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 arriveatthe 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, bestowsehre orhonor 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.
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 period[5] 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 theimprints 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 impactthat 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.
1
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.”[8] Descartes was skeptical of much of what was called human knowledge, but he believed that careful reflection would reveal the self-evident 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.[9]
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.”[10]
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.[11]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.[12]
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 extra-conceptual. 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 philos-ophers 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, 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.”[13] 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 non-Euclidean 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.
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.”[14]
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.[15]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.[16] 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 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 consistentlevels 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 + 2are prime” is not. Moreover, Hilbert proposed that finitary arithmetic is rooted in the intuitive concept of numerical symbols representing plurality.[17]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 mathe-maticians 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.[18] 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 physicist finds the mathematician has been there before him or her.”[19]
1
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 adeception 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.[20] 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.[21] 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 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.[22]
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.[23] 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.