of axioms is inconsistent,

and not that they are consistent

in other words we can never be sure that a set of axioms

does not contain a hidden contradiction.

This shook the mathematical world,

detaching maths from its previous assumed position

as the ultimate arbiter of truth,

Mathematics,

from being something that was essentially

linked to the apparently solid objects of the world,

lost that physicality

as it spread through all manner of human experience

and all manner of intellectual exploration,

until maths became what Bertrand Russell later called

the subject in which we never know what we are talking about,

nor whether what we are saying is true.