Tertium Organum

what really are the dimensions of space
and 
why are there three of them? 
What must strike us as most strange is the fact that it is impossible to 
define 
three-dimensionality
mathematically. 
We are not clear about this, and it seems a paradox to us, because we 
always speak of 
measuring
space; nevertheless, it is a fact that mathematics 
does not feel
the dimensions of space. 
The question arises, how can such a fine instrument of analysis as 
mathematics is, not feel dimensions if they constitute certain real properties 
of space? 
In speaking of mathematics, it is necessary, first of all, to accept as a 
fundamental premise that/or 
every mathematical expression there is a 
corresponding relation of certain realities. 
If this is absent, if this is not so - then there is no mathematics. Expressing
the relations of magnitudes is the task of mathematics; 
this is its main essence, its chief content. But relations must be between 
something. It should always be possible to substitute some reality for the 
algebraical a, 
b
and c. This is the ABC of all mathematics; a, 
b
and 
c
are 
banknotes: they may be genuine, if they have 
something
real behind them, or 
they may be counterfeit, if behind them there is no reality.
'Dimensions' play here a very curious role. If we designate them by the 
algebraic symbols, 
a, b 
and c, these symbols will have the character of 
counterfeit banknotes: they cannot be replaced by any real magnitudes 
capable of expressing the relations of dimensions. 
Usually, dimensions are designated by powers - the first, the 

---

second, the third. That is to say, if a line is called 
a,
then the square, the sides 
of which are equal to this line will be 
a
2
,
and the cube, the sides of which are 
equal to this square, will be a
3
. 
As a matter of fact this is what provided Hinton with a basis for his theory
of 
tessaracts, 
or four-dimensional solids -
a
4
.
But this is sheer fantasy,
because, in the first place, the designation of dimensions by powers is purely 
conventional. All powers may be represented on a line. Let us take a 5­
millimetre segment of the line 
a.
Then a 25-millimetre segment will be its 
square, or 
a
2
;
and a 125-millimetre segment will be its cube, or a
3
. 
How are we to understand that mathematics does not feel dimensions, i.e. 
that the difference between dimensions cannot be expressed mathematically? 
It can be understood and explained in one way only, namely, by the fact 
that 
this difference does not exist. 
Of course we know that all the three dimensions are actually identical, i.e. 
that each of the three dimensions in its turn may be regarded as 
the first,
the 
second,
the 
third,
or vice versa. This by itself proves clearly that dimensions 
are not mathematical magnitudes. All the real properties of a thing can be 
expressed mathematically as magnitudes, i.e. as numbers showing the 
relation of these properties to other properties. 
In the question of dimensions, however, mathematics seems to see more, or 
farther, than we do; certain boundaries which stop us do not seem to hinder 
mathematics from looking 
through
them and seeing that there are no realities 
to correspond to our concepts of dimensions. 
If the three dimensions 
really
corresponded to the three powers, we should 
have the right to say that only three powers refer to geometry, and that all the 
other relations between higher powers, beginning from the fourth, lie beyond 
geometry.
But we have not even got the right to say that. The designation of 
dimensions by powers is absolutely conventional. 
Or, it would be more correct to say that, from the point of view of 
mathematics, geometry is an artificial construction for the purpose of solving 
problems based on conditional 
data,
probably deduced from the 
characteristics of our mentality.
Hinton calls the system of investigation of 'higher space',