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Computers and mathematical philosophies in educational trends.

The purpose of this article is five-fold:

1. To elaborate on mathematical pedagogy, the application of mathematics, the misunderstanding and the myth around this powerful language;

2. To explain the differences between theoretical solutions and constructive solutions;

3. To explain the influence of technology and its pivotal--in many cases, indispensable--role in teaching, and research, and solving real-world problems;

4. To give examples that show the failure of technology to solve several well-posed problems that possess (constructive) solutions, and to give examples indicating that widely-used software sometimes provides the wrong answer; and

5. To discuss the misuse of technology in teaching mathematics, a trend that unfortunately will have adverse effects on learning mathematics.

MATHEMATICS, ITS HISTORY AND PHILOSOPHIES

Mathematics is more than a collection of theorems, definitions, problems, and techniques; it is a way of thought. Understanding mathematics is a complex process. It involves not only following the details of an argument and verifying its correctness, but seeing the overall strategy of arguments, the role played by every hypothesis, and understanding how different theorems and definitions fit together to create the whole. It is a long-term process; in a sense one cannot appreciate the significance of the first result until one has learned about the last result--all of which may or may not be applicable to our daily life.

Mathematics has abstraction, generalization, deductive structures, algorithms, and computation. It is a language that occasionally models reality with remarkable success and occasionally creates its own reality (Pe, 2003, 2004). But most of the times it fails to model natural events and phenomena with accuracy and precision. Most importantly, mathematics has a history, generates philosophies, and as time passes by, it develops.

As an example of the misuse and abuse of mathematics, one can mention the so-called mathematical work that purports to model malignant tumor growth, stock market behavior, climate change, social behavior, and economic development.

To shed some light on the history of mathematics and see how it develops and finds its way in our daily life, we concentrate on "Real Analysis," one of the most important branches of mathematics. Real analysis has found numerous applications, a few of which are in image compression techniques, speech recognition, steganography, cryptography, and modeling natural objects and behaviour (Strichatz, 1995). Analysis has its roots in the work of Archimedes and other ancient Greek geometers, who developed techniques to find areas, volumes, centers of gravity, arc lengths, and tangent to curves. In the seventeen century these techniques were further developed, culminating in the invention of the calculus of Newton and Leibniz. During the eighteen century the calculus was fashioned into a tool of bold computational power and applied to diverse problems of practical and theoretical interest. At the same time the foundation of analysis--the logical justification for the success of the methods--was left in limbo. This had practical consequences: for example, Euler--the leading mathematician of the eighteen century-developed all the techniques needed for the study of Fourier series, but he never carried out the project. On the contrary, he argued in print against the possibility of representing functions as Fourier series, and his argument was based on fundamental misconceptions concerning the nature of functions and infinite series. The proposal that there might be ways to represent functions as Fourier series was put forth by Daniel Bernoulli (Strichartz, 1995).

In the nineteen century, the problem of the foundations of analysis was faced squarely and was resolved. The theory that was developed forms most of the content of the courses that are offered in universities and is known as "Real Analysis." One must mention that the subject is usually described in its logical order, starting from the basic concepts such as sets and numbers, and building up to the more involved concepts of limits, continuity, derivative, and integral. The actual historical order of discovery is almost the reverse; much like peeling a cabbage, mathematicians began with the outermost layers and worked their way inward. Cauchy and Bolzano began the process in the 1820s by developing the theory of functions without defining the real numbers. The first rigorous definition of the real number system came in the work of Dedkind, Weierstrass, and Heine in the 1860s. Set theory came with the work of Cantor, Peano, and Frege (Strichartz, 1995).

The consequence of the nineteenth century foundational work was enormous and is still being felt today. Perhaps the least important consequence was the establishment of a logically valid explanation of the calculus. More importantly, with the resolution of the conceptual difficulties, new problems emerged, which later inspired important new theories such as "Dynamical Systems." Analysis today is a subject of vast scope and beauty, ranging from the abstract to the concrete, characterized both by the bold computation of the eighteenth century, the logical subtlety of the nineteenth century, and the dramatic increase in computer power of twentieth century.

In part, it is the dramatic increase in computer power and sophistication that has influenced some branches of mathematics, giving rise to a new branches. For example, one can mention "Computational Number Theory," "Fractal Geometry," and "Iterated Function System Cryptography" (Khadivi & Pe, in press). The creation of new fields can also lead to interaction and cross-fertilization of seemingly unrelated fields. For example, one can mention the newly-generated field of "IFS cryptography" which builds a bridge between two seemingly different branches of mathematics, namely real analysis (nondiscrete or continuous mathematics) and number theory (discrete mathematics, (Khadivi & Pe, in press).

The indispensability of technology for investigating these newly-created branches of mathematics should not prevent us from raising red flags on the misuse of technologies in teaching and learning mathematics, especially at the senior high-school and at the freshman level in colleges and universities.

THEORETICAL SOLUTION VERSUS CONSTRUCTIVE SOLUTION

In this article we do not deal with ill-posed problems that arise in many aspects of theoretical or applied mathematics, as this subject demands its own research and work. Instead we concentrate on well-posed problems. In general, the well-posed problems can be divided into two categories, namely, type I and type II. For any given type I problem, there is an algorithm for constructing the solution. Type I problems are divided into two classes [I.sub.1] and [I.sub.2]. [I.sub.1] includes those problems, for which the existing algorithm is a polynomial one, and [I.sub.2] (which includes NP-complete problems), those for which no polynomial time algorithms are known (Koblitz, 1998). Type II problems are those for which no known algorithms to construct the solution are known, that is, one may be able to find the solution(s) by trial and error (heuristic).

In this section the crucial and positive effect of technology in constructing the solution of well-posed problems, in particular Type II problems will be surveyed. Also NP-complete problems, their role in our daily life, and the crucial role in technology in their study, will be discussed.

Solution to a well-posed problem. Here we assume that the existence of solutions to a problem has been proven by some method, and one is seeking to construct these solutions. The problems can be divided in two types, Type I, and Type II. Problems of Type I are those for which there is an algorithm for finding the solution. In intuitive terms, an algorithm is a computational or decision-making procedure that can be completely automated (for example, programmed on a computer). Type I problems are divided into two classes [I.sub.1] and [I.sub.2]. The solutions to Type [I.sub.1] problems are found in a "limited time" (feasible times). Type [I.sub.2] problems are those for which applying the algorithm could take indefinitely long (infeasible times). This means that there are problems of this type, for which finding the solution can take infeasibly long using the algorithm. For Type II problems, there is no known algorithm for constructing the solution. For a problem of this type, one can choose a heuristic approach to find a solution. Heuristic approaches to a problem are by no means guaranteed to work or to be correct, but suggest a course of action or suggest what is true trial and error process (Khadivi, 2002).

Most of the real world problems involve problems of Type [I.sub.2] and Type II. In fact one can mention that the security of all public key cryptosystems and digital schemes depend on the fact that there are no known polynomial-time algorithms for Type [I.sub.2] problems or NP-complete problems that are being used in aforementioned examples. In practice, one can call a cryptosystem "computationally secure," if the best known method of breaking the system requires an unreasonably large amount of computer time. This definition or any other (possibly nonequivalent) definition only provides security relative to either computer time or some other problem(s), not an absolute proof of security. So the advances in technology, such as cheap availability of strong processors, can undermine the security of existing cryptosystems (Koblitz, 1998).

Given the aforementioned facts, one must be aware that technology is best suited for solving Type [I.sub.2] (nonfeasible) and Type II problems (the problems with no known algorithm for finding the solutions) as investigated extensively in (Khadivi, 2002).

Examples

Some examples of Type [I.sub.1] problems are as follows:

1. Finding the Greatest Common Divisor and the Least Common Multiple is of Type [I.sub.1]. One can use the Euclidean Algorithm to find the Greatest Common Divisor of two polynomial in F[X], for any field F. For example the G.C.D.(34359738367, 2^64-1)=1, G.C.D.(2x^4+2x^3+x+1,x^4+x^3+2x+2)=2+2x+x^3+x^4 in [Z.sub.3][x], and G.C.D.(2x^4+2x^3+x+1,x^4+x^3+2x+2)=1+x in [Z.sub.5][x], Mathematica returned the results in a fraction of a second.

2. Finding the derivative of a differentiable function is of Type [I.sub.1]. One can use the differentiation definitions and / or rules to find the derivative of functions (if they exist).

3. Simplifying mathematical expressions are of Type [I.sub.1] As an example one can show that 100!/1073741823=43441891469508054127309611719157799418477851447368440845595570402279756083056330870428461563169251669864447275856433377609515008000000000000000000000000/49981.

4. The IFS cryptography scheme involves an algorithm for the purpose of hiding information, such as text and (classical cryptography) keys, in pictures by using the techniques of iterated function systems (IFS, [Khadivi & Pe, in press]). So one can consider this scheme as of Type I, although theoretically, since it involves unlimited iterations (due to the IFS), it is of Type [I.sub.2]. However for practical purposes one can select a large number for the number of iterations to get a pretty accurate picture of the fractal or attractor. So the problem of hiding information could be considered of Type [I.sub.1]! One must appreciate the advances of technology both in hardware and software for implementing schemes such as this and other relatively recent branches of mathematics such as dynamical systems and modeling.

As an example one can use an IFS cryptosystem to hide the message "Send me some money soon." in the following picture (Figure 1). Only the intended individual who knows the encryption key can retrieve the message from the picture. Without the technological advances, the creation of this picture would have been impossible.

[FIGURE 1 OMITTED]

The following theorems provide the foundation for creating Type [I.sub.2] problems.

The Fundamental Theorem of Arithmetic (FTAR) states that any natural number n can be written uniquely (except for the order of factors) as a product of prime numbers. It is customary to write this factorization as a product of distinct primes to the appropriate powers, listing the primes in increasing order. For example: 4200=2^3*3*5^2*7, 1111122222223333344445555566667777788899999=3^2*13^3*13711*34498631*2007708621258793772669830074802849, and 1111122222223333344445555566667777788899999=1493^2*282241*5743382747*307506539925806346613.

These are Type [I.sub.2] problems for which Mathematica provides the factorization in a fraction of seconds. However, Mathematica was is not yet able to factor 1111111111111111111111111122222222222223333333334444444444445555577777777777777577 in more than seventy-two hours on a desktop PC as expected, so that factorization is categorized as Type [I.sub.2] problem. Computing the log of 42 in base 24 in [F.sub.229345007] is also of Type [I.sub.2]. Note that 47^5=229345007.

The following theorems provide uncountably many Type II problems.

1. The Fundamental Theorem of Algebra (FTAL) states that if f(x) is a polynomial of degree n, where n is a positive integer, then f has at least one zero in the complex number system, and f(x) can be written of n linear factor with complex coefficients. This is of Type II. For example: x^4+x^2+1 = (1-x+x^2)(1+x+x^2), and x^5+3x^4-7x^3+2x^2-12x+13 = (-1+x)(-13-x-3x^2+4x^3+x^4). Also x^5+3x^4-7x^3+2x^2-12x+13 = (5+x)(6+x)(3+2x+6x^2+x^3) in [Z.sub.7][x]. Also the solution set to x^4-5x^4+4x^3-3x^2+2 = 0 is {-0.5, 1, 0.25(1-I*Sqrt(15)), 0.25(1+I*Sqrt(15))}. All these problems were done in seconds by Mathematica; however Mathematica was not able to give any solution of closed form to 109x^37-16x^5-172x^4-x^2+x-23 = 0. The software was able to give just one solution of closed form to 109x^37-16x^5-172x^4-x^2+x-78 = 0, and that was {1} as expected. Notice that the solution set to f(x) = 0 must be found, if one needs to find factorization of f(x) over the Complex number system as mandated by FTAL.

2. The Fundamental Theorem of Calculus (FTC) states that if F(t) has a continuous derivative in an open interval I, then for a and b in I, [[integral].sub.[a,b] F'(t)=F(b)-F(a).

This is of Type II. For example, FTC guarantees that [[integral].sub.[0.4, 1]]x^(x cos x) dx exists and represents the area of the region bounded by the graph of the exponential function f(x)=x^(x cos x) and lines x=.4, and x=1; however no one can calculate the exact area. But with advances in technology, one can approximate the area with a high degree of accuracy.

3. Differential Equations are of Type II. Almost all differential equation problems that are attained from modeling natural phenomena such as wave equation, heat equation, pray-preditor equation are of Type II problems. For example dy/dx= Cos(xy)+Sin([x.sup.2]y); x(-3)=3 has he following graphical solution using Euler Method with steps .01 (Figure 2). The following commands in Mathematica yield the graph. The values of f(-3), f(-2.99), f(-2.98), f(-2.97),..., f(2.9), f(3) all are calculated in a fraction of second and then plotted. We do not list all the aforementioned values of f due to space constraints.

eulerstep[f_,{x_,y_},h_]:={x+h,y+hf[x,y]}

euler[f_,{x_,y_},xf_,h_]:=NestList|eulerstep[f,#1,h] &, {x,y}, Ceiling[(xfx)/h]]

f{x_,y_]:=2Sin[x^2+y}+Cos{x*y]

tmp=euler[f,{-3,3},3,.01]

tmpplot=ListPlot[tmp];

{{-3,3}, {-2.99,2.98016}, {-2.98,2.95941}, {-2.97,2.9379}, {-2.96,2.9158}, {-2.95,2.89326},..., {2.96,6.59358}, {2.97,6.60834}, {2.98,6.62098}, {2.99,6.63144}, {3,6.63974}}

[FIGURE 2 OMITTED]

It must be mentioned that one without a strong mathematical background can write well-posed problems of Type II or Type [I.sub.2] for which even the most sophisticated software installed on supercomputers are incapable of finding the solution. As an example, consider the security of public key cryptosystems such as "RSA," "Diffie Hellman," and "AlGamal" Cryptosystem that are constructed by Type II problems. The applications of public key cryptosystems are widely known, and one can refer to (Koblitz) for further readings.

Before we end this section we must emphasize that to tackle daily problems, there is a need to develop new software. Obviously the software and hardware developer must have a very deep knowledge of theoretical solutions and difficulties in constructing the solution(s). We will return to this subject again in an upcoming section.

TECHNOLOGY FAILURE AND WRONG ANSWERS PROVIDED BY WIDELY-USED SOFTWARE

In this section we intend to elaborate on the failure of technology, through some examples, to either provide us with a solution to some well-posed Type II and Type [I.sub.2] problems, or to give us a correct answer to a well posed problem of Type [I.sub.1]. The validity of the claims for the former has been discussed in detail in the previous section. It suffices to say that lack of an algorithm or the NP attributes are the foundations for such a claim. So without further ado we provide examples of Type [I.sub.1] that we have encountered while teaching courses such as Operation Research and Cryptography. These problems were chosen for pedagogical purposes, and hence one could do these problems manually. However when we tried to use Mathematica (as one should use mathematical software to solve problems in such courses) we were provided either with wrong answers or different kinds of error messages. One must not be surprised by this, because such wrong answers could be the result of several factors such as bugs in program, lack of deep knowledge of mathematics on the part of the programmer, lack of attention of the programmer to special cases, pitfall in hardware designs, and most importantly, the round-off errors that result from memory limitation. So while we cannot do without technology to tackle certain problems, we should be careful not to rely totally on the answers provided by the software. A deep knowledge of mathematical concepts is an indispensable tool to analyze the answer provided by software.

Here we provide some examples:

la-Minimize 3x+y+2z

Subject to:

x+2y+3z[greater than or equal to]24

2x+4y+3z=36

y[greater than or equal to]0

z[greater than or equal to]0.

It is easy to see that this problem is unbounded, however Mathematica provides an answer indicating that the minimum is 14 and it is attained at x=0 and y=6, and z=4. This is the command and the answer provided by Mathematica.

ConstrainedMin {3x+y+2z,{x+2y+3z[greater than or equal to]24,2x+4y+3z==36,y[greater than or equal to]0,z[greater than or equal to]0}, {x,y,z}]

Answer={14,{x[right arrow]0,y[right arrow]6,z[right arrow]4}}

More strangely if one uses the LinearProgramming command of Mathematica one gets the correct answer. This is the command and the answer provided by Mathematica.

LinearProgramming[{3,-3,1,2}, {{1,-1,2,3}, {2,-2,4,3}, {-2,2,-4,-3}}, {24,36,-36}] ConstrainedMin ::nbdd: specified domain appears unbounded. (Indeterminate, Indeterminate, Indeterminate, Indeterminate)

However the Mathematica provides the correct answer for:

1b-Minimize 3x+y+2z

Subject to:

x+2y+3z[greater than or equal to]24

2x+4y+3z=36

x[greater than or equal to]0

y[greater than or equal to]0

z[greater than or equal to]0.

This is the Mathematica command and the resulting answer:

ConstrainedMin {3x+y+2z, {x+2y+3z[greater than or equal to]24,2x+4y+3z==36, {x[greater than or equal to]0,y[greater than or equal to]0,z[greater than or equal to]0},}x,y,z}]

Answer={14,{x[right arrow]0,y[right arrow]6,z[right arrow]4}}

2- Maximize x-y+z

Subject to:

x+y[greater than or equal to]2

z-y[greater than or equal to]3

2x+z[less than or equal to]8.

The following is the Mathematica command and the answer:

ConstrainedMax[x-y+z, {x+y[greater than or equal to]2, z-y[greater than or equal to]3, 2x+z[less than or equal to]8}, {x,y,z}]

Answer={6, {x[right arrow]2,y[right arrow]0,z[right arrow]4}}

It is easy to verify that the optimum value is attained at 6 and the answer is attained at more than one point namely at all points on the semi-line x[less than or equal to]3, y=2-x, and z=8-2x. However Mathematica gives (2,0,4) as the optimal solution. Of course the Mathematica answer is incomplete. If we use LinearProgramming command we get (3,-1,2) as an optimal solution that leads to optimum value of 6. Again the answer is incomplete. It must be noticed that in order to use Mathematica command one must use LinearProgramming[c,m,b] that is the syntax for Minimizing cx subject to mx[greater than or equal to]0, and x[greater than or equal to]0. So we write x=[x.sup.+] - [x.sup.-], y=[y.sup.+] - [y.sup.-], z=[z.sup.+] - [z.sup.-] Such that [x.sup.+], [x.sup.-], [y.sup.+] - [y.sup.-], [z.sup.+] - [z.sup.-] are all non-negative variable. Also one must notice that Maximum(F)=-Minimum (-F). Using these facts we end up with:

LinearProgramming[{-1,1,1,-1,-1,1}, {{1,-1,1,-1,0,0}, {0,0,-1,1,1,-1}, {-2,2,0,0,-1,1}}, {2,3,-8}]

Answer={3,0,0,1,2,0}.

CONCLUSION

While mathematics plays a crucial role in our daily lives, this beautiful and powerful language can be easily abused. One must be skeptical of claims that it can model many phenomena with a high degree of accuracy, and in particular, of flawed analytical models that paint a simplistic picture of complex systems.

Similarly, although advances in technology have enhanced the role and applicability of mathematics in our lives, one must be aware that limitations in hardware and software (e.g., memory, round-off error, bugs in the program that result from the programmer's lack of mathematical knowledge) can seriously restrict the validity of the answers we obtain with the help of technological tools.

Ultimately, only with a good theoretical background in mathematics, developed by years of rigorous training, can one be able to sift the relevant, substantial applications and technological aids from the chaff.

References

Khadivi, M. & Pe, J. (in press). An application of IFS to cryptography. World Scientific and Engineering Academy and Society Journal (WSEAS).

Khadivi, M. (2002). Computers for mathematics: Theoretical solution versus constructive solution. Journal of Computers in Mathematics and Science Teaching, 21(3), 279-284.

Koblitz, N. (1998). A course in number theory and cryptography. Berlin: Springer Verlag.

Pe, J. (2003). Ana's golden fractals. Journal of Fractals, 11(4), 309-313.

Pe, J. (2004). The 3x+1 fractal. Compters & Graphics, 28, 431-435.

Strichartz, R. (1995). The way of analysis. Boston: Jones & Bartlett.

Wolfram, S. (1998). Mathematica. [Computer software]. Champaign, IL: Wolfram Research, Inc.

Acknowledgment

The author would like to thank the Jackson State University "Center of University Scholars" and its director professor Mary Coleman for supporting scholarly activities. Special thanks and appreciation also go to Dr. Joseph L. Pe of Motorola for his constructive comments and suggestions for preparation of this work.

M. R. KHADIVI

Jackson State University

USA

mkhadivi@hotmail.com
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Author:Khadivi, M.R.
Publication:Journal of Computers in Mathematics and Science Teaching
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Date:Sep 22, 2006
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