Chapter 14, Deductive And Empirical
Views of Mathematics
This chapter provides several perspectives on mathematics. Some are
slightly at odds. Some are slightly technical. The next chapter
Objective Processes returns to some simpler material.
Empirical Origins of Decimal Arithmetic
The rules of arithmetic and our notation for fractions and decimal
numbers which we use today were created about three hundred years ago.
The popularization of decimal notation began with Simon Stevin (1548
-1620 A.D.) Before the use of decimal notation, our forbears (except
those using the abacus) found arithmetic operations of +, -, ×, and ÷
very awkward to master. Knowledge of arithmetic, like literacy, has
gradually become more widespread since the 15th century. Even
at the start of the 20th century, few people could read, write
and figure. Public education has changed this situation in many
communities.
The rules of addition, subtraction, multiplication and division with
decimal notation had to be discovered or invented. In all this, trial and
error or experimentation, was used to formulate the rules and even the
notation for arithmetic. That is, the rules and methods of arithmetic,
taught in elementary school, are human creations. Despite this, they
work: The results obtained from each arithmetic operation (+, -, ×, and
÷) are reproducible and supposedly not dependent on whom obtains them.
Arithmetic methods were empirically discovered and established. These
methods were invented and then used to solve problems in business and
geometry. Reproducible and repeatable results led to a wide, if not
universal, acceptance of the methods. Calculations, precisely described,
are reproducible.
Note that arithmetic yields an alternative approach to geometry. The use
of coordinates to identify points in the plane, in fact the first
quadrant, by Descartes (1596-1650) eventually led to a geometry based on
rules of arithmetic instead of the assumptions of Euclid. Today, the two
perspectives are often mixed – a departure from the ideal of having only
one basis for geometry. The original approach of Euclid is now labeled as
synthetic geometry while the arithmetic-based approach is labeled
analytic geometry.
Set Theory and Euclidean Model (for Reason)
for a or the deductive codification of mathematics
Philosophers and pure mathematicians are aware of the human origin and
growth of mathematics. In past centuries, the rules and patterns followed
in mathematics were invented and verified in imaginative ad hoc ways —
the reliability of the rules and patterns discovered was sometimes
unclear. In everyday speech, more can be suggested, said or imagined than
proven. Similarly, in mathematics the algebraic way of writing allows
more to be written than shown. This leads to statements for which proofs
of truth or falseness may be of interest and a challenge.
In the middle and late 19th century, members of the then small
mathematical community began to look for a more certain and more rigorous
foundation for the description and manipulation of calculations. In
accordance with the Euclidean Model for Reason, the ideal
foundation consists of a few simple, clear principles on which the rest
of knowledge could be built via rigorous, that is, firm and reliable,
thought free of contradictions. But what assumptions should be made and
what operations should be allowed in mathematical reasoning was not
clear. Despite this, a firm, or nearly firm foundation
Zermelo-Fraenkel set theory for mathematical computations, that
is, arithmetic (analysis, advanced calculus) was formulated around the
period 1905-1920.
The theory itself is almost an accidental outgrowth of investigation by
Georg Cantor (1845-1918) and others of the set concept and what
set-formation rules should be permitted. Too much freedom (too freely
adopted methods for set formation) led to paradoxes.
Prior to the development of set theory, Giuseppe Peano (1858-1932) had
given axioms for the whole numbers. From the whole numbers with the aid
of coordinates (ordered pairs of numbers) and the idea of equivalent
ordered pairs, can be successively developed the integers, rational
numbers (signed fractions), the real numbers and the complex numbers.
Beyond this, the representation of functions by their graphs, here sets
of ordered pairs, implied mathematical operations could be represented
within set theory.
With a selective assumption of rules for set formation, the set
theoretic representation (codification) of ordered pairs and Peano's
axioms for whole numbers become feasible, and this in turn implies a
set theory basis for arithmetic with whole numbers, integers, rational
numbers, real numbers and complex numbers.
The set theoretic foundation relied on the thought-based methods of
logic, that is, on algebraically written rules and patterns, and not on
arguments employing physical concepts or diagrams. The movement from a
previous reliance on diagrams and physical concepts and geometry concepts
or intuition represents the assumption that the most secure chains of
reason are based on the rules and properties of arithmetic or sets.
The Zermelo-Fraenkel set theory provides an axiomatic (assumption and
logic-based) treatment of numbers and arithmetic computations.
These axioms and their arithmetic consequences have no reliance on
geometry or physical arguments or motion. This theory is built on written
implication rules, direct and indirect chains of reason and the algebraic
way of writing and reasoning.
The foundation of knowledge and mathematics was also a concern of
Gottlob Frege (1848-1925), Bertrand Russell (1872-1970) and Alfred
North Whitehead (1861-1947). Ernst Zermelo lived from 1871 to 1953.
Abraham Fraenkel lived from 1891 to 1965.
The development of the set theoretic approach was motivated by the need
to provide an objective, thought-based organization for mathematical
knowledge and arguments with the fewest possible assumptions to avoid
contradictions. The set theoretic foundation gives a framework, a
starting point which is more strict, sure and rigorous than in previous
centuries for mathematical computations. [2]
[2] The set theoretic foundation of mathematics after arithmetic is a
human creation or discovery. There are alternative axiomatic
foundations (frameworks) for mathematics, but I am not familiar with
their details and so cannot discuss them further.
On this set theoretic foundation, mathematical conclusions about sets and
computations can be derived through long chains of deductive reason.
Before and After Set Theory in pure Mathematics
Before the advent of the Zermelo-Fraenkel basis for arithmetic,
mathematics could be viewed as two subjects: geometry and algebra.
Algebraic thought included arithmetic and computations of all forms.
Since the advent, mathematics may be regarded as three subjects:
geometry, algebra and analysis. The modern subject of algebra has been
restricted to a study of the form of computation and of what kind of
objects, if any, will satisfy a given set of rules for computation. The
modern subject of analysis examines numerical computation within the set
theoretic framework for arithmetic.
The Spread of Set-Theory
In universities, the set theoretic foundation for mathematics went (see
[3]) from being a curiosity in the 1920s to an
essential part in the 1950s – more and more areas of mathematics were
viewed from a set theoretic view. And in the 1950s and 1960s, this
approach spread from the universities into high schools.[3]
[3] According to the article Development of Modern Mathematics, an
overview, by R. L. Wilder, in the book Historical Topics
for the Mathematics Classroom, edited and written by J. K.
Baumgart et al. The book was published by the National Council of
Teachers of Mathematics, 1969, second edition 1989, 1906 Association
Drive, Reston, Virginia, USA 22091.
Godel's Disappointment
Zermelo-Fraenkel set theory with its rules (assumptions, restrictions) on
what sets could be formed from others prevented the appearance of
paradoxes, those known at the time of its construction. From the 1920s to
the present (1995) no further paradoxes or contradictions have found.
Thus the theory has stood an empirical test of time for over seventy
years. But this does not guarantee in a thought-based manner that
paradoxes or contradictions within this set theory framework will not be
found.
The reliability and logical self-consistency (absence of contradictory,
mutually exclusive results) of a framework for mathematics was one of the
guiding problems for the mathematical sciences posed by David Hilbert
(1862-1943) at the turn of the 20th century. But in the 1930s, Kurt Godel
showed that a proof of the consistency was not possible in the sense
wanted. In a rule-based system for mathematics large enough to include
counting with the whole numbers, the consistency of the system could not
be deduced, that is, concluded from a chain of implications. Consistency,
the absence of contradictory, mutually exclusive assertions, could not be
proven within such a system.
The foregoing represents an approximate phrasing of Godel's
results.
Godel's conclusion represents a loss for the Euclidean ideal for
reasoning with certainty from first principles: axioms or assumptions.
Our assertion that certain axioms or assumptions could be taken as
self-evident was not enough to guarantee consistency in any framework for
mathematic rich enough to include the whole numbers. This implies perhaps
that the Zermelo-Fraenkel set theoretic framework for mathematics is
presently just a very refined branch of empirical art or science.
Mathematics, the queen of science, is thus but an aloof, yet very
thoughtful, commoner.
Euclidean Model for Physics
Full Axiomatic Foundation or Codification Unlikely
Hilbert also posed the problem of the axiomatic foundation of physics.
The solution is not near. Advances in physics – a description of nature —
have come from experiment and suggestive calculations. These calculations
represent patterns (islands of knowledge) that work well, but
comprehension of why is inadequate. Physics and much of technology form
collections of patterns that work. These subjects remain far from a
complete axiomatic organization in which all their results can be deduced
rigorously from a few clear principles. Islands of knowledge remain.
Operational Views
Today with the aid of electronic computers, engineers, physicists and now
numerical analysts explore, refine and invent mathematical calculations.
They may try to find reproducible calculations in accord with experience.
The calculations are found or discovered through a mixture of deductive
reason, knowledge of the efforts of others, and trial and error
(guesses). This yields an empirical knowledge or view of computation.
This empirical approach to computation may use deductive reason where it
can, and trial and error, after that. Here engineers and physicists,
sometimes using and sometimes in ignorance or defiance of the rigorous,
rule-based parts of mathematics, often find calculations that work. See
[4]
[4] Anyone applying mathematics in their specialty is looking for the
simplest mathematics that can be employed to solve their problems, but
the search for the simplest solution may require a vast knowledge of
mathematics. The simplest solution method that works may be explained
to people with a minimal mathematical background, but such a background
may not be enough to find it in the first instance – exceptions here
are always possible.
Within the empirical approach just described, never-disobeyed rules (or
rules we have confidence in) may be combined with less sure rules to
propose methods that might work. The reliability of a proposed method can
then be examined and the circumstances in which it works or it fails can
be recorded. Of course, reason should be based on the most reliable
implication rules.
In the application of mathematics to other fields, physical observation
and physical theories may suggest or imply the initial equations.
Physical arguments and observation are needed to obtain the equations of
science and technology, but after the formulation of these equations,
their solution and manipulation (the accounting) should rely only on the
more secure assumptions about arithmetic.
To find or determine the motion of objects observed in nature may be the
motivation for constructing an equation. But the construction process may
be inexact. Due to this inexactness, whether or not an equation has
solutions is a question whose answer may be suggested by physical
consideration, but not guaranteed. In contrast to mathematicians,
engineers and scientists, through exposure to calculations that work, may
believe that modeling a system or physical process by equations is
sufficient to guarantee the existence and computability of a solution.
None the less, engineers, scientists and applied mathematicians may
proceed by trial and error; and in this use any equation they have
constructed as a guide. This trial and error may fail or may succeed and
go ahead of the rule-based reason, the formal justification that
might provide a place for its successes in the set theoretic, rule-based
framework for mathematics. In previous centuries and in the absence of
the set-based framework for arithmetic and computation, calculation had
to proceed in a manner secured by geometric arguments, by physical
considerations or by more speculative means.
The discovery or formulation of schemes for the solution of the equations
or problems may be guided by physical considerations or expectations. Yet
the use of physical arguments and mathematical approximations to draw
conclusions of a mathematical nature is suspect — represents weak links
in the chains of reason followed. The best that can be done is to
recognize the weak links and look for replacements. Again chains of
reason, even those with weak links, may provide a guide or a clue to a
more rigorous and secure approach to computations that might work. To
minimize uncertainty, a solution method and the justification of its
steps should depend as much as possible on rigorous mathematical
arguments or accounting. But an engineer or scientist whose calculations
have given results in accord with observation and measurement may feel
that the links, weak or not, in the chain of reasoning are strong enough.
They worked. From this limited (and sometimes very practical)
perspective, there is no immediate need to look for and replace the
weakest links.
Apart from Set Theory
Applied Mathematics andElectric Circuit Theory apart
from pure Mathematics.
A body of mathematical knowledge, a growing acquaintance with
calculations and mathematical reasoning that worked and some which did
not, has been found or built before and besides a single axiomatic
foundation for mathematics. Mathematical knowledge in the 19th century,
pure and applied, was a patchwork of assumptions and consequences. This
patchwork has been extended in the 20th century despite and besides the
presence of the set theoretic framework for mathematics. Three examples
will be briefly given. A more detailed description would require a
knowledge of calculus.
In the 1930s for the study of electricity, Heaviside developed a useful
tool, the concept of a generalized function. His description of it was
apart from the set theoretic concepts and the analytic reasoning then
favored by mathematics. But his use of it worked well. In the 1950s, a
cleaner mathematical theory of generalized functions was developed.
Heaviside's calculations or calculus was essentially incorporated into
and justified within the set theoretic framework – albeit users of
Heaviside calculus, that is, faculty members in electrical
engineering departments, saw no immediate need to master a complicated
mathematical justification for calculations that worked and were familiar
to them. This attitude has slowly faded away with the passage of time,
the retirement of faculty, and the mathematical training of younger
engineers by mathematics departments.
Since the 1920s, the computations of quantum physics have provided
another example of applied mathematics. Physical consideration and
heuristic reasoning led, by trial and error, and the elimination of
approaches that did not work, to equations of motion for
subatomic particles. The physicists developed their own rules of
calculation. They obtained calculations that worked well. Predicted
values of constants agreed amazingly well to those measured to several
decimal places. Quantum physics or mechanics has gone through few
conceptual transformations. With each transformations, theoreticians have
decried that the reasoning that led to some of their previous
computations was fortuitous. So apparently there is a collection of
computations that work, but no strict thought-based framework for it.
Suffice to say that quantum mechanics is still a mysterious subject with
a patchwork of rules for computation that have not (to the best of my
knowledge) been organized in a fully coherent, Euclidean axiomatic
fashion.
Since the 1950s and the advent of the electronic computers, engineers
have developed so called Finite Element computer models for
ships, planes, buildings, etc., via numerical experimentation along with
corroboration via observation. Here, computations were conceived and done
without a rigorous mathematical justification. Mathematical theories have
been develop since the 1960s to offer some refinement or justification.
No Decimals
Within Zermelo-Fraenkel set theory, the decimal representation of real
numbers is not special. There is no direct motivation for decimal
notation or base ten. The discussion of convergence, continuity and
accuracy of computations does not require and can be made independently
of the decimal representation of real numbers. The Zermelo-Fraenkel set
theoretic framework for mathematics provides a path for the comprehension
of mathematics with no reliance on decimal arithmetic, geometry nor
physical senses and abilities – apart from the ability to put pencil or
pen to paper to record chains of reasoning and figuring.
Yet description of the set theoretic foundation of mathematics without
any mention of decimals has separated the higher mathematics from the
common knowledge of arithmetic acquired in elementary school. This has
been perhaps one barrier to the simple explanation of mathematics beyond
the common knowledge or memory of arithmetic, counting and the use of
simple formulas. Our attention now turns to the views of mathematics
found in the classroom. Some critical observations follow. The companion
work Mathematics Curriculum Notes echo them and further offer a
philosophy for mathematics education.
In The Classroom (1995)
Mastery of arithmetic, geometry and further mathematics has been regarded
as a sign of learning and possibly intelligence. Essentially, all
education in mathematics is ended by the completion of schooling or by a
failure or near-failure in a course. The latter leaves a bad impression
of mathematics or one's own ability.
Since people stop their mathematics education at different levels,
mathematics instructors socially fall in the class of people who hear
admissions or confeesions of the form: I hated math or I
liked or studied math until such and such a level. [5]
[5] In college, students typically identify themselves by their field
of studies. In one encounter, I indicated to a new acquaintance John
that my field was one that resulted in my hearing confessions. He was
left to guess the field. Amongst the choices were counseling,
psychology, religious ministry, law and so on. John felt that I had
misled him with the hint I heard confessions. But several minutes
later, after one or two others topics of conversation, John suddenly
indicates that he was good in mathematics until the end of high school.
This was the confession. The hint that I heard confessions was not
misleading.
Ideas described at the end of a math course are typically not
understandable to students until a few moments before their presentation.
This observation applies to courses at all levels in mathematics from
elementary school to university. Taking a course in mathematics is for
many like following a trail with the hope of being able to comprehend
what appears just around the next corner. That has been an insurmountable
obstacle for the explanation and comprehension of mathematics and reason.
In elementary school, instruction is concrete and reassuring.
Explanations of counting, figuring, measurement and geometry may
encourage students to use physical objects and arguments to understand
arithmetic and its basic applications. Confidence may be based on methods
with repeatable and reproducible, and therefore verifiable, results.
Arithmetic methods along with some descriptive geometry [6]
[6] The recognition of simple geometric shapes: circles, rectangles,
squares, pyramids, etc; and the use of standard formulas for
perimeters, areas and volumes.
provides a first most certain view of mathematics. It is the basis of
what was the common knowledge – today some students may forget their
arithmetic knowledge as they pass through high school. And a remedy for
the latter may lie in some drill and repetition.
Most people succeed in mastering figuring with decimal numbers and the
use of simple formulas. These skills are acquired in elementary school
and possibly reinforced at the start of high school, that is, secondary
education. Beyond these skills comes a knowledge of algebra, logic, trig
and/or calculus. In North American high schools, logic unfortunately has
been taught in mathematics only, say in algebra and geometry courses, and
not else where.
Modern Math Curricula
During the late 1950s and mid-1960s modern mathematics instruction in the
form of an axiomatic approach to algebra and geometry, and the
explanation of sets, appeared in high school classes. In contrast to
elementary instruction, in which
- the algebraic way of reasoning and writing is not talked about,
- implication rules do not appear, and
- physical objects and examples were used to introduce and repeat
ideas,
modern mathematics instruction puts aside the use of physical
objects. Instead, it was based on logic, more precisely, the concept of
derivation from first principles or axioms, on a set-based description of
topics, and on a mastery of algebraic reasoning. But the latter, an
algebraic way of writing and reasoning, has been employed in math classes
through generations of students and teachers without a direct
explanation, apart from a few paragraphs in texts to unclearly introduce
the notion of a variable. My book " Three Skills Leading to Algebra"
offers a remedy.
Postscript (September 2006):
(1) The modern mathematics curricula, circa late 1950-80, introduced the
advanced set theory view of pure mathematics in the classroom. While
students learn to count and do arithmetic with the decimal representation
of whole numbers, fractions and/or reals, exactly or approximately, the
axioms for modern mathematics did not mention and hence did not sanction
the decimal representation of numbers, whole to real. The modern
mathematics view of calculus with its epsilon-delta codification of
limits, continuity and convergence was to abstract even for many advanced
students - those that did well in high school and college mathematics.
The modern mathematics curricula provided logical developments with steps
too large or too hard for most to follow or take. The modern mathematics
curricula assumed but did not support the common knowledge in that the
axiom given were not linked to the prior knowledge or experience of
students and teachers, that acquired in say primary school. Masters of
high school and university modern mathematics curricula (analysis
included) may found themselves with two nearly separate views of the
subject - the practical skills that support the common knowledge, and a
theoretical view in the common knowledge of decimals plays no part. All
the foregoing is in addition to the lack of a clear development for
students of the algebraic way of writing and reasoning. Explore the rest
of this site for remedies. Bon Appetit.
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Teachers & Tutors: Site pages offer better or best practices for providing skills -
simpler than expected & comprehensive but for exercises. For your charges, your duty is to study them alone or in
groups and develop skill building exercises & activities to share. Start now. The effort here is the best I can do.
Others are welcome to refine or exceed it. Please do.
Secondary
Mathematics for Ages 11+, A Practical Approach for home-tutoring or -schooling, or for schools & colleges
with local curriculum control. Study how to include site content - its skill development how-TOs and innovations
into present or future lesson plans - some reading required.
Road
Safety Messages and Questions: When and why should you face
traffic when walking along a road or cycle path? Is it a good
idea to hang limbs outside of cars etc? What gives more
protection in a crash: a car, motorbike or bicycle?
See too, the BBC-Belgium story Texting and
Driving - texting & the impossible test - the article links to a gruesome utube video on the subject
The Logic of Injustice:
How Texas sent
an innocent man to his death - The wrong Carlos. Some judgments are irreversible. Procescution: Where and when prosectors play to win rather than for
justice, guilt beyond a reasonable doubt goes unrespected due to prosecutors who putting winning
first, those innocence before the law may be convicted. Some procescutors offices in continuing to accuse after a pardon
due to reasonable doubt or innocent being shown, may sucessfully oppose compensaton for false convictions
by asserting a pardon individual is still under suspicion. Then the pardoned individual or the latter's estate
is not compensation for years or decade
of improper or false imprisonment, or for execution. Site chapters on Logic
and some in Pattern
Based Reason may slowly lead to greater precision in reading, applying and
writing laws.
May 2012, Composition Starting:
Pre-School and Primary Mathematics - Quantitative Skills, An
Intellectual View, Feedback Welcome:
The 8 Most Popular Site Inlinks
Parent Center: Help your child or teen
learn:
Parent-friendly
Work Booklets for ages 3+ to 13 Use these or others to check
or build skills. Other booklets are available but these booklets
allow parents unsure of themselves in mathematics to help their
children. The selection acquired in Canada is published in the
USA. So it has a US orientation. In retrospect, the selection
shows parents what to check with the booklets or by other ways,
the choice is theirs. But in retrospect, the selection does not
cover integral and fractions liquid weights and measures - ask
the publishers to correct that! For ages 9 to 12 say, parents may
compensate by showing boys and girls how to use weights or mass,
and further measures in food preparation. Beyond that children
may be shown how to measure and calculate angles, lengths and
areas [proportional amounts too] directly or by using maps and
plans drawns to scale. Learning how to gather and measure all the
ingredients, pots and pans for a dish or a meal, along with
cleaning up sets the stage for like activities or experiments in
science courses, and in developing organizational skills,
gives boys and girls a head start. Good luck. At the other
extreme, more comprehensive than light, if your motto is
McCainian: drill, drill, drill then Toronto
mathematician and actor John Mighton's jump math organization has jump math
workbooks for at least grades 3 to 8 for at-home and in-school
use - training sessions for teachers available. Jump math has
been expanding to cover older students. Jump Math Samples: plus
Fractions for
Grades 3-4 & Grades 5-6 [Read] Free Resources grades 1 to 8
[unread - likely to be good]. and
Mathematics
Skills For Ages 3 to 14 - technical!
Skills with take
home value - A few ideas
Basic skills include
time-date-calendar Matters; money matters; map, plan and
scale diagram matters;counting, measuring and figuring;
decision making with logic and likelyhood; being careful and
being aware of the domino effect of mistakes; reading and
writing with precision.
Is your child able to add, subtract and multiply amounts
of money, work with fractions, work with clocks and calendars,
work with maps and plans, and measure length, weight-mass and
volume? Schools may promote your son or daughter without
providing basic skills in reading, writing and
arithmetic.
Arithmetic
and Number Theory Skills
Algebra
Starter Lessons
Geometry
- maps plans trigonometry vectors
More
Algebra
70
Calculus Starter Lessons
Calculus Lessons Elsewhere:
-
How to Ace Calculus: Street Wise Guide - Mostly
Text.
-
Flash
Video for Calculus Phobics
They cover basic topics in ways likely to complement your
notes, your textbooks and site material. When Goldilocks
trespassed in the house of the three bears, she found three bowls
of porridge, two not to her liking, and one just right. Different
bears have different tastes. As invited guest here and elsewhere,
if one or more explanations is not to liking, try another. It may
be better or just right.
Unsolicited Advice
Learning to do and high marks if it comes to easy is often
deceptive - light rather than deep. For that reason, students
with learning difficulties determined not to let it get in their
way may go deeper and farther than those with none. High marks,
if the come easy, may be deceptive - provide a too light and not
a deep mastery. That could have been your problem in secondary
school, one that leads to comprehension shock or difficulties in
calculus and more generally in the first year of college. Bon
Appetite.
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