Chapter 6, Rule Based Reason in Mathematics
Volume 1B, Mathematics Curriculum Notes
An Unruled Origin
Decimal notation did not appear overnight. In the past four centuries, it
was invented and refined by many people along with rules for addition,
subtraction, multiplication and division. Decimal notation made the
common knowledge of arithmetic possible. Decimal notation varies from
country to country. In many countries which employ the metric system, a
decimal comma is employed instead of a decimal point.
In Europe, the algebraic way of writing and reasoning started to emerge,
not in finished form of course, about the twelfth century[1]. - that was
a slow, very slow beginning.
[1] Algebra began with examples that provide patterns for
inheritance computations. Following this the use of algebraic shorthand
notation to describe calculation and to change them developed slowly.
The full power of the algebraic shorthand notation for arriving at
conclusions was not fully recognized until the age of Leibniz and
Newton. Leibniz had the idea of a universal algebra for thought, and an
enthusiasm for it. Precursors of algebraic thought existed in ancient
times, but they did not necessarily influence the long gestation or
rebirth of the algebraic way of writing and reasoning in Europe and its
spread around the world. That the rebirth occurred in the geographic
region of Europe is an accident of history.
According to the book The
Historical Roots of Elementary Mathematics, by Bunt, Jones and
Bedient, Dover Publications Inc, New York, 1976 & 1988, the word
algebra is a corruption of the title Al-jebr w’al-muqabala of
an 820 work by Mohammed ibn Musa al-Khowarizimi. His work
amongst other contributions to mathematical thought, included examples
to teach or show the arithmetic patterns followed in the division of
inheritances according to Muslim law. (I suspect the algebraic way of
writing and reasoning with its description of calculations may be
regarded as a refinement of this demonstration of arithmetic patterns.
The proof of this suspicion is a matter for further historical inquiry
or historical hindsight.)
According to the book Algebra began
with examples that provide patterns for inheritance computations.
Following this the use of algebraic shorthand notation to describe
calculation and to change them developed slowly. The full power of the
algebraic shorthand notation for arriving at conclusions was not fully
recognized until the age of Leibniz and Newton. Leibniz had the idea of
a universal algebra for thought, and an enthusiasm for it. Precursors
of algebraic thought existed in ancient times, but they did not
necessarily influence the long gestation or rebirth of the algebraic
way of writing and reasoning in Europe and its spread around the world.
That the rebirth occurred in the geographic region of Europe is an
accident of history.
From then on, algebraic ideas developed, but the rules of procedure were
not clear. Yet the algebraic reasoning including slope calculation
suggested new results and calculations in mathematics and the sciences.
Some were clearly true, some were clearly false and some were doubtful:
more could be stated or suggested than proven with algebra. The
imprecision in algebraic thought was a source of concern. Not all was
certain. In contrast to Euclid’s work on geometry, algebraic thought was
not a model for reason[2]. Despite the latter,
algebra along with say physical intuition provided methods for arriving
at conclusions, where none existed before.
Axiomatic Codification
From the mid-nineteenth century, efforts mostly in Europe began to make
the reasoning in the applications of algebra (including calculus) more
certain like that in Euclid’s work on geometry. Briefly put, from the
mid-eighteenth century to the 1920’s, the formulation of rules for
arithmetic, more precisely for set formation, provided a more certain,
thought-based, set-theoretic foundation [3] for
the algebraic way of arriving at conclusions about sets, numbers and
calculations, and also for geometry.
[3] Technical Note:
Fraenkel in the 1920 modified the 1905 set-theoretic assumptions of
Zermelo. This modification defines what is now called Zermelo-Fraenkel
set-theory foundation for mathematics. Minor variants of this
foundation and some alternative codifications have been considered
since. For a more precise image, advanced college students in
mathematics may consult the non-technical and historical passages in
the work The Logical Foundations
of Mathematics by W. S. Hatcher, 1982, Pergamon Press, ISBN
0-08-025800-X.
The foundation is based on several assumptions, also called axioms, about
set existence and formation. Today, there is a set or arithmetic-based
algebraic approach to geometry that is intellectually more certain than
the treatment of geometry in the work of Euclid. The standards of reason
or persuasion in mathematics have been raised and made more strict with
the passage of time and the accumulation of knowledge of what to do and
avoid.
Thus previously unruled exploration and command of algebraic thought was
axiomatically codified and reorganized in a logical, rule-based fashion.
This rule and thought-based codification further includes methods for
arriving at conclusions used only in mathematical disciplines. The
principle or method of mathematical induction provides a first example
[4].
[4] Further examples are given by
the axioms of
choice
The codification provides a restrictive framework for algebraic and
mathematical thought, less free but more certain or reliable than before.
This almost legal codification provided a more strict logical or
thought-based construction and organization for pure mathematical
knowledge. Logos is a Greek word for thought.
Within the codification, whole numbers, rational numbers and real numbers
are precisely represented by sets, and not by decimals. In the
codification, decimals have no special role nor place. The decimal
representation of numbers can be introduced and their properties derived
from the assumptions. But strange as it may seem, the framework allows
for a precise decimal-free discussion of computations.
The need or desire for precision in the logical codification led to the
writing and selecting of definitions, assumptions and associated chains
of implications, removed from everyday thought and language and remote
from the primary or common knowledge of mathematics. Ease of exposition
and the extension of the common knowledge of mathematics were not
concerns of the codification. Finding a logical, thought-based
sanction for mathematics was[5].
[5] In retrospect, what was found was not
absolute sanction, but a codification, a thought-based framework.
The algebraic way of writing and thinking is present in
all mathematics after arithmetic. Yet in contrast to the wide reach of
algebraic thought, the subject of algebra in mathematics today is more
focused. In college, the study of algebra may drift from the high
school description of operations on real numbers to a description or
study of the various forms of rule-based calculations. Here there is no
concern for the type of numbers, arrows or quantities etc., which
appear in those calculations. The guiding questions in algebra in the
late nineteenth and most of the twentieth centuries have concerned (i)
what is implied by the specification of rules or properties for
operations, and (ii) what kind of objects or numbers, if any, would
satisfy the specified rules or properties.
Before the nineteenth century, algebraic thought and
algebra included calculus and the equations of all mathematical
disciplines – everything that was not in geometry and not in the
recently developed decimal arithmetic. But the concern about the
justification for calculus (slope) related computations and the concern
of how to speak about sets, while avoiding inconsistencies or
contradictions, led to the birth of analysis, a subject separate from
algebra. Within this new subject of analysis, the codification has
yielded a rule-based foundation for the precise description and
handling of set and number based operations. For further information on
the division of mathematics, see the VNR Concise Encyclopedia of
Mathematics. See also the earlier discussion of what is
algebra.
Postscript (Oct 1, 2006): Three Kinds of Reason in mathematics
There appears to be at least three kinds of reason in mathematics: (i)
Pattern Recognition, (ii) Use of methods with repeatable and
reproducible, and thus verifiable results; and (iii) deductive based
chains of implication rules, direct or indirect.
- The recognition of patterns, geometric or arithmetic, is part of
elementary mathematics. There we use patterns. A pattern is tested by
seeing whether or not it fails. If it succeeds we tend to use. If it
fails we reject. Inductive (hands-on, constructive) instruction in
elementary school appears to be based on offering students patterns to
recognize or construct, and then test in the way just described.
- The use of arithmetic and then deductive methods with repeatable and
reproducible results is a basis for learning by rote or with
comprehension in mathematics. Results which are repeatable and
reproducible are verifiable. That leads to confidence in the methods. If
a method does not lead to repeatable and reproducible results, the method
or the user's mastery of it is false.
- The use of deductive reason in mathematics requires two gambles. The
first gamble lies in the assumption of deductive logic, the assumption
that chains of reason, direct or indirect, can be employed to imply
results in a repeatable and reproducible manner. The second gamble lies
in assumptions about numbers. their use and properties, and in the case
of applied or mixed mathematics, their connection to geometry and/or
physics. Every theory provides a good yarn or story or piece of theatre
to follow to fictional or non-fictional conclusions. Theories are refuted
(falsified) when their consequences fail. That being said, tried and
tested, arithmetic, geometric and deductive methods of mathematics
appear to give repeatable and reproducible results.
Mathematics education is a multi-faceted entity. The formal statement of
axioms (arithmetic or geometric patterns) in terms of letters and symbols
, and the consequences of those axioms assume logic and the shorthand
role of letters and symbols in arriving at conclusions, arithmetic,
geometric or deductive. Mathematics instruction need not be rushed.
Students need to learn carefully how to use rules and patterns, or
follow methods, one at a time and one after another, alone or in
combined, to arrive at arithmetic, geometric or formally deductive
results and conclusions. Once students have acquired the patience and
ability to follow or apply methods in a repeatable and reproducible
manner, they are ready to combine methods to arrive at further methods,
and they are ready, patience permitting, to follow theories or
explanations in which understanding why is based on combining methods and
patterns (axioms or assumptions included) to provide explanations. Yet
the ability and patience to carefully follow the steps of methods ,
one step at a time and one after another, with an awareness that an error
in one step leads or most likely leads to incorrect (non-reproducible)
results is required for at least two of the three kinds of reason
mentioned above.
End of Postscript.
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Calculus Starter Lessons
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Learning to do and high marks if it comes to easy is often
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