Download science

CDT403 Research Methodology in Natural Sciences and Engineering
Theory of Science
SCIENCE, KNOWLEDGE, TRUTH, MEANING
Gordana Dodig-Crnkovic
School of Innovation, Design and Engineering
Mälardalen University
1
Theory of Science
Lecture 1 SCIENCE AND KNOWLEDGE: TRUTH, MEANING.
FORMAL LOGICAL SYSTEMS PRESUPPOSITIONS
Lecture 2 LANGUAGE AND COMMUNICATION. CRITICAL
THINKING. PSEUDOSCIENCE - DEMARCATION
Lecture 3 SCIENCE AND RESEARCH: TECHNOLOGY,
SOCIETAL ASPECTS. PROGRESS. HISTORY OF SCIENTIFIC
THEORY. POSTMODERNISM AND CROSSDISCIPLINES
Lecture 4 GOLEM LECTURE. ANALYSIS OF SCIENTIFIC
CONFIRMATION: THEORY OF RELATIVITY, COLD FUSION,
GRAVITATIONAL WAVES
Lecture 5 COMPUTING HISTORY OF IDEAS
Lecture 6 PROFESSIONAL & RESEARCH ETHICS
2
Science, Knowledge, Truth and Meaning
CRITICAL THINKING
WHAT IS SCIENCE?
WHAT IS SCIENTIFIC METHOD?
WHAT IS KNOWLEDGE?
KNOWLEDGE, TRUTH AND MEANING
FORMAL SYSTEMS AND THE CLASSICAL MODEL OF SCIENCE
3
Course Material
– Course web page
http://www.idt.mdh.se/kurser/ct3340/
– Theory of Science Compendium
G Dodig-Crnkovic
– The Golem: What You Should Know about Science
Harry M. Collins & Trevor Pinch
4
Red Thread: Critical Thinking
“Reserve your right to think,
for even to think wrongly
is better than not to think at all.”
Hypatia, natural philosopher and mathematician
5
Haiku – Like Highlights
.遠山が目玉にうつるとんぼ哉
tôyama ga medama
ni utsuru tombo kana
the distant mountain
reflected in his eyes...
dragonfly
Kobayashi Issa (1763-1827)
(Haiku form: 5-7-5 syllables)
6
What Is Science?
Eye
Maurits Cornelis Escher
7
1. Definitions by Goal (Result) and Process (1)
science from Latin scientia, scire to know;
1: a department of systematized knowledge as an
object of study
2: knowledge or a system of knowledge covering
general truths or the operation of general laws
especially as obtained and tested through scientific
method
8
1. Definitions by Goal (Result) and Process (2)
3: such knowledge or such a system of knowledge
concerned with the physical world and its phenomena
: natural science
4: a system or method reconciling practical ends with
scientific laws <engineering is both a science and an
art>
9
2. Science: Definitions by Contrast
To do science is to search for repeated patterns, not
simply to accumulate facts.
Robert H. MacArthur
Religion is a culture of faith; science is a culture of doubt.
Richard Feynman
10
3. Empirical approach.
What Sciences are there?
Dewey Decimal Classification®
http://www.geocities.com/Athens/Troy/8866/15urls.html
000 - & Psychology
200 - ReliComputers, Information & General Reference
100 - Philosophy gion
300 - Social sciences
400 - Language
500 - Science
600 - Technology
700 - Arts & Recreation
800 - Literature
900 - History & Geography
11
3. Dewey Decimal Classification®
500 – Science
510 Mathematics
520 Astronomy
530 Physics
540 Chemistry
550 Earth Sciences & Geology
560 Fossils & Prehistoric Life
570 Biology & Life Sciences
580 Plants (Botany)
590 Animals (Zoology)
12
4. Language Based Scheme
Classical Sciences in their Cultural Context –
Logic
&
Mathematics
1
Natural Sciences
(Physics,
Chemistry,
Biology, …)
2
Culture
(Religion, Art, …)
5
Social Sciences
(Economics, Sociology,
Anthropology, …)
3
The Humanities
(Philosophy, History,
Linguistics …)
4
13
5. Understanding what science is
by understanding what scientists do
"Scientists are people of very dissimilar temperaments doing
different things in very different ways.
Among scientists are collectors, classifiers and compulsive
tidiers-up; many are detectives by temperament and many are
explorers; some are artists and others artisans.
There are poet-scientists and philosopher-scientists and even a
few mystics."
Peter Medawar, Pluto's Republic
14
6. Critique of Usual Naïve Image
of Scientific Method (1)
The narrow inductivist conception of scientific inquiry
1. All facts are observed and recorded.
2. All observed facts are analyzed, compared and classified,
without hypotheses or postulates other than those necessarily
involved in the logic of thought.
3. Generalizations inductively drawn as to the relations,
classificatory or causal, between the facts.
4. Further research employs inferences from previously
established generalizations.
15
Critique of Usual Naïve Image
of Scientific Method (2)
This narrow idea of scientific inquiry is groundless for several
reasons:
1. A scientific investigation could never get off the ground, for a
collection of all facts would take infinite time, as there are infinite
number of facts.
The only possible way to do data collection is to take only relevant
facts. But in order to decide what is relevant and what is not, we
have to have a theory or at least a hypothesis about what is it we
are observing.
16
Critique of Usual Naïve Image
of Scientific Method (3)
A hypothesis (preliminary theory) is needed to give the direction
to a scientific investigation!
2. A set of empirical facts can be analyzed and classified in many
different ways. Without hypothesis, analysis and classification
are blind.
3. Induction is sometimes imagined as a method that leads, by
mechanical application of rules, from observed facts to general
principles. Unfortunately, such rules do not exist!
17
Why is it not possible to derive hypothesis
(theory) directly from the data? (1)
– For example, theories about atoms contain terms like “atom”,
“electron”, “proton”, etc; yet what one actually measures are
spectra (wave lengths), traces in bubble chambers, calorimetric
data, etc.
– So the theory is formulated on a completely different (and more
abstract) level than the observable data!
– The transition from data to theory requests creative imagination!
18
Why is it not possible to derive hypothesis
(theory) directly from the data?* (2)
– Scientific hypothesis is formulated based on “educated guesses”
at the connections between the phenomena under study, at
regularities and patterns that might underlie their occurrence.
Scientific guesses are completely different from any process of
systematic inference.
– The discovery of important mathematical theorems, like the
discovery of important theories in empirical science, requires
inventive ingenuity.
*Here is instructive to study Automated discovery methods in order to see how
much theory must be used in order to extract meaning from the “raw data”
19
Socratic Method
Scientific Method
1. Wonder. Pose a question
(of the “What is X ?” form).
1. Wonder. Pose a question.
(Formulate a problem).
2. Hypothesis. Suggest a plausible
answer (a definition or definiens) from
which some conceptually testable
hypothetical propositions can be deduced.
2. Hypothesis. Suggest a plausible answer (a
theory) from which some empirically testable
hypothetical propositions can be deduced.
3. Elenchus ; “testing,” “refutation,” or
“cross-examination.” Perform a thought
experiment by imagining a case which
conforms to the definiens but clearly fails
to exemplify the definiendum, or vice
versa. Such cases, if successful, are
called counterexamples. If a
counterexample is generated, return to
step 2, otherwise go to step 4.
3. Testing. Construct and perform
an experiment, which makes it possible to
observe whether the consequences specified
in one or more of those hypothetical
propositions actually follow when the
conditions specified in the same
proposition(s) pertain. If the test fails, return
to step 2, otherwise go to step 4.
4. Accept the hypothesis as provisionally
true. Return to step 3 if you can conceive
any other case which may show the
answer to be defective.
4. Accept the hypothesis as provisionally true.
Return to step 3 if there are predictable
consequences of the theory which have not
been experimentally confirmed.
5. Act accordingly.
5. Act accordingly.
20
The Scientific Method
EXISTING THEORIES
AND OBSERVATIONS
HYPOTHESIS
PREDICTIONS
2
3
1
Hypothesis
must be
redefined
Hypotesen
Hypothesis
måste
must
be
justeras
adjusted
SELECTION AMONG
COMPETING THEORIES
TESTS AND NEW
OBSERVATIONS
6
4
Consistency achieved
The hypotetico-deductive cycle
EXISTING THEORY CONFIRMED
(within a new context) or
NEW THEORY PUBLISHED
5
The scientific-community cycle
21
Formulating Research Questions and Hypotheses
Different approaches:
Intuition – (Educated) Guess
Analogy
Symmetry
Paradigm
Metaphor
and many more ..
22
Criteria to Evaluate Theories
When there are several rivaling hypotheses number of criteria can
be used for choosing a best theory.
Following can be evaluated:
– Theoretical scope
– Heuristic value (heuristic: rule-of-thumb or argument
derived from experience)
– Parsimony (simplicity, Ockham’s razor)
– Esthetics
– Etc.
23
Criteria which Good Scientific Theory Shall Fulfill
–
–
–
–
–
–
–
–
Logically consistent
Consistent with accepted facts
Testable
Consistent with related theories
Interpretable: explain and predict
Parsimonious
Pleasing to the mind (Esthetic, Beautiful)
Useful (Relevant/Applicable)
24
Ockham’s Razor (Occam’s Razor)
(Law Of Economy, Or Law Of Parsimony, Less Is More!)
A philosophical statement developed by William of Ockham,
(1285–1347/49), a scholastic, that Pluralitas non est ponenda
sine necessitate; “Plurality should not be assumed without
necessity.”
The principle gives precedence to simplicity; of two competing
theories, the simplest explanation of an entity is to be preferred.
25
What Is Knowledge?
Plato´s Definition
Plato believed that we learn in this life by remembering knowledge
originally acquired in a previous life, and that the soul already
has knowledge, and we learn by recollecting what in fact the
soul already knows.
Plato offers three analyses of knowledge, [dialogues Theaetetus
201 and Meno 98] all of which Socrates rejects.
26
What Is Knowledge?
Plato´s Definition
Plato's third definition:
" Knowledge is justified, true belief. "
The problem with this concerns the word “justified”. All
interpretations of “justified” are deemed inadequate.
Edmund Gettier, in the paper called "Is Justified True Belief
Knowledge?“ argues that knowledge is not the same as justified
true belief. (Gettier Problem)
27
What Is Knowledge?
Descartes´ Definition
"Intuition is the undoubting conception of an unclouded and
attentive mind, and springs from the light of reasons alone; it is
more certain than deduction itself in that it is simpler."
“Deduction by which we understand all necessary inference from
other facts that are known with certainty,“ leads to knowledge
when recommended method is being followed.
28
What Is Knowledge?
Descartes´ Definition
"Intuitions provide the ultimate grounds for logical deductions.
Ultimate first principles must be known through intuition while
deduction logically derives conclusions from them.
These two methods [intuition and deduction] are the most certain
routes to knowledge, and the mind should admit no others."
29
What Is Knowledge?
– Propositional knowledge: knowledge that such-and-such is the
case.
– Non-propositional knowledge (tacit knowledge): the knowing how to
do something.
30
Sources of Knowledge
– A Priori Knowledge (built in, developed by evolution and
inheritance)
– Perception (“on-line input”, information acquisition)
– Reasoning (information processing)
– Testimony (network, communication)
31
Knowledge and Objectivity:
Observations
Observations are always interpreted in the context of an a priori
knowledge. (Kuhn, Popper)
“What a man sees depends both upon what he looks at and also
upon what his previous visual-conceptual experience has taught
him to see”.
32
Knowledge and Objectivity
Observations
– All observation is potentially ”contaminated”, whether by our
theories, our worldview or our past experiences.
– It does not mean that science cannot ”objectively” [intersubjectivity] choose from among rival theories on the basis of
empirical testing.
– Although science cannot provide one with hundred percent
certainty, yet it is the most, if not the only, objective mode of
pursuing knowledge.
33
Perception and “Direct Observation”
34
Perception and “Direct Observation”
35
Perception and “Direct Observation”
36
Perception and “Direct Observation”
37
Perception and “Direct Observation”
"Reality is merely an illusion,
albeit a very persistent one." Einstein
38
Perception and “Direct Observation”
39
Perception and “Direct Observation”
40
Perception and “Direct Observation”
41
Perception and “Direct Observation”
42
43
Perception and “Direct Observation”
Checker-shadow illusion
http://web.mit.edu/persci/people/adelson/checkershadow_illusion.html
See even:
http://persci.mit.edu/people/adelson/publications/gazzan.dir/gazzan.htm
Lightness Perception and Lightness Illusions
http://www.ihu.his.se/~christin/Vetenskapsteori/Vetenskapsteorikurser
44
Direct Observation?!
An atom interferometer, which splits an atom into separate wavelets, can allow
the measurement of forces acting on the atom. Shown here is the laser
system used to coherently divide, redirect, and recombine atomic wave
packets (Yale University).
45
Direct Observation?!
Electronic signatures
produced by collisions of
protons and antiprotons in
the Tevatron accelerator
at Fermilab provided
evidence that the elusive
subatomic particle known
as top quark has been
found.
46
Knowledge Justification
– Foundationalism (uses architectural metaphor to describe the
structure of our belief systems. The superstructure of a belief
system inherits its justification from a certain subset of beliefs –
all rests on basic beliefs.)
– Coherentism
– Internalism (a person has “cognitive grasp”) and Externalism
(external justification)
47
Truth (1)
– The correspondence theory
– The coherence theory
– The deflationary theory
48
Truth (2)
The Correspondence Theory
A common intuition is that when I say something true, my
statement corresponds to the facts.
But: how do we recognize facts and what kind of relation is this
correspondence?
49
Truth (3)
The Coherence Theory
Statements in the theory are believed to be true because being
compatible with other statements.
The truth of a sentence just consists in its belonging to a system of
coherent statements.
The most well-known adherents to such a theory was Spinoza
(1632-77), Leibniz (1646-1716) and Hegel (1770-1831).
Characteristically they all believed that truths about the world could
be found by pure thinking, they were rationalists and idealists.
Mathematics was the paradigm for a real science; it was
thought that the axiomatic method in mathematics could be
used in all sciences.
50
Truth (4)
The Deflationary Theory
The deflationary theory is belief that it is always logically
superfluous to claim that a proposition is true, since this claim
adds nothing further to a simple affirmation of the proposition
itself.
"It is true that birds are warm-blooded " means the same thing as
"birds are warm-blooded ".
For the deflationist, truth has no nature beyond what is captured in
ordinary claims such as that ‘snow is white’ is true just in case
snow is white.
51
Truth (5)
The Deflationary Theory
The Deflationary Theory is also called the redundancy
theory, the disappearance theory, the no-truth theory,
the disquotational theory, and the minimalist theory .
63-70 see Lars-Göran Johansson
http://www.filosofi.uu.se/utbildning/Externt/slu/slultexttruth.htm
and Stanford Encyclopedia of Philosophy
http://plato.stanford.edu/entries/truth-deflationary/
52
Truth and Reality
Noumenon,"Ding an sich" is
distinguished from
Phenomenon "Erscheinung", an
observable event or physical
manifestation, and the two words
serve as interrelated technical
terms in Kant's philosophy.
53
Whole vs. Parts
•
•
•
•
•
tusk  spear
tail  rope
trunk  snake
side  wall
leg  tree
The flaw in all their reasoning is that speculating on the
WHOLE from too few FACTS can lead to VERY
LARGE errors in judgment.
54
The Classical Model of Science
The Classical Model of Scienceis a system S of propositions and
concepts satisfying the following conditions:
• All propositions and all concepts (or terms) of S concern a
specific set of objects or are about a certain domain of being(s).
• There are in S a number of so-called fundamental concepts (or
terms).
• All other concepts (or terms) occurring in S are composed of (or
are definable from) these fundamental concepts (or terms).
55
The Classical Model of Science
• There are in S a number of so-called fundamental propositions.
• All other propositions of S follow from or are grounded in (or are
provable or demonstrable from) these fundamental propositions.
• All propositions of S are true.
• All propositions of S are universal and necessary in some sense
or another.
56
The Classical Model of Science
• All concepts or terms of S are adequately known. A nonfundamental concept is adequately known through its
composition (or definition).
• The Classical Model of Science is a reconstruction a posteriori
and sums up the historical philosopher’s ideal of scientific
explanation.
• The fundamental is that “All propositions and all concepts (or
terms) of S concern a specific set of objects or are about a
certain domain of being(s).”
Betti A & De Jong W. R., Guest Editors, The Classical Model of Science I: A MillenniaOld Model of Scientific Rationality, Forthcoming in Synthese, Special Issue
57
Science and Truth
– Science as controversy (new science, frontiers)
– Science as consensus (old, historically settled)
– Science as knowledge about complex systems
– Opens systems, paraconsistent logic
58
Scientific Truth (1)
– Physics professor is walking across campus, runs
into Math professor. Physics professor has been
doing an experiment, and has worked out an
empirical equation that seems to explain his data,
and asks the Math professor to look at it.
59
Scientific Truth (2)
– A week later, they meet again, and the Math
professor says the equation is invalid. By then, the
Physics professor has used his equation to predict
the results of further experiments, and he is getting
excellent results, so he asks the Math professor to
look again.
– Another week goes by, and they meet once more.
The Math professor tells the Physics professor the
equation does work, ”but only in the trivial case
where the numbers are real and positive."
60
Proof
The word proof can mean:
• originally, a test assessing the validity or quality of something.
Hence the saying, "The exception that proves the rule" -- the
rule is tested to see whether it applies even in the case of the
(apparent) exception.
• a rigorous, compelling argument, including:
– a logical argument or a mathematical proof
– a large accumulation of scientific evidence
– (...)
(from Wikipedia)
61
Mathematical Proof
In mathematics, a proof is a demonstration that, given certain
axioms, some statement of interest is necessarily true.
(from Wikipedia)
62
Mathematical Proof
Proofs employ logic but usually include some amount of natural
language which of course admits some ambiguity. In the context
of proof theory, where purely formal proofs are considered, such
not entirely formal demonstrations are called "social proofs".
The distinction has led to much examination of current and historical
mathematical practice, quasi-empiricism in mathematics, and socalled folk mathematics (in both senses of that term).
The philosophy of mathematics is concerned with the role of
language and logic in proofs, and mathematics as a language.
(from Wikipedia)
63
Mathematical Proof
Regardless of one's attitude to formalism, the result that is proved
to be true is a theorem; in a completely formal proof it would be
the final line, and the complete proof shows how it follows from
the axioms alone. Once a theorem is proved, it can be used as
the basis to prove further statements.
The so-called foundations of mathematics are those statements
one cannot, or need not, prove. These were once the primary
study of philosophers of mathematics. Today focus is more on
practice, i.e. acceptable techniques.
(from Wikipedia)
64
Pressupositions and Limitations
of Formal Logical Systems
Axiomatic System of Euclid: Shaking up Geometry
Euclid built geometry on a set of few axioms/postulates (ideas
which are considered so elementary and manifestly obvious that
they do not need to be proven as any proof would introduce
more complex ideas).
When a system requires increasing number of axioms (as e.g.
number theory does), doubts begin to arise. How many axioms
are needed? How do we know that the axioms aren't mutually
contradictory?
65
Pressupositions and Limitations
of Formal Logical Systems
Axiomatic System of Euclid: Shaking up Geometry
Until the 19th century no one was too concerned about
axiomatization.
Geometry had stood as conceived by Euclid for 2100 years.
If Euclid's work had a weak point, it was his fifth axiom, the axiom
about parallel lines. Euclid said that for a given straight line, one
could draw only one other straight line parallel to it through a
point somewhere outside it.
66
EUCLID'S AXIOMS (1)
1. Every two points lie on exactly one line.
2. Any line segment with given endpoints may be continued in
either direction.
3. It is possible to construct a circle with any point as its center and
with a radius of any length. (This implies that there is neither an
upper nor lower limit to distance. In-other-words, any distance,
no mater how large can always be increased, and any distance,
no mater how small can always be divided.)
67
EUCLID'S AXIOMS (2)
4. If two lines cross such that a pair of adjacent angles are
congruent, then each of these angles are also congruent to any
other angle formed in the same way. (Says that all right angles
are equal to one another.)
5. (Parallel Axiom): Given a line l and a point not on l, there is one
and only one line which contains the point, and is parallel to l.
68
NON-EUCLIDEAN GEOMETRIES (1)
Mid-1800s: mathematicians began to experiment with different
definitions for parallel lines.
Lobachevsky, Bolyai, Riemann: new non-Euclidean geometries by
assuming that there could be several parallel lines through the
outside point or there could be no parallel lines.
69
NON-EUCLIDEAN GEOMETRIES (2)
Two ways to negate the Euclidean Parallel Axiom:
– 5-S (Spherical Geometry Parallel Axiom): Given a line l and a
point not on l, no lines exist that contain the point, and are parallel
to l.
– 5-H (Hyperbolic Geometry Parallel Axiom): Given a line l and a
point not on l, there are at least two distinct lines which contains
the point, and are parallel to l.
70
Reproducing the Euclidean World
in a model of the Elliptical Non-Euclidean World.
71
Spherical/Elliptical Geometry
In spherical geometry lines of latitude are not great circles (except for
the equator), and lines of longitude are. Elliptical Geometry takes the
spherical plan and removes one of two points directly opposite each
other. The end result is that in spherical geometry, lines always intersect
in exactly two points, whereas in elliptical geometry, lines always
intersect in one point.
72
Properties of Elliptical/Spherical Geometry
In Spherical Geometry, all lines intersect in 2 points. In elliptical
geometry, lines intersect in 1 point.
In addition, the angles of a triangle always add up to be greater
than 180 degrees. In elliptical/spherical geometry, all of Euclid's
postulates still do hold, with the exception of the fifth postulate.
This type of geometry is especially useful in describing the Earth's
surface.
73
Hyperbolic Cubes
74
DEFINITION:
Parallel lines are infinite lines in the same plane that do not intersect.
Hyperbolic Universe
Flat Universe
Spherical Universe
Einstein incorporated Riemann's ideas into relativity theory to
describe the curvature of space.
75
MORE PROBLEMS WITH
AXIOMATIZATION…
Not only had Riemann created a system of geometry which put
commonsense notions on its head, but the philosophermathematician Bertrand Russell had found a serious paradox
for set theory!
He has shown that Frege’s attempt to reduce mathematics to
logical reasoning starting with sets as basics leads to
contradictions.
76
HILBERT’S PROGRAM
Hilbert’s hope was that mathematics would be reducible to finding
proofs (by manipulating the strings of symbols) from a fixed
system of axioms, axioms that everyone could agree were true.
Can all of mathematics be made algorithmic, or will there always be
new problems that outstrip any given algorithm, and so require
creative mind to solve?
77
AXIOMATIC SYSTEM OF PRINCIPIA:
PARADOX IN SET THEORY
Mathematicians hoped that Hilbert's plan would work because
axioms and definitions are based on logical commonsense
intuitions, such as e.g. the idea of set.
A set is any collection of items chosen for some characteristic
common for all its elements.
78
RUSSELL'S PARADOX (1)
There are two kinds of sets:
– Normal sets, which do not contain themselves, and
– Non-normal sets, which are sets that do contain themselves.
The set of all apples is not an apple. Therefore it is a normal set. The
set of all thinkable things is itself thinkable, so it is a non-normal set.
79
RUSSELL'S PARADOX (2)
Let 'N' stand for the set of all normal sets.
Is N a normal set?
If it is a normal set, then by the definition of a normal set it
cannot be a member of itself. That means that N is a non-normal
set, one of those few sets which actually are members of
themselves.
80
RUSSELL'S PARADOX (3)
But on the other hand…N is the set of all normal sets; if we
describe it as a non-normal set, it cannot be a member of itself,
because its members are, by definition, normal.
81
RUSSELL'S PARADOX (4)
Russell resolved the paradox by redefining the meaning of 'set' to
exclude peculiar (self-referencing) sets, such as "the set of all
normal sets“.
Together with Whitehead in Principia Mathematica he founded
mathematics on that new set definition.
They hoped to get self-consistent and logically coherent system …
82
RUSSELL'S PARADOX (5)
… However, even before the project was complete, Russell's
expectations were dashed!
The man who showed that Russell's aim was impossible was Kurt
Gödel, in a paper titled "On Formally Undecidable Propositions
of Principia Mathematica and Related Systems."
83
GÖDEL: TRUTH AND PROVABILITY (1)
Kurt Gödel actually proved two extraordinary theorems. They have
revolutionized mathematics, showing that mathematical truth is
more than bare logic and computation.
Gödel has been called the most important logician since Aristotle.
His two theorems changed logic and mathematics as well as the
way we look at truth and proof.
84
GÖDEL: TRUTH AND PROVABILITY (2)
Gödels first theorem proved that any formal system strong enough
to support number theory has at least one undecidable
statement. Even if we know that the statement is true, the
system cannot prove it. This means the system is incomplete.
For this reason, Gödel's first proof is called "the incompleteness
theorem".
85
GÖDEL: TRUTH AND PROVABILITY (3)
Gödel's second theorem is closely related to the first. It says that no
one can prove, from inside any complex formal system, that it is
self-consistent.
"Gödel showed that provability is a weaker notion than truth, no
matter what axiomatic system is involved.
In other words, we simply cannot prove some things in
mathematics (from a given set of premises) which we
nonetheless can know are true. “ (Hofstadter)
86
TRUTH VS. PROVABILITY
ACCORDING TO GÖDEL
After: Gödel, Escher, Bach - an Eternal Golden Braid
by Douglas Hofstadter.
87
TRUTH VS. PROVABILITY
ACCORDING TO GÖDEL
Gödel theorem is built upon Aristotelian logic.
So it is true within the paradigm of Aristotelian logic.
However, nowadays it is not the only logic existing.
88
LOGIC (1)
The precision, clarity and beauty of mathematics are the
consequence of the fact that the logical basis of classical
mathematics possesses the features of parsimony and
transparency.
Classical logic owes its success in large part to the efforts of
Aristotle and the philosophers who preceded him. In their
endeavour to devise a concise theory of logic, and later
mathematics, they formulated so-called "Laws of Thought".
89
LOGIC (2)
One of these, the "Law of the Excluded Middle," states that every
proposition must either be True or False.
When Parminedes proposed the first version of this law (around
400 B.C.) there were strong and immediate objections.
For example, Heraclitus proposed that things could be
simultaneously True and not True.
90
NON-STANDARD LOGIC
FUZZY LOGIC (1)
Plato laid the foundation for fuzzy logic, indicating that there was a
third region (beyond True and False).
Some among more modern philosophers follow the same path,
particularly Hegel.
But it was Lukasiewicz who first proposed a systematic alternative
to the bi-valued logic of Aristotle.
91
NON-STANDARD LOGIC
FUZZY LOGIC (2)
In the early 1900's, Lukasiewicz described a three-valued logic,
along with the corresponding mathematics.
The third value "possible," assigned a numeric value between True
and False.
Eventually, he proposed an entire notation and axiomatic system
from which he hoped to derive modern mathematics.
92
NON-STANDARD LOGICS
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
Categorical logic
Combinatory logic
Conditional logic
Constructive logic
Cumulative logic
Deontic logic
Dynamic logic
Epistemic logic
Erotetic logic
Free logic
Fuzzy logic
Higher-order logic
Infinitary logic
Intensional logic
Intuitionistic logic
Linear logic
•
•
•
•
•
•
•
•
•
•
•
•
•
•
Many-sorted logic
Many-valued logic
Modal logic
Non-monotonic logic
Paraconsistent logic
Partial logic
Prohairetic logic
Quantum logic
Relevant logic
Stoic logic
Substance logic
Substructural logic
Temporal (tense) logic
Other logics
93
NON-STANDARD LOGICS
http://www.earlham.edu/~peters/courses/logsys/nonstbib.htm
http://www.math.vanderbilt.edu/~schectex/logics/
94
The Limits of Reason - G J Chaitin
The limits of reason
Scientific American 294, No. 3 (March 2006), pp. 74-81.
Epistemology as information theory: from Leibniz to Ω
Collapse 1 (2006), pp. 27-51.
Reprinted in Teoria algoritmica della complessità, 2006.
Meta Math!
first paperback edition
Vintage, 2006.
Speculations on biology, information and complexity
Bulletin of the European Association for Theoretical Computer
Science 91 (February 2007), pp. 231-237.
95
Meaning (1)
All meaning is determined by the method of analysis where the
method of analysis sets the context and so the rules that are
used to determine the “meaningful” from “meaningless”.
C. J. Lofting
96
Meaning (2)
At the fundamental level meaning is reducible to distinguishing
• Objects (the what) from
• Relationships (the where)
which are the result of process of
• Differentiation or
• Integration
97
Meaning (3)
Human brain is not tabula rasa (clean slate) on birth but rather
contains
• behavioral patterns to particular elements of environment (genebased)
• template used for distinguishing meaning based on the distinctions
of “what” from “where”
• Meaning as use implies holistic rationality, and value systems
(hence ethical views) are integrated in the aims of a rational agents.
98
http://ndpr.nd.edu/review.cfm?id=12083 Notre Dame Philosophical Reviews
99
Assignments
– Assignment 2: Analyze Pseudoscience vs. Science
(in groups of two)
– Discussion of Assignment 2 - compulsory
– Assignment 2-extra (For those who miss the
discussion of the Assignment 2)
– Assignment 3: GOLEM: Analyze Three Cases of
Theory Confirmation (in groups of two)
– Discussion of Assignment 3 - compulsory
– Assignment 3-extra (For those who miss the
discussion of the Assignment 3)
100