Human Evolution
IV Session
Formal Sciences
A multidisciplinary anthropic focus
Creating formal sciences
Interdisciplinarity leads us to formal activity
of the mind.
The human brain is gifted with functional skills to:
•
perceive and analyze a world of structures
• abstract certain structural features of reality
• conceive and imagine new forms and structures
Why is the human mind capable of creating
formal sciences?
Could mental forms, formalization, formal
sciences, exist without brains? (Hofstadter)
2
Structures and rationality
Our human evolutive sensitive experience
is structured by the perception of:
 Differentiated
objects as a unity (elements
of a set).
 Structures, operations and relations
between them
 Consequential, logical, unity
3
Structures and rationality
The world is experienced
as a variety of structures
logically involved in other
structures of superior level.
We do not know neither
the final micro-physic
nor the final macrophysic structures, but we
know a multiplicity of
consequential intermediate
connections.
4
Counting and measurements
The first formal abstraction was probably
that of numbers and geometric figures.
Human mind was capable:
To
perceive differentiated entities: two fish,
three stones…measure changes in time: days,
seasons, years… compare size, shape and
relative position of geometrical figures.
To formally represent these entities: by
abstract numerical structures concerning
numbers and operations on them and abstract
geometrical structures with metrical properties.
5
Counting and measurements
Mayan numerals
6
Classical arithmetic and geometry
Formal Sciences seemed to be firmly
established on the simple foundation of
numbers and geometry:
 The
formal language of mathematics allowed
counting and space-time measurements.
 Developments in formal logic and set theory led
to questions about mathematical certainty:
•
•
•
What is the cause of mathematical certainty?
Why does mathematical reason work like it actually
does?
What is the ontological status of formal entities?
7
Formal theories
8
Foundation of mathematics
 The
relationship between mathematics
and logic.
 Philosophers
of mathematics began to
divide into various schools of thought.
 Logic
apriorism, formalism and
intuitionism emerged partly in response
to the search for the causes of
mathematical certainty.
9
Apriorism

Platonism suggests that mathematical entities
exist independently of the human mind.

E. Kant believes that the objectivity of
mathematics is based in space and time as
an a priori forms of sensibility.

Different forms of apriorism remain present
among today mathematicians.

For Roger Penrose some mathematical
assertions belong to an unchanging world
of essences.
10
Logicism
Mathematics can be known a priori because it is
part of logic. Logic is the proper foundation of
mathematics. Logicism becomes strong with the
formalization of logic.
G. Frege constructed a formal logical system that
made it possible to represent the logical inferences
as formal operations.
This program was continued by Russell and
Whitehead.
11
Formalism
Formal logic is a part of formal mathematics.
Can mathematics rationally justify itself as a
purely formal science?
The meta-mathematical Hilbert program
intends to justify mathematics as a pure formal
science.
Meta-mathematics uses mathematics as a formal
language to speak about mathematics as a formal
object.
12
Intuitionism/constructivism
Gödel caused a crisis in the Hilbert’s programme
proving that, if the formal system of arithmetic is
consistent, then it is incomplete.
Intuitionism rejected the meta-mathematical formal
foundation of mathematics. Only the mathematical
entities which can be explicitly constructed are
admitted.
Intuitionist logic does not contain the law of
excluded middle. Constructivism regards the sets
with infinite elements.
13
Objective reality
14
Natural aposteriorism
What is the origin of human reason’s
formal capacities?

Aposteriorism responds to the evolutive
adaptation of consciousness to an objective
and structural reality.
 Neurological anthropology, evolutive
epistemology, and authors like J. Piaget and
X. Zubiri do support this point of view.
 It is congruent with the paradigm of evolution
in modern Science.
15
Natural aposteriorism
Aposteriorism states the:
 structural construction of an objective
physical world.
 emergent properties of mind to adapt
behaviourally to this structural world.
 emergent structural representation of reality
and capacity to abstract specific structural
features.
 emergent skills to imagine created structural
forms in order to open new possibilities for
knowledge and technology
16
Computationalism

Computing is formal mathematics applied to the
development of algorithms.

From ancient times algorithmic processes have
been used in algebra and formal logic.

The old mechanist ideal consisted of obtaining
a mechanical artifice by which it would be
possible to execute all the deductions.

In 1936 Turing specified the informal idea of an
algorithm through what we call the Turing
machine.
17
Computationalism
By means of his machine, Turing showed that
there is no general solution to the
Decidability’s Problem: Given a formal statement in
a formal system, there is not always a general algorithm
which decides if the statement is valid or not.
The incompleteness of arithmetic caused
disappointment against the mechanicist ideal,
but the negative solution to the problem of
decidability by the Turing machine, in some way,
involves a deepening in this disappointment.
18
Liberation and extension of FFSS
19
Liberation and extension of FFSS
The applicability of computing has
highlighted applied dimensions of
languages and formal models.
This has led to the use of a plurality of
logics, suited to several finalities.
The existence of a plurality of logics
opens up new perspectives for
scientific language.
20
Anthropic perspectives
Reality:

Formal languages allow a high degree of
objectivity. Is this objectivity total? Are formal
sciences really objective and independent of
the subject that formulates them?
 The pluralism of formal systems and their
undecidability leads us to ask about the
rationality of inevitable non formal decisions.
 Are formal sciences a mask disfiguring the real
world?
 How does mathematics access the real world?
21
Anthropic perspectives
Technology:
 We
can say that we control a scientific theory
when we have expressed it in formal
language. Formal language, as it is objective,
permits the technological implementation of
scientific theories.
 Do
formal sciences qualify natural sciences for
a new design of theoretical frameworks of new
instrumental machinery to formalize and
control reality?
22
Anthropic perspectives
Metaphysics:
 Do
formal sciences formalize open or
closed systems?
 Will they establish insurmountable limits
to human reason?
 How far will they be able to formalize
open or closed natural systems?
 How does philosophy of formal sciences
behave in face of metaphysics?
23
Expressing reality
24
Knowing reality
New forms to know reality:

Classical formal sciences intended to be
mechanicist.
 Do we have new formalisms for new
holistic ontologies? Formal sciences are
analytical, not holistic. They can analytically
interpret holistic properties.
 The human mind: Do we have new classicalquantum forms to formally describe the
functioning of our human mind?
25
Creating technology
New forms for manipulating technologically
reality:



Classical technology intends to be mechanistic.
Technology is based in our formal control of reality,
and classically the ideal of reality control was
mechanist.
Quantum technology: quantum properties such as
superposition and entanglement can be used to
represent and structure formal data.
Classical-quantum technology for information
processing: new forms to implement new
technologies of the mind?
26
Speculating in metaphysics
The main problem of metaphysics: Universe’s
sufficiency/insufficiency
Physical knowledge is constructed by applying
formal models. Do formal sciences offer to
physics a closed and self consistent formal
system to organize natural knowledge?
Philosophy should think about internal
possibilities of formal sciences
Gödel’s theorem shows that if formal systems
are consistent, they are incomplete and
therefore they are intrinsically open.
Consistency is an unrenounceable value of
formal systems
27
Speculating in metaphysics
Do formal sciences qualify
human reason for an open or a
closed metaphysics?
Will formal sciences empower
human reason for an absolute
consistent dominium over natural
reality?
28