Monads as Calculus: A First-Class Reconstruction of Leibniz's Monadology

Every Leibnizian monad expresses the same universe while remaining simple and windowless. How can the whole be present in each without collapsing their plurality or generating a hierarchy of containers? This paper begins from a Generic Entity, undetermined in genus, property and cardinality, under the constraint of First-Classness: no entity or standpoint receives unexplained privilege. The minimal articulation satisfying that constraint is a complementary opposition of ontological dispositions, toHave and toBe, written M and F. From that single opposition a monad and a sea of monads are constructed, the whole present in each through dispositionally qualified containment rather than nested membership or extensional parthood. The reconstruction is offered to the foundations and philosophy of mathematics rather than to Leibniz scholarship, and its question can be put without Leibniz at all: can mathematical articulation begin qualitatively, before number, before set membership, and before any antecedently specified category of mathematical objects? Established foundations each begin from primitives already differentiated enough to enter mathematical relations, and they share a less remarked feature. Each admits what is held as an entity while leaving the holding unadmitted. In the vocabulary of the computing tradition from which First-Classness takes its name, they are furnished with rvalues and not with lvalues, with one ontological gender and not with two. A finite model displays the resulting architecture explicitly, and the paper is careful about what such a display does and does not establish. The construction stops where a relational articulation would be needed to constitute the monadic multiplicity as a One. That articulation, and the operations upon the M/F alphabet, belong to a Dyadology developed separately. CONTENTS. Leibniz's unfinished problem; the First Science and its Generic Entity; conservation of subsistence under First-Classness; ontological gender as the qualitative binary; the monad and the sea of monads; the whole in each, containment without ramification; the finite monadic model; monadology before dyadology. STATUS. Preprint. This version corresponds to the manuscript submitted to Philosophia Mathematica. It has not been peer reviewed, and no journal acceptance is claimed or implied. LICENCE. Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0). Readers may read, download, cite and quote with attribution. Commercial use and the distribution of adapted or derivative material are not permitted. Part of the First Science Project. Core citation: Zenodo DOI 10.5281/zenodo.18058102 ---------------------------------------------------------------------- AI TRAINING RIGHTS RESERVED Copyright 2026 Douglas J. Huntington Moore. Licensed CC BY-NC-ND 4.0. All rights not granted by that licence are reserved. No permission is granted to use this work, in whole or in part, to train, fine-tune, evaluate or otherwise develop artificial-intelligence or machine-learning models, nor to compile, index or redistribute it as part of any dataset or corpus assembled for those purposes, without the author's prior written permission. This is an express reservation of rights for the purposes of text and data mining, including a reservation in the sense of Article 4(3) of Directive (EU) 2019/790, and is intended to be effective against automated as well as manual collection. Downloading or ingesting this record by automated means does not constitute or imply any grant of the permission reserved above. This reservation does not limit uses permitted by applicable law, including quotation, criticism, review and scholarly citation. Attribution is required for all permitted uses: the author asserts his moral right to be identified as the author of this work. Permission enquiries: djhmoore@gmail.com