‘To Infinity and Beyond!’

Bernhard Bolzano and the rigors of analysis

Paper 2

History of Mathematics

December 6, 2001

Problem 14.30

ABSTRACT

Every college math student despairs during the required first course in analysis. Professors everywhere hear the popular complaint, “Why do we have to prove this again?” from students, the author of this work included, who can’t see the mathematical forest for the trees. Analysis is more than the rigor of complicated proofs that beginning mathematicians take for granted. Analysis is philosophy – mathematics philosophy. By studying not only the proofs but the ideas behind them, it is much easier to come to an appreciation of the thought put into the development of this rigor by Bernhard Bolzano and other mathematicians of the 18th and 19th centuries

It’s 5:25, Monday afternoon, and I’m sitting in Dr. Goldberg’s Advanced Calc I class.

Like any good math major, I’m thinking about numbers. I’m exactly one-third of the way

through the 75-minute class. In five minutes, I’ll be two-fifths of the way through, and in 10

minutes…

At least when I look at the clock, it looks like I might be looking at the board. It seems

more polite than looking the other way and staring out the window.

At the top of my notes, I’ve written the word continuity. I draw a curvy line all around it.

My line is continuous. But it’s not a function, because I’ve got more than one y for each x. We

learned that in high school. So why am I sitting here, proving the intermediate value theorem,

after six semesters of college math? I look back at the clock. Three-fifths done. I hear the word

“quiz” and sit up straighter. We have a quiz? Oh, well. I wouldn’t understand the proofs any

better if I had studied anyway.

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On the board, Dr. Goldberg writes “Let, for two reals a and b, a<b, a function f be

continuous on a closed interval [a,b] such that f(a) and f(b) are of opposite signs. Then there

exists a number x0 in the interval [a,b] with f(x0)=0. Prove.”

Well, he always says, when in doubt, draw a pretty picture. So I draw this:

It’s a function. F(a) and f(b) are of opposite signs.

Then I have x0 in [a,b] such that f(x0)=0. Fine. Nobody else

has handed in their quiz yet, so I start daydreaming. The

funniest thing is that what comes to mind is Dr. Troutman’s

math history presentation from the previous class. He was talking about Bernhard Bolzano and

the development of analysis, and the intermediate value theorem. I remember that the theorem

Dr. Goldberg is asking us to prove is the Bolzano theorem, and it’s a special case of the

intermediate value theorem. If I can remember all this, why can’t I remember this proof?

I give up and look out the window. Leave these proofs to Bolzano… I don’t need any of

this analysis stuff. I can just look it up if it ever becomes essential that I prove the continuity of a

real-valued function, which seems doubtful at best. My friend Jessi looks over at me and

whispers, “Joan!” But it’s too late. I’ve already dozed off and begun to dream. Again, I’m back

in Math History, and it looks like I’m giving a presentation on one of my papers.

“Bernhard Bolzano was born in 1781 in Prague, which was then part of Bohemia, now

Czechoslovakia,” I hear myself say. I remember reading that in Eves. “On page 486,” I

continue, “Eves claims that Bolzano was so fully aware of the need for rigorization in analysis

that Felix Klein later referred to him as ‘The Father of Arithmetization.’”

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f(a)

f(b)

I walk into the class and sit down, all the while listening to myself present. No one else

seems to be aware of me. This might be useful, I think to myself, so I get out a pen and paper

and begin to write, taking notes from my own presentation:

Bernhard Bolzano, born Oct. 5, 1781, in Prague, was one of the greatest

mathematicians to produce work in the 19th century. Bolzano was one of the first

mathematicians to delve into the concept of infinity, working with infinite sets and describing the

boundedness of functions (Eves 486). In fact, a special case of the intermediate value theorem so

important to calculus is known as the Bolzano theorem in his honor (Bogomolny).

The saddest aspect of Bolzano’s life is that most of his great contributions to

mathematics were not recognized until years after his death. Most of his peers ignored his work,

and many proofs that were rightly his were claimed by Cantor, Weierstrauss and others. Even

today, most books on math history seem to gloss over Bolzano’s contributions, although the

Internet provides more information, kept by those mathematicians who do understand the

importance of Bolzano’s work.

One problem leading to Bolzano’s lack of recognition was his name – Bernardus

Placidus Johann Nepomuk Bolzano. In 1805, when he was ordained as a priest, Bolzano took

the more modern Bernhard as his first name, and texts today often modernize it further to

Bernard. However, even these simple changes were confusing in the inaccurate system of

records kept by the Catholic church. It was not until 1871, 30 years after Bolzano’s death, that

mathematicians realized that Bernhard Bolzano the Catholic priest was the same man as

Bernardus Bolzano the mathematician (Wachsmuth).

Many of Bolzano’s contributions to mathematics were actually produced from his

study of philosophy and religion. When Bolzano first entered the University of Prague in 1796,

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he studied philosophy, religion and mathematics (Golba). Because his parents were both very

pious Christians, Bernhard grew up with a very high moral code and felt strongly about sticking

to his principles (Golba). The Catholic Encyclopedia claims that Bolzano’s mathematical studies

were “notably successful,” but that his strongest work came in philosophical theory. Later, in

1805, Bernhard went against the wishes of his father (who hoped that he would continue his

study in mathematics), joined the ecclesiatic order and was ordained as a priest (Knight).

So great were Bolzano’s scholarly achievements that he was invited to be a

professor of the philosophy of religion at the University of Prague the same year that he was

ordained, a great honor (Knight). Strangely enough, Bolzano battled doubts about his religion

for much of his early life. It was not until after his ordination that Bolzano adopted the view that

a religious doctrine is justified “if it can be shown that believing it promises greater moral good

than not believing it” (Simons). This sort of rationalism was very evident throughout much of

Bolzano’s work, both in religion and in mathematics, and it was not well-accepted by the church

(Knight).

One of Bolzano’s duties as chair of the religious department at the university was to deliver an “exhortatory sermon” every Sunday to all the students (Simons). These lectures, although popular with the students, soon led to trouble for Bolzano. He openly criticized the prescribed theology textbook of the church, and was very progressive in his religious and philosophical ideas. According to Simons, this “brought him to the unwelcome notice of conservatives in the church.” In 1819, after several investigations, Bolzano was removed from his post and forbidden to publish any of his works (Golba).

According to Simons’ Web site, the restrictions were later loosened, so that Bolzano could

publish works of mathematics and other writings that had no relationship to the church. He also

was allowed to continue to lead Mass as a priest, but was not allowed to hear confessions.

It was during this time that Bolzano once again had time to focus on mathematics, which

he had not worked seriously on for perhaps 15 years. According to Simons and Hieke, from

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1823 until 1830, Bolzano spent every summer in the town of Techobuz at the home of his friends

Josef and Anna Hoffmann. Because Bolzano had always been in somewhat poor health, Anna

Hoffmann spent much time trying to care for Bolzano. From 1830 until 1841, Bolzano lived

permanently with the Hoffmann family. During this period, Bolzano wrote much of his major

work, Grossenlehre. It was a studious attempt on his part to “put the whole of mathematics on a

logical foundation,” as O’Connor and Robertson’s Web site says. This work was published in

parts, and Bolzano always had hoped that his students would finish and publish the complete

work after his death.

In this work, Bolzano expanded upon the idea that had made up his doctoral dissertation

at the University of Prague many years before. In the preface to his doctoral thesis, Bolzano

wrote: “I could not be satisfied with a completely strict proof if it were not derived from

concepts which the thesis to be proved contained, but rather made use of some fortuitous, alien,

ntermediate concept, which is always an erroneous transition to another kind” (O’Connor).

His later works were an attempt to derive a new system of mathematics that was in

keeping with this philosophy. In fact, in Der binomische Lehrsatz… and Rein analytischer

Beweis…, “Pure Analytical Proof,” Bolzano makes an attempt to free calculus from the concept

of the infinitesimal. In the preface of the first book, Bolzano clearly states his intention: to

provide “a sample of a new way of developing analysis.” (O’Connor). A later mathematician,

when analyzing Bolzano’s goals, said the following:

“In this work, Bolzano did not wish only to purge the concepts of limit, convergence and

derivative of geometrical components and replace them by purely arithmetical concepts. He was

aware of a deeper problem: the need to refine and enrich the concept of number itself.”

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In Bolzano’s writings, one of his major contributions was the proof of the intermediate

value theorem using this new approach. He also wrote a book called Paradoxien des

Unendlichen, a study of “paradoxes of the infinite,” according to O’Connor. In this work,

published posthumously in 1851, the first use of the word “set” appears, and Bolzano gives

examples of one-to-one correspondences between the elements of an infinte set and the elements

of a proper subset, using a concept that is now known as 0א , or aleph-null, for the cardinality of

certain infinite sets.

The first of these concepts, the proof of the intermediate value theorem, may seem

simple, but Bolzano’s method of direct proof posed one problem: The notion of continuity was

necessary for the first time, and only Bolzano recognized its importance (Stillwell 196). The

intermediate value theorem is as follows:

Suppose f:[a,b] into the reals is continuous. Suppose f(a)<0 and f(b)>0. Then there

exists some x0 in the interval (a,b) such that f(x0)=0 (Fitzpatrick 49). Bogomolny’s Web site

gives a thorough proof of this theorem by a mathematician named Scott Brodie. It is as follows:

Let f(x) be a continuous function on the closed interval [a,b], with f(a) < 0 < f(b). We are

to show that there is a real number c, between a and b, such that f(c) = 0.

We first need to understand what is meant by a continuous function. This is usually taken to

mean that for real numbers near the number x, the function values lie near f(x). This is usually

stated in terms of a "challenge" and "answer" dialogue:

We say "f is continuous at x" if, for each open interval J containing f(x), we can find an open

interval I containing x so that for each point y in I, f(y) lies in the interval J. We say "f(x) is

continuous" if it is continuous in the sense given above at every point x of its domain.

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As a simple application of this notion of continuity, we may note the "Sign-Preserving property

of Continuous Functions": If f is continuous at a, and f(a) < 0, then there is an open interval I

containing a such that f(x) < 0 for every x in I. It is also necessary to have what is sometimes

called the Axiom of Completeness. One common version of that axiom which is suitable for this

purpose is the "Least Upper Bound" axiom: Every non-empty set of real numbers which is

bounded above has a least upper bound.

Now, the proof: Let S be the set of numbers x within the closed interval from a to b where f(x) <

0.

Since S is not empty (it contains a) and S is bounded (it is a subset of [a,b]), the Least Upper

Bound axiom asserts the existence of a least upper bound, say c, for S. We show that this number

c satisfies the requirements of Bolzano's theorem:

There are three possibilities for f(c): either f(c) < 0, f(c) > 0, or f(c) = 0. We show the first two

choices lead to contradictions.

Suppose f(c) < 0, so that c is a member of S. Then the Sign Preserving Property of Continuous

Functions asserts the existence of an open interval I containing c where f takes only negative

values -- that is, I is a subset of S. But I (and thus S) contains points greater than c, so c cannot be

an upper bound (let alone the least upper bound) for S. This contradiction forces us to reject the

possibility that f(c) < 0.

Suppose instead that f(c) > 0. The argument is similar but not identical to the previous case.

Since f is continuous, there is an open interval H containing c where f takes only positive values.

But H contains points less than c, which must also be upper bounds for S, since no point of H can

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lie in S. This contradicts our choice of c as the least upper bound for S, so we must reject the

possibility that f(c) > 0.

We conclude that f(c) = 0. (From Bogomolny, http://www.cut-the-knot.com/fta/brodie.html).

Using this proof, many further theorems of analysis can be developed, such as those dealing

with the concept of boundedness for sets of real numbers.

Eves’ text states: “A real number a is called an upper bound of a nonempty set M of real

numbers if for each number m of M we have m is less than or equal to a, and a is called a least

upper bound of M if a<b whenever b is any other upper bound of M. A basic and important

property of the real number system asserts that if a nonempty set of real numbers has an upper

bound, then it has a least upper bound.” This is what was stated above in Brodie’s proof as the

Least Upper Bound, or supremum, axiom.

14.30 (a): Give a definition of a lower bound and a greatest lower bound of a nonempty set of

real numbers.

In this case, most of the question can be answered by “flipping around” the definition of an upper

bound and least upper bound. A lower bound of a set M, denoted a, means that, for all m in M, a

is less than or equal to m. Also, a is the greatest lower bound, or infimum, if b<a whenever b is

any other upper bound of M.

14.30 (b): Prove that a nonempty set of real numbers can have at most one least upper bound

and at most one greatest lower bound.

For this proof, we will prove that the least upper bound is unique and then use the same ideas in

reverse to show that the same is true for the greatest lower bound. First, assume that there exist

two upper bounds for a set M, call them a and b, with a and b unequal, and assume that they are

actually both LEAST upper bounds. Without loss of generality, assume a>b. Then, from our

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definition of least upper bound, b is a least upper bound of M, but a is not. However, if a>b and

a is a least upper bound of M, which is our assumption, then for some >0, a->b is an element

of M, and then, by the definition of being an upper bound, b is not an upper bound for M because

there is an element of M greater than b. Then, by definition, there is only one least upper bound

of M. The same will hold for greatest lower bounds; if the greatest lower bound is not unique,

then one of the two greatest lower bounds must be greater than the other, and there will exist

some number between the two bounds that is less than the smaller of the two bounds. In that

case, the smaller one really is not a greatest lower bound or even a lower bound at all, because

some element of the set M is less than it.

14.30 (c): Give an example of a nonempty set M of real numbers that has the following: Both an

upper and a lower bound. An example of a set like this is the set {(-1)n} or {-1,1,-1,1,-1,…}.

The upper bound of this set is 1, and the lower bound is –1. An upper bound but no lower

bound. An example of a set like this is the set {-1/n2} where the upper bound is 0 but the

function goes to negative infinity. A lower bound but no upper bound. An easy example of a set

like this will be {1/n2} which has a lower bound of 0 but goes off to positive infinity. Neither an

upper nor a lower bound. The set of integers has no upper nor lower bound; it goes to infinity in

both directions. A least upper bound that is in the set M. The previously given set of {(-1)n} has

as its least upper bound 1, which is an element of the set M={(-1)n}. A least upper bound that is

not in the set M. The previously given set of {-1/n2} has a least upper bound of 0, but 0 is not an

element fo the set M={-1/n2}.

14.30 (d): Prove that if a nonempty set M of real numbers has a lower bound, then it has a

greatest lower bound.

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Proof: For this proof, we will consider the set of “negatives” of a set S, where S is bounded

below. That is, let T={real numbers x such that –x is an element of S}. Then for any number x, b

is less than or equal to x if and only if –x is less than or equal to –b. Thus, a number b is a lower

bound for S if and only if the number –b is an upper bound for T. Since the set S has been

assumed to be bounded below, it follows that the set T is bounded above. What Brodie called

the Completeness Axiom, above, says that there must be a least upper bound for T, say c. Since

lower bounds of S occur as negatives of upper bounds of T, the number –c is the greatest lower

bound for S.

Suddenly, I sat straight up. Looking around the room, I realized that only Dr. Goldberg

was left. I hurriedly handed him my paper containing the doodle I had drawn of the graph.

Embarrassed to have been sleeping, I rushed out of the room.

The next class, when Dr. Goldberg gave our quizzes back, I sat with my usual

resignation, expecting my typical 7 out of 15 points. Imagine my surprise at the great 25 out of

20 I saw at the top of the page! Scrawled in the margins was this simple comment from Dr.

Goldberg: “Only Bolzano could have given this much background on this proof! Good job!”

I don’t think I’ll tell him that it was math history paper that showed me just how

important the rigors of analysis are to all of mathematics. I’ll let him think I knew that all along.

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BIBLIOGRAPHY

Bell, E.T. Men of Mathematics. New York: Simon and Schuster, 1937.

Bogomolny, Alexander. “Intermediate Value Theorem – Bolzano Theorem,” “Remarks on Proving the Fundamental Theorem of Algebra.” http://www.cut-the-knot.com/fta/brodie.html and http://www.cut-the-knot.com/Generalization/ivt.html. Copyright 1996-2001.

Eves, Howard Whitley. An Introduction to the History of Mathematics. With cultural connections by Jamie H. Eves. Philadelphia: Saunders College Publications, 1992.

Fitzpatrick, Patrick M. Advanced Calculus: A Course in Mathematical Analysis. Boston: PWS Publishing Co., 1996.

Golba, Paul. “Bolzano, Bernhard.” http://www.shu.edu/html/teaching/math/reals/history/bolzano.html. Copyright 1994-2000.

Knight, Kevin. “Bernhard Bolzano.” New Advent Catholic Encyclopedia. http://www.newadvent.org/cathen/02643c.htm. Copyright 1999.

Muir, Jane. Of Men and Numbers. New York: Dover Publications Inc., 1996.

O’Connor, J.J., and E.F. Robertson. “Bernhard Placidus Johann Nepomuk Bolzano.” http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Bolzano.html. December 1996.

Pappas, Theoni. Mathematical Scandals. San Carlos, California: Wide World Publishing/Tetra, 1997.

Simons, Peter, and Alexander Hieke. “The Bernard Bolzano Pages.” http://www.sbg.ac.at/fph/bolzano/bolzano-life.html. Copyright 2000.

Stillwell, John. Mathematics and its History. New York: Springer-Verlag Publishing, 1989.

Wachsmuth, Bert G. “Bolzano Theorem.” Interactive Real Analysis. http://www.shu.edu/html/teaching/math/reals/cont/proofs/bolzthm.html. Copyright 1994-2000.

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