Mathematics and Logic - Skill and Concept Development

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 15^th century. Even at the start of the 20^th 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.

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 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.

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]

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]

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.

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]

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]

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]

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

1. the algebraic way of reasoning and writing is not talked about,
2. implication rules do not appear, and
3. 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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