The rigorous
articulation of experience dictates that experience is bounded
by its forms; that is to say, each element of experience is a
form. Particular forms lie in relational fields and are to be
considered a possibility or mode of forms to be connected to a
given one.
Recursive synthesis, simply
described, consists in replacing the many with one, an element
to be used in further syntheses. The synthesized element
is related to the replaced ones through the very relation of
collecting (synthesis). It both ‘hides’ and ‘opens up into’ the
collected elements. The totality therefore is the subset in sum
of all parts that comprise the one.

With multiple
generations, there comes forth the notion of the finite state.
In recursion, this notion of things being finite yields to a
transitive nature. The problem to model is the exact universe of
subsets that do not exceed the range of any conceivable
dimensional assessment. It turns out that the synthetic ‘form’
of the totalities is found in their topological ‘structure’. The
concrete forms of mathematical experience are anything that can
be given a finite alphabet and finite rules of generation; thus,
we have a generative relation or generational construct.

Each part of
mathematics has its own productive rules for the forms, and its
internal relation which make the forms into a relational field.
Thus, the forms in a propositional calculus are its sentences,
generated in the standard manner from the logical connectives
and propositional symbols. Any boundary that exists is always
between an interior and exterior. This is the natural point of
contact with topology and modal logic. The axioms for
topology are defined for ‘parts’ of a topological ‘space’.
We consider chains of the form (f,R) consisting of a form ‘f’
and a relation ‘R’ connecting ‘f’ to other forms.

Sets are
replaced by concepts represented by forms in the given
relational system. The radical standpoint of formal experience
should be contrasted with that of epistemology, which considers
objects of knowledge, not those of experience. The concept of
‘set’ is changed when we move from finite to infinite
collections. The decisive step in pure epistemology is that no
algorithm exists to create what can not be created. Standard
mathematical practice requires that there be a ‘power object’.
The crucial question in set theory hinges on the power axiom;
that is, do we make the power object a set- especially in light
of the fact that set theory requires mathematical objects such
as the set of all Real numbers to be represented as sets.

Axiomatic set
theory cannot provide a full account of the set of real numbers.
There exists the possibility of treating the object as the
totality, in unity, different from the mere algorithm that
created it. This is where set theory fails as a foundation for
mathematics as a consequence of the metaphysical layering it has
transitively extended across finite boundaries to the formalism
of mathematics. This is evidenced in the calculation of a
radical in irrational form to a value that can not be
recursively synthesized to its original form. The reason for
this is very simple. The finite structure of logical space does
not solve the recursively synthetic problem of movement beyond
the limits of reason defined by boundaries.

It must
therefore be stipulated that the collecting process of synthesis
must be grounded on a simple basis which reduces the general
field of such problems. Uniformizaton is essential for
mathematics of recursive synthesis. In order to make collecting
and computing effective, a uniform scale of recursive levels
(powers of 10) was introduced. Products, or conversely
division, are uniform reformulations of syntheses, and uniform
scales of products (powers) make a uniform recursive synthesis
possible. Recursive synthesis is made possible by the condition
that every result of finite collecting is again an element of
the domain. Uniformization consists in the finding the units
which provide the common ground of the things considered.

The introduction
of coordinates then allows for a recursive hierarchy of forms
transformed into a homogenous field of calculus in which
individual operations are simply ‘coordinate-wise arithmetic
operations’. Descartes was very much a part of the merging of
coordinate geometry with arithmetic computation. He understood
that in order to make the movement of thought ‘continuous and
nowhere interrupted’ the parts must be arranged in series with
simple relations connected clearly by replacing real values with
truth values. Boolean operators present the dilemma of
propositional truth as analyzed into the truth distribution
1,0,1,1 over (0,0),(1,0),(0,1),(1,1). Aside from the
analysis there is the truth that it is any combination of (1,0)
then (0,1) that have the highest probability of propagation.
Still classical logic was replaced by binary logic.