Download Introduction In the frigid air of an East Prussian morning, a young

Phil φsophy
and
Mat π ematics
~
Two Men of Königsberg
Samuel Otten
Grand Valley State University
2006
Introduction
In the frigid air of an East Prussian morning, a young boy sets out upon the
streets of Königsberg as he does six days a week. He walks briskly to arrive at the
archway of the Collegium Fridericianum by 7:00 am. The approaching schoolyard
leaves much to be desired in the way of intellectual freedom and social
interaction. The home from which he has departed, however, will be described
later in his life as a warm, understanding, and supportive environment. The
boy’s father, a harness maker, bestows ehre or honor upon the family and
provides a manageable living. His mother, whom he will thank for “an education
that could not have been better,” has a special relationship with her young son,
and she lovingly refers to him as Manelchen.1 The mother and son often take
long walks together, sharing in the beauty of the world and the joy of their
company. Sadly, these walks were not destined to continue as death robbed the
thirteen-year-old of his mother, and robbed the mother of the opportunity to
witness the heights and far-reaching influence her son would achieve in his
professional career.
Another young boy from Königsberg began his education at the same
institution – the
Friedrichskolleg Vorschule – as an eight-year-old, two years
behind most of his classmates. It was fortunate that his mother had initiated
informal studies with him at their home, located “only a few blocks from the
river.”2 This home-based education did not cease while the child was attending
school, as his mother was known to write assigned essays for him. The young
boy, sharing the sentiment of the lad above, found the intellectual environment
at Friedrichskolleg extremely restrictive and would later remember his days there
as unhappy ones. Thankfully, things improved years afterward when he entered
the University of Königsberg.3 There he could focus solely on the subject that
piqued his interest, though it was not himself but a classmate who soon won
international regard.4 Nevertheless, by the end of his career our young man
would be the foremost figure in his field.
1
Immanuel Kant, the subject of the former paragraph, and David
Hilbert, the latter, are prominent and historic figures in the academic
world. Kant was a defining voice of the Enlightenment period5 and
perhaps the greatest philosopher with regard to metaphysical inquiry. His
critiques of pure and practical reason, especially, ushered in a new era of
philosophical treatment. Hilbert made important contributions in the
fields of invariant theory, algebraic number theory, geometry, and
functional analysis. He was the champion of the Formalist school and was
recognized as the motivator behind most mathematical efforts of the 20th
Century, due to his presentation of the Paris Problems during the
International Congress of Mathematicians in 1900.
The current paper could scarcely scratch the surface of these two men,
their work, or their influence. It will focus instead on a few connections
between the individuals as they relate to the philosophy of mathematics.
First, a development of historical context through an overview of the
imprints they made on mathematical foundations. Second, an account of
how Kant and Hilbert both successfully uncovered latent forces at work
with regard to human awareness and understanding. Third, a glimpse of
the perspectives the men take with regard to the limits of knowledge and
the implications these have for mathematical truths.
It is the goal of this essay to cultivate in its readers an appreciation of
two facts: that a young man who struggled mightily in his own education
with the lowest levels of mathematics could later impact that same
discipline down to its very foundations,6 and that a college student who
was not even the leading mathematician among his group of friends could
later set the tone for an entire century of mathematical discourse.7
Moreover, this work is intended to serve as a reminder of the healthy and
intricate union that once existed between philosophy and mathematics.
2
Synthesis and Formalism
As the 17th Century was drawing to a close, Europe was nursing its
wounds and contemplating the religious wars that had inflicted them. It
seemed as though the violence stemmed from an impassioned reliance on
irrationality, superstition, and intellectual tyranny, propped up by the
notion of tradition. The subsequent period, known as the Enlightenment,
was dominated by thinkers who sought to progress away from the bloody
consequences of irrationality. Instead, they worked to employ rationality
as a means of systematizing epistemology, ethics, and aesthetics.
Concerning epistemology – the study of knowledge – in particular, the
Enlightenment contained a marked effort to move away from a reliance on
personal revelation and mysticism, in favor of the axiomatic method. A
similar shift was taking place in science through the work of Isaac Newton.
René Descartes, one of the foremost figures of the Enlightenment,
famously stated Cogito ergo sum – I think, therefore I am. This utterance
was the result of his effort to find a “clear and distinct foundation for
thought.”1 Descartes was skeptical of much of what was called human
knowledge, but he believed that careful reflection would reveal the selfevident axioms for any system of thought. His own meditation on the
subject led him to the conclusion that he was a thinking being, and this
formed the basis of his subsequent knowledge.2
Descartes and others, such as Hume and Voltaire, were striving to
bring reason and thorough contemplation to society as a beneficial
alternative to traditional superstition. Immanuel Kant entered the fray in
the latter half of the 18th Century and brought with him the motto Sapere
Aude! – dare to know. For Kant, the Enlightenment was characterized by
“the courage to use your own intelligence.”3
3
So it was that as philosophers and scientists alike worked toward a
more reasoned approach in their disciplines, they looked to the firm field
of mathematics and its deductive technique as their model.4 Kant realized
that this reliance on a mathematical approach to the
systemization of knowledge made it a chief priority to
understand the state of mathematical knowledge itself. He
addressed this topic in his most important work, the Critique
of Pure Reason.5
According to Kant, knowledge is divided along two dimensions. First, it
can be classified as a priori or a posteriori regarding its appeal to
experience. For instance, a logical tautology – it is raining or it is not
raining – is known to be true without empirical support and would be
classified as a priori knowledge. On the other hand, a claim about
spatiotemporal reality – it is currently raining in Allendale, MI – requires
empirical support to determine its truth-value and would be classified as a
posteriori knowledge. Second, Kant makes the distinction between
analytic and synthetic knowledge. Without delving too deeply into Kant’s
propositional philosophy, it will suffice for our purposes to view analytic
judgments as an elucidation of that which is inherently contained (or
excluded) by concepts, and synthetic judgments as claims that are extraconceptual. The statement “all bachelors are men” is analytic because the
concept of being a man is enclosed within the definition of a bachelor.
Contrastingly, the statement “bachelors enjoy food” is synthetic because
the affinity for food is not found in the definition of bachelor.
If readers are feeling as though the paragraph above simply describes
different labels for the same distinction, they are echoing the sentiments
of countless philosophers, namely, those prior to Kant. It is true that
analytic knowledge is necessarily a priori, and Kant noted that philosophers spend the largest part of their time working in this realm. It is also
the case that most synthetic judgments rely on experience and are thus
a posteriori, this being the territory of the empirical sciences. However,
4
Kant posited the existence of synthetic a priori judgments, the crucial case
where new knowledge can be constructed that is necessarily true without
appeal to experience. It is at this vital frontier that we find mathematics.
Consider the concept of a triangle and you will be able to reach several
conclusions – a triangle is a geometric figure, it has three angles, etc.
These propositions are merely explicating characteristics that are inherent
to any triangle. Kant argued that mathematicians are able to generate
propositions that are qualitatively different from those above. For
instance, mathematicians are able to show, without any reliance upon
empirical information, that the interior angles of a Euclidean triangle sum
to two right angles. Mathematicians do this by calling upon several other
concepts, such as extension, parallelism, and correspondence, and
bringing them into concert with the concept of a triangle.
Because
propositions
like
the
Triangle
Sum
theorem
can
be
demonstrated non-empirically, and because they provide knowledge that
is extra-conceptual, we see Kant’s reasoning for classifying the majority of
mathematical judgments as synthetic a priori. Kant himself writes that
these judgments “cannot be found within the concepts” but instead are
found “a priori in the intuition corresponding to the concept, and can be
connected with it synthetically.”6 This notion of intuition is a vital
component of Kant’s framework. It can be thought of as a means through
which singular and immediate knowledge is achieved in relation to
objects, and the intuitions derive from mental, not empirical, activity.
Soon after the Critique of Pure Reason was making its impact, the nonEuclidean revolution took place. Its effect on Kant’s work will be discussed
in a later section. At present, it is sufficient to note that the emergence of
consistent non-Euclidean geometries, along with the development of an
expansive yet ungrounded analysis, led to a foundational crisis in
mathematics near the beginning of the 20th Century. Three foundational
schools – logicism, intuitionism, and formalism – attempted to resolve the
crisis, and each was effectively a response to the work of Kant.
5
Logicism, as developed by Frege, Russell, and others, comprises the
view that mathematical knowledge is not synthetic a priori but analytic a
priori; indeed, all of mathematics may be reduced to logic and is in
essence a complex, 2000-year-old, ever-growing logical tautology. The
supposed logical and analytic basis of mathematics led Bertrand Russell
to conclude that mathematical propositions were not only true in worlds
consisting of Kantian intuition, but true in “all possible worlds.”
Intuitionism, as expounded by Brouwer, is more affirming of Kant than
logicism. Intuitionists argue that mathematical knowledge is the result of
an actively constructive (i.e., synthetic) process in the human mind and is
the product of intuition, not a glimpse of some Platonic realm nor an
analytic tautology. They require mathematical entities to be explicitly
constructed and reject the law of excluded middle so that all concepts may
lie directly within the scope of intuition. It is the belief of intuitionists (and
also Kant, as we shall see shortly) that mathematics is mind-dependent,
“concerning a specific aspect of human thought.”7
Formalism, which is where we reunite with David Hilbert,
is based upon the observation that mathematics is largely the
manipulation of characters, a game utilizing formal rules
(rooted in logic) and arbitrary axioms.8 Indeed, Hilbert
famously
stated
that
“mathematics
is
a
game
played
according to certain simple rules with meaningless marks on paper.” It is
the job of mathematicians to play the game, formulating vast arrays of
implications following from the definitions, axioms, and postulates.9 If
scientists or any other interested party were to find an application or a
real-world model that corresponds to an axiomatic theory, which has
happened constantly throughout history, they would be free to reap the
benefits of the mathematical community’s work. However, this is of no
concern to the formalist mathematician.
Hilbert, the leading proponent of formalism in the early 20th
Century, championed the goal of a complete and completely consistent
6
axiomatization of all branches of mathematics. He did not wish to show
that the entirety of mathematics was a necessary logical truth, but instead
that it was composed entirely of logically consistent implications, that is, if
the axioms of a particular theory are assumed to be true, then every
theorem within that theory necessarily follows. Neither did Hilbert wish to
say that mathematics was meaningless and arbitrary (despite his
comment above). His “finitary arithmetic” was set upon an intuitive base
and served as a building point for consistent levels of higher mathematics.
Finitary arithmetic is a subsystem of general arithmetic consisting of
statements that involve the natural numbers and can be decided in a
finite number of steps. For example, the statement “there is a number p
greater than 100 and less than 100! such that p is prime” is finitary, while
the statement “there is a number p greater than 100 such that both p and
p + 2 are prime” is not. Moreover, Hilbert proposed that finitary arithmetic
is rooted in the intuitive concept of numerical symbols representing
plurality.10 This finitary foundation was considered meaningful by Hilbert
and offered ontological and Kantian credence to mathematics, though he
conceded that the greater part of the field was occupied by deductions
from axioms that may be considered arbitrary.
Formalism is the dominant philosophical position among mathematicians in the early 21st Century. One of the reasons for this is the
aversion that many mathematicians have toward philosophy altogether,
and formalism allows for many probing questions to be side-stepped.11 At
the same time, formalism grants mathematical researchers a great deal of
freedom because any axiomatic system that sparks interest, however
“pure”, may be the subject of investigation. This freedom has permitted
mathematics to be at the forefront of human knowledge, and to be
hauntingly predictive of the advances in science. According to Steven
Weinberg,
“it
is
positively
spooky
how
the
mathematician has been there before him or her.”12
7
physicist
finds
the
Latent Forces Revealed
Prior to Copernicus, it was commonly held by thinkers and laypeople
alike that the sun revolved around the earth. Certainly it appeared that
way at the time, as it does today. However, an historic figure considered
the role of the observer in the situation and the result was the Copernican
Revolution of science. Centuries later, the West’s greatest thinkers
remained entrapped within a deception of a different kind, and it would
take another historic figure – Immanuel Kant – to shed light on the
problem and bring about a “Copernican Revolution” of philosophy.
Philosophers before Kant principally believed that the careful use of
pure reason could lead to objective knowledge about the world as it is,
about objects in themselves. In the Critique of Pure Reason, Kant
considered the role of the observer (i.e., the knower) in the process of
gathering knowledge about the world.1 He made explicit the role of the
knower’s mind in the formulation of knowledge by distinguishing between
that which is known by the mind and that which exists independent of it. 2
Kant’s revolutionary proposition was that the mind is not a tabula rasa,
but an active originator of its mental images. Perceptual input must
necessarily be processed and organized or else it is meaningless noise
and, as Kant puts it, “nothing to us.”
Though it seems contradictory prima facie, an artist’s canvas is a
useful analogy for the mind of the knower in Kant’s theory. This notion of
a canvas is distinct from a tabula rasa because the latter implies a passive
recipient of perceptions that displays things exactly as it receives them,
while the former alludes to an active and integral component of the result.
Just as a canvas influences the brushstrokes, interacts with the paint,
and contributes texture to the artwork, so does the knower’s mind
8
interpret perceptions and organize them according to inherent structures.
The canvas is not itself the work of art, yet the painting does not exist
apart from it.3
For Kant, the formal characteristics and structures of the world (as far
as pure reason can conclude) are there because the mind of the knower
has put them there. Space and time are two such characteristics
underlying human judgments with regard to objects. Since the knower’s
mind necessarily places its perceptions onto a canvas of space and time,
mathematics can offer meaningful knowledge by investigating the canvas
itself, taking up the study of space with geometry, and of time with
arithmetic and analysis.4 If mathematics can determine truths with regard
to the canvas that is the backdrop of all human perception, then certainly
it has offered a consequential kind of knowledge. The nature of this
knowledge may be understood more clearly through the presentation of an
analogy:
A television station employs an anchorman for the purpose
of delivering the nightly news. It is currently an hour before air,
and he will soon be selecting a tie and suit jacket from a
seemingly endless supply in the dressing room. Meanwhile, a
woman across town is eating her supper and knows with
certainty that she will see the anchorman present the headlines
wearing a gray tie. How can such knowledge exist? She has a
black-and-white television set.
Clearly, knowledge can be gained from an understanding of the
medium through which perceptions are delivered. In the same way that
the woman above understood her television set, mathematicians can offer
understanding of the mental structures of space and time. Even though
the knowledge is not concerning the world as it is (she doesn’t know the
actual tie color), Kant would contend that it is genuine knowledge and the
only type that we can truly have any certainty over.
9
With this in mind, Kant turned to Euclidean geometry as an example.
He reasoned that the human mind is inherently Euclidean and we
therefore contemplate all spatio-objects in this environment. It was for this
reason that the two-thousand-year development of geometry, dating back
to Euclid of Alexandria, had proved so fruitful and so applicable. It was
fruitful because judgments could be intuited and synthesized regarding
the structure of the human mind, and applicable because all perceived
objects must adhere to this structure. However, it later emerged that
throughout the entirety of this two-thousand-year development, assumptions were being made unbeknownst to the geometers. Bertrand Russell
noted, concerning Euclid himself, that “his demonstrations require many
axioms of which he is quite unconscious.”5
Just prior to the 20th Century, David Hilbert released Grundlagen der
Geometrie.6 The centerpiece of the work was twenty axioms of Euclidean
geometry, a far cry from Euclid’s original five. Hilbert was able to make
explicit the assumptions that Euclid had indeed used in his proofs but
had failed to include among his axioms and postulates. For instance, in
Proposition 1 of Book I, Euclid presents the construction of an equilateral
triangle on a given line segment. The crux of the
demonstration rests on two intersecting circles (see
figure), but how does Euclid know that the circles
indeed intersect? Relying solely upon his axioms and
postulates, he doesn’t, though it seems reasonable
enough
given
the
depiction.
But
The construction of an
equilateral triangle.
mathematical
conclusions (and Kant’s synthetic a priori judgments) cannot be based
upon diagrams since the standards of rigor exclude such appeals (and it
may be construed as empirical in nature). To correct these errors of
omission committed by generations and generations of geometers, Hilbert
included previously concealed axioms of incidence, order, congruence, and
continuity.7 His work set a new standard for axiomatization and
mathematical proof-writing.
10
Limits and Optimism
The previous section discussed Kant’s view that the human mind
imposes its own characteristics onto the perceived world, especially with
regard to space and time. Hence, much can be learned through
investigation of the human mind and its intuitions (e.g., the synthetic a
priori judgments of mathematics), but certainty about the world in itself is
unattainable. For Kant, the human mind is the great limiting factor of our
knowledge. The same entity that allows us to contemplate and
comprehend, ironically, stands in the way of the achievement of absolute
knowledge. This is because nothing can be grasped by the mind that does
not enter the realm of the mind, thus encountering the structures and the
frameworks that characterize it. It follows that pure reason is impotent
when it comes to judgments concerning the world per se, and is confined
to the conclusions it may draw regarding the world as perceived by
humans.
For more than two-thousand years, mathematicians in general, and
geometers in particular, did not see it this way. With Euclid’s Elements in
hand, they felt they were formulating and proving propositions that were
true of the world as it is. How could it have been any other way? The
theorems of geometry followed directly from axioms and postulates that
were so obviously true they didn’t require proof. Then, in the early 19th
Century, Lobachevsky and Bolyai produced consistent non-Euclidean
geometries, with another to follow from the hand of Riemann. It became
apparent to the mathematical community that the standard-bearer of
certainty, Euclidean geometry, was not, in fact, generating absolute
knowledge concerning the world in itself, but only one interpretation of it.1
11
At this point, things become even more subtle. The development of
non-Euclidean geometries, as presented above, seems to vindicate Kant’s
argument that pure reason (viz., mathematics) does not inform us about
the real world as such, but instead the world as perceived by human
minds. Viewed in this light, non-Euclidean geometries seem to bolster
Kant’s case. Historically, and quite to the contrary, they were a near
death-blow to the Kantian perspective.2 Kant had posited the fact that
the canvas of the human mind was distinctly and solely Euclidean,
and mathematics, as the source of synthetic a priori judgments,
should only be capable of reaching “truths” within a Euclidean
framework. Non-Euclidean geometries were a blatant counterexample to
this claim, essentially showing that mathematicians could fruitfully work
within both the Euclidean framework and a separate, incompatible
framework. In other words, the mathematician could exchange her mental
canvas at will.
Hilbert’s Grundlagen der Geometrie, in addition to supplying a complete
set of geometric axioms, articulated the way in which mathematicians
were able to overcome Kant’s Euclidean constraint. The solution came in
the form of the modern-day interpretation of the axiomatic method. No
longer were theoretical axioms an expression of unproven truths, nor
where they a representation of the mind’s unalterable structure; instead,
they became assumptions made as the basis of deductive reasoning. In
this manner, Hilbert made room for Euclidean and non-Euclidean
geometries, as they are both systems of logical deductions following from
distinct (though related) sets of axioms. Within Hilbert’s axiomatic theory,
the canvas of the mind is not predetermined, but is somewhat controllable
by the mathematician.3
This view of axiomatics, as presented and promoted by Hilbert,
overcame another of Kant’s limitations on human reason. Kant, in
his Transcendental Aesthetic from the Critique of Pure Reason, argues
that intuition consists entirely of the representation of appearances and
12
contains nothing but relations, and since nothing can be known in
itself simply through relations, we cannot know the inner properties
of objects. Hilbert, contrarily, emphasizes the profound effect of relations when defining mathematical objects. Indeed, these relationships
are all that can be provided because an attempt to explicitly define
each fundamental term is necessarily futile. According to Hilbert,
“a concept can be fixed logically only by its relations to other concepts.
These relations, formulated in certain statements I call axioms, thus
arriving at the view that axioms...are the definitions of the concepts.”4
Where Kant holds that knowledge of objects as they exist is unattainable,
Hilbert contends that as long as consistency is present within a system
of axioms (i.e., the implicit definers), then “the things defined by
them exist.”5
Hilbert’s optimism in the face of Kant’s substantial limitations was
pervasive. The Hilbert programme had as its goal the proof of the
consistency and the completeness of every branch of mathematics, an
optimistic objective if ever there was one.6 Hilbert even believed he had a
means for accomplishing this meta-mathematical task, namely, the
finitary arithmetic discussed above. By rooting each system to the
intuitively fertile ground of Hilbert’s finitary mathematics, consistency and
completeness would be assured.7 If this were achieved, every mathematician could rest easy knowing that the theory they work within will not
lead them astray and their engagement with particular problems is not in
vain. On a smaller, yet no less significant scale, Hilbert firmly believed
that any clearly formulated mathematical problem was capable of being
solved (or being proven unsolvable). This confidence in the power of
mathematics provided, and still provides, inspiration for mathematicians
around the world.8
13
Conclusion
Immanuel Kant began his five year descent toward death with these
words: “My gentlemen, I am old and weak, and you must consider me as a
child.”1 And as his mind was leaving him Kant made sure to walk daily,
just as the child who set out to Fridericianum every morning. This old
man of Königsberg was responsible for a revolution of thought. His work
directed a vast amount of philosophical inquiry in the centuries that
followed him, and he played an integral part in the drama that led to the
movement of mathematical foundations.
As David Hilbert neared the end of his life, he watched disappointedly
as Germany entered the Third Reich and many of his professional
colleagues and friends were forced from their positions. “How beautiful the
old days were, how ugly the present.”2 Indeed, missing from his old age
was the time spent with close friends on college campuses contemplating,
discussing, and living mathematics. Thinking back to his early days, Hilbert
maintained “the most beautiful town in Germany is still Königsberg!”3 And
it was this town that produced one of the foremost mathematicians of the
early 20th Century. He contributed important work to a wide range of
fields, his presentation of the Paris Problems spurred mathematical
inquiry for decades, and his leadership in the Formalist foundational
school is largely responsible for the continuing popularity of its tenets.
In Königsberg, on the tomb of Immanuel Kant, these words are
inscribed: “Two things fill the mind with ever new and increasing
admiration and awe, the more often and perseveringly my thinking
engages itself with them: the starry heavens above me and the moral law
within me.” Matching the depth of this wonder for the human intellect is
David Hilbert’s confidence in it. His epitaph reads: Wir müssen wissen. Wir
werden wissen – We must know. We will know.
14
INTRODUCTION
1
Kuehn, p 31.
Reid, p 2. The river referred to is the Pregel, which encompasses a small island within Königsberg called
the Kneiphof, or beer garden. The seven bridges leading to and from this island are famous in mathematics as
the basis for the Königsberg Bridge Problem. Solved by Leonhard Euler, this problem and similar
considerations led to the development of graph theory and topology.
2
Interestingly, Kant and Hilbert both attended the University of Königsberg as well as Friedrichskolleg.
They entered the university in 1740 and 1880, respectively. Kant began lecturing for the University of
Königsberg in 1755; Hilbert began in 1886.
3
4
This classmate was Hermann Minkowski who, as a student, was awarded the Grand Prix des Sciences
Mathématiques from the Paris Academy.
5
See Kant’s Answer to the Question: ‘What is Enlightenment?’
6
Kuehn, p 50, describes Kant’s mathematical ability in grade school as “dismal.”
Reid, p 12, notes that Hilbert became friends with the gifted Minkowski in spite of his father’s objection
that taking up an acquaintance with “such a famous man” would be impertinent. Indeed, Hilbert may have
resided at third in the young group after the addition of the talented Adolf Hurwitz, p 13.
7
SYNTHESIS AND FORMALISM
1
Wikipedia, the Enlightenment.
Some hold that Descartes’ strikingly simple statement contains the conclusion in its premise, and amounts
to nothing more than circular reasoning, albeit a small circle. The concept of I inherently contains an
assumption of existence, so the famous quote may be reduced to “I, therefore I.”
2
3
Kant, Answer to the Question: ‘What is Enlightenment?’
4
Euclid of Alexandria is responsible for the most widely celebrated example of the axiomatic method on
display in his Elements. From five common notions and five accepted postulates, Euclid deductively developed
and codified a wealth of geometric results that remain true today. In a similar manner, Kant often wrote in a
proposition-proof format and Newton used a minimal set of natural laws to derive his physical mechanics.
5
Published in 1781, the Critique of Pure Reason was followed by the Critique of Practical Reason (1788)
and the Critique of Judgment (1790), these constituting the core of his writings.
6
Kant, p 91.
7
Shapiro, p 175.
8
Most formalists would hold that the axioms of mathematics are not arbitrary, but they may as well be
arbitrary. Mathematics has been extremely successful in its various applications, especially the sciences,
precisely because its axioms are not arbitrary – they are either chosen to coincide with observed reality or
they are formulated after the applications have been developed. However, formalism maintains that
meaningful mathematics can take place with arbitrary sets of axioms because the rules can still be followed
and implications can still be found.
9
This description concerns the largest branch of formalism, referred to as deductivism.
Shapiro, p 161. These numerical symbols may be viewed as |, | |, | | |, | | | |, … Hilbert appeals to the highly
intuitive nature of these symbols and then contends that the familiar Arabic numerals simply serve to more
efficiently communicate statements regarding this intuition.
10
Shapiro, p 150, writes the following from the formalist perspective: “What is mathematics about? Nothing,
or it can be regarded as about nothing. What is mathematical knowledge? It is knowledge of what follows
from what. Mathematical knowledge is logical knowledge. How is a branch of mathematics applied? By
finding interpretations that make its axioms true.”
11
12
Shapiro, p 39.
15
LATENT FORCES REVEALED
1
Like the revolutions of Copernicus and Kant, the Einsteinian revolution of the early 20 th Century resulted
from taking the observer’s velocity into account with regard to the speed of light. For more, see Otten,
Appendix C. Einstein cites the Critique of Pure Reason as a great influence that aided him in his
development of relativity theory.
2
Kant employs two distinct terms in this regard. The noumenal world refers to objects in themselves, as they
truly are. About this world the human mind can truly know very little. The phenomenal world consists of
objects as perceived and represented by the human mind within its various frameworks. It is within this realm
that human judgments and knowledge flourish.
3
If this analogy does not fully emphasize the active nature of the mind, consider a musical instrument as it
relates to music. The instrument is not the music itself, but the music must pass through the instrument thus
inheriting its tone and timbre.
4
This type of mathematical knowledge applies to the phenomenal world only.
5
Dunham, p 38.
Interestingly, Hilbert quotes Kant in the epigraph of Grundlagen: “All human knowledge begins with
intuitions, then passes to concepts and ends with ideas.”
6
7
See Hilbert’s Axioms of Geometry on page 18 of the current work.
LIMITS AND OPTIMISM
1
For more, see Kline, chapter 4.
2
En route to his dissertation, Hilbert was required to engage in a defense of two theses of his own choice.
One of his selections was to defend Kant’s notion of a priori mathematical knowledge. Non-Euclidean
geometries had toppled this claim with regard to space (Euclidean geometry), but Hilbert defended Kant’s
position with regard to time (arithmetic). A college-aged Hilbert said, “the objections to Kant’s theory of the
apriori nature of arithmetical judgments are unfounded.” Reid, p 17.
3
The canvas, or the set of axioms, is not completely controllable. It is uninteresting to adopt contradictory
axioms since anything can be proven true within such a system. Also, some sets of axioms are utterly
unfruitful.
4
Shapiro, p 157.
5
Shapiro, p 156.
6
The emergence of Kurt Gödel’s Incompleteness theorems rendered these attempts futile.
7
Conversely, Frege believed that the production of a model for a mathematical system guaranteed that
system’s truth and, subsequently, its consistency.
8
Richard Courant cited Hilbert’s “contagious optimism” as a source of “vitality for mathematics.”
CONCLUSION
1
Kuehn, p 414.
2
Reid, p 205.
3
Reid, p 211.
16
Timeline
1637
René Descartes publishes the Discourse on Method.
1650
Descartes dies.
1687
Isaac Newton’s Philosophiae Naturalis Principia Mathematica is released.
1689
John Locke completes An Essay Concerning Human Understanding, arguing that the
mind begins as a “blank slate.”
1724, Apr 22
Immanuel Kant is born to Johann Georg and Anna Regina Kant.
1737
Kant’s mother dies.
1740
Kant enrolls in the University of Königsberg.
1770
The University of Königsberg admits Kant as a full professor.
1781
The Critique of Pure Reason is released.
1788
The Critique of Practical Reason is released.
1789
The lower classes of France overthrow Louis XVI.
1790
The Critique of Judgment is released.
1804, Feb 12
Kant dies.
1829
A treatise of hyperbolic geometry is published by the Russian Nikolai Lobachevsky.
1832
János Bolyai, a Hungarian, independently develops hyperbolic geometry.
1854
Bernhard Riemann establishes elliptic geometry through a lecture.
1862, Jan 23
David Hilbert is born to Otto and Maria Hilbert.
1880
Hilbert enrolls in the University of Königsberg.
1886
Hilbert begins lecturing at the University of Königsberg.
1895
Felix Klein arranges for Hilbert to become the Chairman of Mathematics at the
University of Göttingen.
1899
Hilbert publishes Grundlagen der Geometrie.
1900
Hilbert presents the Paris Problems at the International Congress of Mathematics.
1905
Albert Einstein experiences his “annus mirabilis” and introduces the special theory of
relativity as well as the equation e  mc 2 .
1920
Hilbert proposes his meta-mathematical programme.
1931
Kurt Gödel releases his incompleteness theorem.
1943, Feb 14
Hilbert dies.
17
Hilbert’s Axioms of Geometry
Undefined Terms






Axioms of Congruence
Points
Lines
Planes
Lie on, Contains
Between
Congruent
13. If A, B are two points on a line a, and A' is a
point on the same or on another line a' then
it is always possible to find a point B' on a
given side of the line a' such that AB and
A'B' are congruent.
14. If a segment A'B' and a segment A"B" are
congruent to the same segment AB, then
segments A'B' and A"B" are congruent to
each other.
15. On a line a, let AB and BC be two segments
which, except for B, have no points in
common. Furthermore, on the same or
another line a', let A'B' and B'C' be two
segments which, except for B', have no
points in common. In that case if AB ≈ A'B'
and BC ≈ B'C', then AC ≈ A'C'.
16. If ABC is an angle and if B'C' is a ray,
then there is exactly one ray B'A' on each
Axioms of Incidence
1.
2.
3.
4.
5.
6.
7.
8.
For every two points A, B there exists a
line a that contains each of the points A, B.
For every two points A, B there exists no
more than one line that contains each of the
points A, B.
There exists at least two points on a line.
There exist at least three points that do not
lie on a line.
For any three points A, B, C that do not lie
on the same line there exists a plane α that
contains each of the points A, B, C. For every
plane there exists a point which it contains.
For any three points A, B, C that do not lie
on one and the same line there exists no
more than one plane that contains each of
the three points A, B, C.
If two points A, B of a line a lie in a plane
α then every point of a lies in the plane α.
If two planes α, β have a point A in
common then they have at least one more
point B in common.
There exist at least four points which do
not lie in a plane.
"side" of line B'C' so that ∠A'B'C'≅∠ABC.
Furthermore, every angle is congruent to
itself.
17. If for two triangles ABC and A'B'C' the
congruences AB ≈ A'B', AC ≈ A'C' and
∠BAC ≈ ∠B'A'C' are valid, then the
congruence ∠ABC ≈ ∠A'B'C' is also
satisfied.
Axiom of Parallels
18. Let a be any line and A a point not on it.
Then there is at most one line in the plane
that contains a and A that passes through A
and does not intersect a.
Axioms of Order
9.
If a point B lies between a point A and a point
C then the points A, B, C are three distinct
points of a line, and B lies between C and A.
10. For two points A and C, there always exists
at least one point B on the line AC such
that C lies between A and B.
11. Of any three points on a line there exists no
more than one that lies between the other two.
12. Let A, B, C be three points that do not lie
on a line and let a be a line in the plane
ABC which does not meet any of the points
A, B, C. If the line a passes through a point
of the segment AB, it also passes through a
point of the segment AC, or through a point
of the segment BC.
Axioms of Continuity
19. If AB and CD are any segments, then there
exists a number n such that n copies of CD
constructed contiguously from A along the
ray AB willl pass beyond the point B.
20. An extension of a set of points on a line with
its order and congruence relations that
would preserve the relations existing among
the original elements as well as the
fundamental properties of line order and
congruence (Axioms I-III and V-1) is
impossible.
18
Bibliography
“You will not learn from me philosophy, but philosophizing,
not thoughts merely for repetition but thinking.”
Immanuel Kant
“We hear within us the perpetual call:
There is a problem. Seek its solution.”
David Hilbert
 Dunham, W. (1990). Journey through Genius: The Great Theorems of Mathematics.
John Wiley & Sons, Inc., Indianapolis, IN.
 Heath, T. (1956). The Thirteen Books of Euclid’s Elements (3 Vols.). Dover
Publications, Inc., New York, NY.
 Greenberg, M. (1993). Euclidean and Non-Euclidean Geometries: Development and
History. W.H. Freeman & Company, New York, NY.
 Hilbert, D. (1965). Foundations of Geometry [trans. Townsend]. Open Court Press,
La Salle, IL.
 Kant, I. (1965). Critique of Pure Reason [trans. Smith]. St. Martin’s Press, New York,
NY.
 Kline, M. (1980). Mathematics: The Loss of Certainty. Oxford University Press, New
York, NY.
 Kuehn, M. (2001). Kant: A Biography. Cambridge University Press, Cambridge, UK.
 Otten, S. (2005). Philosophy and Mathematics: A Collection of Conversations.
[undergraduate paper] Grand Valley State University, Allendale, MI.
 Reid, C. (1970). Hilbert. Springer-Verlag, New York, NY.
 Shapiro, S. (2004). Thinking About Mathematics: The Philosophy of Mathematics.
Oxford University Press, New York, NY.
 Wikipedia Online Encyclopedia. [online resource] David Hilbert. Retrieved
1/27/2006 from http://en.wikipedia.org/wiki/David_Hilbert.
 Wikipedia Online Encyclopedia. [online resource] The Enlightenment. Retrieved
2/11/2006 from http://en.wikipedia.org/wiki/The_Enlightenment.
 Wikipedia Online Encyclopedia. [online resource] Immanuel Kant. Retrieved
1/27/2006 from http://en.wikipedia.org/wiki/Immanuel_Kant.
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