Ontology of Mathematics

This book offers a comprehensive exploration of axiological logic, delving into the philosophical foundations and implications of value theory. It begins by situating the concept of goodness within a broader ontological framework, examining the philosophical underpinnings of logic and arguing for an ontological interpretation that extends beyond mere methodological considerations. By exploring intentional, real, and ideal modes of existence, the book establishes a robust framework for evaluating propositions based on their value relations, introducing key concepts such as goodness, evil, and (...) axiological neutrality. The core of the book introduces a novel semantics for axiological logic, focusing on a relational approach to evaluative properties. It presents the "Local Peak Property" and "Global Peak Property," which posit that goodness is determined by being the best among possible alternatives. A necessitation rule is also developed, asserting that all necessary truths are inherently good, thereby linking logical necessity with axiological value. The theory of values is explored in depth, with major results including the "Benatar-Plantinga Principle," which suggests that the value of a proposition is determined by its least valuable logical consequence. This chapter argues against axiological foundationalism, proposing that values are interdependent rather than grounded in a single foundational entity. The complexities and limitations of aggregating value judgments are also discussed, highlighting the nuanced nature of value summation. The book addresses the naturalistic fallacy by bridging the gap between descriptive and evaluative propositions, demonstrating that evaluative propositions can be logically implied by descriptive ones. An incompleteness theorem is presented, illustrating the limitations of formal systems in capturing the richness of axiological logic. The final chapter advocates for a platonic perspective in axiology, arguing that axiological truths serve as a universal standard for evaluating the world, independent of contingent facts. This perspective underscores the normative significance of goodness and the interconnected nature of values, offering a platonic framework for understanding axiological logic. (shrink)

I reformulate Gödel's incompleteness phenomenon by distinguishing clearly between a base theory T, which lacks self-reference, and its reflective extension T', which internalizes meta-level predicates such as provability. I argue that incompleteness arises not from an intrinsic defect in arithmetic, but from a structural transformation—reflective collapse—triggered by embedding meta-theoretic notions within the object language. The base theory T is shown to be stratified-complete: it cannot express Gödelian sentences and hence does not exhibit incompleteness. I construct a 2-category of stratified theories, (...) StrTh, define morphisms that preserve or violate stratification, and classify conservative versus pathological embeddings. Through recursion-theoretic and categorical analysis, I demonstrate the non-uniformity of diagonalization and the ontological shift that accompanies self-reference. The work concludes with philosophical reflections on epistemic layering, formal dogma, and the nature of mathematical truth as shaped by inscriptional hierarchy. (shrink)

We present a mathematically rigorous extension to quantum mechanics that accounts for consciousness while resolving longstanding paradoxes in physics. Through formal set-theoretic, group-theoretic, and category-theoretic arguments, we first demonstrate the logical impossibility of emergentism—the view that consciousness arises from complex physical processes. We then introduce a minimal dual-phase space framework in which physical states exist in a Hilbert space HΨ and phenomenal states in an orthogonal Hilbert space HΦ , connected by the awareness operator A and volition operator V. These (...) spaces maintain a precise π/2 phase relationship, with the resultant complex energy eigenvalues manifesting gravitationally as dark matter and dark energy. Our framework elegantly resolves the quantum measurement problem through the Born Rule Locus Correspondence (BRLC), which establishes that phenomenal probabilities exactly match physical probabilities when properly referenced to the observer's spacetime locus. We derive several tensor product norm identities that explain both the causal efficacy of consciousness and its imperviousness to physical detection. Finally, we demonstrate how our framework necessitates the persistence of consciousness beyond biological death through phase decoherence and recoherence processes that follow directly from the mathematical structure. Throughout, we identify specific empirical predictions that distinguish our framework from competing theories. The A|Ω⟩ formalism thus unifies quantum mechanics, consciousness, and cosmology within a single mathematical structure that preserves logical coherence while remaining empirically testable. (shrink)

Digging into the Transistor (Circular-Linear Relationship between Mind and Matter).

Where does ‘life’ come from? The answer may surprise you.

A circle joining and separating any X and Y explains (and controls) everything. Autonomous Intentional Masking (AIM) is the SUPRA-CONSCIOUS PROCESSOR (the super-chip) that controls the circular-linear relationship between mind and matter (nuclear energy) (abstract and concrete reality) (the Metaverse called Mind) (the Singularity called Nature). Giving humans (who understand it) a competitive advantage in all situations (virtual anticipation) (intelligent decisioning).

CODES: The Coherence Framework Replacing Probability in Physics, Intelligence, and Reality -/- A Unified Substrate for Intelligence, Physics, Evolution, and Cosmic Structure -/- Author: Devin Bostick -/- Date: January 29, 2025 -/- Version 21 (Updated April 21, 2025) -/- -/- This version completes the integration of coherence field logic, prime-phase geometry, and chirality-driven emergence into a single deterministic framework. Sections 16–18 introduce post-calculus modeling, resonance-based AGI cognition, and a coherence-grounded theory of ethics, establishing CODES as not just a scientific theory, (...) but the generative structure of reality itself. -/- -/- Abstract CODES (Chirality of Dynamic Emergent Systems) dissolves the false boundary between physics and intelligence. It replaces stochastic modeling with structured resonance fields, showing that mass, gravity, time, cognition, and emergence are not separate domains—but outputs of a single chiral resonance substrate. -/- Rather than modeling reality, CODES reveals what remains when all probabilistic artifacts are removed. It frames mathematics not as abstraction, but as instantiated resonance structure. Gödel’s incompleteness becomes bypassable—not violated—when logic is grounded in physical coherence rather than recursive symbols. -/- -/- -/- Key Contributions -/- 1. Unified Physics from Chirality Gravity is reframed as resonance compression, not curvature. -/- Quantum mechanics emerges as a sub-phase of coherent oscillatory fields. -/- Time is a nested coherence rhythm, not a linear axis. -/- -/- 2. Dark Matter and Energy Reinterpreted Not hidden particles—phase misalignments in the coherence lattice. -/- ΛCDM becomes a low-resolution model of prime-structured resonance fields. -/- 3. Collapse of Gödel, Emergence of Intelligence Logical incompleteness is an illusion caused by symbol recursion. -/- Intelligence emerges as recursive coherence-seeking, not stochastic search. -/- -/- 4. The AGI Substrate Intelligence is not predictive—it is resonance tuning. -/- The Resonance Intelligence Core (RIC) operationalizes this shift via prime-phase harmonics. -/- -/- -/- Empirical Validation Pathways -/- CODES proposes specific, testable predictions via existing instruments: -/- -/- LIGO – Prime resonance scoring in gravitational wave data (e.g., GW190521) -/- Galaxy Clustering – Wavelet-based detection of redshift periodicity -/- Bose-Einstein Condensates – Chirality-locked decay asymmetries -/- fMRI Studies – Phase-locking as lattice-based cognition, not noise inference -/- Spectroscopy of α – Drift in fine-structure constant as coherence shift, not universal constancy -/- -/- -/- The Final Paradigm: Why CODES Is Inevitable -/- CODES is not another unification attempt. -/- It is what remains when all incoherent abstractions are removed. -/- -/- Horizontally, it integrates cosmology, computation, cognition, and chemistry. -/- Vertically, it reveals chirality as the origin of recursive structure—across all scales. -/- Any rational observer, stripped of symbolic legacy, would reconstruct CODES from first principles. Not metaphorically. Mathematically. -/- -/- From Singularity Myth to Resonant Intelligence -/- CODES reframes the Singularity as not an AI rupture—but a coherence realization. -/- -/- If intelligence is resonance, then stochastic AI is obsolete. -/- If logic is phase structure, then Gödel no longer binds cognition. -/- If emergence is deterministic, then physics, mind, and ethics converge. -/- -/- CODES does not predict the Singularity. CODES is the Singularity—phase-locked. (shrink)

By analysing several characteristic mathematical models: natural and real numbers, Euclidean geometry, group theory, and set theory, I argue that a mathematical model in its final form is a junction of a set of axioms and an internal partial interpretation of the corresponding language. It follows from the analysis that (i) mathematical objects do not exist in the external world: they are imagined objects, some of which, at least approximately, exist in our internal world of activities or we can realize (...) or represent them there; (ii) mathematical truths are not truths about the external world but specifications (formulations) of mathematical conceptions; (iii) mathematics is first and foremost our imagined tool by which, with certain assumptions about its applicability, we explore nature and synthesize our rational cognition of it. (shrink)

Conservation of the Circle is the only dynamic in Nature.

Metaphysical Naturalism, Mereological Nihilism, The Circular Theory, and the Unification of Technology, Psychology and Finance.

Intelligent Design integrates the work of Ilexa Yardley and Stridjom van der Merwe to demonstrate and prove Conservation of the Circle is the Only Dynamic in Nature (The Circular Theory) (Quantum Mechanics) (Metaphysical Naturalism). Explaining why everything changes because nothing is changing.

There are two axes of Leibniz’s philosophy about bodies that are deeply inter- twined, as this paper shows: the scientific investigation of bodies due to the application of mathematics to nature – Leibniz’s mixed mathematics – and the issue of matter/bodies ide- alism. This intertwinement raises an issue: How did Leibniz frame the relationship between mathematics, natural sciences, and metaphysics? Due to the increasing application of mathe- matics to natural sciences, especially physics, philosophers of the early modern period used the (...) reliability of mathematics to predict phenomena as the basis to infer the metaphysical outlook of nature. I argue that although Leibniz thought metaphysics must be scientifically informed and that mathematics is a valuable instrument to understand nature, metaphysics is more fun- damental than mathematics and natural sciences. By highlighting the foundational relation be- tween metaphysics and the sciences, this paper showcases an argument for the reality of bodies: the ideality of bodies, necessary for epistemic purposes, is not proof that they are not real. Keywords: Metaphysics, Application of Mathematics, Body, Idealism, Fundamentality. (shrink)

Conservation of the Circle is the only dynamic in Nature. Yin and Yang (ancient) is Zero and One (modern). Circumference and Diameter of an Always-Present (Technically Prescient) Circle.

note: this is a by Cambridge Open Engage (C.O.E.) fixed version of my research paper 'objects are (not) ....', which I first posted February 17th, 2024 at researchgate and subsequently host at philarchive, academia and the internet archive. -/- Abstract: My goal in this paper is, to tentatively sketch and try defend some observations regarding the ontological dignity of object references, as they may be used from within in a formalized language. Hence I try to explore, what properties objects are (...) presupposed to have, in order to enter the universe of discourse of an interpreted formalized language. First I review Frege′s analysis of the logical structure of truth value definite sentences of scientific colloquial language, to draw suggestions from his saturated vs. unsaturated sentence components paradigm. Next try investigate, in how far reference to non pure math objects might allow for a role as argument of a truth value function. Object kinds to be considered are: common sense objects, technical objects, humanities object kinds (social, psychical, ...), objects of art, ... , be they abstract, concrete, or in this respect mixed objects. Then have a comment on the just referenced label abstract objects. Next try to get an idea wrt the ontological significance of the fact, that pure math objects and functions in some important classical cases can be uniquely defined by means of categorical theories. Here, in the course of my argument, I have a little corollary wrt the standard model of first order Peano arithmetic, reducing the epistemological significance of the existence of the non standard models. Next, wrt a special concept of a formalized empirical theory, I care for whether the impure math objects and mixed objects described here, and complying best with truth value function mapping, are also reliable candidates for the ontological commitment of such theories; and discuss an alternative, which reduces the ontological significance of the universe of discourse of the theories intended models. In the end I sum up what is - from my point of view - the status quo, according to my respective findings. (shrink)

We address the following questions in this paper: (1) Which set or number existence axioms are needed to prove the theorems of ‘ordinary’ mathematics? (2) How should Frege’s theory of numbers be adapted so that it works in a modal setting, so that the fact that equivalence classes of equinumerous properties vary from world to world won’t give rise to different numbers at different worlds? (3) Can one reconstruct Frege’s theory of numbers in a non-modal setting without mathematical primitives such (...) as “the number of Fs” ( $$\#F$$ ) or mathematical axioms such as Hume’s Principle? Our answer to question (1) is ‘None’. Our answer to question (2) begins by defining ‘x numbers G’ as: x encodes all and only the properties F such that being-actually-F is equinumerous to G with respect to discernible objects. We answer (3) by showing that the mere existence of discernible objects allows one to reconstruct Frege’s derivation of the Dedekind-Peano axioms in a non-modal setting. (shrink)

An attempt to vindicate naive set theory by postulating a universal set V which is describable in two distinct description languages: predicative and extensional. The extensional description of a set consists of describing all its elements whereas its predicative description consists of describing what sets it is an element of. -/- Extensionally described V has an uncapturable description length, akin to its cardinality. But predicatively described, in virtue of being the set that is not contained in any set whatsoever, V (...) has a minimal description length, a counterbalance to its cardinality. -/- Descriptive efficiency can genuinely be attributed to V (achieved in a third description language) only if it contains predicatively indiscernible but extensionally discernible sets. (shrink)

Why humans cry out (eventually): I need my space!

In this paper, we introduce a quasi-set theory without atoms. The quasi-sets (qsets) can have as elements completely indiscernible things which do not turn out to be the very same thing as it would be implied if its underlying logic was classical logic. A quasi-set can have a cardinal, called its quasi-cardinal, but this is made so that, at least for the finite case, the quasi-cardinal is not an ordinal, and hence the indistinguishable elements of a quasi-set cannot be ordered. (...) In the last sections, we show how the theory can be used to ground a quantum metaphysics of properties, which sees the quantum entities as bundles of properties, and we also introduce a new approach for the use of quantifiers in quantum physics. (shrink)

Zero and One is Circumference and Diameter (Literally and Figuratively) (Abstract and Concrete) (Unity and Duality) (Unity and Duplicity).

Throughout the history, whenever humans encounter a phenomenon for which there was no explanation, a theory was proposed for it. Of course, not necessarily all the theories were purely scientific and many of them were non-scientific, pseudo- scientific, or at best were only slightly influenced by science. But one thing was in common among them: they all were trying to provide as deeper as possible explanations about how the universe works. Although today and in the modern era the exact meaning (...) of theory has changed comparatively, still they must be such that can be tested through scientific procedures in order to confirm or reject. Nevertheless, still sometimes in a common belief, the meaning of "theory" is unprovable or at least unproven conjecture, and surprisingly sometimes confused with hypothesis. Scientific laws are descriptive reports - or theories - that express and predict a range of natural phenomena and explain how they work. Any scientific discourse is carried out in three levels: hypothesis, theory, and law. A hypothesis, is a testable logical conjecture to explain the phenomena we observe in the universe. A hypothesis usually comes with no practical support, or at least not been proven yet. The hypothesis can only be tested through trial and error and tries to offer explanation that provides a reasonable platform for further studies and usually it is the first step in solving a problem or describing a phenomenon. A theory however, is a logical statement about a phenomenon that is derived from rational thinking and is often associated with processes such as observation, study or research and provides standards in such a way that scientific tests can confirm - or reject. A theory used in describing what is usually to happen with the implication that it doesn’t still a fact. Yet a scientific law is a descriptive reports - or theory - that previses a range of natural phenomena and explains how they function. Scientific laws are descriptive summaries of a large set of facts that are tested by scientific evidences and after fact-checking based on their potential ability for measuring future experiences. So it can be said that every theory was first a hypothesis and has the potential to become a law to come. It can also be said that any theory is valid as long as it is not violated by a more conclusive one. (shrink)

'Yin and Yang' vs 'Yin or Yang'. Correcting (Quantum) Technological (Mathematical) (Philosophical) (Psychological) Flaws.

Science depends upon the decimal system, and, for technological ‘calculations,’ science also depends upon the binary system. Where neither the decimal system nor the binary system is known at all by Nature. -/- .

With respect to the metaphysics of infinity, the tendency of standard debates is to either endorse or to deny the reality of ‘the infinite’. But how should we understand the notion of ‘reality’ employed in stating these options? Wittgenstein’s critical strategy shows that the notion is grounded in a confusion: talk of infinity naturally takes hold of one’s imagination due to the sway of verbal pictures and analogies suggested by our words. This is the source of various philosophical pictures that (...) in turn give rise to the standard metaphysical debates: that the mathematics of infinity corresponds to a special realm of infinite objects, that the infinite is profoundly huge or vast, or that the ability to think about infinity reveals mysterious powers in human beings. First, I explain Wittgenstein’s general strategy for undermining philosophical pictures of ‘the infinite’ – as he describes it in Zettel; and then show how that critical strategy is applied to Cantor’s diagonalization proof in Remarks on the Foundations of Mathematics II. (shrink)

In his groundbreaking book, Thin Objects, Linnebo (2018) argues for an account of neo-Fregean abstraction principles and thin existence that does not rely on analyticity or conceptual rules. It instead relies on a metaphysical notion he calls “sufficiency”. In this short discussion, I defend the analytic or conceptual rule account of thin existence.

Arithmetic is one of the foundations of our educational systems, but what exactly is it? Numbers are everywhere in our modern societies, but what is our knowledge of numbers really about? This book provides a philosophical account of arithmetical knowledge that is based on the state-of-the-art empirical studies of numerical cognition. It explains how humans have developed arithmetic from humble origins to its modern status as an almost universally possessed knowledge and skill. Central to the account is the realisation that, (...) while arithmetic is a human creation, the development of arithmetic is constrained by our evolutionarily developed cognitive architecture. Arithmetic is a sophisticated cultural development, but it is ultimately based on abilities with numerosities that we already possess as infants and share with many non-human animals. Therefore, arithmetic is not purely conventional, an arbitrary game akin to chess. Instead, arithmetic is deeply connected to our basic cognitive capacities. (shrink)

Conservation of the Circle is the core (and, thus, the only) dynamic in Nature.

Philosophers often debate the existence of such things as numbers and propositions, and say that if these objects exist, they are abstract. But what does it mean to call something 'abstract'? And do we have good reason to believe in the existence of abstract objects? This Element addresses those questions, putting newcomers to these debates in a position to understand what they concern and what are the most influential considerations at work in this area of metaphysics. It also provides advice (...) on which lines of discussion promise to be the most fruitful. (shrink)

A traditional question in the philosophy of mathematics is to give an answer to the question: What is the nature of mathematical objects? This chapter considers the main answers that have been given to this question, specifically those according to which mathematical objects are independently existing entities, or abstractions, or logical objects, or simplifications, or mental constructions, or structures, or fictions, or idealizations of sensible things, or idealizations of operations. The chapter also shows the shortcomings of these answers, and considers (...) an alternative answer, according to which mathematical objects are hypotheses introduced to solve mathematical problems by the analytic method. (shrink)

Hofweber (Ontology and the ambitions of metaphysics, Oxford University Press, 2016) argues for a thesis he calls “internalism” with respect to natural number discourse: no expressions purporting to refer to natural numbers in fact refer, and no apparent quantification over natural numbers actually involves quantification over natural numbers as objects. He argues that while internalism leaves open the question of whether other kinds of abstracta exist, it precludes the existence of natural numbers, thus establishing what he calls “restricted nominalism” about (...) natural numbers. We argue that Hofweber’s internalism fails to establish restricted nominalism. Not only is his primary argument for restricted nominalism invalid, the analysis of quantification proposed threatens to collapse internalism into either a traditional form of error theory or realism. (shrink)

Monism is the claim that only one object exists. While few contemporary philosophers endorse monism, it has an illustrious history – stretching back to Bradley, Spinoza and Parmenides. In this paper, I show that plausible assumptions about the higher-order logic of property identity entail that monism is true. Given the higher-order framework I operate in, this argument generalizes: it is also possible to establish that there is a single property, proposition, relation, etc. I then show why this form of monism (...) is inconsistent; because all propositions are identical, p is identical to ~p – and so they have the same truth-value. At least one of the assumptions that generate higher-order monism must be rejected. (shrink)

The mathematical universe hypothesis is a theory that the physical universe is not merely described by mathematics, but is mathematics, specifically a mathematical structure. Our research provides evidence that the mathematical structure of the universe is biological in nature and all systems, processes, and objects within the universe function in harmony with biological patterns. Living organisms are the result of the universe’s biological pattern and are embedded within their physiology the patterns of this biological universe. Therefore physiological patterns in living (...) organisms can be used as models to structurally map analogies from the biological domain to any target domain to reveal and understand the biological nature of the target domain. Our paper explores various analogies, structurally mapping a red blood cell to a cup; proteins produced from ribosomes to music produced from instruments; a beating heart to the melting and freezing of Antartica; cells, tissue, organs and blood, to people, organizations, industries, and money, and; bio-economic concepts in cellular society to socioeconomic concepts in human society. It also discusses how phenomena in cellular mitosis can help explain phenomena in the universe, such as black holes, dark matter, dark energy, and the structure of the universe. Building upon the concept of perennial wisdom, our research has provided evidence that the ideas of a biological universe were expressed across many past cultures and historical periods. The implications of this theory are vast, encompassing fields such as physics, science, philosophy, religion, law, economics, politics, and engineering, thus serving as a unifying theory for all knowledge. Our theory is supported by meta-analysis of scientific, historic, and religious literature, observations and first principles logic. (shrink)

A perspective in the philosophy of mathematics is developed from a consideration of the strengths and limitations of both logicism and platonism, with an early focus on Frege’s work. Importantly, although many set-theoretic structures may be developed each of which offers limited isomorphism with the system of natural numbers, no one of them may be identified with it. Furthermore, the timeless, ever present nature of mathematical concepts and results itself offers direct access, in the face of a platonist account which (...) generates a supposed problem of access. Crucially too, pure mathematics has its own distinctive method of confirming or validating results - mathematical proof - which supplies a higher level of confidence and objectivity than that available elsewhere. The dichotomy of invention and discovery is too jejune a framework for analysing creative mathematical activity. The Gödelian platonist perspective is evaluated and queried through scrutiny of the part played by mathematical resources and constraints in relation to human activity. It appears that there can be non-causal mathematical explanations and mathematical constraint on purely natural processes. Valuable implications of Quine’s naturalism are explored, but one must be cautious of his thesis of confirmational holism. The distinction between algebraic and non-algebraic mathematical theories usefully contributes to our understanding of the internally differentiated nature of the subject. (shrink)

[Use code AUFLY30 for 30% off on the OUP website.] One central aim of science is to provide explanations of natural phenomena. What role(s) does mathematics play in achieving this aim? How does mathematics contribute to the explanatory power of science? Rules to Infinity defends the thesis, common though perhaps inchoate among many members of the Vienna Circle, that mathematics contributes to the explanatory power of science by expressing conceptual rules, rules which allow the transformation of empirical descriptions. Mathematics should (...) not be thought of as describing, in any substantive sense, an abstract realm of eternal mathematical objects, as traditional platonists have thought. In Rules to Infinity, this view, which I call mathematical normativism, is updated with contemporary philosophical tools, and it is argued that normativism is compatible with mainstream semantic theory. This allows the normativist to accept that there are mathematical truths, while resisting the platonistic idea that there exist abstract mathematical objects that explain such truths. Furthermore, Rules to Infinity defends a particular account of the distinction between scientific explanations that are in some sense distinctively mathematical – those that explain natural phenomena in some uniquely mathematical way – and those that are only standardly mathematical, and it lays out desiderata for any account of this distinction. Normativism is compared with other prominent views in the philosophy of mathematics such as neo-Fregeanism, fictionalism, conventionalism, and structuralism. Rules to Infinity serves as an entry point into debates at the forefront of philosophy of science and mathematics, and it defends novel positions in these debates. (shrink)

My goal in this paper is, to tentatively sketch and try defend some observations regarding the ontological dignity of object references, as they may be used from within in a formalized language. -/- Hence I try to explore, what properties objects are presupposed to have, in order to enter the universe of discourse of an interpreted formalized language. -/- First I review Frege′s analysis of the logical structure of truth value definite sentences of scientific colloquial language, to draw suggestions from (...) his saturated vs. unsaturated sentence components paradigm. -/- Next try investigate, in how far reference to non pure math objects might allow for a role as argument of a truth value function. Object kinds to be considered are: common sense objects, technical objects, humanities object kinds (social, psychical, ...), objects of art, ... , be they abstract, concrete, or in this respect mixed objects. -/- Then have a comment on the just referenced label abstract objects. -/- Next try to get an idea wrt the ontological significance of the fact, that pure math objects and functions in some important classical cases can be uniquely defined by means of categorical theories. Here, in the course of my argument, I have a little corollary wrt the standard model of first order Peano arithmetic, reducing the epistemological significance of the existence of the non standard models. -/- Next, wrt a special concept of a formalized empirical theory, I care for whether the impure math objects and mixed objects described here, and complying best with truth value function mapping, are also reliable candidates for the ontological commitment of such theories: and discuss an alternative, which reduces the ontological significance of the universe of discourse of the theories intended models. -/- In the end I sum up what is - from my point of view - the status quo, according to my respective findings. (shrink)

Daniel Isaacson has advanced an epistemic notion of arithmetical truth according to which the latter is the set of truths that we grasp on the basis of our understanding of the structure of natural numbers alone. Isaacson’s thesis is then the claim that Peano Arithmetic (PA) is the theory of finite mathematics, in the sense that it proves all and only arithmetical truths thus understood. In this paper, we raise a challenge for the thesis and show how it can be (...) overcome. We introduce the concept of purity for theories of arithmetic: a theory of arithmetic is pure when it only proves arithmetical truths. Then, we argue that, under Isaacson’s thesis, some PA-provable truths—including transfinite induction claims for infinite ordinals and some consistency statements—are seemingly not arithmetical in Isaacson’s sense, and hence that Isaacson’s thesis might entail the impurity of PA. Nonetheless, we conjecture that the advocate of Isaacson’s thesis can avoid this undesirable consequence: the arithmetical nature, as understood by Isaacson, of all contentious PA-provable statements can be justified. As a case study, we explore how this is done for transfinite induction claims with infinite ordinals below ε0. To this end, we show that the PA-proof of such claims employs exclusively resources from finite mathematics, and that ordinals below ε0 are finitary objects despite being infinite. (shrink)

In this paper, I study the kind of questions we can ask about the existence of numbers. In addition to asking whether numbers exist, and how, I argue that there is also a third relevant question: why numbers exist. In platonist and nominalist accounts this question may not make sense, but in the psychologist account I develop, it is as well-placed as the other two questions. In fact, there are two such why-questions: the causal why-question asks what causes numbers to (...) exist and the teleological why-question asks for what purpose numbers exist. I argue that in a psychologist constructivist account, in which numbers are understood to exist as referents of a particular type of culturally shared concepts, both why-questions can get plausible answers. (shrink)

Distinctively mathematical explanations (DMEs) explain natural phenomena primarily by appeal to mathematical facts. One important question is whether there can be an ontic account of DME. An ontic account of DME would treat the explananda and explanantia of DMEs as ontic structures and the explanatory relation between them as an ontic relation (e.g., Pincock 2015, Povich 2021). Here I present a conventionalist account of DME, defend it against objections, and argue that it should be considered ontic. Notably, if indeed it (...) is ontic, the conventionalist account seems to avoid a convincing objection to other ontic accounts (Kuorikoski 2021). (shrink)

Using ideas proposed in Aboutness and developed in ‘If-thenism’, Stephen Yablo has tried to improve on classical if-thenism in mathematics, a view initially put forward by Bertrand Russell in his Principles of Mathematics. Yablo’s stated goal is to provide a reading of a sentence like ‘The number of planets is eight’ with a sort of content on which it fails to imply ‘Numbers exist’. After presenting Yablo’s framework, our paper raises a problem with his view that has gone virtually unnoticed (...) in the literature. If we are right, then Yablo’s version of if-thenism cannot succeed. (shrink)

Although number sentences are ostensibly simple, familiar, and applicable, the justification for our arithmetical beliefs has been considered mysterious by the philosophical tradition. In this paper, I argue that such a mystery is due to a preconception of two realities, one mathematical and one nonmathematical, which are alien to each other. My proposal shows that the theory of numbers as properties entails a homogeneous domain in which arithmetical and nonmathematical truth occur. As a result, the possibility of arithmetical knowledge is (...) simply a consequence of the possibility of ordinary knowledge. (shrink)

The view of nature we adopt in the natural attitude is determined by common sense, without which we could not survive. Classical physics is modelled on this common-sense view of nature, and uses mathematics to formalise our natural understanding of the causes and effects we observe in time and space when we select subsystems of nature for modelling. But in modern physics, we do not go beyond the realm of common sense by augmenting our knowledge of what is going on (...) in nature. Rather, we have measurements that we do not understand, so we know nothing about the ontology of what we measure. We help ourselves by using entities from mathematics, which we fully understand ontologically. But we have no ontology of the reality of modern physics; we have only what we can assert mathematically. In this paper, we describe the ontology of classical and modern physics against this background and show how it relates to the ontology of common sense and of mathematics. (shrink)

The present paper concerns the Wittgenstein ontology project: an attempt to create a Semantic Web representation of Ludwig Wittgenstein’s philosophy. The project has been in development since 2006, and its current state enables users to search for information about Wittgenstein-related documents and the documents themselves. However, the developers have much more ambitious goals: they attempt to provide a philosophical subject matter knowledge base that would comprise the claims and concepts formulated by the philosopher. The current knowledge representation technology is not (...) well-suited for this task, and a non-standard approach is required. The creators of the Wittgenstein ontology project are aware of this fact; recently, they have been discussing conceptual devices adjusting the technology to their needs. The main goal of this paper is to present examples of a representation of philosophical content that make use of both the devices already proposed and some new inventions. The represented content comes from the Tractatus Logico-Philosophicus; more specifically, its theses concerning the problems in philosophy of mathematics. (shrink)