Of all the activities that can be used to historically enrich mathematics teaching and learning, one of the easiest and most fruitful is the use of student research projects. Students obtain a deeper understanding of the human and personal side of mathematics by researching the lives and works of individual mathematicians and/or groups of mathematicians. Identifying an individual name with a person who had a specific place and date of birth, a record of schooling, a participation in family life and an advocation to mathematics, helps demystify the subject. In many respects, mathematics was developed by ordinary men and women who through their persistence and special insights on the subject formulated the discipline we know today.

Reports on the lives and work of mathematicians can be very brief. See some of the samples given below. Such reports can either be oral or written, submitted as an individual or group project and shared with classmates. Many excellent mathematical biographies exists for student consultation for example: Lawrence Young’s, Mathematicians and Their Times, 1981; E.T. Bell’s, Men of Mathematics 1937; Dunnington’s Carl Friedrich Gauss, Titan of Science, 1955 and Lewis Moore’s Isaac Newton, A Biography, 1962. Reports can be made more challenging by assigning themes to be investigated, for example: “Sixteenth Century Algebraists”: “The Circle Squarers”, or cooperative relations between mathematicians explored such as “Teacher and Pupil” or “Father and Son” etc. Another approach is to assign topics of investigation involving particular mathematical problems, ones which within themselves have a long history, for example: “Euclid’s Fifth Postulate” or the “Constructability of Regular Polygons”. Several examples of undergraduate and secondary school students report situations are given below.

• Undergraduate Assignment Formats
• An Undergraduate Report
• Secondary School Assignment Formats
• A Brief Secondary School Student Report

The following pages are examples of assignments from a college course in the history of mathematics (from Professor R. M. Davitt, University of Louisville):

 

Assignment:  Topics for Classroom Presentations

 

Note: Your instructor will provide the student who chooses a specific topic for her/his classroom presentation with at least one solid resource for that subject.

Note: The topics are listed in the order in which they will be covered. The earlier ones will definitely be presented by students. However, depending upon the final, stabilized population of the course, it is quite possible that the last few topics will not be the subjects of classroom presentations.

Pithy Descriptions of the Topics

1.         The Pythagorean Theorem and Pythagorean triples

2.         The three classical Greek construction problems

3.         The Golden Section

4.         The history and properties of π

5.         The history and properties of 0

6.         The history and properties of e

7.         The history and properties of i

8.         Theories of the infinite

9.         Hilbert’s Tenth Problem

10.       The Four Color Theorem

11.       The Continuum Hypothesis

12.       Kepler’s Conjecture

13.       Fermat’s Last Theorem

14.       The Riemann Hypothesis

15.       The Prime Number Theorem

16.       The Goldbach Conjecture

17.       The Poincaré Conjecture

18.       Just six numbers - The deep forces that shape the universe

19.       The question of NP-completeness

20.       The Fundamental Theorem of Algebra

 

The instructor always reserves the right to direct students to “more appropriate” topics based upon his perceptions of their mathematical backgrounds and mathematical maturity. In other words, he doesn’t want anyone getting in over her/his head. The level of sophistication needed to understand and discourse intelligently upon some of the latter topics is quite high.

 

As the first essay assignment, please respond in some detail to the following prompts:

 

1. Identify an “aspect” of mathematics not volunteered by a student in class and write a paragraph or two on why you believe that “aspect” is indeed a feature of mathematics as you see it. This aspect can be one you have personally thought of outside of class or one of the ones on the master list of aspects with which you have been supplied after the in-class discussion of answers to the question “What is Mathematics” in the form “Mathematics is:...”.

 

2. Identify the one aspect of mathematics on the master list which most appeals to you and explain the reason(s) for your choice. Examples, experiences, personal insights, and the like are quite appropriate points of discussion in explaining your choice to your instructor.

 

3. List three mathematicians and a mathematical accomplishment of each from each (That is a total of 15 people and 15 claims to fame.) Periods II A, II B, III, IV A, and IV B of the Ontology Recapitulates Phylogeny overview of the chronological evolution of mathematics discussed in class during the first week of the course. Your Period IV B mathematicians must have something to do with mathematics either inspired by the computer or mathematics which is accomplished via a computer.

 

         Essay Assignment:    Humanizing Mathematics

 

An effective viewpoint for overviewing mathematics and its history is to concentrate on mathematics as a human endeavor. Textbooks written for “liberal arts” (LA) mathematics general education courses often reflect this perspective. For example, Harold Jacob’s LA text is entitled Mathematics, A Human Endeavor and Sherman Stein’s is called Mathematics, The Man-Made (sic) Universe. Another good source for delving into the human side of mathematics is the book Essays in Humanistic Mathematics, edited by Alvin M. White and published by The Mathematical Association of America, Washington, D.C., 1993.  Copies of this collection of essays are available from your instructor on a first-come-first-served basis. Two very dependable sources for information on mathematical scientists are The Dictionary of Scientific Biography (DSB) and Women of Mathematics (WoM), which are in the Reference Sections of the Library. And then there is E. T. Bell's classic, (in)famous Men of Mathematics which consists of 28 melodramatic but enticing biographical essays (sometimes of more than one person at a time) of important personages in the history of mathematics who died before 1920. The principal purpose of this essay on an individual mathematical scientist and her/his work is to help humanize the admittedly abstract field of mathematics and to develop richer humanistic contexts in which you can place your mathematical studies.

 

In this essay, you will investigate the biography and mathematical/scientific career of an important mathematical scientist whose story we will not hear about as part of some book report in the last days of the course. For the record, during the book report classroom presentations on 12/3 and 12/5, the lives and work of Georg Cantor, Leonhard Euler, Galileo Galilei, Kurt Gödel, David Hilbert, Sir Isaac Newton, Emmy Noether, George Polya, and John von Neumann will be discussed. The list below contains 18 additional mathematical scientists of some historical import, one of whom you will be assigned on 9/19 as the “subject” for your essay by the Churchill Down's pill-pull method for assigning post positions for the Derby. You will report on your mathematician on Mathematicians” Day, 10/8.

 

The Mathematical Scientists are:

Archimedes (287-212 BCE) [IIA]

Girolamo Cardan (1501-1576) [IIB]

Augustin Louis Cauchy (1789-1857) [IVA]

Sir Arthur Cayley (1821-1895) [IVA]

René Descartes (1596-1650) [III]

Paul Erdős (1913-1996)

Pierre de Fermat (1601-1665) [III]

Carl Friedrich Gauss (1777-1855) [IVA]

Sophie Germain (1776-1831) [IVA]

Godfrey Harold Hardy (1877-1947) [IVA]

Grace Murray Hopper (1906-1992) [IVB]

Sophia Kovalevskaia [Sonya Kovalesky] (1850-1891)

Joseph Louis Lagrange (1736-1813) [III/IVA]

Gottfried Leibniz (1646-1716) [III]

Nicolai Lobachevsky (1793-1856) [IVA]

Julia Robinson (1919-1985) [IVA]

Claude Shannon (1916-2001) [IVB] (NY Times obituary available - he died recently)

Alan Turing (1912-1945) [Precursor of Period IVB]

 

          Essay Assignment:  Philosophy   

 

Mathematicians and scientists have long sought to discover the most unquestionable, impregnable logical foundations on which to base the validity (“truth”) of mathematics. The field of study in which this question is investigated is called the Philosophy of Mathematics or Metamathematics. In the 1930s the work of Kurt Gödel established that no one particular philosophy of mathematics is “better” than any other in the sense that we cannot prove that one philosophical viewpoint on metamathematics is superior to all the others. It is simply a matter of taste which philosophy of mathematics a given mathematician adopts as her or his most fundamental set of beliefs on the subject. There are currently at least 7 Schools of Metamathematics. These primary schools are listed below with a cryptic phrase describing each school.

 

Pythagorean School  – “All is number.”

Euclidean/Aristotelian School – “The essence of mathematics is to begin with certain declared quotidian and mathematical definitions and premises and then to build up the collection of legitimate theorems in the mathematical system via arguments based upon Aristotelian logic.”

Platonic School – “Mathematical objects already exist and are waiting to be ‘discovered’ in an imperfect ‘shadowy’ fashion by humans.”

Kantian School – “Certain mathematical concepts (e.g., Euclidean geometry) are ‘hard-wired' into the human brain a priori and cannot be circumvented.”

School of Logicism – “To establish a basis for mathematics humans must first develop a ‘sufficiently rich’ system of logic so that mathematics then becomes a branch of that logic.”

School of Formalism – “Mathematics is chiefly concerned with formal symbolic systems which are devoid of content. Mathematics is mostly concerned with questions of the logical ‘consistency,’ ‘independence,’ and ‘completeness’ of its various branches.”

School of Intuitionism (Constructivism) – “Mathematics is to be built solely by finite constructive methods based upon the intuitively given sequence of natural numbers.”

 

In this essay you are to:

 

(1) Write a one paragraph description elaborating on the basic tenets of each school listed above concerning its conception of the philosophical foundations of mathematics (which are merely hinted at above by your instructor).

(2) List two or more individuals who were or are strong proponents of the tenets of each Metaphysical School listed above and describe their efforts to promote that school. [In writing this part of the essay, it is certainly OK to use any persons after whom a particular school is named if you so choose.]

(3) Choose the one school (philosophy) of mathematics listed above which you most subscribe to and write a paragraph or two detailing why you prefer that metaphysical viewpoint over the other listed schools’ beliefs.

 

Hilbert's Tenth Problem

 

David Hilbert was born on January 23rd, 1862 in Königsberg, Prussia. He worked in many areas of mathematics including geometry, algebra, number theory, and mathematical physics to just name a few. In 1899, Hilbert greatest work of the time was his book Grundlagen der Geometrie (Foundations of Geometry), which put Euclidian geometry in a formal axiomatic setting and removed the flaws from Euclidean Geometry. It was of Hilbert's belief, or inspiration, that all of mathematics could be placed in a formal axiomatic setting and thus all mathematical theorems could be proven. He began what is known as the "formalist school" of mathematics.

In 1900, he was selected as the leading mathematician of the time to present the keynote speech at the 2nd International Congress of Mathematics in Paris. In his introduction of his lecture he states:

“Who of us would not be glad to lift the veil behind which the future lies hidden; to cast a glance at the next advances of our science and at the secrets of its development during future centuries? What particular goals will there be toward which the leading mathematical spirits of coming generations will strive? What new methods and new facts in the wide and rich field of mathematical thought will the new centuries disclose?”

Hilbert went on to state the nature of mathematical problems, in that they should be difficult, but not completely inaccessible, and what mathematicians should do about them, then he went on to state 23 problems that he felt were relevant to the foundation and future research of mathematics as seen at the turn of the century.

“The great importance of definite problems for the progress of mathematical science in general... is undeniable. ... [for] as long as a branch of knowledge supplies a surplus of such problems, it maintains its vitality.... every mathematician certainly shares.. the conviction that every mathematical problem is necessarily capable of strict resolution ... we hear within ourselves the constant cry: There is the problem, seek the solution. You can find it through pure thought...”

The 23 Paris problems became the backbone for mathematical research in the 20th century. Many of the problems were standing open problems of the time. Many of the problems have been proven. Several are still open problems.

Without doubt, Hilbert was the dominant influence on 20'' century mathematics. Such a grand body of work and new areas of mathematics have evolved from attacking his challenges and visions. His greatness can be summed up by his first student Otto Blumenthal:

“In the analysis of mathematical talent one has to differentiate between the ability to create new concepts that generate new types of thought structures and the gift for sensing deeper connections and underlying unity. In Hilbert’s case, his greatness lies in an immensely powerful insight that penetrates into the depths of a question. All of his works contain examples from far flung fields in which only he was able to discern an interrelatedness and connection with the problem at hand.  From these, the synthesis, his work of art, was ultimately created.  Insofar as the creation of new ideas is concerned, I would place Minkowski higher, and of the classical great ones, Gauss, Galois, and Riemann. But when it comes to penetrating insight, only a few of the very greatest were the equal of Hilbert.” - Otto Blumenthal, Hilbert's first student.

 

Hilbert’s 10th Problem

 

Hilbert’s 10th problem was the only one of the 23 Paris Problems that was a decision problem. It was to devise an algorithm to determine if a given Diophantine equation was solvable. It was to specify a procedure which, in a finite number of steps, enables one to determine whether or not a given Diophantine equation with an arbitrary number of indeterminates and with integral coefficients has a solution in rational integers.

The formal statement of the 10th Problem as given by Hilbert:

10. Determination of the solvability of a Diophantine equation. Given a Diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: To devise a process according to which it can be determined by a finite number of operations whether the equation is solvable in rational integers.

The 10th problem was proven impossible by Julia Robinson and Martin Davis in 1970.

In 1948, after receiving her doctorate at the University of California at Berkeley, Julia Robinson began her work on Hilbert’s 10th problem. This problem occupied most of her professional career. Along with Martin Davis and Hilary Putman, she gave a fundamental result which contributed to the solution to Hilbert's Tenth Problem. The Davis-Putnam-Robinson paper was presented in 1961. She worked on the problem for over twenty years, building a foundation which Yuri Matiyasevic used in 1970 to prove that there is not a general method for determining solvability. She also did important work on that problem with Yuri Matiyasevic after he gave the solution in 1970.

Thus, Hilbert's 10th problem was solved in the negative, in that there is no algorithms to determine if a given diophantine equation was solvable.

In 1930, Hilbert retired and the city of Königsberg made him an honorary citizen of the city. He gave an address which ended with six famous words showing his enthusiasm for mathematics and his life devoted to solving mathematical problems: “Wir mussen wissen, wir werden wissen” – “We must know, we shall know.”

 

Diophantine Equations

 

Diophantus, often known as the “father of algebra,” was a Greek mathematician in approximately 250 AD. He was the first to attempt algebraic notation. In his Arithmetica, a work on solutions of algebraic equations and the theory of numbers, he showed how to solve simple and quadratic equations. Diophantus was interested in exact solutions rather than approximate solutions considered perfectly appropriate. Diophantus found interest in polynomial equations in one or more variables for which it is necessary to find a solution in integer form. He found solutions to equations that are negative or irrational square roots to be useless.

Diophantine Equation - An equation in which the coefficients are integers and the solutions are also required to be integers.

Examples of Diophantine Equations:

x + y = 5          linear

x²+7=2ⁿ

(2x–1)²=2ⁿ –7

x² + p = 2ⁿ

x³ + 3 = 4ⁿ

xⁿ + yⁿ = zⁿ, for n = 2    Pythagoras’ Theorem,

for n ≥ 3 Fermat's Last Theorem (that there is no integer solution)

x² – dy² = 1      Pell's equation

ax + by = 1      Bezout's identity

A⁴ + B⁴ = C⁴ + D⁴

 

Hilbert’s 23 Paris Problems

 

A set of (originally) unsolved problems in mathematics proposed by Hilbert. Of the 23 total, ten were presented at the Second International Congress in Paris on August 8, 1900. Furthermore, the final list of 23 problems omitted one additional problem on proof theory (Thiele 2001).

Hilbert's problems were designed to serve as examples for the kinds of problems whose solutions would lead to the furthering of disciplines in mathematics, and are summarized in the following list.

 

1a. Is there a transfinite number between that of a denumerable set and the numbers of the continuum? This question was answered by Godel and Cohen to the effect that the answer depends on the particular version of set theory assumed.

1b. Can the continuum of numbers be considered a well ordered set? This question is related to Zermelo's axiom of choice. In 1963, the axiom of choice was demonstrated to be independent of all other axioms in set theory, so there appears to be no universally valid solution to this question either.

2. Can it be proven that the axioms of logic are consistent? Godel's incompleteness theorem indicated that the answer is “no,” in the sense that any formal system interesting enough to formulate its own consistency can prove its own consistency iff it is inconsistent.

3. Give two tetrahedra which cannot be decomposed into congruent tetrahedra directly or by adjoining congruent tetrahedra. Max Dehn showed this could not be done in 1902 by inventing the theory of Dehn invariants, and W. F. Kagon obtained the same result independently in 1903.

4. Find geometries whose axioms are closest to those of Euclidean geometry if the ordering and incidence axioms are retained, the congruence axioms weakened, and the equivalent of the parallel postulate omitted. This problem was solved by G. Hamel.

5. Can the assumption of differentiability for functions defining a continuous transformation group be avoided? (This is a generalization of the Cauchy functional equation.) Solved by John von Neumann in 1930 for bicompact groups. Also solved for the Abelian case, and for the solvable case in 1952 with complementary results by Montgomery and Zipin (subsequently combined by Yamabe in 1953). Andrew Gleason showed in 1952 that the answer is also “yes” for all locally bicompact groups.

6. Can physics be axiomatized?

7. Let α be algebraic and β irrational. Is α^β then transcendental (Wells 1986, p. 45)? α^β is known to be transcendental for the special case of β an algebraic number, as proved in 1934 by Aleksander Gelfond in a result now known as Gelfond's theorem (Courant and Robins 1996). However, the case of general irrational β has not been resolved.

8. Prove the Riemann hypothesis. The conjecture has still been neither proved nor disproved.

9. Construct generalizations of the reciprocitv theorem of number theory.

10. Does there exist a universal algorithm for solving Diophantine equations? The impossibility of obtaining a general solution was proved by Julia Robinson and Martin Davis in 1970, following proof of the result that the relation n = F_2m (where  F_2m   is a Fibonacci number) is Diophantine by Yuri Matiyasevich (Matiyasevich 1970; Davis 1973; Davis and Hersh 1973; Davis 1982; Matiyasevich 1993; Reid 1997, p.107). More specifically, Matiyasevich showed that there is a polynomial P in n, m, and a number of other variables x, y, z, … having the property that n = F_2m iff there exist integers x, y, z, …such that P(n, m, x, y, z,…) = 0.

11. Extend the results obtained for quadratic fields to arbitrary integer algebraic fields.

12. Extend a theorem of Kronecker to arbitrary algebraic fields by explicitly constructing Hilbert class fields using special values. This calls for the construction of holomorphic functions in several variables which have properties analogous to the exponential function and elliptic modular functions (Holzapfel 1995).

13. Show the impossibility of solving the general seventh degree equation by functions of two variables.

14. Show the finiteness of systems of relatively integral functions.

15. Justify Schubert's enumerative geometry (Bell 1945).

16. Develop a topology of real algebraic curves and surfaces. The Tanivama-Shimura conjecture postulates just this connection. See Gudkov and Utkin (1978), Ilyashenko and Yakovenko (1995), and Smale (2000).

17. Find a representation of definite form by squares.

18. Build spaces with congruent polyhedra.

19. Analyze the analytic character of solutions to variational problems.

20. Solve general boundary value problems.

21. Solve differential equations given a monodromy group. More technically, prove that there always exists a Fuchsian system with given singularities and a given monodromy group. Several special cases had been solved, but a negative solution was found in 1989 by B. Bolibruch (Anasov and Bolibruch 1994).

22. Uniformization.

23. Extend the methods of calculus of variations.

 

** from Mathworld - http://mathworld.wolfram.com/HilbertsProblems.html

MEN OF MATHEMATICS   

 

Archimedes

Apollonius

al-Khowarizmi

al-Kashi

Aryabhata the Elder

Charles Babbage

Jakob Bernoulli

Bhaskara

Janos Bolyai

George Boole

Brahmagupta

Georg Cantor

Girolamo Cardano

Nicholas Copernicus

Augustus De Morgan

Rene Descartes

Diophantus

Albert Einstein

Eratosthenes

Euclid

Leonhard Euler

Pierre Fermat

Fibonacci

Gottlob Frege

Galileo Galilei

Evariste Galois

Carl Friedrich Gauss

Sophie Germain

Kurt Godel

W.R. Hamilton

David Hilbert

Hypatia

Johannes Kepler

Felix Klein

Sonya Kovalevsky

Joseph Louis La Grange

Gottfried W. Leibniz

Liu Hui

Marin Mersenne

John Napier

Isaac Newton

Emmy Noether

Omar Khayyam

Blaise Pascal

Pythagoras

S. Ramanujan

Regiomontanus

Bernhard Riemann

Bertrand Russell

S. Stevin

James Sylvester

Thales

Evangelista Torricelli

Tsu Ch'ung-chih

Francois Viete

John von Neumann

A.N. Whitehead

Yang Hiu

Zeno of Elea

 

Format for Biography Assignment

         

MATHEMATICIAN'S NAME:______________________________________________  

 

DATE OF BIRTH:______________   PLACE OF BIRTH:    _______________________

DATE OF DEATH:_____________   PLACE OF DEATH:________________________

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THREE EVENTS IN THE MATHEMATICIAN'S LIFE - DATE ALSO IF KNOWN

 

1.

 

2.

 

3.

 ************************************************************************

THREE ACCOMPLISHMENTS IN MATHEMATICS

 

1.

 

2.

 

3.

 

************************************************************************

THREE FACTS OF INTEREST ABOUT YOUR MATHEMATICIAN

 

1.

 

2.

 

3.

 

************************************************************************

SOURCES

 

1.

 

2.

 

3.

 

 

Biography Report

 

MATHEMATICIAN'S NAME: Carl Friedrich Gauss

 

DATE OF BIRTH: April 30. 1777       PLACE OF BIRTH: Brunswick, West Germany

 

DATE OF DEATH: February 23, 1855                        PLACE OF DEATH: Göttingen

 

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THREE EVENTS IN THE MATHEMATICIAN'S LIFE - DATE ALSO IF KNOWN

 

1.  From 1795-98 he went on to secondary school and studied even further at the University of Göttingen, granted by the Duke of Brunswick.

 

2.  In 1799 he obtained his doctorate in absentia from the University of Helmstedt. His thesis showed the first proof of fundamental algebra which states every algebraic equation has a root of a+bi (i being the square root of –1).

 

3.  In 1807 he became professor of astronomy and director of the new observatory at the University of Göttingen where he remained for the rest of his life.

 

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THREE ACCOMPLISHMENTS IN MATHEMATICS

 

1. He discovered the method of least squares.

 

2. He discovered a Non-Euclidean Geometry.

 

3. He contributed to the theory of numbers.

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THREE FACTS OF INTEREST ABOUT YOUR MATHEMATICIAN

 

1. Gauss was deeply religious, aristocratic in bearing and conservative.

 

2. Gauss was acknowledged to be one of the 3 leading mathematicians of all times: the others being Archimedes and Newton.

 

3. Gauss was an exceptionally precocious child. And at 3 he detected an error in his father’s bookkeeping. At 10 he amazed his fellow students by summing the integers from 1 to 100. ************************************************************************

SOURCES

 

1. Encyclopedia Americana

 

2. Encyclopedia Britannica

 

3. The Papers of Carl Gauss