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FOUNDATIONS OF MATHEMATICS
ABSTRACTIONS POSTULATE
In "STRUCTURES OF MIND" we have defined the Abstractions
Postulate (AP) as condition of meaningfulness of abstract
constructs:
**
Abstract, symbolic constructs may be justified solely by
their capacity to coordinate events which represent their
unique meaning and justification, where coordination of
events implies considering them in their context, i.e.
upon their background of continuum.
**
Mathematical theories are abstract structures and, as such,
may get their meaning and justification only by being founded
in AP. Yet, this criterion, although necessary, is not
sufficient to distinguish mathematics from other rational
abstract structures also founded in AP. Therefore we shall
start by trying to define mathematics before discussing
-foundation of mathematics in continuum,
-falsifiability of mathematical axioms,
preceded by a brief revue of the concurrent foundational
crisis.
MATHEMATICS
Precise lexicographic definition of "mathematics" is
unattainable for the following reasons:
By virtue of AP, words and other symbolic linguistic constructs
are not known or meaningful by themselves, but point to their
underlying events which embody their meaning.
Dictionaries try to define them either with help of presumably
better known synonyms, or intensionally - by giving a superclass
and specific characteristic, defining for instance a bicycle as
a two-wheeled velocipede.
Now, mathematics does not have a better known synonym and
is too general and too vast to admit a precise superclass
and specific characteristic.
As all generalities it admits only a vague extensional
definition by enumerating some of its typical subclasses and
some characteristics shared by them. Thus, mathematics is,
or consists of, geometry, arithmetic, algebra, algebraized
geometry, calculus, vector and tensor calculi, topology, etc.,
dealing with eventtual qualities of shape, quantity and
order, founded in the Abstractions Postulate and axiomatic
in the sense of "DOGMATIC THEORIES AND AXIOMATIC MODELS"
based upon the ERN Logic.
One may perhaps imagine arbitrary, unfounded chains of
symbols and call it "mathematics", but we dismiss it and
reserve the term "mathematics" for abstract structures
compatible with the above extensional definition.
Review of the foundational crisis
The crisis of foundations of mathematics is universally
recognized and seen as the conflict of three mainstream
views: Goedel's pseudo-platonic reifications, Brouwer's
pseudo-intuitionism and Hilbert's games with arbitrary chains
of characters. However, they all share the foundations in
discreteness embodied in the set theory.
Fraenkel honestly laid the crisis at the door of the very
concept of the set theory, rather than blaming internal
quarrels among its shades.
As we have shown in "SET THEORY", the crisis manifests itself
-by innumerable versions and shades of the set theory, all
equally inconclusive,
-by the failure to define the most fundamental concepts of
the set theory, such as "set" and "number", in spite of over
100 years of efforts of the logical establishment, including
Cantor, Zermelo, Fraenkel, Russell, Frege, Quine, Church,
Tarski, Goedel, Hilbert, Brouwer and innumerable others,
-by the incongruous concept of continuum and fallacious
procedures supporting it. As continuum is a crucial issue of
the present chapter, we shall review the way the set theory
deals with it.
Set theories define "continuum" as the set of real numbers.
It certainly has nothing to do with the intuitive continuum
of time/awareness which we posited in "TIME, AWARENESS AND
EVENTS" as the ultimate foundation of the human universe,
nor with the infinite continuous space of physical reality
which we defined in "NATURAL MODEL". In order to distinguish
the set theoretical numeric gimmick from the ontological
foundation of human universe and physical reality, we shall
put the "continuum" of real numbers in quotes.
Let's note, by the way, that "set" and "number" staying
as yet undefined, the definition of "continuum" as the
"set of (real) numbers" does not have a leg to stand upon.
The "continuum" is supported by the famous "Continuum
Hypothesis": Calling A0 and A1 respectively the transfinite
cardinals of sets of natural and real numbers, Cantor proved
that A1 is greater than A0 and postulated the "Continuum
Hypothesis" stating that there is no set whose cardinality
falls strictly between A0 and A1.
Poincare considered transfinity as a disease and Kronecker
as scientific charlatanry. We cannot but agree with them
and see the "Continuum Hypothesis" together with its
underlying transfinity as delusional humbug.
Ontological foundations
In the next section we postulate that mathematics is founded
in continuum which is the cornerstone of Second Enlightenment's
ontology - the Relativistic Dialectic (RD). It seems therefore
advisable to justify RD as the pertinent foundation of scientific
models.
Most ontologies are dogmatic, consisting of whimsical
speculations a priori, aspiring to absolute truth, and
high-handedly snubbing science, know-how and, above all,
facts. "If the facts disagree with me then so much worse
for the facts." - this Hegel's declaration may serve as
motto of Dogmatism.
Yet, there exist rational ontologies, endeavoring to found
their contemporary science and know-how and considering
their presumptions as axioms verifiable or falsifiable by
facts.
Kant's ontology springs to the mind. It has been briefly
introduced in the "PREFACE". Here we wish to consider one
of its axioms derived from the science of the First
Enlightenment culminating in Newton's model:
Science was concurrently considered as absolute and certain,
thus necessary. On the other hand, science is constructed by
"synthetic" statements and only apriori statements may be
necessary. Consequently, Kant postulated the axiom of
existence of "synthetic" statements apriori. It persisted
till the Second Enlightenment which falsified it by revealing
the fuzzy and uncertain fabric of science.
Consequently, the particular assumptions of Kant's view
became for us obsolete and falsified, but his method of
deriving an axiomatic ontology from the current know-how
stays a topical and well-advised example.
Einstein followed it conceiving his "Physical Reality",
which is the kernel of the Relativistic Dialectic (RD) put
forth in the present essay. It has still deeper and more
intimate relations with the cutting edge of concurrent
physics than those of Kant with Newton. Kant conceived
a fair and pertinent ontology, which generalized and founded
a posteriori the Newton's model. Einstein's "Physical Reality"
for the first time in the history precedes and concretely
underlies physics, which uses directly some ontological
assertions as axioms fit for rigorous processing. Such is,
for instance, the case of the ontological Covering Principle
("see NATURAL MODEL"), underlying the derivation of the
General Relativity by means of the mental experiment of
"Rotating Disk". The Extended Relativity would be unthinkable
without being concretely founded in the ontology of the
"Physical Reality" and in particular in its Covering
Principle.
Arithmetics founded in continuous geometry
In "NATURAL MODEL" we have asserted:
**
mind's faculty of putting every body situated in any arbitrary
manner into contact with the quasi rigid continuation of a
chosen body of relation B0 is the basis of our intuition of space.
In pre-scientific thinking, the solid earth's crust plays the role
of B0. The very name geometry indicates that the idea of space is
mentally connected with the earth considered as the relation body.
We shall find the intuition of space founded by rigid bodies
at the base of Einstein's Covering Principle which requires
physical distance to be measured, also in mental experiments,
with physical rods complying with physical rules, such as
the Lorentz Contraction. The Covering Principle underlies
directly or indirectly the entire Extended Relativity.
Continuous space, its primacy and the discrete covering
measurement rods are aspects of the fundamental Polarity
Continuum/Discreteness (CD) defined in "TIME, AWARENESS
AND EVENTS" as the elementary structure of human Universe.
In mathematical terms geometry symbolizes the basic intuitive
image of continuous space and arithmetic - its discrete
covering measurements. The ultimate foundation of mathematics
is the continuum. Discrete concepts starting with that of
"number" are founded in continuum and symbolize its covering
measurements. Shortly, arithmetic is founded in geometry.
**
If mathematics were restricted to arithmetic, we could stop
here. But it extends beyond arithmetic over innumerable
interrelated theories. In order to exemplify their ultimate
foundation in continuum we shall recall the concept of
foundation hierarchies.
Foundation hierarchies
Besides being founded in their own, "local" axioms, models
may be founded in other "founding" models. By definition,
a model or a discipline is "founded" in a "founding" one,
when it accepts the latter's axioms and theorems as its own
axioms.
Thus, physics is founded in mathematics and does not derive
the principles of calculus, of vectors, tensors, etc. but
considers their mathematical formulations as axioms of its
own models.
This foundation hierarchies stems from the ontological
intuition of continuum, the primary aspect of the Polarity
Continuum/Discreteness (CD), the fundamental construct of
the physical and human reality ("NATURAL MODEL"). Hence,
rational axiomatic models are ultimately founded in continuum.
In "SET THEORY" we saw the fallacies resulting from attempting
to found mathematics in discreteness. In Tome 2 "FOUNDATIONS
OF QUANTUM PHYSICS" we shall discuss the controversy between
quantizing the fundamental continuum of field and attempts
to found quantum physics in sheer discreteness without
considering SPACE or field continuum.
Briefly, the Relativistic Dialectic posits rational models
as ultimately founded in the intuitive continuum. Abstract
structures lacking this foundation fall in the domain of
irrationality, beyond the human universe of discourse.
In the following section we shall illustrate the reduction
to continuous foundation with help of Euler's Formula,
sufficiently complex and high up in the foundation hierarchy
to be considered as typical.
We don't pretend that whole established mathematics is
rational in the above sense. On the contrary, we see a lot
of theories or theorems as irrational and meaningless plays
with symbols, e.g. Banach-Tarski Paradox or Goedel's Theorem.
The following example may help to decide if some established
theory can be considered as a rational model, or disregarded.
Euler's Formula
e**(ix)=cos(x) + i*sin(x)
Feynman called Euler's formula "our jewel" and "one of the
most remarkable, almost astounding, formulas in all of
mathematics". Indeed, on the one hand it relates completely
dissimilar concepts of exponential and trigonometric functions
with help, moreover, of apparently out-of-the-way imaginary
numbers. On the other hand, in spite of its concise form it
involves astounding variety and extent of mathematical domains,
to mention complex numbers, exponential function, trigonometry
hyperbolic and circular, algebra, algebraic (analytic) geometry,
calculus with differentiation, integration and infinite series,
etc.
Sharing Feynman's admiration, we note that the ramifications
of Euler's Formula make it particularly suitable to exemplify
our postulate of the ultimate founding of rational models in
the ontological continuum. The demonstration is surprisingly
simple:
1.The proof of Euler's Formula consists in the infinite series
of both sides being equal.
2.Infinite series are founded in calculus.
3.Calculus is the symbolic, abstract map of the fundamental
structure of human reality - the CD (Continuous/Discrete)
polarity - differentiation representing continuum and
integration - its discretization.
Besides the Euler's Formula, there are innumerable rational
mathematical constructs clearly regressing to continuum.
Let's mention the Gauss-Ostrogradsky theorem, which allows
to express several "continuity equations" in alternative
differential or integral forms.
Falsifiability of mathematical axioms
Having justified the postulate of rational mathematical
models being founded in continuum, we are still facing the
open problem of falsifiability of mathematical axioms. While
axioms of natural sciences and applied mathematics are
falsifiable by physical facts, one often objects that pure
mathematics are entirely mental and don't deal with any facts
suitable to falsify them.
Our ontology rebuts this objection with two arguments
postulated in "NATURAL MODEL" in accord with Einstein's
"PHYSICS AND REALITY":
1.In spite of the illusion of "objective reality" human
Physical Reality is entirely immanent and mental, which
puts it at parity with mathematics.
2. ...imagery supports "mental experiments", whose events
are projections of abstract concepts, of emotional impressions
or images brought about recursively by analogy with known ones.
Mental experiments support most, if not all human creativity.
As it may appear rather complex, we shall illustrate it with
the example of hyperbolic geometry.
During 2000 years Greek, Persian, Arab and modern European
mathematicians tried to derive the Parallel Postulate from
other Euclidean axioms. All those trials failed, but several,
especially those proceeding by reductio ad absurdum, came up
with strange byproducts, which finally added up to the
complete and consistent Hyperbolic Geometry published by
Lobachevsky and Bolyai. It replaces the Euclidean Postulate
of a single parallel with any number of distinct parallels
higher than one. One of its theorems states that the ratio
of circumference to diameter of a disk is greater than pi.
In concert with covering principle and Lorentz Transformation,
the mental experiment of rotating disk results in this ratio
geater than pi, thus postulating the hyperbolic SPACE of
spinning cosmic referentials and creating the General Relativity.
Further development generalized the hyperbolic SPACE to Riemann
SPACE defining pointwise flat, elliptic and hyperbolic geometries,
representing centrifugal and centripetal areas of Cosmos as
well as their transitions.
So far it was a pure mathematical mental experiment including
the purely abstract concept of Cosmos. Physical factual
verifications such as Einstein's lens came years later emphasizing
the ontological equivalence of mathematical and physical mental
experiments.
Postface
We believe to have shown the equivalence of foundations of
mathematics and physics in ontological continuum. In other
terms, mathematics does not have, nor need particular
foundations, different from the general foundations of
rational symbolic constructs.
History seems to confirm our view. Euclides, Pytagoras,
Muhammad ibn Musa al-Khwarizmi, Descartes, Newton, Leibnitz,
Euler, Gauss, Riemann, Dirac did their mathematics without
needing any special foundations, did not wait for the so
called analytic philosophers to create them, nor noticed
them, if contemporary.
Another confirmation of our view may be found in the
ultimate Frege's disappointment. In 1923 he came to the
conclusion that the aim he had set himself throughout most
of his career, namely to found arithmetic in (predicate)
logic, was wrong. He decided instead (like ourselves) that
one had to base the whole of mathematics on geometry.
He began to work on these ideas but had not progressed far
by the time of his death.
