What are the foundations of mathematics? Early answers to this question were closely related to geometry, and historically, the philosophy of mathematics and the mathematics of geometry maintained a unique connection for more than two thousand years. During this period absolute certainty reigned, and here we shall survey major developments in the evolution of geometry and metamathematics in relation to certitude. We will begin with the origins of the belief in mathematical certainty in Classical Greece, then survey its connection to science through to the seventeenth-century. In closing, we will examine the decline of certainty in the early nineteenth-century, when the discovery of non-Euclidean geometry forced uncertainty on to mathematics and philosophy.
Perhaps the first inquiry in to mathematical foundations was by the Greek philosopher Thales (c. 624 - 547 BCE). Thales saw that in counting and measuring, the practices of unconnected regions coincided, and the practices of one region applied to others. This coincidence enabled different groups to make calculations in the same way, for example when working with physical spaces that approximated elementary mathematical shapes, such as rectangular grain fields. Observing that geographically diverse peoples treated numbers and numeric operations similarly, Thales asked: why?
The practices Thales observed had developed independently, but appeared to share the same general form, and to be generally applicable and accurate, and this was a remarkable fact when compared to the non-generality of other regional practices, for example in politics and religion. In attempting to account for his observations, Thales approached his explanation empirically and universally, and his mode of explanation differed dramatically from the prevalent mode of explanation, which was pre-deductive (and which we refer to as pre-deductive precisely because of the power and prevalence of deduction, after Thales).
Pre-deductive discourse, as seen for example in the religious texts of Thales' era, presented claims in a de facto manner, and presented idealized assertions and idealized consequences, while Thales attempted to arrive at conclusions about observations, and also inquired about the very basis of his observations. Thales was therefore grasping towards a new mode of discourse that we might describe as proto-deductive.
Owing to the nature of his investigations, Thales introduced the term "geometry," meaning "earth measurement," in reference to land plotting and similar activities. The term "mathematics" meaning "knowledge," was introduced after Thales by his mathematical successors, the Pythagoreans. With respect to metamathematics, the origin of these terms is important, being an indicator of the reason geometry and mathematics came to be well-defined fields of inquiry. Geometry arose to organize regionally diverse but conceptually united practices, and approached the real world in terms of magnitudes, and elementary operations that related those magnitudes; and mathematics arose to treat of magnitudes and operations more generally.
Enthralled by the incredible utility and uniformity of mathematics, the Pythagoreans developed a mystical belief system based on the idea that mathematical associations were the framework within which the physical world unfolded. In their framework the concept of number was central, and the Pythagoreans equated math and numbers with metaphysical genesis, as can be seen from one of their oaths; "Bless us, divine number, thou who generates gods and men!"
The Pythagoreans made a number of discoveries that correlated nature closely with mathematics, such as the discovery that musical harmonies may be represented in terms of whole number ratios. This provided fodder for the idea that mathematics was not merely the prism through which nature could be understood, but that nature was mathematics; that "all things are numbers." This metamathematical idea led the Pythagoreans to categorize nature hierarchically, such that math was the source of the universe, and expressed itself in terms of the discrete and the continuous, where the discrete gave rise to the absolute (arithmetic) and the relative (music), and the continuous gave rise to the static (geometry) and the moving (astronomy). Mathematics was the fountainhead, prior to both "gods and men," and generated and organized all of nature; an important claim, because it made mathematics more basic than gods, and was therefore connected to Thales' reasoning process, in that both reassessed religious thinking.
In sum, Thales considered the practices of mathematics generally, and approached math in a way that prefigured deduction, and the Pythagoreans took the universality of mathematics to indicate that the universe was identifiable with mathematics. Thus, mathematical practices had directly spurred metaphysical reflections, and those reflections yielded metamathematical conclusions that led to realignments in existing philosophies. Although claims that appealed to God in pre-deductive modes of explanation still dominated, by the era of the Pythagoreans they were increasingly challenged by mathematical considerations.
Like the Pythagoreans, the Greek philosopher Plato (c. 424 - 347 BCE) believed mathematics was fundamental to being, however, unlike the Pythagoreans, Plato did not believe a hierarchy of categories such as the discrete and continuous captured the foundations of mathematics. For Plato, mathematics existed in the eternal world of Forms, while humans lived in the temporal world, in an ever-changing process of becoming. The Forms effected the universe, and the universe's physical forms were constantly undergoing change, and because of this the real world presented only a shadow of the Forms to humans, meaning humans had limited access to the perfect Forms of mathematics. Mathematics did underpin nature, but natural sensations presented nature and math to humans incompletely.
Because mathematical Forms existed independently of human experience and could not be properly perceived via the senses, Plato eschewed the incompleteness of sensation, turned inwards, and concluded true knowledge of the Forms was to be achieved through cogitation. Because mathematics transcended human experience, it was a natural truth that could be established by transcendent thought. Thus, Plato accepted the Pythagorean belief in mathematics as a basic reality that exists independently of humans, and combined it with Thales' concern for understanding the connections between ideas in a universally consistent manner.
Responding to Plato's metamathematical deliberations, his student Aristotle (384-322 BCE) took up the project of formalizing Thales' reasoning procedure, and elaborated on the relationship between claims and conclusions, and denied that mathematical truth corresponded to the contemplation of ideal mathematical Forms. For Aristotle, Forms inhered within physical existence, and the foundation of mathematics was forms inhering in the world. True mathematics were indeed arrived at by reasoning, however reasoning was to be based on observations of the Forms in nature, rather than arguing from purely intellective premises about the Forms. Physical experience was the foundation for arriving at accurate mathematics: observing the world, analyzing those observations generally, and categorizing those analyses produced truth. Only thus could humans draw objective and accurate conclusions about the mathematical Forms.
Building on the work of Thales, the Pythagoreans, Plato, and Aristotle (and others), the Greek expositor Euclid (c. 300 BCE) set forth in his Elements a series of mathematical proofs using the recently developed logico-deductive format, beginning with mathematical axioms and postulates, combining these with mathematical rules, and setting out the conclusions that followed from these combinations.
In the Elements, Euclid exhibited the mathematics of his era, which were primarily concerned with geometrical results, by taking mathematical truths that were seemingly self-evident, and using precise, repeatable procedures, that any reader could reapply to develop the exact same theorems. Metamathematically, the Elements is important philosophically and historically, because if its reader accepted the mathematical axioms and operations as defined within -- as they apparently had to -- they were also forced to accept its conclusions. For this reason, the Elements possessed a finished quality; there was no room for further development of the theorems laid out, because none found a reason to disagree with them. Hence, in a sense, the Elements completed the project Thales' started, in its development and presentation of an apparently universally applicable and accurate mathematics.
Mathematics, then, was not seen like other subjects such as politics and religion, which permitted contention and ceaseless disputation and were therefore a collection of claims that were in at least some degree vague or indefinite. It seemed that in mathematics, one observed reality as it was, by universally proving the validity of a theorem. All observers could reproduce a theorem, and thus be certain they shared in the knowable reality of that theorem in exactly the same way as all other observers.
Therefore, as the end of Classical Greek civilization approached, mathematics was regarded as a domain that advanced certain knowledge, because of the metamathematical belief that math's foundations were perfectly natural, and that math's theorems were equivalent to natural relations, as revealed through systematic observation and testable manipulation.
The enduring power of this metamathematical certitude was captured in the results of the Greek mathematician Archimedes (c. 287 - 212 BCE), who combined physical motion with mathematics in such an innovative and lasting manner that many regard his proper intellectual successor to be Isaac Newton (1642 - 1727 CE). Addressing the ancient problem of squaring the circle, Archimedes provided an extraordinary geometric solution that synthesized circular and linear motions. Although these motions were acceptable in Euclidean geometry their synthesis was unprecedented, and though Archimedes' results were not strictly Euclidean, they were rigorous and had all the certainty of a Euclidean result.
This was of singular importance in the history of metamathematics, for after Euclid and Archimedes, the development of geometry, and advances in the investigation of metamathematical certainty languished, for nearly two millennia. Looking forward, we find it was not until the seventeenth-century that new and significant progress occurred in the study of geometry; and, pursuant to the progress of geometry, it was only in the eighteenth-century that significant progress occurred in the study of the foundations of mathematics.
With respect to geometry, the objective of Galileo Galilei (1564 - 1642 CE) was to apprehend the algebra of objects moving in space. In Particular, Galileo's goal was to determine which properties of natural objects and motion could be measured and related to each other mathematically. Accordingly, he came to focus on physical features such as weight, velocity, acceleration, and force. Investigating the foundations of mathematics was not one of Galileo's direct concerns, as he noted in his Discourses and Mathematical Demonstrations Concerning Two New Sciences (1638); "The cause of the acceleration of the motion of falling bodies is not a necessary part of the investigation."
Nonetheless, though Galileo aimed at practical explanations and not foundational ones, he did comment on natural philosophers that developed systems based on mere argumentation, rather than systems based on physical experimentation. Importantly, though Galileo was catholic, and his metamathematics reflected his metaphysics -- God was the basis of existence, and therefore math -- Galileo felt God had no immediate place in physical explanations of the world, because "the universe ... is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures." Proportionately, nature was revealed to humanity by direct study of the world, rather than otherworldly speculation.
This practical bent was shaped by Galileo's metamathematical belief that there was a fundamental difference between idealizing on the one hand, and measuring and then idealizing on the other. In terms of historical continuity, the importance of Galileo was that he took up the methods of Aristotle and Euclid, and picked up the physically oriented studies of Archimedes, in order to develop mathematical equations that correlated natural properties to natural regularities.
In connection to foundations, René Descartes (1596 - 1650 CE) agreed that God was the source of reality and the designer of mathematics, and that God was the reason humanity was able to perceive truths about reality. For Descartes, the fact that God had designed reality mathematically was evident in the patterns we observed, and, as a perfect being, God presented patterns to humans only if they represented truth, and therefore we could be sure of our observations.
Like Plato, Descartes posited a world of perfection that was partially accessible via the senses, and like Aristotle and Galileo, Descartes believed sense datum should be analyzed to arrive at true mathematical theorems. Combining natural patterns with intellective analysis, Descartes associated the properties of lines and points with the symbolic mode of representation, and revolutionized the study of nature by introducing the concepts of variable magnitude and coordinate geometry -- the latter having also been developed by Pierre de Fermat (c. 1607 - 1665 CE), independently of Descartes.
Using Euclidean theorems as a basis, coordinate geometry correlated geometric properties to general algebraic statements that related those properties, and defined curves using symbolic relations. Like the equations of Galileo, coordinate geometry tied physical phenomena to quantitative relations, and, when taken altogether, the works of Galileo, Descartes, and Fermat redefined both the purpose and content of natural philosophy, by grounding it in mathematics. This was a new science imbued with a new type of certainty, based on the authority of God through the certainty of his mathematics.
Adopting both the foundations and practices of the new science, Isaac Newton (1642 - 1727 CE) also maintained that God was the foundation of the universe, and therefore mathematics. In contrast to Galileo and Descartes however, Newton's religion was primary, and was a personal motivation for his mathematical work.
Like Galileo and Descartes, Newton regarded his mathematical intuitions and discoveries as confirmation of his religious ideals, and like Galileo, Newton's emphasis was practical. Building on coordinate geometry, Galileo's studies of motion, and Descartes' conception of variable magnitude, Newton developed the calculus, which approached a curve as a flowing quantity that moved across time, thus defining a close relationship between time and motion. The calculus was a sort of procedural algebra that could be used to manage and understand relations between changing variables, per real world examples such as planetary orbits. For Newton, the harmony of his algebraic mechanics with real world mechanics demonstrated that the universe proceeded along its course mathematically, and the calculus was a testament to its supernatural designer.
Motivated by religion and drawing religious conclusions from his science, Newton's mentality was reminiscent of the Pythagoreans, and his esoteric declarations and studies mark him as somewhat of a mathematical mystic. This fact is easily understandable, in reference to the historical milieu he lived in, but salient metamathematically, because for Newton, Galileo, Descartes, Fermat, and a preponderance of their contemporaries, there was an essential accord between the qualities of God and the quantitative relations of mathematics.
Considering the transformation of natural philosophy from the period beginning immediately before Galileo, and ending with Newton, we observe that science underwent a mathematical reformulation. Before Galileo, natural philosophers concerned themselves with testing ideas against other ideas. By the time of Newton, scientific investigations were concerned with scrutinizing experience, and collating results mathematically. This was crucial in the history of metamathematics, because with the advent of Galileo's equations of motion, Descartes' and Fermat's coordinate geometry, the calculus, and Newtonian mechanics, the goal of science became aligned with the early mathematical goal of defining axioms that were self-evident. Much like Euclid's Elements, if one accepted the physical axioms and postulates of science as well as the rules and equations that related them -- as they apparently had to -- they were also forced to accept the conclusions of science. Unlike the controversies permitted by natural philosophy before Galileo, the experiments and conclusions of science were now repeatable and testable, and there was an air of inevitability and certainty about the new science, because it presented a universally applicable physics based on a universally applicable mathematics. With respect to its algebraic and geometric foundations, there appeared to be no room for disagreement, whether mathematical or metamathematical, because through science mathematics clearly represented nature.
The new science (specifically the calculus), was in fact attacked, on religious grounds, by the influential philosopher George Berkeley (1685 - 1753 CE), the Bishop of Cloyne, in Ireland. However, Berkeley's attack yielded no immediate metamathematical consequences, and this is relevant because the incredible practical utility of algebra and geometry in science continued to be interpreted as proof positive of the correctness of mathematics, and its foundation, God.
The next major development that concerned the relationship between geometry and the foundations of mathematics was the philosophy of Immanuel Kant (1724 - 1804 CE), whose epistemology maintained the content of mathematics, but radically altered its foundations. For Kant, the essence of mathematics was not simply nature as it is, because nature as it is, is unknowable for humans. Human minds possess an architecture that systematizes observations and perceptions by its own internal rules, rather than apprehending the foundations of the universe, and we can never know a thing in itself, independent of our mental architecture. That architecture is natural, but it is does not capture nature, and the well-ordered certainty of math and mathematical science arises from the prescripts of the mind, which include a non-empirical form of knowledge about temporality and spatiality, which we express in the form of our self-evident axioms of mathematics. Geometry and therefore mathematical science were not valid because they were built on proper observation and reflection, but because they rested atop valid spatio-temporal intuitions.
Here, Kant vouchsafed the soundness of Euclidean geometry in a new way, and united his philosophy of mind with Euclid's axioms, postulates, and theorems. Not long after Kant passed away however, this aspect of Kantian philosophy and the long-standing certainty of Euclidean geometry were invalidated by the discovery of non-Euclidean geometries, when it was realized the Euclidean system was not the one system, but only one system among many.
In the first half of the nineteenth-century, János Bolyai (1802 - 1860 CE) and Nikolay Lobachevsky (1792 - 1856 CE) independently demonstrated geometries that were consistent, and did not respect Euclid's fifth postulate, that;
"If a straight line incident to two straight lines has interior angles on the same side of less than two right angles, then the extension of these two lines meets on that side where the angles are less than two right angles."
Contary to the fifth postulate, Bolyai's and Lobachevsky's geometries permitted the construction of multiple "parallel" lines for any given line through a given point. This can be seen, for example, by considering a plane in the shape of a circle, thus enabling one to draw an arc line across the diameter of the circle, and then selecting a point inside the circle that is not on the diameter line, such that numerous lines pass through that point, on angles such that these lines never meet the diameter, because all lines are terminated by the boundary of the circle.
The existence and features of non-Euclidean geometries completely undermined metamathematical certainty, and foisted uncertainty on all scientific and metaphysical suppositions that rested on mathematics. This sparked vigorous attempts to retrieve certainty, including many non-geometric programs such as logicism and formalism, all aimed at rigorously explicating and certifying the foundations of mathematics. Ultimately however, the long-term result of these efforts was only to further separate mathematics from certainty in unexpected ways, and this gave rise to the post-modern perception of mathematics as rooted in reality and internally cohesive, but not certain in any absolute physical or metaphysical sense.
Reflecting on the rise and fall of certainty in geometry and metamathematics from Thales to Lobachevsky, we see that when mathematics first arose it was taken straightforwardly, as a practical device that solved problems in the real world. In prehistory and Classical history, mathematics was approached as a device that simply was and simply worked, much like a door or field plough. When Thales took up mathematics however, he latched on the fact that mathematics was not quite like other devices, and he observed its physical manifestations, and speculated on it supra-physically. This mode of speculation was instrumental in generating Classical Greek metaphysics, and culminated in the logico-deductive method, and the incredibly powerful Euclidean system.
The Euclidean system reigned with certainty for millennia, and though mathematics continued to evolve, and explanations for its certainty changed, the fact of certainty remained. Attempts to explain the basis and correctness of mathematics ranged from Forms and God, to nature and mental architectonics, but even though metamathematical claims varied, mathematical claims did not. Whatever its metamathematics, mathematics itself was absolutely accurate.
The discovery of non-Euclidean geometries instantly destroyed the possibility of absolute mathematical certainty, and this is an extraordinary fact, because for millennia brilliant mathematicians were exactly wrong in their metamathematical certitude. Looking back to the end of certainty, it appears certainty was as much a goal as a hypothesized feature of mathematics; that mathematicians undertook mathematics because they wanted to work with something that was guaranteed.
At a fundamental level, the rise and subsequent fall of mathematical certainty was central to the philosophical and scientific recognition of human fallibility. Today it is believed that nature exists, but because of the peculiarities of our experience of it, there always remains the possibility that our metamathematical and metaphysical claims are inaccurate and perhaps entirely false. Thus, the end of mathematical certainty has given rise to a new kind of certainty, that regardless of its foundations, mathematics remains the most powerful tool humans possess for mediating between themselves and nature, and that the development of mathematics enables us to expose falsities -- such as the absolute certainty of mathematics -- and thus allows us to work towards the refinement and extension of better justified, if not certain beliefs.
Part of the series: UWO