# Quantitative Inferential Reasoning

## A history of mathematical thinking

Number intuition is evident throughout the animal kingdom, with most organisms seeming to count, demonstrated by their movements, vocalizations, and additional sounds. Animals have a sense of proportion, for their sound production swells, recedes or otherwise modulates with continuity that signifies the integrating of relativities: pressure, volume, speed and the like. Human conceptualizations and grammatical constructions also proportion, consisting of hybrid structures organized as relativities nestled within relativities.

Proportion is ubiquitously intuited in complex ways, but the ability to count seems more limited, which is clearly the case for humans as research shows perception of quantity in sets of multiple objects is increasingly delayed after exceeding only a couple items, and ancient languages such as those of hunter-gatherer tribes have words for ‘one’, ‘two’ and ‘many’, with even ancient Greek leaving etymological signs of similar origins despite the fact that this culture became mathematically advanced. It is probable that, though the intuited law of the excluded middle applies to any sized quantity of particulars as a nearly unconscious, immediate perception, being simply the quality of distinction between multiple objects, counting is added on to human thought as a pragmatic and cultural construct, its acquisition linked to the scrutiny of specific percepts. Unlike almost fully a priori space and time, comprehension of even small numbers and related mathematical operations depends on brain plasticity, requiring practice with fingers (‘digits’), objects, symbols or a calculator, after which it becomes internalized like language.

Synthesis of number counting with perceptions of proportion were of course crucial to the success of early humans, as our species is incapable of achieving apex status without production of technology that requires both quantity, as in amount of wood or stone to build a dwelling, and symmetry as well as balance for the sake of a functional projectile, working snare mechanism or the like. Technology involves interrelated, counterbalanced objects and forces along with their numerical definition, the conceiving of which is ingrained in human thinking as memes.

Proportion is not only a matter of utility but also aesthetics, and some of the most notable examples of mathematical precision, its roots and social niche, come from art. Cave paintings of Europe, dances of prehistoric and historic cultures, music providing the backdrop or centerpiece of so many gatherings all reveal the love of form and contrast, order amongst variety, so central to our psyches. Words do not do justice to impact on the mind of experiencing talented artists conjure their creations godlike out of nothing, effortlessly when well-practiced. It floods the body with pleasure and jolts the observer into intense focus, like a cathartic miracle. In prehistoric youths of a less sublimated time, artistry probably resulted in more than a little infatuation. The art itself inspires wonder, awe, respect, seeming almost supernatural; one yearns for union with it, a connection to its beauty. The spirituality of art probably engendered disproportionate or prioritized reproduction, with the development of an elite artist a key event for the tribe conferring prestige upon that individual.

As cultures began to apply math in more complex ways, advancement in techniques of abstract counting was driven along to supplement mensuration rather than as the independent logic puzzle it became in modern academics, which put the first artists in an important position as supreme measurers, the masters of structural precision. Art, numinosity and technical innovation tended to initially be entwined as religious, symbolic and functional beauty, the expression of what was most valued in a society and its impressions of the cosmos, with strong impetus to exceed the boundaries of previous achievement in competition with rival craftsmen and cultures. Incarnation of proportion was progress and one-upmanship, so math had to make huge strides to meet snowballing demand for greater quantities of more precise measurement.

Civilized division of labor took shape and mathematical expertise became a specialized professional role having a specific training regimen, coinciding with formulation of methods for difficult calculating to assist the counting of lengths and dimensions. Humans had used hands, feet and other roughly standardized objects to estimate quantitative properties in the building of structures and tool construction for at least tens of thousands of years, the primitive seeds of formal geometry, but with introduction of the requisite social class, techniques were improved and measurement apparatuses honed to minute precision in the service of obtaining exact numbers and figures.

Ancient Greeks were the first to construct a system of principles in mathematics, derived from geometrical methods in common usage, a conceptual synthesis in their literature that culminated analysis spanning centuries. The Greek historian Herodotus speculated that formal math originated with rope-stretchers of ancient Egypt who used their devices to survey land after Nile floods, reestablishing boundaries between agricultural plots in which identifying markers had been disturbed or washed away. Aristotle hypothesized that math as a distinct discipline emerged from priestly classes of Egypt who used the leisure afforded by their social standing to measure and calculate movements of celestial bodies, the passing of time, record the history of their politics, and peer into properties of ideal shapes like polygons and circles. From modern investigation, including the deciphering of cuneiform script, it seems likely that ancient Mesopotamia, the most heavily populated region of early antiquity, had math as advanced as trigonometry even before Egyptian forms, from where it spread to Egypt, and later to Greece after in excess of a thousand more years, by about the 6th century B.C.E. or even prior.

In building the first large-scale structures, the main variables to be considered were length, width, height, angle, weight, and by calculative extension area and volume. The more massive constructions were typically made of a small collection of materials with reliably known, resilient hardness, the basic property in structural engineering. Knowledge of the range of possibilities for antiquity’s narrow selection of materials and chemistries blended over many generations with growth in understanding mathematical figures to facilitate more geometrically complex city planning.

Early Mesopotamian math analyzed to completion the triangle, the most basic geometrical object, establishing trigonometric ratios of every applied triangle’s component sides and angles. Since all polygonal shapes can be broken down into a right angled figure such as a square or rectangle and some combination of triangles, it became possible to scale up the proportions of any angular structural plan to dimensions as majestic as materials made practicable starting with a minimal amount of guestimation, trial and error. For instance, a hexagon is simply a square and two identical isosceles triangles, and an octagon a square and four identical isosceles triangles. Even many irregular triangles were mastered, their constituent ratios approximated to as many as six decimal places, so that weight was the only barrier to construction with straight lines.

While formal principles of calculus such as limit laws had not come available yet in antiquity, some calculuslike techniques were in use as a means to define properties of circles. The area of a circle could be determined by the method of exhaustion, circumscribing a polygon or inscribing a circle inside of it, measuring and calculating polygonal properties such as lengths of sides trigonometrically, then incrementally increasing the number of sides and recalculating until properties of the circle could be adequately extrapolated from a trend in the shrinking ratio between successive values. This mathematical procedure had been perfected by the height of ancient Greece, and building with circles, spheres and round columns was common practice. The nature of arcs — combinations of straight lines and curves — was also studied, and by Roman times arches were regularly included in building designs such as the famous Roman coliseum, parts of which are still standing today.

All the math of antiquity was very spatial since Hindu-Arabic numerals were not yet in ordinary usage and even basic operations such as multiplication and division proved difficult tasks. Finding unknown quantities was accomplished with geometry, converting sets of values into proportionalities of diagrams, then applying properties of figures, and many treatises were written to consolidate and generalize the characteristics of ideal objects. Most influential was ancient Greek Euclid’s Elements, the majority of which survived antiquity, providing the basis for Western geometry all the way until Europe’s Early Modern mathematics of the 17th century.

Arab mathematics of the late first millennium C.E. assimilated ancient math and combined it with the use of Hindu-Arabic numerals to invent linear algebra, solving for unknown values with expressions rather than figures. These techniques simplified arithmetical and algebraic calculation enough that they became accessible to even young children; math was no longer a rarefied technical field. By the Renaissance, the latest developments in Arab scholarship had become fully available to Europe, and linear algebra was extensively analyzed by the most brilliant mathematicians on the continent in order to exploit its full theoretical and practical potential.

The next big event in quantification was effected by Early Modern Europeans, their synthesis of geometry and algebra. Frenchman Rene Descartes inspired the Cartesian coordinate plane with the ‘x’ and ‘y’ axis still in use today, enabling description of two dimensional geometric figures by two variable equations of linear algebra, the basic variety of what is termed a ‘function’. Geometers outline the form of either angular or curved shapes by plugging values into these languagelike expressions consisting of mathematical symbols and then solving for two or more ‘variable’ unknowns, with each set of solutions converging on a particular point within a coordinate system, oriented in space. This launched analysis of more complicated figures, as any constituent properties — points, lengths, areas, volumes — could be easily calculated from their location within an ideally precise quantitative matrix, then materially instantiated to extents constrained only by technological capabilities, which were improving rapidly. 17th century mathematicians had rendered numerical definition of geometrical form much more finely grained, both in theory and practice, as maximally as contemporaneous data acquisition could require, opening up exponential change, elliptical orbits, and reflection from arcs to examination.

Even with coordinate systems, the method of exhaustion was a huge challenge to apply to complex curved shapes because of the overwhelming permutability in their local behavior. Math was no longer a matter of perfecting knowledge of a few basic shapes while disregarding anything asymmetrical or of complex symmetry. Some saw the acute relevance of geometry for defining and predicting patterns in the natural world, and sought to fashion a generalized account of ‘limits’, solutions to the method of exhaustion, which would both address 17th century theoretical problems and prepare humanity for a future of quantitative modeling.

The English Isaac Newton and Germany’s Gottfreib Leibnitz independently developed calculus, a set of laws, theorems and formulas that continued to expand after completion of their foundational systems, which allowed the method of exhaustion to be applied by analytically processing functions, without slavery to laborious measuring. Leibniz’s formulation was primarily in the abstract, as pure math, but Newton was a physicist and used calculus operations to analyze functions as descriptions of complex structure in previously inaccessible phenomena of the physical world, a joining of math and empiricism to educe the first quantitative laws of nature, most influentially the dynamics of gravitational attraction, sufficient for explaining all motion of macroscopic objects in the era’s known universe. Along with eventual modeling of the wave nature of electromagnetic radiation, sound waves, liquid waves, and the friction-induced shock waves of energy transmission through solid matter, the loose ends were tied together and a complete theory of force dynamics amongst macroscopic objects in our solar system finalized, what we know as classical physics.

Alchemy was transitioning into the scientific discipline of chemistry, and calculus provided assistance in defining changes in matter, for collections of experimental data regarding proportions of substance and rates of chemical reaction could be fitted with the most statistically accurate shapes to make these phenomena more intelligible and predictable. With precision measurement of minuscule weights, as well as the concept of heat and various other types of radiation transfer, energetic changes of material form and motion became quantitative, and intuitions about the macroscopic world were found applicable to features of microscopic substantiality in innovative modeling. Scientists pioneered theoretical descriptions of vanishingly minute concentrations and amounts of substance by utilizing mensurative concepts of nanoscale ‘volume’, ‘mass’, ‘density’, ‘pressure’, ‘moles’, envisioning these phenomena as produced by movements and fluctuations of postulated atoms. Particle chemistry provided the basis for structural conceptualizings of ‘temperature’, ‘kinetic energy’, ‘potential energy’ and ‘electrical charge’. As matter became more modeled and predictable in laboratory settings, experimental setups were standardized, then scaled up in order to industrially mass produce chemicals with a large array of properties defined to quantitative exactness using infinitesimally detailed, calculus-based analysis of data sets. The core, thermodynamic paradigm was a picture of atoms exchanging energy via collisions and achieving relatively stable or unstable arrangements by way of electromagnetic attractive and repulsive forces, sufficing to represent in images almost all the math of earthbound chemistry.

Even with calculus-based processing of functions, the method of exhaustion in analytic geometry was a monumental task to employ in real world situations, as massive amounts of calculation with complex decimals had to be performed, schematics drawn, instructions for immensely long procedures written, and double-checking of all the elements in an engineering project — quantitative specifications for materials and structures — carried out for quality control, a process that could take months if not years, requiring intricate division of labor and prolonged collaboration. Electronic calculators simplified calculation, but it was not until advanced computing came on the scene that engineering reached its full potential. Programs such as CAD (computer-aided design) translate mathematical expressions into computer code, with inferencing needed to proceed step by step to the solution of a problem enacted automatically by a visual interface on the computer monitor calibrated to this programming, sliding seamlessly between any numerical values. Engineers could interact directly with spatiality — the model of a physical structure or data set — while the computer processed quantification submerged beneath the display. With an immediate link between conceptual creativity and material instantiations in an elimination of calculative delay, the accelerated rapidity of modification and completion granted engineering a dexterity to be markedly more original, one might even say artistic. A whole new world of structural aestheticism opened up to technologists at the automation of mathematical reasoning.

Math began with instinctual and practical love of proportion, breaking through a succession of technical barriers that were placing limitations on functional possibility to come full circle and once again be almost pure artistry as computers dissolved restraints on form. Mathematical inference provides humanity the means to not only envision precision structure but write and talk it as language, more than natural grammar’s rough representation overlaying something of fundamentally different configuration, but further a direct correspondence of symbols to qualitative properties that when quantified are identical to the symbols, a lexemic method which greatly reduces ambiguity and error while wholly eliminating inexactitude. Our techniques of mathematical inference harmonize proportion, counting, the brain’s capacity for plasticity and the human aptitude for understanding concepts in the form of figures and linguistic linearities, ascending to categorically greater levels of both preciseness and complexity of expression.