The elementary theory of the category of sets arose from a purely practical educational need. When I began teaching at Reed College in 1963, I was instructed that first-year analysis should emphasize foundations, with the usual formulas and applications of calculus being filled out in the second year. Since part of the difficulty in learning calculus stems from the rigid refusal of most textbooks to supply clear, explicit, statements of concepts and principles, I was very happy with the opportunity to oppose that unfortunate trend. Part of the summer of 1963 was devoted to designing a course based on the axiomatics of Zermelo-Fraenkel (ZF) set theory (even though I had already before concluded that the category of categories is the best setting for “advanced mathematics”). But I soon realized that even an entire semester would not be adequate for explaining all the (for a beginner bizarre) membership-theoretic definitions and results, then translating them into operations usable in algebra and analysis, then using that framework to construct a basis for the material I planned to present in the second semester on metric spaces.
However I found a way out of the ZF impasse and the able Reed students could indeed be led to take advantage of the second semester that I had planned. The way was to present in a couple of months an explicit axiomatic theory of the mathematical operations and concepts (composition, functionals, etc.) as actually needed in the development of the mathematics. Later, at the ETH in Zurich, I was able to further simplify the list of axioms to an elementary theory of the category of sets.
One cannot add up length and area.
P.S.
mass x acceleration = force
mass + acceleration = (cardinal sin 😉
Product is not a shorthand for sum!
Professor F. William Lawvere, 18 October 2017
Dear Friends and Colleagues,
I am deeply sad about the loss of Myles, my friend and a pillar of the community surrounding Eilenberg and Mac Lane. I highly respected him as a creative collaborator. Whenever we met at meetings, or spoke on the phone, even if we had not seen each other for a long time, we knew each others way of thinking.
Thanks to Andre Joyal for his obituary and the wealth of information about the work of Myles Tierney. I elaborate below on our collaboration.
Myles and I had independently recognized the need for an axiomatic theory of sheaves and related matters, and at a gathering in Albrecht Dold’s house near Heidelberg, we agreed to collaborate on the construction of such a theory during our upcoming stay at Dalhousie University. (Myles had agreed to join our group of researchers for a year.) Already in the first days of our seminar Myles made important advances, such as formulating the axioms of exactness which we knew would have to be theorems of a correct theory of topos.
He emphasized in general that Grothendieck had made the category (rather than the space) the central aspect that we should explicitly axiomatize. We had a pre-publication copy of SGA4 that we consulted frequently. In that work there was a significant advance over previous formulations of the sheaf condition: In a pre-sheaf topos, a covering specified by a Grothendieck topology, was no longer an infinite family of subobjects, but a single subobject R, (which of course might be imagined to arise as the union of a family); this made possible the formulation of notions in finitary terms, indeed in terms of a single operator whose properties Myles made precise. (Some people object to calling such operators topologies; the same objection applies to Grothendieck’s use of the term topologies for his equivalent notion, which is of course also no literally a topology in the classical sense. Later we referred to such an operator as a localness operator, as a modal operator it is locally the case that, or as a Tierney closure operator which – as I pointed out to Kuratowski on his visit to Dalhousie – is not a Kuratowski closure operator since it preserves intersections, rather than unions. Similar operators arise in other parts of mathematics where they are sometimes called ‘nuclei’.)
Another unique feature of SGA4 is that it contains no definition of topos; indeed every rigorous mention is of U-topos. This parameter U was essentially a model of set theory, and previous work on the category of sets showed clearly that for mathematical purposes, the use of composition of mappings is more effective than towers of membership. Thus Myles and I decided to replace U itself by an arbitrary topos (in our determination of that term), and indeed a general U-topos E could now be seen as structured by a morphism E –> U. This provided a suitable codomain for the 2-functor from internal U-sites to U-toposes, whose image consisted of those E which contain a bound in U. That, our fundamental preliminary goal, was proved in his thesis by the remarkable student of Myles, the late Radu Diaconescu.
We also had available a preliminary version of Monique Hakims thesis applying Grothendieck’s notion of classifier topos to parameterized complex analysis. This notion can be seen as a key step of Model Theory, except that the use of conjunction and disjunction and existential quantification is replaced by Grothendieck’s direct recognition of which classes of structures are defined in terms of finite limits and small colimits. Of course, a logician will expect or want primitive predicates and basic axioms to present such notions of structure, and to facilitate such recognition. Our initial work to make explicit such a notion of theory-presentation was carried out by several people. It became evident that such a construction does not work for general elementary theories, because the negative operations of universal quantification and implication are not preserved by the geometrical morphisms of toposes, even though these operations are well-defined and exist within each particular topos. (Restricting to open geometric morphisms or to positive presentations can partly circumvent this limitation.) The positive presentations need to use the classical idea of sequent, rather than the mere specification of a class of formulas that aim to be true.
The classifier toposes are useful for mathematical questions other than logical ones, for example in combinatorial topology, as first pointed out in detail by Andre Joyal. The relation between combinatorial schemes and the spaces they generate is an adjoint functor. Specifically, any structure carried by a unit interval or by other basic space in a topos of spaces gives rise to the adjoint between the topos of spaces itself and the classifying topos for such structures, for example, distributive lattice, Boolean algebra, total ordering. Myles foresaw such clarifying applications, and in many other ways his knowledge of topology supplemented my own background.
A useful construction made explicit by Myles is still not widely recognized:
The category of co-algebras for a given left-exact comonad in a given topos, is indeed another topos.
This can be seen as an essential step in the construction of a pre-sheaf topos.
A significant advance in the world of logic was the result of Diaconescu, showing that the axiom of choice implies Boolean logic. His professor Myles Tierney had proved the independence of the continuum hypothesis, making essential use of the notion of sheaf (LNM 274). It was in preparing his talks for our seminar 1969-1970 that Myles formulated most of the logic needed for that result. Sabah Fakir who took part in these seminars sent notes each week to Jean Benabou who also gave a seminar simultaneously. Anders Kock was a very active participant, whose later Aarhus seminar with Gavin Wraith also contributed to the rapid dissemination of the results and the general point of view.
After I had presented our results at the 1970 ICM, we organized an international meeting in January 1971; Myles returned from Rutgers to Halifax to explain the continuum hypothesis to the 70 participants.
Myles and I were gratified in the ensuing months and years to see that our general point of view was taken up by many mathematicians and logicians. Further applications, for example the role of Boolean sheaves in algebraic geometry, still await detailed publication. Several works by Michael Barr initiated such applications.
In 1974 Myles Tierney and Alex Heller organized a meeting in New York City in honor of Samuel Eilenberg’s 60th birthday; they edited the proceedings as Algebra, Topology, and Category Theory. That book contains an influential article by Myles on the construction of classifier toposes for internal sites.
Myles Tierney, Fred Linton, Jon Beck and I were roughly the same age when we started our studies at Columbia University; Mike Barr, Peter Freyd, John Gray, and Barry Mitchell were senior to us, and collaborated with us too. Sammy had attracted a group, all of whom continued to contribute to category theory.
In the ensuing years Myles organized a regular seminar in New York on weekends. It was attended by categorists from several states. I attended the seminar frequently from Buffalo, and I took along several students including Kimmo Rosenthal and Phil Mulry. Thanks to the generosity of Myles and Hanne, everybody was invited to sleep on the floor of their loft.
When I last spoke with Myles on the phone just a few months ago he explained to me the specific topological intuition underlying the notion of Kan complex.
I know that Myles would have joined me in welcoming any future mathematics on the horizon for which our work might have provided a stepping stone.
Toposes, Algebraic Geometry and Logic, LNM 274, Springer, 1972.
Algebra, Topology, and Category Theory, Academic Press, 1976.
Everyday human activities such as building a house on a hill by a stream, laying a network of telephone conduits, navigating the solar system, require plans that can work. Planning any such undertaking requires the development of thinking about space. Each development involves many steps of thought and many related geometrical constructions on spaces. Because of the necessary multistep nature of thinking about space, uniquely mathematical measures must be taken to make it reliable. Only explicit principles of thinking (logic) and explicit principles of space (geometry) can guarantee reliability. The great advance made by the category theory / theory of naturality invented 60 years ago by Eilenberg and MacLane permitted making the principles of logic and geometry explicit; this was accomplished by discovering the common form of logic and geometry so that the principles of the relation between the two are also explicit. They solved a problem opened 2300 years earlier by Aristotle with his initial inroads into making explicit the Categories of Concepts. In the 21st century, their solution is applicable not only to plane geometry and to medieval syllogisms, but also to infinite-dimensional spaces of transformations, to “spaces” of data, and to other conceptual tools that are applied thousands of times a day. The form of the principles of both logic and geometry was discovered by categorists to rest on “naturality” of the transformations between spaces and the transformations within thought.
PALABRAS DE BIENVENIDA
Todos comenzamos a coleccionar ideas matem´aticas en los primeros a˜nos de la ni˜nez, cuando descubrimos que nuestras manos son reflejo la una de la otra y, despu´es, cuando aprendemos que otros ni˜nos tambi´en tienen abuelas —de manera que ´esta es una relaci´on abstracta que un ni˜no puede tener con una persona mayor—; y, luego, cuando nos damos cuenta de que las relaciones “t´ıo” y “primo” son tambi´en de este tipo; cuando nos cansamos de perder en el juego de tres en l´ınea (o gato) y lo analizamos completamente para nunca m´as perder; cuando tratamos de descifrar por qu´e las cosas se ven m´as grandes conforme se van acercando o si el contar alguna vez termina.
Conforme el lector avanza, este libro puede agregar algunos tesoros a la colecci´on pero ´este no es su objetivo. En lugar de esto, esperamos mostrar c´omo poner la gran bodega en orden y encontrar la herramienta adecuada en el momento en que se la necesita, de manera que las ideas y m´etodos nuevos que se coleccionan y desarrollan a lo largo de la vida puedan tambi´en encontrar sus lugares apropiados. Hay en estas p´aginas conceptos generales que trascienden las fronteras artificiales que dividen aritm´etica, l´ogica, ´algebra, geometr´ıa, c´alculo, etc´etera. Habr´a poca discusi´on sobre c´omo llevar a cabo c´alculos especializados pero mucha sobre el an´alisis que sirve para decidir qu´e pasos es necesario hacer y en qu´e orden. Cualquiera que haya batallado con un problema genuino sin que se le haya ense˜nado un m´etodo expl´ıcito sabe que ´esta es la parte m´as dif´ıcil.
Este libro no podr´ıa haber sido escrito hace sesenta a˜nos; el lenguaje preciso de conceptos que utiliza apenas estaba siendo desarrollado. Cierto, las ideas que estudiaremos han sido empleadas por miles de a˜nos pero aparecieron primero solamente como analog´ıas apenas percibidas entre temas. Desde 1945, cuando la noci´on de “categor´ıa” fue formulada con precisi´on por primera vez, estas analog´ıas han sido afinadas y se han convertido en maneras expl´ıcitas de transformar un tema en otro. La buena fortuna de los autores les ha permitido vivir en estos tiempos interesantes y observar c´omo la visi´on fundamental de categor´ıas ha llevado a una comprensi´on m´as clara, y de all´ı a organizar de mejor manera y, a veces, dirigir el crecimiento del conocimiento matem´atico y sus aplicaciones.
Este libro ha sido usado en clases de preparatoria y universidad, seminarios de posgrado y por profesionales en varios pa´ıses. La respuesta ha reforzado nuestra convicci´on de que personas de diversa formaci´on pueden dominar estas importantes ideas.
International Conference on Theoretical Neurobiology
Enormous progress has been made in experimentally explicating the neural processes underlying various brain functions such as conscious perception, conceptual thinking, and emotional feelings. Unraveling the mathematical content of these experimental findings can help us impart the precision of engineering design to therapeutic treatments of dementia, schizophrenia, depression and many other debilitating neuropsychiatric disorders. But building models of the brain that accord with data, especially the contextual quintessence of cognitive neuroscience, is particularly challenging. Theoreticians have been trying to extend the reductionist elementism that has hitherto been extremely successful in physical sciences to encompass neurocognitive phenomena only to discover its limitations. Is the current state-of affairs indicative of the limitations of science and mathematics or do we have more sophisticated methods at our disposal?
Mathematical experience has shown that set-theoretic elementism fails to accurately model geometry also, which led to the development of category theory. The relevance of category theory for brain sciences can be readily discerned by noting that in category theory objects are described not in terms of their constituent elements but in terms of their relations to other objects reminiscent of the contextual influences in cognitive neuroscience. Category theory can be thought of as a marriage of holism with reductionism; it incorporates the insights of holism while furthering the reductionist program of calculation with certainty. This takes on added significance in the light of Thomas Albright and colleagues’ keen observation that the advancement of neuroscience depends on methods that combine holistic insights with reductionist concreteness: “the issue is whether we can succeed in developing new strategies for combining reductionist and holistic approaches in order to provide a meaningful bridge between molecular mechanism and mental processes: a true molecular biology of cognition” (Neuron 25:S1, 2000). Category theory provides a means to “put together”, an antidote to reductionist breaking down, which has also been ranked high on neuroscience agenda. In the words of Carla Shatz, “the challenge now is to put the molecules back into the cells, and the cells back into the [neural] systems and the systems back into [the brain trying to really understand behaviour and perception” (Nature 414:4, 2001). Realizing the import of the considered assessments of Albright and Shatz, National Brain Research Centre, New Delhi, India has taken the initiative to exploit the philosophically profound and methodologically concrete category theory to catalyze the maturation of neural science into brain science.
Acknowledging the primacy of experimental data, the conference sessions begin with lectures on experimental findings, followed by computational neuroscience talks, and conclude with expositions of category theory. This format is designed to see how far the methods of computational neuroscience go to meet the demands of data, and to see how category theory relates to the specific concerns of the brain sciences. Each session is followed by an hour-long triangular debate with experimental and computational neuroscientists and category theoreticians to bring into sharp focus the strengths and limitations of these approaches, which in turn can help us see how they can complement one another.
The Legacy of Steve Schanuel!
My friend and collaborator Steve Schanuel died a year ago on July 21, 2014.
Steve was a mathematicians’ Mathematician. He loved the many facets of mathematics and loved to solve problems that colleagues and students presented to him. He was generous and patient with young students and happiest when he could solve interesting problems that made him sparkle with joy. The students loved him, and so did we, his colleagues.
A free man with no wish for fame or fortune, unencumbered by politics, history, society, gossip, he did not get distracted by philosophy. He gave his time and energy to the problems that presented themselves, he loved to discuss them and spin further solutions. He did not like to write, and seemed to be happy when scribbling and thinking. But when a real mathematical problem presented itself, he was the most serious and hardworking scientist. That applied in particular to the real problem of passing knowledge on to young people.
We both shared the passion of teaching and the belief that large numbers of students could benefit from some explicit knowledge of conceptual methods. It had originally been proposed to us to write a text book on ‘Discrete mathematics’, to which Steve immediately replied ‘no, we will emphasize methods that are applicable to both the continuous and the discrete‘.
In the mathematical world he is best known for Schanuel’s Lemma, and for Schanuel’s Conjecture. Steve discovered the Lemma when he was still a graduate student at Chicago; it became a key instrument for those who participated in the development of Grothendieck’s linear ‘Klassentheorie’ (K-Theory). The brilliant Schanuel’s Conjecture, concerning transcendental number theory, has given rise to several advances due to the efforts of Steve himself and of dedicated logicians and number theorists, but it has still not been proved.
But there are further contributions, of relevance to all branches of mathematics, that bear the stamp of elementary clarity so characteristic of Steve’s work. For example, his ‘What is the Length of a Potato?‘ presents original contributions in the process of a supremely elementary exposition of the classical subject of geometric measure theory.
When I first met Steve in 1974, he explained to me a way of presenting the theory of affine-linear spaces in terms of the category of vector spaces. We developed that idea for 20 years, during which Steve proved several new mathematical results that I explain in my 1994 contribution to the historical analysis of the work of the great geometer Hermann Grassmann.
Partly in response to some remarks in Federico Gaeta’s notes on Grothendieck’s 1973 Buffalo course, and partly as a necessary basis for his 1990 theory of Negative Sets, Steve devised the notion of extensive category as a natural relativization of the notion of distributive category. His insight was that the spaces in such categories have both Euler characteristic and dimension, that both of these quantities can be derived from a single ‘rig’, and that moreover the two quantities alone sometimes determine the space up to isomorphism. This remarkable non-linear Klassentheorie became a key thread in what we came to call ‘Objective Number Theory‘. Some of the results, which Steve derived from his theory of rigs, later turned out to be important in the study of O-minimality, in particular in the work of his student Adam Strzebonski on semi-algebraic groups.
In retrospect, it may seem astonishing that the term ‘rig’ had not been proposed decades earlier: we constantly come across examples of commutative algebraic systems with two constants 0, 1, and two binary operations, +, x, which do not necessarily have negatives and hence become rings only upon tensoring with Z. Thus omitting the ‘n’ for negatives, such algebras seem to deserve the name ‘rigs’. The previously available name ‘Commutative semi-rings with 1’ is unwieldy and even carries a faint suggestion that these objects are only half-legitimate. We were amused when we finally revealed to each other that we had each independently come up with the term ‘rig’. Thorough algebraist that he was, Steve went on to determine the simple objects in the category of rigs and to develop part of the needed theory of finite presentation; naturally, he also investigated modules over rigs, projective and otherwise.
Steve’s basic construction, passing from a distributive category to its rig of isomorphism classes, and then tensoring with standard rigs, gave crucial additional information in some basic examples. Tensoring with Z to obtain a ring provided Euler characteristics for the spaces in the category. But tensoring instead with the rig ‘2’ (in which 1 + 1 = 1) measures the dimension of the spaces. In some crucial cases, such as his ‘negative sets’ (where X = 1 + 2X) those two invariants are sufficient to determine the space up to isomorphism. The ‘number theory of objects‘ turns out to be more subtle than either the theory of small infinite cardinals or of recursive sets (both of the latter have the same resulting rig, consisting of natural numbers, together with one infinite element satisfying too many equations, of course the theory of recursive SUB-sets is by contrast very rich).
While teaching Conceptual Mathematics, we had noted that in the category of directed graphs, the arrow A satisfies the quadratic equation A^2 = A + 2D (with dot D), and hence reasoned that algebraic equations have further uses in combinatorics. Noting that Steve’s equation for a negative set is a special case of the equation describing the data type ‘lists in the alphabet A’, namely X = 1 + AX, we tried simple substitution in order to get a description of ‘lists of lists’, namely X = 1 + X^2, which is also known as the binary tree equation for data types. This led easily to the conclusion that the tree data type has the primitive 6th root of unity as its Euler characteristic, and that in fact the more precise form X^7 = X of this conclusion gives an interesting object of dimensions. This led to the conjecture that in some combinatorial categories there are no isomorphisms other than those which are rig-theoretic consequences of the defining equations, in other words, that they objectify the rig presented; that simply means that the proof of entailments is just high school algebra, except that negatives and cancellation are not used; in such calculations the expressions may become longer and longer under repeated substitutions, but then suddenly collapse due to the use of the defining equations. Our friend Andreas Blass dubbed this case of the conjecture ‘Seven Trees in One’, and proved it in a brilliant paper motivated by the idea of the classifying topos for the equation in question.
Subsequently, several other cases were proved by Robbie Gates. It became clear that equations of the fixed-point kind were appropriate candidates for objectification, at least from the data type point of view, with the right-hand side of the polynomial equation involving a signature in the sense of universal algebra; a fixed-point bijection holds for free algebras over free theories (also known as ‘Peano algebras’). Matias Menni’s recent work has succeeded to remove restrictive conditions on the signature. Extensions to fixed-point equations in several variables (corresponding to signatures for multi-sorted universal algebra), had also been proposed by Steve.
For many winter vacations Steve accompanied Fatima and me to Oaxaca, Mexico, where we worked on extending and recording the results of Objective Number Theory. Steve and I studied deep into the nights; sometimes Fatima heard us giggle, because we had discovered how simply some results could be proved. In the morning she typed the notes that I had left on her table. Whenever unfinished trains of thought occurred in the manuscript, we wrote SHOULD in large letters… We felt that our work received an additional inspiration from the ancient Zapotec city that could be glimpsed from our roof top.
Although Steve is gone, his work and his guiding spirit live on.
F. William Lawvere
July 21, 2015
Dear All,
What is the mathematical content of the preposition OF?
For example, product and sum correspond to the mathematical content of AND and OR, respectively. It is interesting to note that, going by the universal mapping property definitions of product and sum, sum is defined in terms of product (as in: if both sandwich and pizza are on a menu, then I can choose sandwich or pizza; see Lawvere and Schanuel, Conceptual Mathematics, p. 327, 354).
In thinking about ‘of’, we think of properties (e.g., height of posina).
Properties are properties of objects represented as structures (in a sturctureless background). So, structure-property relation seems to be relevant in abstracting the mathematical content of ‘of’. Along these lines, with
f: A –> B
as
B-valued property of A
(ibid., pp. 81-85), ‘of’ appears algebraic (ibid., pp. 370-371).
This is how far I got in extracting the mathematical content of ‘of’. I’d be truly grateful to you for any pointers you may have. Pardon me if this is silly.
Happy Tuesday 🙂
Thanking you, posina
Consider
boundary(C) = C AND (NOT(C))
e.g., boundary(India) = India AND (NOT(India))
and
core(C) = NOT(NOT(C))
(color of a surface can be considered as its core).
We obtain the object by putting together its boundary and core:
C = (C AND (NOT(C))) + (NOT(NOT(C)))
which can be thought of as a qualitative generalization of the (familiar) quantitative:
1 = 0 + 1
Dual to this Leibniz core, we have the Poincaré conjecture:
all of which is reminiscent of:
Mon principal guide dans mon travail a été la recherche constante d’une cohérence parfaite, d’une harmonie complète que je devinais derrière la surface turbulente des choses, et que je m’efforçais de dégager patiemment, sans jamais m’en lasser. C’était un sens aigu de la “beauté”, sûrement, qui était mon flair et ma seule boussole. Ma plus grande joie a été, moins de la contempler quand elle était apparue en pleine lumière, que de la voir se dégager peu à peu du manteau d’ombre et de brumes où il lui plaisait de se dérober sans cesse. Certes, je n’avais de cesse que quand j’étais parvenu à l’amener jusqu’à la plus claire lumière du jour. J’ai connu alors, parfois, la plénitude de la contemplation, quand tous les sons audibles concourent à une même et vaste harmonie. Mais plus souvent encore, ce qui était amené au grand jour devenait aussitôt motivation et moyen d’une nouvelle plongée dans les brumes, à la poursuite d’une nouvelle incarnation de Celle qui restait à jamais mystérieuse, inconnue — m’appelant sans cesse, pour La connaître encore…
Récoltes et Semailles
My main guide in my work was the constant search for a perfect coherence, a complete harmony that I guessed behind the turbulent surface of things, and that I endeavored to release patiently, without ever getting tired of it. It was a keen sense of “beauty”, surely, which was my flair and my only compass. My greatest joy was, less to contemplate it when it had appeared in full light, than to see it free itself little by little from the cloak of shadow and mist where it liked to hide constantly. Of course, I only stopped when I had managed to bring it to the clearest light of day. I then experienced, sometimes, the fullness of contemplation, when all the audible sounds contribute to the same and vast harmony. But more often still, what was brought to light immediately became the motivation and means of a new plunge into the mists, in pursuit of a new incarnation of She who remained forever mysterious, unknown – calling me constantly, to know her again…
Speaker: Posina Venkata Rayudu
Chairperson: Nithin Nagaraj
Associate Professor, Consciousness Studies Programme, NIAS
Date: 15 March 2023
Time: 9:30 AM
Venue: Lecture Hall, NIAS
Abstract:
There is no mind-matter problem in practice: we go with ease between the material world of things and the mental realm of ideas as we think about things and make things we think of (e.g., this abstract). Here we solve the attendant theoretical problem, i.e. we present a conceptual repertoire needed to develop a declarative understanding of conscious participation in the practice of everyday life. We begin with: res extensa vs. res cogitans, and the corresponding dual: change vs. unity. Next, we show how to objectify change, which accounts for the perceptual objectification of physical contrasts into people, things, and their peccadilloes populating our conscious experience. Change, in turn, is natural (or not-a-miracle), i.e., unity-respecting change. We find, in the light of our theory, that the effectiveness of mathematics—understood as the broad objective logic of qualities resulting from quantitative variations—in the natural sciences, which include social, psychological, biological, and physical, is reasonable. We conclude with an application of our theory: show how artificial intelligence (AI) can advance by bridging across the hitherto insurmountable schism: statistical vs. mathematical.
About the speaker:
Mr. Posina Venkata Rayudu is a neuroscientist. Currently Rayudu is engaged in 1. making mathematics universally comprehensible and usable, 2. abstracting quality zero (zero change and zero cohesion) needed for the advancement of mathematics in particular and science in general, 3. compounding epistemology and ontology into which reality is resolved, and 4. explicating Dharma—Becoming consistent with Being—as the Zeroth Law of Motion (that Newton failed to abstract).
Conceptual Mathematics
Conceptual Mathematics 