UNIVERSITY OF THE PHILIPPINES (LOS BAÑOS) MATH 1 - QUANTITATIVE REASONING

2. We describe the inter-relationships among logic,
mathematics and science, which open the way to
understanding the scientific method, the
principal means by which knowledge is
acquired today.
Hopefully with this,
you will be skeptical
about the untested claims of
pseudo-science.
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1.3 The Search For Truth And Knowledge

3. William Paley (Natural Theology, 1802)
“Suppose I pitch my foot against a stone, and were asked
how this stone came to be there; I might possibly answer,
that for anything I knew to the contrary, it had lain there
forever… But suppose I had found a watch upon the
ground, and it should be inquired how the watch happened
to be in the place; I should hardly think of the answer
which I had given before, that, for anything I knew, the
watch might have always been there… The inference, we
think, is inevitable; that the watch must have had a maker.”
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1.3.1 The Search for Truth
Can the question of the existence of God, or Creator,
be settled through logic?

4. Paley’s argument:
Premise: The existence of something as
complex and functional as a
watch implies the existence of a
watchmaker.
Premise: Biological systems are more
complex and functional as a
watch.
Conclusion:
Biological systems imply the
existence of a Creator.
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5. • This provides us an example of how
philosophers have tried to use logic to
ascertain ultimate truths.
• It also demonstrates the inter-relationships
among logic, mathematics and science.
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6. Classical Logic and Mathematics
Mathematics and logic always have been
intertwined.
â The study of logic is considered to be a
branch of mathematics.
â Just as logic is used to test the validity of
arguments, mathematics is used to
establish the truth of mathematical
propositions.
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7. âMathematics establishes the truth of a
theorem by constructing a proof.
âConsider, for example, the famous
Theorem of Pythagoras (580-500 BC)
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Pythagorean Theorem. For any right
triangle whose legs measure a and b
units and whose diagonal measures c
units, a2 + b2 = c2.

8. Proof of Bhaskara (12th century Hindu
mathematician)
8
a
b
c
Given any right triangle
with bases a and b and
hypotenuse c, construct
a square whose length of
a side measures c units.
c
c
c
c

9. 9
c
c
c
c
a
a - b
The area of the outer
square region is c2.
This is the same as the
sum of the areas of the 4
triangular regions and
the inner square region.
The area of a triangular
region is .ab
2
1
The area of the inner
square region is . 2
ba 
b
b

10. 10
Aristotle (384-322 B.C.)
Source: www.uah.edu/colleges/liberal/philosophy/heikes/302/time/rembrant/aristotle-homer.jpg

11. • The similarity between logic and mathematics explains
why many philosophers were also considered
mathematicians.
• Aristotle is a well-known Greek philosopher, tutor of
Alexander the Great
- probably the first person who attempted to
give logic a rigorous foundation.
- believed that truth could be established from
three basic laws:
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12. Three Basic Aristotelian Laws
• The law of identity
A thing is itself.
• The law of excluded middle
A statement is either true or false.
• The law of non-contradiction
No statement is both true and false.
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13. • Aristotle’s laws were the basis of the logic
used by the Greek mathematician Euclid to
establish the foundations of geometry (in his
famous treatise The Elements (300 BC).
• Euclid began with only 5 postulates or
premises from which he derived all of
classical geometry, also known as Euclidean
geometry.
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14. • There is great trust in the validity of classical
or Aristotelian logic.
• This led directly to the development of the
modern scientific method and the
accompanying advances in human
knowledge and technology.
• This interplay of logic and mathematics may
have been the single greatest factor in the
rise of the Western world, beginning in the
Renaissance, as the center of scientific,
industrial, and technological development.
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15. Leibniz’s Dream
• Recognizing that logic could be
used to establish mathematical
truths, could logic also be used to
establish other truths? Could it be
used to determine “universal
truths”?
15
Gottfried Leibniz
 Leibniz (1646-1716) attempted to establish a calculus of
reasoning which can be used to decide all arguments;
suggested that an international symbolic language for
logic be developed with which equations of logic could
be written and used to calculate a “solution” to any
argument.

16. What happened with Leibniz’ dream?
• Leibniz had little progress. Real work on
creating a symbolic logic had to wait
nearly 200 years until George Boole
published “The Laws of Thought” in 1854.
• Boole tried to treat logic as a mechanical
process akin to algebra and developed the
fundamental ideas for using mathematical
symbols and operations to represent
statements and to solve problems in logic.
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17. George Boole
The success of Boole’s work led to the
development of symbolic logic.

18. • Bertrand Russell and Alfred North Whitehead in their work
“Principia Mathematica” (published 1910-1913) sought to
put all of mathematics into a standard logical form by
attempting to derive all known mathematics from symbolic
laws of thought.
• Russell hoped that this would lead to Leibniz’s dream of
creating a system of logic in which all truths could be
derived from a few basic principles.
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19. So what really became of Leibniz’s dream?
• Kurt Gödel in 1931 proved that
the dream could never be
achieved.
• Leibniz’s dream was shattered!
• But this ushered in a new period
in the relationship between logic
and mathematics, often termed
the period of modern logic.
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Kurt Gödel

20. The History of Logic
20
Classical Logic
(300 BC to mid 1800’s)
Symbolic Logic
(mid 1800’s to 1931)
Modern Logic
(since 1931)
Aristotelian
logic
Euclidean
Geometry
Algebra
of Sets
Godel’s
Theorem

21. 1.3.2 The Limitations of Logic
Gödel’s Theorem
Mathematicians believed that for an
ultimate system of logic to be realized, a
first step is to show that mathematics could
be wholly understood as a system of logic.
Only then could mathematical logic be
developed into Leibniz’s dream of a
calculus of reasoning.
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22. Mathematics as an Axiom System
David Hilbert sought to formalize
mathematics as a system in which all
mathematical truths, or theorems, could be
derived from a few basic assumptions called
axioms, by applying rules of logic.
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23. Required Properties of an Axiom System
• It must be finitely describable, that is, the number of basic
axioms should be limited.
• It must be consistent, that is, it should have no internal
contradictions (statements that are both true and false).
• It must be complete, that is, the basic axioms should allow
analysis of every possible situation.
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24. In 1931, Kurt Gödel, an Austrian
mathematician, proved that no formal
system of logic can possess all three
required properties. He proved that no
system can be simultaneously complete,
consistent and finitely describable.
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25. Implications of Gödel’s Theorem
Gödel’s theorem spawned entirely new
branches of mathematics and philosophy.
Some of its consequences are:
• Some true mathematical theorems can never
be proven.
• Some mathematical problems can never be
solved.
• No systematic approach to mathematics can
answer all mathematical questions.
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26. In other words, no absolute way exists to
define the concept of truth.
Gödel’s theorem virtually hit the nail on
the coffin of Leibniz’s dream. A
calculus of reasoning that resolves all
kinds of arguments can never be found.
Gödel’s theorem may well be one of the
most important discoveries of human
history.
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27. The Value of Logic
If no system of logic can be perfect, what
good is logic then?
• Logic allows the discovery of new
knowledge and the development of new
technology.
• Logic provides ways to address disputes,
even if it cannot always ensure their
resolution.
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28. 28
• Through logic, you can study
your personal beliefs and societal
issues.
• Logic can help you study the
nature of truth, though logic
cannot ultimately answer all
questions.

29.  Though logic alone may fail under
some circumstances, logical reasoning
is an excellent tool for understanding
and acquiring knowledge.
 Finding the proper balance
between logic and other processes
of decision making is one of the
greatest challenges of being human.
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30. 1.3.3 Logic and Science
What is science?
Lat. scientia which means “having
knowledge” or “to know”.
Science is knowledge acquired through
careful observation and study;
knowledge as opposed to ignorance or
misunderstanding.
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31. What is the scientific method?
• It is a set of principles and procedures, based on
logic, for the systematic pursuit of knowledge.
• It depends on logical analysis both in
determining how to pursue knowledge and in
testing and analyzing proposed theories.
• It depends on mathematics, not only in the close
historical ties between mathematics and science,
but also in the demand for quantitative
measures.
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32. Fact, law, hypothesis and theory
Fact - a simple statement that is indisputably or
objectively true, or close as possible to being so.
Law - a statement of a particular pattern or order in
nature
Hypothesis - a tentative explanation for some set of
natural phenomena, sometimes called “an educated
guess”
Scientific theory - an accepted (that is, extensively
tested and verified) model that explains a broad
range of phenomena
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33. The Scientific Method
(an idealization of the process used to discover or
construct new knowledge)
1. Recognition and formulation of a problem
2. Construction of a hypothesis
3. New predictions
4. Unbiased and reproducible tests of new predictions
5. Modification of hypothesis
6. Hypothesis passes many tests and becomes a theory
7. Theory continually challenged and re-tested for
refinement, expansion, and/or replacement
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34. Science, Nonscience and Pseudoscience
Many people still seek knowledge through ways that
do not follow the basic tenets of the scientific method.
Nonscience - any attempt to search for knowledge
that knowingly does not allow the scientific method
Pseudoscience - that which purports to be science
but, under careful examination, fails tests conducted
by the scientific method
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35. What distinguishes science from these?
• The central claims of non-science and pseudo-
science are not borne out in scientific tests; tests
are either unimportant (for non-science) or
biased (for pseudo-science).
• The distinguishing quality of the scientific
method is its unbiased and reproducible testing.
• As the scientific method is an idealization,
boundaries between them are not always clear.
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36. Is science objective?
• By their very definition, scientific theories must be
objective, not subject to individual interpretation or
biases.
• However, individual scientists are always biased.
• Biases can show up in the following ways:
1. Opinion may matter in the choice of the
hypothesis.
2. Commission of the so-called “scientific fraud”
in hypothesis testing.
• The scientific method allows continued testing by
many people, thus mistakes based on personal
biases will eventually be discovered.
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37. 1.3.4 Paradoxes
• A paradox is a situation or statement that seems to violate
common sense or to contradict itself.
• Paradoxes allow for the recognition of problems which may
lead to new principles, to new facts, or to a new scientific
theory.
• Paradoxes may or may not be resolved.
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38. 38
The “up-and-down” paradox
Up
Down
This person will tend to
“fall off”.
North America: we’re ok.
Australia: we’re gonna fall
off.

39. ?Is Sx
,If Sx ion.contradicta,then Sx
,If Sx ion.contradicta,then Sx
Let  .SxxS  :
The “set” paradox

40. 40
The barber’s paradox
http://www.google.com.ph/imgres?imgurl=http://content.artofmanliness.com/uploads/2008/05/barber3.jpg&imgrefu
rl=http://artofmanliness.com
In a certain
town, the
barber shaves
those and only
those who don’t
shave
themselves.
Who shaves the
barber?

41. If the barber doesn’t shave
himself, then the barber
shaves him, a contradiction.
If the barber shaves himself,
then the barber does not
shave him, a contradiction.

42. 42
The Paradox of Light
Is light a particle or wave?
In the 20th century, physicists have
shown that light is both particle and
wave.
This discovery led to the a new area
of physics known as quantum
mechanics.

43. Zeno’s Paradox
(baffled mathematicians for 2,000 years but
resolved already today)

44. Imagine a race between the
warrior Achilles and a tortoise.
The tortoise is given a small
head start. As Achilles is much
faster, he will soon overtake
the tortoise and win the race.
Or will he?

45. 

  
  

46. Conclusion
• The process of discovering new knowledge is
rarely straightforward.
• The process of resolving paradoxes can provide
insight into an idea or even lead to a new
discovery.
• The concept of truth is complex.
46

47. The
end!!!