Arithmochora — the Mathematical domain
Terminology
The term "mathematics" may draw its origins all the way back to the Pythagorean school. Its root, mathema, means "disposed to learn, to study"; although the Pythagoreans may have consciously circumscribed its use to their study of numbers and geometry, later usage veered back into full generality, and as late as 1700, "mathematics" would refer to astronomy or another domain of natural philosophy rather than to what we would expect today. For that reason, I would rather abandon that root in a refurbished terminological system. The choice to derive my neologisms from Ancient Greek is quite restrictive in this case, as ancient equivalents to modern concepts like structures, shapes or sets do not exist, and proximate translations can be in turn too generic or confusing. I have then chosen to use the one root that indisputably refer to an ever-identifiable concept, and robustly tied to modern mathematics still: that of number. Arithmo- is the Greek translation of the Latin-derived "number", hence the name of this mathematical domain: arithmochora.
Arithmatoms (mathematical objects)
Mathematical objects have a specific status in this classification. Unlike planets, persons or artworks, they are neither natural kinds nor artificial kinds. In a previous chapter, we’ve explored at length the ontology of this domain, concluding in favor of ante rem structuralism: the idea that mathematical objects are "places" in a mathematical structure (that we called "theory"), defined by a network of true propositions and the constraint of logical inference. We have then not set the stage for a pure Platonist view where mathematical objects would form a clear realm that can be rigorously distinguished from concepts, fictions and other abstractions. The ontological framework that allows to define arithmetic theories is the same as can be used to establish Balzac’s literary world. The difference may reside in our ability to use neo-logicist foundations to dervie core parts of mathematics from logic itself. As we have seen, that is very much an open question and here is not the place to re-litigate the foundations of mathematics.
If we don’t attempt to be exhaustive, we can easily outline a basic list of objects that do belong to the mathematical domain. That includes would-be foundational objects like sets or categories, as well as numbers (perhaps the most natural archetype of a mathematical object), algebraic structures (groups, rings, modules, fields, vectors, matrices, tensors...), topological structures (open and closed sets, nets, filters, neighborhoods, manifolds...), discrete structures (permutations, combinations, graphs, trees, lattices, formal languages, automatons, algorithms...), geometrical structures (points, lines, shapes, volumes...) and operations (functions, maps, morphisms, transformations, functors...). It also includes mathematical theories and logical calculi themselves.
Arithmoplases (mathematical events)
Mathematical objects and structures are abstract and are nowhere "in time". Humans instinctually picture mathematical transformations as an analog to physical processes, but that is misleading since transformations do not form an ontologically distinct category of mathematical objects (e.g. a function is also a sub-set of two sets’ product). There are no "mathematical events".
Arithmenoia (mathematical concepts)
Mathematical objects are abstract, so the risk here is to propose concepts like number, quantity, shape, which are simple abstractions of mathematical objects proper. More generally, the conceptual idea associated to the mathematical domain is that of a structure or system that can be transported from one realization to the next. We will describe this idea under the guise of formalism. Of course, formalism does not originate in mathematics: it derives from the formalization of ontology which is at the root of this philosophical system. But in terms of the levels below sophia’s theoretical philosophy, the mathematical domain is the one whose study enables the formalization of all others.
Arithmosophy (mathematical philosophy)
The discipline commonly known as "philosophy of mathematics" can be summed up as the ontology of mathematical objects, and a quest for foundations of mathematics (which is a form of metaphysics of mathematical objects, since it investigates the derivation of mathematical objects from first principles, just like abstract objects are generated by algebraic rules). We have already encountered mathematical ontology and foundational questions, as part of the "formal ontology" storey. Based on the correspondence we’ve just established, this makes the philosophy of mathematics a part of the sophia plane: of theoretical philosophy. The same can be said of the "philosophy of logic" (really, logical ontology), a part of that same storey from a more fundamental standpoint.
Other apparent candidates could be the "philosophy of statistics" or "the philosophy of computing", but closer examination shows those topics to be focused on the actual uses of these theories and their epistemic, ethical, aesthetic implications, making them a part of philosophy of science.
In conclusion there is no phronesis level for the mathematical domain, a consequence of its specific nature.
Arithmology (mathematical science)
It is worth wondering why arithmology, the term I propose to cover mathematics, is even a part of episteme, a theoretical science. After all, I seem to contradict a principle I’ve established before, that sophia, theoretical philosophy, should be divided following the 8 storeys that start with the simplest formal structures and end with complex concepts like norms or values, whilst episteme, theoretical science, would be mapped on the 13 levels of the scale of nature. That would seem to exclude mathematics from episteme, since mathematical objects do not exist in nature. The objects studied by other sciences may not be concrete directly (they can be ordinary objects, like an animal, or artificial kinds, like a musical composition), but they all have their ultimate ground in the physico-mental universe, which is not the case of mathematical structures.
But the practice of mathematics and its ever-astounding "unreasonable effectiveness" suggest that there should be an exception. Indeed, sophia can be seen as a "meta-theory of the world". It deals in generalities rather than in general objects: metaphysics tell us what ordinary objects are, but doesn’t care much for actual ordinary objects like a dog or the sun — sciences do, however. Similarly, the formal ontology part of sophia is firmly set at a meta level: it helps make clear what logic is, the status of "other logics", how we should define mathematical theories and objects, etc. But it doesn’t do any mathematics. Numbers, sets and functions are just examples, not an object of study.
As I recalled above, the philosophies of logic and mathematics are really about all formal systems. Their objects can be diverse, and belong to the natural or human world; but if we define a formal system of solar system objects or of Balzac characters, there are episteme disciplines dedicated to their kind. If our formal system describes group theory or complex function analysis though, there is no natural nor human science the relevant objects can call home. I then need to introduce a single formal science at episteme level, calling attention to the special status of mathematical objects, that special class of abstract particulars that are wholly untethered from the concrete universe.
Sub-disciplines
Within each natural science, it is fair to say that many sub-fields are well-defined, fairly independent disciplines in themselves. This is not the case anymore in mathematics; we may even heartily endorse Henri E. Bliss’s conclusion that "the relations of the branches are so intricate, so plastic, so recondite, that it is well-nigh impossible to define them or to comprehend them." This would surprise the Ancients: the traditional dyad of geometry and arithmetic seemed clear-cut: the science of Space and that of Number; the continuous and the discrete.
But from then on, each new route for mathematical enquiry would entail some degree of co-mingling. The development of algebra in the Hellenistic and Islamic worlds as a study of equations and a generalised arithmetic has increasingly relied on geometrical methods to solve major problems. Descartes walked that route in the opposite direction, following Viète’s footsteps, using algebraeic methods to create "analytical geometry". The subsequent development by Leibniz and Newton of infinitesimal calculus established a focus on infinites and limits that would set analysis apart for a time. Last of the major high-level branches to mature, topology started out as an exotic form of geometry, until the emphasis on concepts of distance and limits underlined the deep connection to analysis.
The onomastics of major topics show that the branches do not just share common roots, their connections have only grown: algebraic geometry, algebraic topology, topological algebra, geometric analysis, arithmetic algebra, and countless more combinations are testament to that. Keeping in mind, then, that we’re talking more of clusters than proper, independent sub-disciplines here, the best way to identify major areas would be to rely on the fundamental distinction between the discrete (separate, distinct, detached) and the continuous (uninterrupted, smooth, unbroken).
Discrete mathematics
Arithmetic is the theory of number (whole and rational numbers).
Combinatorics is (loosely) the theory of counting and studies enumeration as well as graphs, networks and lattices.
Algebra is concerned with general structures that display symmetry, patterns, and can be seen as a generalization of the above.
Continuous mathematics
Geometry is the study of spaces using concepts like shape, distance, angle, curvature, etc.
Analysis is the study of change, and more specifically of functions with real variables (or generalizations, like complex numbers, vectors, tensors...).
Topology is concerned with general spaces equipped with a notion of closeness. It is a generalization of both geometry and analysis.
Again: this breakdown may seem quite neat, but that is far from the case in practice. The distinction between structure and space is designed to invoke some of our instinctual understanding via Umwelt analogies, but there is in fact no way to properly distinguish the two, hence the infinite range of combinations, with one "family" borrowing the other’s methods, and hybrid objects being constructed in every way imaginable.
In addition to those various domains which grew out of ancient ideas of mathematics, there is a more recent "meta" layer.
Metamathematics
Mathematical logic
Set theory
Category theory
The terminology is a bit muddled by historical vagaries. In particular, "mathematical logic" can easily appear to be ambiguous. In one sense, it could denote the formalization of logic using mathematical techniques and structures; that is the "algebra of logic" tradition initiated by Boole and Schröder. In another, mathematical could be the study of mathematical reasoning through logical investigation (being more accurately described as the logic of mathematics, like there is a logic of law or a logic of ethics, for instance). That is the approach explored by Frege, Peano and Russell.
Modern mathematical logic is the confluence of both traditions, together with the axiomatic project in mathematics. It reframes a logic as a formal system characterized as follows:
We start from a formal language that is clearly distinguished from the metalanguage used to talk about it. That distinction is actually quite straightforward when talking of natural languages: if I read the Cambridge Grammar of Classic Greek, it is quite clear I’m learning about a language (ancient Greek) by reading sentences couched in a different metalanguage (modern English). That distinction tended to get lost in logic textbooks; formalization certainly helps to set these boundaries. A formal language is defined by:
a formal vocabulary, comprising all the symbols that can be used as part of the language.
a formal grammar that comprises formation rules, defining which formulas (strings of symbols) are well-formed, and thus part of the language.
A formal language becomes a formal system with the adjunction of:
a proof system that comprises a deductive apparatus, which defines the rules of valid inference between well-formed formulas, and a system of axioms. A theorem is either an axiom or a formula that can be inferred from another theorem. A chain of deductions from axioms to a theorem is called a formal proof. The proof system completes the grammar for form the full syntax of the language. The rules of inference represent a notion of logical consequence from a syntactic perspective.
an interpretation which assigns to every formula of the language a meaning (for instance, a term could be associated to a mathematical or a concrete object. Often, an interpretation outlines a sub-set of formulas called sentences which are interpreted as truth-values. Sentence which are true in one interpretation at least are called satisfiable and that interpretation is a model of the formula. Sentences which are true in all interpretations are valid. Interpretations introduce a semantic notion of logical consequence. The clear separation between formal syntax and semantics is a hallmark of formal systems.
Mathematical logic is then classically divided in proof theory (which studies proof systems and syntactic consequence) and model theory (which studies interpretations and semantic consequence).
That resolves one part of the classificatory question: since formal systems are mathematical objects, and since we use mathematical reasoning to manipulate them (or more clearly, we use mathematical reasoning in the metatheory to talk about formal systems (and therefore about mathematical reasoning in the theory)), mathematical logic belongs to mathematics. Then, why put it under the header of metamathematics? Both concepts have been intertwined to the point of being indistinguishable, and indeed, that seems fair. Formal systems can be used to describe "a logic", for instance the propositional logic of Aristotle can be represented by a formal language, axioms, rules of inference, and an interpretation. But it also appeared that any mathematical theory could be represented as a formal system. This just required the introduction of non-logical symbols in the formal language, and of non-logical axioms. The study of metamathematics then becomes inseparable from mathematical logic, since the formal description of mathematics involves both the use of formal systems and the choice of a logic (to us moderns, there is not just "arithmetic" but "an" arithmetic, which could be built on 1st-order logic, 2nd-order logic, modal logic, and so on).
What of another term we haven’t evoked here yet: metalogic? Stricto sensu, this should be an endeavor to make metatheorical statements about logic. But as is apparent with its identification to metamathematics, mathematical logic already seeks to prove metatheorems about formal systems. And since logics are also mathematical objects in this framework, the distinction betwen metamathematics and metalogic does not seem to be useful anymore.
As to set theory and category theory, they are part of mathematics in a similar way to algebra or topology. But they developed to their current breadth in the context of metamathematical investigations as would-be foundations for all other theories. Their ubiquity gives them a natural place in that section (set theory having a deep link with formal semantics, category theory starting as an explicit abstraction from mathematical structures).
Arithmotechny (mathematical technique)
Because of the specific nature of the mathematical "science", the question of what constitutes "applied mathematics" is a bit awkward. Clearly, mathematical knowledge is developed in service of concrete problems all the time, but since those fall into a taxonomy of concrete objects and events, they will correspond to cross-pollination between mathematics and another discipline.
To distinguish between "pure, theoretical mathematics" and a corpus of coherent knowledge that can be identified as "applied mathematics" but is still unadulterated by other specific disiplines is a fine line to draw. Yet we should mention the following traditional topics:
Statistics is connected to probability theory, a transversal subject in mathematics that aims to represent randomness. Statistics applies its results and harnesses other mathematical fields to synthetize, analyze, predict, and draw inferences from actual data.
Numerical analysis studies the methods to best calculate or approach the numerical value of mathematical expressions.
Computer science: the study of computation writ large is a significant branch of mathematics borne out of recursion theory. It gives rise to a broad, transversal field of mathematical informaticsthat encompasses algorithmics, the theory of coding, automata theory and the study of data structures. Its more practical aspects include the study of computer hardware designs of software designs, or of programming languages.
Information science is a closely related topic that relies on the mathematical representation of information, its acquisition, quantification, communication, validation, encryption, etc.
Cybernetics: the study of systems is of course an extremely generic concept. It is still an applied field since it is geared for the analysis of concrete structures, their dynamics, self-regulation, optimization and control.
Cross-disciplines
Cross-disciplinarity between mathematics and other sciences is of course unlike any other since mathematics, the "language of science" according to one Galilei Galileo, has been an underpinning of the modern scientific method. To the unrelenting mathematization of vast swathes of the natural sciences, we can add two observations: (1) even in the realm of human sciences, mathematical reasoning and formalization is a prevalent mindset (think of Durkheim introducing statistical methods in sociology, or Jakobson’s algebraic approach to the analysis of Baudelaire’s poetry), and (2) the rise of the aforementioned applied fields has created even more avenues for mathematical applications. Because of that ubiquity of mathematics, we will only provide a list to introduce our neologisms for these cross-disciplines.
Dynarithmology: mathematics applied to physics.
Cosmoarithmology: mathematics applied to astronomy.
Hylearithmology: mathematics applied to chemistry.
Geoarithmology: mathematics applied to geology.
Ecoarithmology: mathematics applied to ecology.
Bioarithmology: mathematics applied to biology.
Topoarithmology: mathematics applied to geography.
Chremarithmology: mathematics applied to economics.
Phyloarithmology: mathematics applied to sociology.
Dikaiarithmology: mathematics applied to legal theory.
Politarithmology: mathematics applied to political science.
Chronoarithmology: mathematics applied to history.
Psychoarithmology: mathematics applied to psychology.
Glossoarithmology: mathematics applied to linguistics.
Tropoarithmology: mathematics applied to anthropology and cultural theory.
Theoroarithmology: mathematics applied to metascience.
Hermeneoarithmology: mathematics applied to literature theory.
Kaloarithmology: mathematics applied to art theory.
We should note that in common usage, the various components of these cross-disciplines are named according to re-occurring templates, of which we’ll list the main ones:
"Mathematical X" is the one we should expect of course. "Mathematical physics" and "mathematical economics" in particular, cover significant subjects in mathematics, the discovery of which got its momentum from the corresponding disciplines.
"Statistical X" relates to the use of statistical techniques in other disciplines, obviously related to the growing significances of data. The natural sciences had relied on observations since the pre-Socratics, and found a further source of data in rigorous experiments. The collection of what we would now call demographic, econometric, epidemiologic or geostatistic data, often for governance purposes started an expansion of statistical applications that has continued to this day.
"Computational X" (or occasionally "Numerical X") applies computational techniques to solves problems in a given discipline. That is not necessarily connected to the modern computer: many a differential equation does not have exact solutions and "manual" computational techniques were developed to yield numerical solutions. Since then, advances in genomics, chemistry and large swathes of modern physics can be credited to modern computational power and methods.
"Systems X" refers to the application of systems science concepts to a given discipline. In geology or in biology, which have splintered into more and more specialized sub-disciplines, the approach has allowed to restore a top-down view of extremely complex structures as systems.
"Theoretical X" is a passe-partout adjective that etymologically should be mostly meaningless, but in practice vaguely refers to either of the preceding options. "Theoretical astronomy" appears to be essentially computational astronomy; "theoretical ecology" is indistinguishable from systems ecology; "theoretical chemistry" can be likened to mathematical chemistry since it puts the emphasis on formalization and the rigorous usage of chemical concepts.
Arithmurgy (mathematical activities)
The human activites that we can associate with this domain are expectedly broad and vague, due to the pervasiveness of mathematics and their application.
Analytics cover the countless activities whose practitioner can be said to perform analysis of some sort. This can entail the use of mathematics or statistical theory, but also much more informal usages of systems or information science.
Logistics & Operations are essentially the converse operation, consisting in a reverse use of analytics to create systems that will fulfil specific objectives. In practice, it often refers to activities like procurement, maintenance, distribution, transportation, disposition, or in the case of military logistics, planning and carrying out the deployment of forces.
Information technology is an umbrella term that, today, covers the design and programming of computer software, or the management of data structures. Similar activities before the moder computer arose would likely have fallen under the analytics of logistics umbrellas.
Communications is an activity sector focusing on the effective transmission of information. Starting from optical signals (beacons and semaphores) or acoustic ones (drums...), the sector is dominated by telecommunication technology, such as the telegraph, telephone, radio, television, and lastly computer networking (the Internet).
Library and archive management are activities that use a body of knowledge with various names over the decade: "documentation science", "library science", "archival science", and that today is best described as a sub-set of information science that focuses on knowledge organization, information architecture, information retrieval, curation, cataloguing, indexing, etc.
Arithmergons(mathematical artifacts)
The mathematical domain is not rife with artifacts, since the objects it manipulates are not concrete, and it is best learnt in books, with pen and paper, or perhaps some chalk and a board. Its main practical application though, which is to compute has engendered interesting objects:
Analog calculators, like the abacus or slide rules, rely on continuous physical parameters to represent numerical values, and help with calculation.
Digital calculators harness mechanisms to delineate algorithmic steps and only allow discrete values, starting from gears and levers (calculating clocks, the arithmometer, the Pascaline, or Leibniz’s stepped reckoner...) to electromechanical machines like the tabulator, and lastly to electronic calculators using transistor circuitry.
Computers are not qualitatively different, but must be distinguished from the breadth of their applications. Calculators implement a number of set functions, while a computer is supposed to realize any algorithm allowed by its machine programming language. Charles Babbage’s analytical machine, which Ada Lovelace wrote code for, is considered to be the first mechanical computer. Computer evolution experienced exponential growth in the second half of the twentieth century, when the first, vacuum-tube-based, electronic programmable computers (like ENIAC), quickly made way to a second generation (transistor-based), a third (integrated circuits) and a fourth (microprocessor-based). That latest generation enabled the appearance of personal computer, portable computers, and ultimately, ubiquitous computers.
Arithmodomes (mathematical edifices)
For similar reasons as for artifacts, it is not easy to delineate what "mathematical edifices" might be. One could "do" mathematics anywhere - too broad a category. Universities and research centers are about research activities by professional, not about mathematical activities by humans (and we want to avoid category mistakes). I would mention the following:
Supercomputing facilities: computers used to be whole rooms rather than devices. Supercomputers are still buildings.
Data centers are also building that house computers but their purpose is storage rather than heavy-duty computation.
Arithmanthrope (mathematical personas)
In this classification, "persona" does not refer to a specific sub-group of human beings who act in the mathematical domain (that would be mathematicians, statisticians... but this is not part of a professional nomenclature). It denotes a facet present in all human beings, albeit to wildly different degrees. In this case, it is the cognizer, representing the cognitive ability to reason, compute and calculate.
Arithmopraxy (mathematical experience)
The lived experience, or praxis, that stems from this domain, is abstraction, as a human experience. Subjective experience is not just perceptions and thoughts, it’s inseparable from a tendency to develop a model, a representation, a formalism, a systematization of that experience.