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

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 TruthCan the question of the existence of God, or Creator,

be settled through logic?

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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• 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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Classical Logic and MathematicsMathematics and logic always have been

intertwined. The study of logic is considered to be a

branch of mathematics. 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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Mathematics establishes the truth of a theorem by constructing a proof.

Consider, 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.

Proof of Bhaskara (12th century Hindu mathematician)

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a

bc

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

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c

c

c

c

aa - 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 .ab21

ab214

The area of the inner square region is . 2ba

2ba 2c

222 22 cbabaab 222 cba

b

b

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Aristotle (384-322 B.C.)

Source: www.uah.edu/colleges/liberal/philosophy/heikes/302/time/rembrant/aristotle-homer.jpg

• 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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Three Basic Aristotelian Laws

• The law of identityA thing is itself.

• The law of excluded middleA statement is either true or false.

• The law of non-contradictionNo statement is both true and false.

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• 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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• 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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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”?

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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.

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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George Boole

The success of Boole’s work led to the development of symbolic logic.

• 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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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

The History of Logic

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Classical Logic(300 BC to mid 1800’s)

Symbolic Logic(mid 1800’s to 1931)

Modern Logic(since 1931)

Aristotelianlogic

EuclideanGeometry

Algebraof Sets

Godel’sTheorem

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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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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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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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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Implications of Gödel’s TheoremGödel’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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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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The Value of LogicIf 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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• 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.

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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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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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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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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The Scientific Method(an idealization of the process used to discover or construct new knowledge)

1. Recognition and formulation of a problem2. Construction of a hypothesis3. New predictions4. Unbiased and reproducible tests of new predictions5. Modification of hypothesis6. Hypothesis passes many tests and becomes a theory7. Theory continually challenged and re-tested for

refinement, expansion, and/or replacement

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Science, Nonscience and PseudoscienceMany 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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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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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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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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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.

? Is Sx

, If Sx ion.contradict a ,then Sx

, If Sx ion.contradict a ,then Sx

Let .SxxS :

The “set” paradox

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The barber’s paradox

http://www.google.com.ph/imgres?imgurl=http://content.artofmanliness.com/uploads/2008/05/barber3.jpg&imgrefurl=http://artofmanliness.com

In a certain town, the

barber shaves those and only those who don’t

shave themselves.

Who shaves the barber?

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.

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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.

Zeno’s Paradox (baffled mathematicians for 2,000 years but

resolved already today)

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?

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.

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The end!!

!