Some have argued for a division of epistemic labor in which mathematicians supply truths and philosophers supply their necessity. We argue that this is wrong: mathematics is committed to its own necessity. Counterfactuals play a starring role.

Einstein, like most philosophers, thought that there cannot be mathematical truths which are both necessary and about reality. The article argues against this, starting with prima facie examples such as "It is impossible to tile my bathroom floor with regular pentagonal tiles." Replies are given to objections based on the supposedly purely logical or hypothetical nature of mathematics.

An increasing amount of contemporary philosophy of mathematics posits, and theorizes in terms of special kinds of mathematical modality. The goal of this paper is to bring recent work on higher-order metaphysics to bear on the investigation of these modalities. The main focus of the paper will be views that posit mathematical contingency or indeterminacy about statements that concern the `width' of the set theoretic universe, such as Cantor's continuum hypothesis. Within a higher-order framework I show that contingency about (...) the width of the set-theoretic universe refutes two orthodoxies concerning the structure of modal reality: the view that the broadest necessity has a logic of S5, and the "Leibniz biconditionals" stating that what is possible, in the broadest sense of possible, is what is true in some possible world. Nonetheless, I suggest that the underlying picture of modal set-theory is coherent and has attractions. (shrink)

There is a major debate as to whether there are non-causal mathematical explanations of physical facts that show how the facts under question arise from a degree of mathematical necessity considered stronger than that of contingent causal laws. We focus on Marc Lange’s account of distinctively mathematical explanations to argue that purported mathematical explanations are essentially causal explanations in disguise and are no different from ordinary applications of mathematics. This is because these explanations work not by appealing to (...) what the world must be like as a matter of mathematical necessity but by appealing to various contingent causal facts. (shrink)

It seems possible to know that a mathematical claim is necessarily true by inspecting a diagrammatic proof. Yet how does this work, given that human perception seems to just (as Hume assumed) ‘show us particular objects in front of us’? I draw on Peirce’s account of perception to answer this question. Peirce considered mathematics as experimental a science as physics. Drawing on an example, I highlight the existence of a primitive constraint or blocking function in our thinking which we (...) might call ‘the hardness of the mathematical must’. -/- This is a 3-page abstract for the DIAGRAMS2018 conference, summarising prior work in papers such as (“Things Unreasonably Compulsory”, Cognitio 15 (1): 89-112 (2014), and "The Hardness of the Iconic Must", Philosophia Mathematica 20 (1): 1-24 (2012)). (shrink)

In many diagrams one seems to perceive necessity – one sees not only that something is so, but that it must be so. That conflicts with a certain empiricism largely taken for granted in contemporary philosophy, which believes perception is not capable of such feats. The reason for this belief is often thought well-summarized in Hume's maxim: ‘there are no necessary connections between distinct existences’. It is also thought that even if there were such necessities, perception is too passive (...) or localized a faculty to register them. We defend the perception of necessity against such Humeanism, drawing on examples from mathematics. (shrink)

This paper aims to clarify Merleau-Ponty’s contribution to an embodied-enactive account of mathematical cognition. I first identify the main points of interest in the current discussions of embodied higher cognition and explain how they relate to Merleau-Ponty and his sources, in particular Husserl’s late works. Subsequently, I explain these convergences in greater detail by more specifically discussing the domains of geometry and algebra and by clarifying the role of gestalt psychology in Merleau-Ponty’s account. Beyond that, I explain how, for Merleau-Ponty, (...) mathematical cognition requires not only the presence and actual manipulation of some concrete perceptible symbols but, more strongly, how it is fundamentally linked to the structural transformation of the concrete configurations of symbolic systems to which these symbols appertain. Furthemore, I fill a gap in the literature by explaining Merleau-Ponty’s claim that these structural transformations are operated through motor intentionality. This makes it possible, in turn, to contrast Merleau-Ponty’s approach to ontologically idealistic and realistic views on mathematical objects. On Merleau-Ponty’s account, mathematical objects are relational entities, that is, gestalts that necessarily imply situated cognizers to whom they afford a specific type of engagement in the world and on whom they depend in their eventual structural transformations. I argue that, by attributing a strongly constitutive role to phenomenal configurations and their motor transformation in mathematical thinking, Merleau-Ponty contributes to clarifying the worldly, historical, and socio-cultural aspects of mathematical truths without compromising what we perceive as their universality, certainty, and necessity. (shrink)

In “What Makes a Scientific Explanation Distinctively Mathematical?” (2013b), Lange uses several compelling examples to argue that certain explanations for natural phenomena appeal primarily to mathematical, rather than natural, facts. In such explanations, the core explanatory facts are modally stronger than facts about causation, regularity, and other natural relations. We show that Lange's account of distinctively mathematical explanation is flawed in that it fails to account for the implicit directionality in each of his examples. This inadequacy is remediable in each (...) case by appeal to ontic facts that account for why the explanation is acceptable in one direction and unacceptable in the other direction. The mathematics involved in these examples cannot play this crucial normative role. While Lange's examples fail to demonstrate the existence of distinctively mathematical explanations, they help to emphasize that many superficially natural scientific explanations rely for their explanatory force on relations of stronger-than-natural necessity. These are not opposing kinds of scientific explanations; they are different aspects of scientific explanation. (shrink)

The paper discusses Hilbert mathematics, a kind of Pythagorean mathematics, to which the physical world is a particular case. The parameter of the “distance between finiteness and infinity” is crucial. Any nonzero finite value of it features the particular case in the frameworks of Hilbert mathematics where the physical world appears “ex nihilo” by virtue of an only mathematical necessity or quantum information conservation physically. One does not need the mythical Big Bang which serves to concentrate (...) all the violations of energy conservation in a “safe”, maximally remote point in the alleged “beginning of the universe”. On the contrary, an omnipresent and omnitemporal medium obeying quantum information conservation rather than energy conservation permanently generates action and thus the physical world. The utilization of that creation “ex nihilo” is accessible to humankind, at least theoretically, as long as one observes the physical laws, which admit it in their new and wider generalization. One can oppose Hilbert mathematics to Gödel mathematics, which can be identified as all the standard mathematics until now featureable by the Gödel dichotomy of arithmetic to set theory: and then, “dialectic”, “intuitionistic”, and “Gödelian” mathematics within the former, according to a negative, positive, or zero value of the distance between finiteness and infinity. A mapping of Hilbert mathematics into pseudo-Riemannian space corresponds, therefore allowing for gravitation to be interpreted purely mathematically and ontologically in a Pythagorean sense. Information and quantum information can be involved in the foundations of mathematics and linked to the axiom of choice or alternatively, to the field of all rational numbers, from which the pair of both dual and anti-isometric Peano arithmetics featuring Hilbert arithmetic are immediately inferable. Noether’s theorems (1918) imply quantum information conservation as the maximally possible generalization of the pair of the conservation of a physical quantity and the corresponding Lie group of its conjugate. Hilbert mathematics can be interpreted from their viewpoint also after an algebraic generalization of them. Following the ideas of Noether’s theorem (1918), locality and nonlocality can be realized both physically and mathematically. The “light phase of the universe” can be linked to the gap of mathematics and physics in the Cartesian organization of cognition in Modernity and then opposed to its “dark phase”, in which physics and mathematics are merged. All physical quantities can be deduced from only mathematical premises by the mediation of the most fundamental physical constants such as the speed of light in a vacuum, the Planck and gravitational constants once they have been interpreted by the relation of locality and nonlocality. (shrink)

It may be a myth that Plato wrote over the entrance to the Academy “Let no-one ignorant of geometry enter here.” But it is a well-chosen motto for his view in the Republic that mathematical training is especially productive of understanding in abstract realms, notably ethics. That view is sound and we should return to it. Ethical theory has been bedevilled by the idea that ethics is fundamentally about actions (right and wrong, rights, duties, virtues, dilemmas and so on). That (...) is an error like the one Plato mentions of thinking mathematics is about actions (of adding, constructing, extracting roots and so on). Mathematics is about eternal relations between universals, such as the ratio of the diagonal of a square to the side. Ethics too is about eternal verities, such as the equal worth of persons and just distributions. Mathematical and ethical verities do both constrain actions, such as the possibility of walking over the seven bridges of Königsberg once and once only or of justly discriminating between races. But they are not themselves about action. In principle, neither mathematical nor ethical verities are subject to historical forces or disagreement among tribes (though they can be better understood as time goes on). Plato is right: immersion in mathematics induces an understanding of the necessities underpinning reality, an understanding that is essential for distinguishing objective ethics from tribal custom. Equality, for example, is an abstract concept which is foundational for both mathematics and ethics. (shrink)

This paper presents a Peircean take on Wittgenstein's famous rule-following problem as it pertains to 'knowing how to go on in mathematics'. I argue that McDowell's advice that the philosophical picture of 'rules as rails' must be abandoned is not sufficient on its own to fully appreciate mathematics' unique blend of creativity and rigor. Rather, we need to understand how Peirce counterposes to the brute compulsion of 'Secondness', both the spontaneity of 'Firstness' and also the rational intelligibility of (...) 'Thirdness'. (This is a written version of a presentation I gave at the “Peirce’s Mathematics” conference, Universidad Nacional de Colombia, November 25-27, 2015, which was organized by Professor Fernando Zalamea. The piece owes much to the inspiration of Prof. Zalamea's writings on philosophy of mathematics.). (shrink)

View more Abstract If logical truth is necessitated by sheer syntax, mathematics is categorially unlike logic even if all mathematics derives from definitions and logical principles. This contrast gets obscured by the plausibility of the Synonym Substitution Principle implicit in conceptions of analyticity: synonym substitution cannot alter sentence sense. The Principle obviously fails with intercepting: nonuniform term substitution in logical sentences. ‘Televisions are televisions’ and ‘TVs are televisions’ neither sound alike nor are used interchangeably. Interception synonymy gets assumed (...) because logical sentences and their synomic interceptions have identical factual content, which seems to exhaust semantic content. However, intercepting alters syntax by eliminating term recurrence, the sole strictly syntactic means of ensuring necessary term coextension, and thereby syntactically securing necessary truth. Interceptional necessity is lexical, a notational artifact. The denial of interception nonsynonymy and the disregard of term recurrence in logic link with many misconceptions about propositions, logical form, conventions, and metalanguages. Mathematics is distinct from logic: its truth is not syntactic; it is transmitted by synonym substitution; term recurrence has no essential role. The ‘=’ of mathematics is an objectual relation between numbers; the ‘=’ of logic marks a syntactic relation of coreferring terms. -/- . (shrink)

This paper is on Descartes’ account of modality and, in particular, his account of the necessity of the laws of nature. He famously argues that the necessity of the “eternal truths” of logic and mathematics depends on God’s will. Here I suggest he has the same view about the necessity of the laws of nature. Further, I argue, this is a plausible theory of laws. For philosophers often talk about something being nomologically or physically necessary because (...) of the laws of nature, but this necessity is thought to be metaphysically contingent. However, they struggle to explain how the laws could be genuinely necessary while being metaphysically contingent. The chief advantage of Descartes view, I argue, is that God’s will can plausibly explain both the necessity of the laws and the contingency of the laws. So, Descartes’ theistic account of laws provides a plausible explanation, perhaps the best explanation, of the contingent-necessity of laws of nature. (shrink)

In this dissertation on Hilary Putnam's philosophy, I investigate his development regarding meaning and necessity, in particular mathematical necessity. Putnam has been a leading American philosopher since the end of the 1950s, becoming famous in the 1960s within the school of analytic philosophy, associated in particular with the philosophy of science and the philosophy of language. Under the influence of W.V. Quine, Putnam challenged the logical positivism/empiricism that had become strong in America after World War II, with influential (...) exponents such as Rudolf Carnap and Hans Reichenbach. Putnam agreed with Quine that there are no absolute a priori truths. In particular, he was critical of the notion of truth by convention. Instead he developed a notion of relative a priori truth, that is, a notion of necessary truth with respect to a body of knowledge, or a conceptual scheme. Putnam's position on necessity has developed over the years and has always been connected to his important contributions to the philosophy of meaning. I study Hilary Putnam's development through an early phase of scientific realism, a middle phase of internal realism, and his later position of a natural or commonsense realism. I challenge some of Putnam’s ideas on mathematical necessity, although I have largely defended his views against some other contemporary major philosophers; for instance, I defend his conceptual relativism, his conceptual pluralism, as well as his analysis of the realism/anti-realism debate. (shrink)

In this paper I discuss Mark Steiner’s view of the contribution of mathematics to physics and take up some of the questions it raises. In particular, I take up the question of discovery and explore two aspects of this question – a metaphysical aspect and a related epistemic aspect. The metaphysical aspect concerns the formal structure of the physical world. Does the physical world have mathematical or formal features or constituents, and what is the nature of these constituents? The (...) related epistemic question concerns humans’ cognitive ability to reach the formal structure of the physical world. Among other things, I explore the interaction of mathematical and non-mathematical cognition in physical discovery. (shrink)

So-called ‘distinctively mathematical explanations’ (DMEs) are said to explain physical phenomena, not in terms of contingent causal laws, but rather in terms of mathematical necessities that constrain the physical system in question. Lange argues that the existence of four or more equilibrium positions of any double pendulum has a DME. Here we refute both Lange’s claim itself and a strengthened and extended version of the claim that would pertain to any n-tuple pendulum system on the ground that such explanations are (...) actually causal explanations in disguise and their associated modal conditionals are not general enough to explain the said features of such dynamical systems. We argue and show that if circumscribing the antecedent for a necessarily true conditional in such explanations involves making a causal analysis of the problem, then the resulting explanation is not distinctively mathematical or non-causal. Our argument generalises to other dynamical systems that may have purported DMEs analogous to the one proposed by Lange, and even to some other counterfactual accounts of non-causal explanation given by Reutlinger and Rice. (shrink)

Many philosophers are baffled by necessity. Humeans, in particular, are deeply disturbed by the idea of necessary laws of nature. In this paper I offer a systematic yet down to earth explanation of necessity and laws in terms of invariance. The type of invariance I employ for this purpose generalizes an invariance used in meta-logic. The main idea is that properties and relations in general have certain degrees of invariance, and some properties/relations have a stronger degree of invariance (...) than others. The degrees of invariance of highly-invariant properties are associated with high degrees of necessity of laws governing/describing these properties, and this explains the necessity of such laws both in logic and in science. This non-mysterious explanation has rich ramifications for both fields, including the formality of logic and mathematics, the apparent conflict between the contingency of science and the necessity of its laws, the difference between logical-mathematical, physical, and biological laws/principles, the abstract character of laws, the applicability of logic and mathematics to science, scientific realism, and logical-mathematical realism. (shrink)

This article had its beginning with Einstein's 1919 paper "Do gravitational fields play an essential role in the structure of elementary particles?" Together with General Relativity's statement that gravity is not a pull but is a push caused by the curvature of space-time, a hypothesis for Earth's ocean tides was developed that does not solely depend on the Sun and Moon as Kepler and Newton believed. It also borrows from Galileo. The breakup of planets and asteroids by white dwarfs, neutron (...) stars or black holes is popularly ascribed by today's science to tidal forces (gravitation emanating from the stellar body and having a greater effect on the near side of a planet/asteroid than the farthest side). Remembering Einstein's 1919 paper, it was apparent that my revised idea of tidal forces improves on current accounts because it views matter and mass as unified with space-time whose curvature is gravitation. Unification is a necessity for modern science's developing view of one united and entangled universe – expressed in the Unified Field Theory, the Theory of Everything, String theory and Loop Quantum Gravity. The writing of this article was also assisted by visualizing the gravitational fields forming space-time being themselves formed by a multitude of weak and presently undetectable gravitational waves. The final part of this article concludes that the section BITS AND TOPOLOGY will lead to the conclusions in ETERNAL LIFE, WORLD PEACE AND PHYSICS' UNIFICATION. The final part also compares cosmology to biological enzymes and biology's substrate of reacting "chemicals" - using virtual particles, hidden variables, gravitation, electromagnetism, electronics’ binary digits, plus topology’s Mobius strip and figure-8 Klein bottle. The product is mass - enzyme, substrate and product are all considered mathematical in nature. Also, gravitation and electromagnetism are united using logic and topology – showing there’s no need in this article for things like mathematical formalism, field equations or tensor calculus. (shrink)

The INBIOSA project brings together a group of experts across many disciplines who believe that science requires a revolutionary transformative step in order to address many of the vexing challenges presented by the world. It is INBIOSA’s purpose to enable the focused collaboration of an interdisciplinary community of original thinkers. This paper sets out the case for support for this effort. The focus of the transformative research program proposal is biology-centric. We admit that biology to date has been more fact-oriented (...) and less theoretical than physics. However, the key leverageable idea is that careful extension of the science of living systems can be more effectively applied to some of our most vexing modern problems than the prevailing scheme, derived from abstractions in physics. While these have some universal application and demonstrate computational advantages, they are not theoretically mandated for the living. A new set of mathematical abstractions derived from biology can now be similarly extended. This is made possible by leveraging new formal tools to understand abstraction and enable computability. [The latter has a much expanded meaning in our context from the one known and used in computer science and biology today, that is "by rote algorithmic means", since it is not known if a living system is computable in this sense (Mossio et al., 2009).] Two major challenges constitute the effort. The first challenge is to design an original general system of abstractions within the biological domain. The initial issue is descriptive leading to the explanatory. There has not yet been a serious formal examination of the abstractions of the biological domain. What is used today is an amalgam; much is inherited from physics (via the bridging abstractions of chemistry) and there are many new abstractions from advances in mathematics (incentivized by the need for more capable computational analyses). Interspersed are abstractions, concepts and underlying assumptions “native” to biology and distinct from the mechanical language of physics and computation as we know them. A pressing agenda should be to single out the most concrete and at the same time the most fundamental process-units in biology and to recruit them into the descriptive domain. Therefore, the first challenge is to build a coherent formal system of abstractions and operations that is truly native to living systems. Nothing will be thrown away, but many common methods will be philosophically recast, just as in physics relativity subsumed and reinterpreted Newtonian mechanics. -/- This step is required because we need a comprehensible, formal system to apply in many domains. Emphasis should be placed on the distinction between multi-perspective analysis and synthesis and on what could be the basic terms or tools needed. The second challenge is relatively simple: the actual application of this set of biology-centric ways and means to cross-disciplinary problems. In its early stages, this will seem to be a “new science”. This White Paper sets out the case of continuing support of Information and Communication Technology (ICT) for transformative research in biology and information processing centered on paradigm changes in the epistemological, ontological, mathematical and computational bases of the science of living systems. Today, curiously, living systems cannot be said to be anything more than dissipative structures organized internally by genetic information. There is not anything substantially different from abiotic systems other than the empirical nature of their robustness. We believe that there are other new and unique properties and patterns comprehensible at this bio-logical level. The report lays out a fundamental set of approaches to articulate these properties and patterns, and is composed as follows. -/- Sections 1 through 4 (preamble, introduction, motivation and major biomathematical problems) are incipient. Section 5 describes the issues affecting Integral Biomathics and Section 6 -- the aspects of the Grand Challenge we face with this project. Section 7 contemplates the effort to formalize a General Theory of Living Systems (GTLS) from what we have today. The goal is to have a formal system, equivalent to that which exists in the physics community. Here we define how to perceive the role of time in biology. Section 8 describes the initial efforts to apply this general theory of living systems in many domains, with special emphasis on crossdisciplinary problems and multiple domains spanning both “hard” and “soft” sciences. The expected result is a coherent collection of integrated mathematical techniques. Section 9 discusses the first two test cases, project proposals, of our approach. They are designed to demonstrate the ability of our approach to address “wicked problems” which span across physics, chemistry, biology, societies and societal dynamics. The solutions require integrated measurable results at multiple levels known as “grand challenges” to existing methods. Finally, Section 10 adheres to an appeal for action, advocating the necessity for further long-term support of the INBIOSA program. -/- The report is concluded with preliminary non-exclusive list of challenging research themes to address, as well as required administrative actions. The efforts described in the ten sections of this White Paper will proceed concurrently. Collectively, they describe a program that can be managed and measured as it progresses. (shrink)

Gravitation is interpreted to be an “ontomathematical” force or interaction rather than an only physical one. That approach restores Newton’s original design of universal gravitation in the framework of “The Mathematical Principles of Natural Philosophy”, which allows for Einstein’s special and general relativity to be also reinterpreted ontomathematically. The entanglement theory of quantum gravitation is inherently involved also ontomathematically by virtue of the consideration of the qubit Hilbert space after entanglement as the Fourier counterpart of pseudo-Riemannian space. Gravitation can be (...) also interpreted as purely mathematical or logical “force” or “interaction” as a corollary from its ontomathematical (rather than physical) realization. The ontomathematical approach to gravitation is implicit in general relativity equating it to operators in pseudo-Riemannian space obeying the Einstein field equation and also well-known by the “geometrization of physics”. Quantum mechanics shares the same by the separable complex Hilbert space and defining “physical quantity” by the Hermitian operators on it. One can interpret special Minkowski space involved by special relativity and the qubit Hilbert space of quantum information as Fourier counterparts immediately noticing that general relativity means gravitation as the Fourier counterpart of non-Hermitian operators implying non-unitarity and the violation of energy conservation and thus destroying Pauli’s particle paradigm. Since the Standard model obeys it, this explains the impossibility of “quantum gravitation” in any framework conservatively generalizing the Standard model so that it would include gravitation along with electromagnetic, weak, and strong interactions. Einstein’s geometrization of gravitation can be continued into a purely mathematical theory of it following Euclid’s realization for geometry to be exhaustively built in a deductive and axiomatic way as well as Riemann’s parametrization of all the class of Euclidean and non-Euclidean geometries by “space curvature”, then being generalized to Minkowski space as the operators on pseudo-Riemannian space as the Einstein field equation means gravitation. The transition from mathematical gravitation to logical one can rely on the historical lesson of the pair of Lobachevski’s and Riemann’s approaches now “reversely”, i.e., from the latter to the former. Logical gravitation is linkable to Hegel’s dialectical logic and ontological dialectics abandoning their interpretations as a new zero logic substituting classical propionyl logic. The approach of ontomathematics generalizing that of ontology, traceable even to Aristotle’s reformation of Plato’s doctrine, needs Hegel’s doctrine to be formalized as a first-order logic naturally containing Boolean algebra, isomorphic to both classical propositional logic and set theory being the class of all first-order logics, as a sub-logic along with Peano arithmetic as another sub-logic. The first-order logic at issue is called Hilbert arithmetic and elaborated in detail in other papers. It allows for both self-foundation of mathematics to be internally proved as complete and furthermore, quantum mechanics reinterpreted as quantum information to be included by the qubit Hilbert space interpretable in turn as a dual and physical counterpart of Hilbert arithmetic in a narrow sense, that is, both counterparts constitute Hilbert arithmetic in a wide sense, being mathematical and physical simultaneously and thus overcoming the Cartesian dualism of “body” gapped from “mind” by an abyss. Then, the proper philosophical interpretation of gravitation to be the fundamental ontomathematical force or interaction overcomes the ridiculous belief of the Big Bang wrongly alleged to be a scientific theory. Ontomathematical gravitation suggests an omnipresent and omnitemporal medium of “God’s” creation “ex nihilo” following only the natural necessity of quantum-information conservation particularly and locally manifested as energy conservation. (shrink)

Properties and relations in general have a certain degree of invariance, and some types of properties/relations have a stronger degree of invariance than others. In this paper I will show how the degrees of invariance of different types of properties are associated with, and explain, the modal force of the laws governing them. This explains differences in the modal force of laws/principles of different disciplines, starting with logic and mathematics and proceeding to physics and biology.

In a lot of domains in metaphysics the tacit assumption has been that whichever metaphysical principles turn out to be true, these will be necessarily true. Let us call necessitarianism about some domain the thesis that the right metaphysics of that domain is necessary. Necessitarianism has flourished. In the philosophy of maths we find it held that if mathematical objects exist, then they do of necessity. Mathematical Platonists affirm the necessary existence of mathematical objects (see for instance Hale and (...) Wright 1992 and 1994; Wright 1983 and 1988; Schiffer 1996; Resnik 1997; Shapiro 1997 and Zalta 1988) while mathematical nominalists, usually in the form of fictionalists, hold that necessarily such objects fail to exist (see for instance Balaguer 1996 and 1998; Rosen 2001 and Yablo 2005). In metaphysics more generally, until recently it was more or less assumed that whatever the right account of composition—the account of under what conditions some xs compose a y—that account will be necessarily true (for a discussion of theories of composition see Simons 1987 and van Inwagen 1987 and 1990; the modal status of the composition relation is explicitly addressed in Schaffer 2007; Parsons 2006 and Cameron 2007). Similarly, it has generally been assumed that whatever the right account is of the nature of properties, whether they be universals, tropes, or whether nominalism is true, that account will be necessarily true (though see Rosen 2006 for a recent suggestion to the contrary). In considering theories of persistence it has been widely held that whether objects endure or perdure through time is a matter of necessity (Sider 2001; though see Lewis 1999 p227 who defends contingent perdurantism). And with respect to theories of time it is frequently held that whichever of the A- or B-theory is true is necessarily true. A-theorists often argue that there is time in a world only if the A-theory is true at that world (see for instance McTaggart 1903; Markosian 2004; Bigelow 1996; Craig 2001) while B-theorists often argue that the A-theory is internally inconsistent (Smart 1987; Mellor 1998; Savitt 2000 and Le Poidevin 1991). Once again, we find a few recent contingentist dissenters. Bourne (2006) suggests that it is a contingent feature of time that it is tensed, and thus that the A-theory is contingently true. Worlds in which there exist only B-theoretic properties are worlds with time, it is just that time in those worlds time is radically different to the way it is actually. Other defenders of the B-theory, though not expressly contingentists, do offer arguments against versions of the A-theory that try to show that such A-theories theories are inconsistent with the actual laws of nature (see for instance Saunders 2002 and Callender 2000); these arguments, at least, leave room for the possibility that the A-theories in question are contingently false (at least on the assumption that the laws of nature are themselves contingent, an assumption that not everyone accepts). Despite some notable exceptions, necessitarianism has flourished in many, if not most, domains in metaphysics. One such exception is Lewis’ famous defence of Humean supervenience as a contingent claim about our world. Lewis does not argue that necessarily, the supervenience base for all matters of fact in a world is nothing but a vast mosaic of local matters of particular fact. Rather, he thinks that we have reason to think that our world is one in which Humean supervenience holds (see Lewis 1986 p9-10 and 1994). Another exception to the necessitarian orthodoxy is to be found in the lively debate about the modal status of the laws of nature. Here, if anything, contingentism has been the dominating force, with it generally being held that there are possible worlds in which different laws of nature hold (this view is defended by, among others, Lewis 1986 and 2010; Schaffer 2005 and Sidelle 2002). Necessitarian dissenters hold that the laws of nature are necessary, frequently because they think it is necessary that fundamental properties have the causal or nomic profiles they do (see for instance Shoemaker 1980 and 1988; Swoyer 1982; Bird 1995; Ellis and Lierse 1994). Nevertheless, when it comes to thinking about the nature of the laws themselves, the necessitarian presumption is back on firm footing. Though there is disagreement about whether the laws are generalisations that feature in the most virtuous true axiomatisation of all the particular matters of fact (often known as the Humean view of laws and defended by Ramsey 1978; Lewis 1986 and Beebee 2000) or whether laws are relations of necessity that hold between universals (a view defended by Armstrong 1983; Dretske 1977; Tooley 1977 and Carroll 1990) no one has seriously suggested that it might be a contingent matter which of these is the right account of laws. The necessitarian orthodoxy is not surprising since metaphysics is largely an a priori process. While a priori reasoning may be used to determine whether a proposition is necessary or contingent, it is not well placed (in the absence of a posteriori evidence) to determine whether a contingent proposition is actually true or false. Since metaphysicians aim to tell us which principles are true in which worlds, on the face of it the discovery that metaphysical principles are contingent seems to make part of the task of metaphysics epistemically intractable. In what follows I consider two reasons one might end up embracing contingentism and whether this would lead one into epistemic difficulty. The following section considers a route to contingent metaphysical truths that proceeds via a combination of conceptual necessities and empirical discoveries. Section 3 considers whether there might be synthetic contingent metaphysical truths, and the final section raises the question of whether if there were such truths we would be well placed to come to know them. (shrink)

The INBIOSA project brings together a group of experts across many disciplines who believe that science requires a revolutionary transformative step in order to address many of the vexing challenges presented by the world. It is INBIOSA’s purpose to enable the focused collaboration of an interdisciplinary community of original thinkers. This paper sets out the case for support for this effort. The focus of the transformative research program proposal is biology-centric. We admit that biology to date has been more fact-oriented (...) and less theoretical than physics. However, the key leverageable idea is that careful extension of the science of living systems can be more effectively applied to some of our most vexing modern problems than the prevailing scheme, derived from abstractions in physics. While these have some universal application and demonstrate computational advantages, they are not theoretically mandated for the living. A new set of mathematical abstractions derived from biology can now be similarly extended. This is made possible by leveraging new formal tools to understand abstraction and enable computability. [The latter has a much expanded meaning in our context from the one known and used in computer science and biology today, that is "by rote algorithmic means", since it is not known if a living system is computable in this sense (Mossio et al., 2009).] Two major challenges constitute the effort. The first challenge is to design an original general system of abstractions within the biological domain. The initial issue is descriptive leading to the explanatory. There has not yet been a serious formal examination of the abstractions of the biological domain. What is used today is an amalgam; much is inherited from physics (via the bridging abstractions of chemistry) and there are many new abstractions from advances in mathematics (incentivized by the need for more capable computational analyses). Interspersed are abstractions, concepts and underlying assumptions “native” to biology and distinct from the mechanical language of physics and computation as we know them. A pressing agenda should be to single out the most concrete and at the same time the most fundamental process-units in biology and to recruit them into the descriptive domain. Therefore, the first challenge is to build a coherent formal system of abstractions and operations that is truly native to living systems. Nothing will be thrown away, but many common methods will be philosophically recast, just as in physics relativity subsumed and reinterpreted Newtonian mechanics. -/- This step is required because we need a comprehensible, formal system to apply in many domains. Emphasis should be placed on the distinction between multi-perspective analysis and synthesis and on what could be the basic terms or tools needed. The second challenge is relatively simple: the actual application of this set of biology-centric ways and means to cross-disciplinary problems. In its early stages, this will seem to be a “new science”. This White Paper sets out the case of continuing support of Information and Communication Technology (ICT) for transformative research in biology and information processing centered on paradigm changes in the epistemological, ontological, mathematical and computational bases of the science of living systems. Today, curiously, living systems cannot be said to be anything more than dissipative structures organized internally by genetic information. There is not anything substantially different from abiotic systems other than the empirical nature of their robustness. We believe that there are other new and unique properties and patterns comprehensible at this bio-logical level. The report lays out a fundamental set of approaches to articulate these properties and patterns, and is composed as follows. -/- Sections 1 through 4 (preamble, introduction, motivation and major biomathematical problems) are incipient. Section 5 describes the issues affecting Integral Biomathics and Section 6 -- the aspects of the Grand Challenge we face with this project. Section 7 contemplates the effort to formalize a General Theory of Living Systems (GTLS) from what we have today. The goal is to have a formal system, equivalent to that which exists in the physics community. Here we define how to perceive the role of time in biology. Section 8 describes the initial efforts to apply this general theory of living systems in many domains, with special emphasis on crossdisciplinary problems and multiple domains spanning both “hard” and “soft” sciences. The expected result is a coherent collection of integrated mathematical techniques. Section 9 discusses the first two test cases, project proposals, of our approach. They are designed to demonstrate the ability of our approach to address “wicked problems” which span across physics, chemistry, biology, societies and societal dynamics. The solutions require integrated measurable results at multiple levels known as “grand challenges” to existing methods. Finally, Section 10 adheres to an appeal for action, advocating the necessity for further long-term support of the INBIOSA program. -/- The report is concluded with preliminary non-exclusive list of challenging research themes to address, as well as required administrative actions. The efforts described in the ten sections of this White Paper will proceed concurrently. Collectively, they describe a program that can be managed and measured as it progresses. (shrink)

This paper aims to show that—and how—Plato’s notion of the receptacle in the Timaeus provides the conditions for developing a mathematical as well as a physical space without itself being space. In response to the debate whether Plato’s receptacle is a conception of space or of matter, I suggest employing criteria from topology and the theory of metric spaces as the most basic ones available. I show that the receptacle fulfils its main task–allowing the elements qua images of the Forms (...) to exist as sensible things by being that in which the elements appear, change and move–in virtue of being pure continuity. All further qualifications required for a full notion of space are derived solely from the content of the receptacle. (shrink)

Kant argued that Euclidean geometry is synthesized on the basis of an a priori intuition of space. This proposal inspired much behavioral research probing whether spatial navigation in humans and animals conforms to the predictions of Euclidean geometry. However, Euclidean geometry also includes concepts that transcend the perceptible, such as objects that are infinitely small or infinitely large, or statements of necessity and impossibility. We tested the hypothesis that certain aspects of nonperceptible Euclidian geometry map onto intuitions of space (...) that are present in all humans, even in the absence of formal mathematical education. Our tests probed intuitions of points, lines, and surfaces in participants from an indigene group in the Amazon, the Mundurucu, as well as adults and age-matched children controls from the United States and France and younger US children without education in geometry. The responses of Mundurucu adults and children converged with that of mathematically educated adults and children and revealed an intuitive understanding of essential properties of Euclidean geometry. For instance, on a surface described to them as perfectly planar, the Mundurucu's estimations of the internal angles of triangles added up to ∼180 degrees, and when asked explicitly, they stated that there exists one single parallel line to any given line through a given point. These intuitions were also partially in place in the group of younger US participants. We conclude that, during childhood, humans develop geometrical intuitions that spontaneously accord with the principles of Euclidean geometry, even in the absence of training in mathematics. (shrink)

In Pantsar, an outline for an empirically feasible epistemological theory of arithmetic is presented. According to that theory, arithmetical knowledge is based on biological primitives but in the resulting empirical context develops an essentially a priori character. Such contextual a priori theory of arithmetical knowledge can explain two of the three characteristics that are usually associated with mathematical knowledge: that it appears to be a priori and objective. In this paper it is argued that it can also explain the third (...) one: why arithmetical knowledge appears to be necessary. A Kripkean analysis of necessity is used as an example to show that a proper analysis of the relevant possible worlds can explain arithmetical necessity in a sufficiently strong form. (shrink)

The essential idea of Leibniz’s Theodicy was little understood in his time but has become one of the organizing themes of modern mathematics. There are many phenomena that are possible locally but for purely mathematical reasons impossible globally. For example, it is possible to build a spiral staircase that is rising at any given point, but it is impossible to build one that is rising at all points and comes back to where it started. The necessity is mathematically (...) provable, so not subject to exception by divine power. Leibniz’s Theodicy argues that God could improve the universe locally in many ways, but not globally. This paper defends Leibniz, giving positive reasons for believing that there are so many necessary interconnections between goods and evils that God is faced with a choice like the classic Trolley case, where all of the scenarios that could be chosen upfront contain evils, but some more than others. Local changes for the better seem easy to imagine, but a proper understanding of global constraints undermines the initial impression that they can be done without global cost. The paper concludes by explaining how the context of the Leibnizian argument makes it reasonable to pursue the issue of whether there are no worlds better than this one. (shrink)

When this article was first planned, writing was going to be exclusively about two things - the origin of life and human evolution. But it turned out to be out of the question for the author to restrict himself to these biological and anthropological topics. A proper understanding of them required answering questions like “What is the nature of the universe – the home of life – and how did it originate?”, “How can time travel be removed from fantasy and (...) science fiction, to be made scientific and practical?”, and “How can the proposed young age of genus Homo be made to actually be reasonable – when simply stating it would be solid ground for instant rejection and dismissal?” The result is that the article also talks about subjects like Artificial Intelligence, General Relativity, and cosmology. -/- From where did life originate? God? Evolution? Panspermia? If the tendency of humans and scientists to regard undiscovered science as pseudoscience can be overcome, Einstein gave another alternative to consider when he introduced General Relativity. Time isn’t linear – progressing in a straight line from past to present to future. That assumption ignores Relativity which states that space AND TIME are curved. Where did life and the genetic code come from? Can the answer build AI? -/- The first question can be answered by the section of this article titled SETI, Evolution, and Time which says life (possibly multicellular and intelligent) and the genetic code came from humans acquiring knowledge of these things over the centuries, then applying that knowledge – via terraforming, accumulation of raw materials like amino acids and nucleic acids, genetic engineering - to a time in the past when life didn’t exist. From that origin, life evolved through innumerable mutations and adaptations, with humans once again acquiring knowledge of it in cyclic (nonlinear) time. -/- The second question is answered by saying artificial intelligence (AI) as the product of life is only half of the equation. The other half refers to Relativity’s curved space-time and violation of the notion that time always travels from past to future. We have always lived in an artificially intelligent, non-probabilistic universe where everything in time and space is connected into one thing by quantum entanglement – making the brain and genes products of binary-digit activity or artificial intelligence (life is not merely dependent on biology’s “lock and key” mechanisms but also possesses AI). -/- The earliest documented representative of the genus Homo is Homo habilis, which evolved around 2.8 million years ago. Scientists used to believe there was a straight line from H. habilis to us, Homo sapiens. This article will use the “advanced” waves loved by Physics Nobel laureate Richard Feynman, view the history of science through the lens of Conic Sections applied to Relativity’s curved space-time, and incorporate the necessity of so-called imaginary time * – popularized by Prof. Stephen Hawking. While the evolutionary proposals are more in agreement with this early straight line than with modern theories, Albert Einstein’s General Relativity is used to transform the straight line into a curved line, ultimately concluding that Homo habilis (H. habilis) originated only (and unbelievably, as far as today’s science and technology is concerned) ~250,000 years ago. Other branches and dead ends of Homo – e.g. Neanderthals – are the result of mutations and adaptations, with the resultant modifications to anatomy and physiology. The surprisingly young age of H. habilis allows nearly 200,000 years for habilis, or one of its descendants, to reach Australia … if this country’s indigenous Aboriginal population did, as claimed, reach this “island continent” 60,000 years ago. -/- * The ultraviolet catastrophe, also called the Rayleigh–Jeans catastrophe, is a failure of classical physics to predict observed phenomena: it can be shown that a blackbody - a hypothetical perfect absorber and radiator of energy - would release an infinite amount of energy, contradicting the principles of conservation of energy and indicating that a new model for the behaviour of blackbodies was needed. At the start of the 20th century, physicist Max Planck derived the correct solution by making some strange (for the time) assumptions. In particular, Planck assumed that electromagnetic radiation can only be emitted or absorbed in discrete packets, called quanta. Albert Einstein postulated that Planck's quanta were real physical particles (what we now call photons), not just a mathematical fiction. From there, Einstein developed his explanation of the photoelectric effect (when quanta or photons of light shine on certain metals, electrons are released and can form an electric current). So it appears entirely possible that another supposed mathematical trickery (imaginary time and the y-axis of Wick rotation) will find practical application in the future. -/- The article includes mathematical references to cosmology (spoiler alert – you’ll read about things like Vector-Tensor-Scalar Geometry, topology, the “eternal present”, Einstein’s Unified Field, the inverse-square law, and there being no Big Bang and no multiverse - but there will also be no equations). -/- The other subheadings in this essay are – -/- NONLINEAR TIME AND ELECTRICAL ENGINEERING (about a 2009 electrical engineering experiment at America’s Yale University, and cosmic wormholes) -/- BITS AND TOPOLOGY (base-2 maths aka Binary digiTS, Mobius strips, and figure-8 Klein bottles) -/- WICK ROTATION, CAUSALITY, AND UNITING TIME (do past, present and future co-exist in an “eternal present”?) -/- DIGITAL BRAIN, DIGITAL UNIVERSE (if the brain and the universe are ultimately composed of binary digits, we'll someday be able to do the same things with the brain and universe that we now do with computers) -/- PROPOSAL: HUMAN AND ANIMAL INSTINCTS ARE THE RESULT OF THE UNIVERSE BEING UNIFIED BY BINARY DIGITS (AND TOPOLOGY) (If everything in the universe is ultimately composed of electronic BITS, then the universe must possess Artificial Intelligence - some prefer the term Cosmic Consciousness) -/- INFORMATION THEORY CONQUERS A RED GIANT (preserving Earth by keeping the Sun near today’s level of activity forever). 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Charles Peirce's diagrammatic logic — the Existential Graphs — is presented as a tool for illuminating how we know necessity, in answer to Benacerraf's famous challenge that most ‘semantics for mathematics’ do not ‘fit an acceptable epistemology’. It is suggested that necessary reasoning is in essence a recognition that a certain structure has the particular structure that it has. This means that, contra Hume and his contemporary heirs, necessity is observable. One just needs to pay attention, not (...) merely to individual things but to how those things are related in larger structures, certain aspects of which relations force certain other aspects to be a certain way. (shrink)

Although the invariance criterion of logicality first emerged as a criterion of a purely mathematical interest, it has developed into a criterion of considerable linguistic and philosophical interest. In this paper I compare two different perspectives on this criterion. The first is the perspective of natural language. Here, the invariance criterion is measured by its success in capturing our linguistic intuitions about logicality and explaining our logical behavior in natural-linguistic settings. The second perspective is more theoretical. Here, the invariance criterion (...) is used as a tool for developing a theoretical foundation of logic, focused on a critical examination, explanation, and justification of its veridicality and modal force. (shrink)

The mental rotation ability is an essential spatial reasoning skill in human cognition and has proven to be an essential predictor of mathematical and STEM skills, critical and computational thinking. Despite its importance, little is known about when and how mental rotation processes are activated in games explicitly targeting spatial reasoning tasks. In particular, the relationship between spatial abilities and TetrisTM has been analysed several times in the literature. However, these analyses have shown contrasting results between the effectiveness of Tetris-based (...) training activities to improve mental rotation skills. In this work, we studied whether, and under what conditions, such ability is used in the TetrisTM game by explicitly modelling mental rotation via an ACT-R based cognitive model controlling a virtual agent. The obtained results show meaningful insights into the activation of mental rotation during game dynamics. The study suggests the necessity to adapt game dynamics in order to force the activation of this process and, therefore, can be of inspiration to design learning activities based on TetrisTM or re-design the game itself to improve its educational effectiveness. (shrink)

Corcoran, J. 2005. Counterexamples and proexamples. Bulletin of Symbolic Logic 11(2005) 460. -/- John Corcoran, Counterexamples and Proexamples. Philosophy, University at Buffalo, Buffalo, NY 14260-4150 E-mail: corcoran@buffalo.edu Every perfect number that is not even is a counterexample for the universal proposition that every perfect number is even. Conversely, every counterexample for the proposition “every perfect number is even” is a perfect number that is not even. Every perfect number that is odd is a proexample for the existential proposition that some (...) perfect number is odd. Conversely, every proexample for the proposition “some perfect number is odd” is a perfect number that is odd. As trivial these remarks may seem, they can not be taken for granted, even in mathematical and logical texts designed to introduce their respective subjects. One well-reviewed book on counterexamples in analysis says that in order to demonstrate that a universal proposition is false it is necessary and sufficient to construct a counterexample. It is easy to see that it is not necessary to construct a counterexample to demonstrate that the proposition “every true proposition is known to be true” is false–necessity fails. Moreover the mere construction of an object that happens to be a counterexample does not by itself demonstrate that it is a counterexample–sufficiency fails. In order to demonstrate that a universal proposition is false it is neither necessary nor sufficient to construct a counterexample. Likewise, of course, in order to demonstrate that an existential proposition is true it is neither necessary nor sufficient to construct a proexample. This article defines the above relational concepts of counterexample and of proexample, it discusses their surprising history and philosophy, it gives many examples of uses of these and related concepts in the literature and it discusses some of the many errors that have been made as a result of overlooking the challenging subtlety of the proper use of these two basic and indispensable concepts. (shrink)

Newton frequently characterized his methodology as distinctive and capable of achieving greater evidential support than that of his contemporaries, due to its mathematical character. Newton's pronouncements reflect a striking position regarding the role of mathematics in natural philosophy. We can give an initial characterization of his position by considering two questions central to seventeenth century debates about the applicability of mathematics. First, how are we to understand the distinctive universality and necessity of mathematical reasoning? One common way (...) to preserve the demonstrative character of mathematics was to restrict its domain, as far as possible, to pure abstractions. The subject matter of mathematics is then taken to be abstracted from the changeable natural world, consisting of quantity and magnitude themselves rather than the objects bearing quantifiable properties. Yet restricting the domain in this way leaves it difficult to see how mathematics relates to natural phenomena. How could the book of nature be written in a language of pure abstractions? Second, what is the proper role of mathematical reasoning in natural philosophy? Many followed Aristotle in consigning mathematics to a subordinate role. Mathematics could not fulfill the main aim of natural philosophy, namely to provide demonstrations reflecting the essences of things and nature's causal order. A related concern was also pressing for the mechanical philosophers: a merely mathematical demonstration fails to provide an intelligible mechanical explanation. The very title of Newton's masterpiece, Philosophiae Naturalis Principia Mathematica, would have been extremely perplexing: how could natural philosophy be based on mathematical principles? My aim is to articulate Newton's position regarding the mathematization of nature in response to these two concerns. (shrink)

⦿ In my dissertation I introduce, motivate and take the first steps in the implementation of, the project of naturalising modal metaphysics: the transformation of the field into a chapter of the philosophy of science rather than speculative, autonomous metaphysics. -/- ⦿ In the introduction, I explain the concept of naturalisation that I apply throughout the dissertation, which I argue to be an improvement on Ladyman and Ross' proposal for naturalised metaphysics. I also object to Williamson's proposal that modal metaphysics (...) --- or some view in the area --- is already a quasi-scientific discipline. -/- ⦿ Recently, some philosophers have argued that the notion of metaphysical modality is as ill defined as to be of little theoretical utility. In the second chapter I intend to contribute to such skepticism. First, I observe that each of the proposed marks of the concept, except for factivity, is highly controversial; thus, its logical structure is deeply obscure. With the failure of the "first principles" approach, I examine the paradigmatic intended applications of the concept, and argue that each makes it a device for a very specific and controversial project: a device, therefore, for which a naturalist will find no use for. I conclude that there is no well-defined or theoretically useful notion of objective necessity other than logical or physical necessity, and I suggest that naturalising modal metaphysics can provide more stable methodological foundations. -/- ⦿ In the third chapter I answer a possible objection against the in-principle viability of the project: that the concept of metaphysical modality cannot be understood through the philosophical analysis of any scientific theory, since metaphysical necessity "transcends'' natural necessity, and science only deals with the latter. I argue that the most important arguments for this transcendence thesis fail or face problems that, as of today, remain unsolved. -/- ⦿ Call the idea that science doesn't need modality, "demodalism''. Demodalism is a first step in a naturalistic argument for modal antirealism. In the fourth chapter I examine six versions of demodalism to explain why a family of formalisms, that I call "spaces of possibility'', are (i) used in a quasi-ubiquitous way in mathematised sciences (I provide examples from theoretical computer science to microeconomics), (ii) scientifically interpreted in modal terms, and (iii) used for at least six important tasks: (1) defining laws and theories; (2) defining important concepts from different sciences (I give several examples); (3) making essential classifications; (4) providing different types of explanations; (5) providing the connection between theory and statistics, and (6) understanding the transition between a theory and its successor (as is the case with quantisation). -/- ⦿ In fifth chapter I propose and defend a naturalised modal ontology. This is a realism about modal structure: my realism about constraints. The modal structure of a system are the relationships between its possible states and between its possible states and those of other systems. It is given by the plurality of restrictions to which said system is subject. A constraint is a factor that explains the impossibility of a class of states; I explain this concept further. First, I defend my point of view by rejecting some of its main rivals: constructive empiricism, Humean conventionalism, and wave function realism, as they fail to make sense of quantum chaos. This is because the field requires the notion of objective modal structure, and the mentioned views have trouble explaining the modal facts of quantum dynamics. Then, I argue that constraint realism supersedes these views in the context of Bohm's standard theory and mechanics, and underpins the study of quantum chaos. Finally, I consider and reject two possible problems for my point of view. -/- ⦿ A central concern of modal metaphysicians has been to understand the logical system that best characterises necessity. In the sixth chapter I intend to recover the logical project applied to my naturalistic modal metaphysics. Scientists and philosophers of science accept different degrees of physical necessity, ranging from purely mathematically necessary facts that restrict physical behaviour, to kinetic principles, to particular dynamical constraints. I argue that this motivates a multimodal approach to modal logic, and that the time dependence of dynamics motivates a logic of historical necessity. I propose multimodal propositional (classical) logics for Bohmian mechanics and the Everettian theory of many divergent worlds, and I close with a criticism of Williamson's approach to the logic of state spaces of dynamic systems. (shrink)

It is generally agreed that using lethal or otherwise serious force in self-defense is justified only when three conditions are satisfied: first, there are some grounds for the defender to give priority to his own interests over those of the attacker (whether because the attacker has lost the protection of his right to life, for example, or because of the defender’s prerogative to prefer himself to others); second, the harm used is proportionate to the threat thereby averted; third, the harm (...) is necessary to avert that threat. The first and second conditions have been exhaustively discussed, but the third has been oddly neglected. Meanwhile a prominent school of thought has arisen, in the ethics of war, which seeks to ground the justification of killing in war in principles of individual self-defense. They too have failed to offer any substantive analysis of necessity, while at the same time appealing to it when it suits them to do so. In this paper, I attempt a detailed analysis of the necessity constraint on defensive force, and explore the implications of that analysis for the attempt to transpose principles of individual self-defense into the context of warfare. (shrink)

The literature on mathematics suggests that intuition plays a role in it as a ground of belief. This article explores the nature of intuition as it occurs in mathematical thinking. Section 1 suggests that intuitions should be understood by analogy with perceptions. Section 2 explains what fleshing out such an analogy requires. Section 3 discusses Kantian ways of fleshing it out. Section 4 discusses Platonist ways of fleshing it out. Section 5 sketches a proposal for resolving the main problem (...) facing Platonists—the problem of explaining how our experiences make contact with mathematical reality. (shrink)

Much mainstream analytic epistemology is built around a sceptical treatment of modality which descends from Hume. The roots of this scepticism are argued to lie in Hume’s (nominalist) theory of perception, which is excavated, studied and compared with the very different (realist) theory of perception developed by Peirce. It is argued that Peirce’s theory not only enables a considerably more nuanced and effective epistemology, it also (unlike Hume’s theory) does justice to what happens when we appreciate a proof in (...) class='Hi'>mathematics. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Newton's Metaphysics: Essays by Eric SchliesserMarius StanEric Schliesser. Newton's Metaphysics: Essays. Oxford: Oxford University Press, 2021. Pp. 328. Hardback, $99.90.Newton owes his high regard to the quantitative science he left us, but his overall picture of the world had some robustly metaphysical threads woven in as well. Posthumous judgment about the value of these threads has varied wildly. Christian Wolff thought him a metaphysical rustic, as did Hans (...) Reichenbach some two centuries later ("Die Bewegungslehre bei Newton, Leibniz und Huygens," Kant-Studien 29 [1924]: 416–38). In the 1960s, the tide would turn, as Howard Stein and James Edward McGuire separately began to show that Newton's metaphysics was not just sophisticated, but often more compelling than its early modern alternatives (see respectively "Newtonian Spacetime," Texas Quarterly 10 [1967]: 174–200; and Tradition and Innovation: Newton's Metaphysics of Nature [Dordrecht: Kluwer Academic Publishers, 1995]). Eric Schliesser's book unfolds in that same register of appreciative, high scholarship, and it attests to the enduring attraction of its subject.The book grew out of previous discrete papers, to which he added a few more for this occasion. Accordingly, it does not come with a master argument. Rather, it is an in-depth exploration of some key metaphysical themes in Newton. As such, it befits its subject figure, who reflected on themes and concepts while stopping short of working out systems.The first major theme is Newton and Spinozism. The latter does not denote Spinoza's metaphysics. In fact, Schliesser explains, Newton shared with Spinoza some weighty commitments, for example, to space being actually infinite, and to substance monism: "for Newton, there is strictly speaking only one genuine substance," namely, God (33). Rather, "Spinozism" is an interpreter's category for a bundle of three theses: identifying God and nature; denying final causation in the physical world; and the assumption of "blind" metaphysical necessity. Some counted Hobbes and John Toland as Spinozist in this sense, and even Epicurus, avant la lettre. Newton and his followers argued vehemently against this package, as Schliesser shows in chapters 4, 5, and 8. Their chief complaint was that Spinozism is unable to account for the "origin of motion," for a certain type of order, and for the stability of cosmological structure—and to do so in a way that "meets the standards of Newtonian mechanics" (122). This interpretive lens allows Schliesser to draw Kant in, by reading his youthful Theory of Heaven as a possible Spinozist reply to their objections (chapter 3). [End Page 157]A second theme is the metaphysics of mechanics, where Schliesser weaves together several strands. One is the ontology of gravity, for which he offers a novel construal: for Newton, gravity counts as real, but in a qualified sense. Namely, it is not essential and not intrinsic: "even after creation, a lonely partless particle of matter in the universe would not be said to gravitate." And gravity is relational: a "shared quality" of two or more bits of matter, which obtains in virtue of their sharing a nature (19). Another strand is Newton's ontology of time—really, a subtle, difficult topic long neglected. Famously, Newton in Principia defends absolute, true, and mathematical time. The question is what these three qualifiers denote, and whether Newton thought they were synonymous. Schliesser in chapter 7 argues for the provocative view that "absolute" and "true" time are not identical concepts; rather, they pick out different entities. These entities share a common structure, namely, metric and topological. That makes them species of "mathematical" time (179). But, Schliesser contends, Newton has unequal warrant for his two concepts of time.Yet another theme is formal causation. It comes to the fore in chapter 5, which grapples with Newton's opaque idea that space is an "emanative effect." Schliesser explains that here, "emanation picks out a species of formal causation" (147), but in a novel, early modern sense that comes from Bacon, not Aristotle. If space has a formal cause, then so has its twin brother, time, the topic of chapter 7. God is their common formal cause. Then in chapter 6 (coauthored with Zvi Biener) Schliesser adds that, for Newton, laws... (shrink)

This article will consider imagination in mathematics from a historical point of view, noting the key moments in its conception during the ancient, modern and contemporary eras.

The article considers the problem of ensuring the economic security of trade enterprises by forming an optimal pricing policy. The methodology of formation the minimum and maximum selling prices of trade enterprise, maintenance of margin of economic security, which is based on research of turnover costs and working capital of trade enterprise is offered. Based on statistical data of trade enterprises, the types of prices by product range are determined, which form a stable economic situation and ensure economic security of (...) trade enterprise. The necessity of forming the optimal price policy and selling prices of trade enterprises with the use of Pareto efficiency tools and construction of single-criteria and multi-criteria tasks, indicators of quality of commodity resources of trade enterprises is substantiated. Using the Excel software processor and the «Regression» function, economic-mathematical models of optimal prices for the product range of trade enterprises are built. The results testified to the effectiveness of the proposed model of the optimal pricing policy of trade enterprises, as the obtained values are within the minimum selling prices, which provides economic security of trade enterprises. The process of forming the pricing policy of trade enterprises to ensure economic security is proposed, which is based on the principles of pricing, appropriate methodological tools and monitoring of market environment factors. Organizational, economic, legal, social and market (marketing) mechanisms for the formation of effective pricing policy of trade enterprises aimed at ensuring economic security are identified. This study is practically interesting for personnel of trade enterprises, regardless of organizationallegal forms of ownership and activities, and theoretically – for researchers dealing with pricing. (shrink)

Logic has pride of place in mathematics and its 20th century offshoot, computer science. Modern symbolic logic was developed, in part, as a way to provide a formal framework for mathematics: Frege, Peano, Whitehead and Russell, as well as Hilbert developed systems of logic to formalize mathematics. These systems were meant to serve either as themselves foundational, or at least as formal analogs of mathematical reasoning amenable to mathematical study, e.g., in Hilbert’s consistency program. Similar efforts continue, (...) but have been expanded by the development of sophisticated methods to study the properties of such systems using proof and model theory. In parallel with this evolution of logical formalisms as tools for articulating mathematical theories (broadly speaking), much progress has been made in the quest for a mechanization of logical inference and the investigation of its theoretical limits, culminating recently in the development of new foundational frameworks for mathematics with sophisticated computer-assisted proof systems. In addition, logical formalisms developed by logicians in mathematical and philosophical contexts have proved immensely useful in describing theories and systems of interest to computer scientists, and to some degree, vice versa. Three examples of the influence of logic in computer science are automated reasoning, computer verification, and type systems for programming languages. (shrink)

I identify two reasons for believing in the objectivity of mathematical knowledge: apparent objectivity and applications in science. Focusing on arithmetic, I analyze platonism and cognitive nativism in terms of explaining these two reasons. After establishing that both theories run into difficulties, I present an alternative epistemological account that combines the theoretical frameworks of enculturation and cumulative cultural evolution. I show that this account can explain why arithmetical knowledge appears to be objective and has scientific applications. Finally, I will argue (...) that, while this account is compatible with platonist metaphysics, it does not require postulating mind-independent mathematical objects. (shrink)

Formal logic has often been seen as uniquely placed to analyze mathematical argumentation. While formal logic is certainly necessary for a complete understanding of mathematical practice, it is not sufficient. Important aspects of mathematical reasoning closely resemble patterns of reasoning in nonmathematical domains. Hence the tools developed to understand informal reasoning, collectively known as argumentation theory, are also applicable to much mathematical argumentation. This chapter investigates some of the details of that application. Consideration is given to the many contrasting meanings (...) of the word “argument”; to some of the specific argumentation-theoretic tools that have been applied to mathematics, notably Toulmin layouts and argumentation schemes; to some of the different ways that argumentation is implicated in mathematical practices; and to the social aspects of mathematical argumentation. (shrink)

I want to make sense of the view that singular causation involves a metaphysical necessary connection. By this I understand, where A and B are particulars, that ifA causes B then in every possible world in which A (or an A-indiscernible) or B (or a B-indiscernible) occurs, A (or an Aindiscernible) and B (or a B-indiscernible) occur. In the singularist approach that I will favour causal facts do not supervene on laws, causal relata are best understood as tropes, causation is (...) founded on the nature of its terms, and the necessity thus involved does not entail essentialism, determinism, and other usual problems. (shrink)

Disagreements that resist rational resolution, often termed “deep disagreements”, have been the focus of much work in epistemology and informal logic. In this paper, I argue that they also deserve the attention of philosophers of mathematics. I link the question of whether there can be deep disagreements in mathematics to a more familiar debate over whether there can be revolutions in mathematics. I propose an affirmative answer to both questions, using the controversy over Shinichi Mochizuki’s work on (...) the abc conjecture as a potential example of both phenomena. I conclude by investigating the prospects for the resolution of mathematical deep disagreements in virtue-theoretic approaches to informal logic and mathematical practice. (shrink)

The term ‘continuous’ in real analysis wasn’t given an adequate formal definition until 1817. However, important theorems about continuity were proven long before that. How was this possible? In this paper, I introduce and refine a proposed answer to this question, derived from the work of Frank Jackson, David Lewis and other proponents of the ‘Canberra plan’. In brief, the proposal is that before 1817 the meaning of the term ‘continuous’ was determined by a number of ‘platitudes’ which had some (...) special epistemic status. (shrink)

A textbook on proof in mathematics, inspired by an Aristotelian point of view on mathematics and proof. The book expounds the traditional view of proof as deduction of theorems from evident premises via obviously valid steps. It deals with the proof of "all" statements, "some" statements, multiple quantifiers and mathematical induction.

Causal platonism asserts that mathematical objects cause neural states in human brains. I raise the following four objections to it. (i) Quantum entanglement does not show that one object can causally affect another, although one is nontemporal, nonspatial, and unchanging. (ii) Causal platonism can neither be justified a posteriori nor a priori. (iii) To postulate mathematical media to flesh out mathematical causation is to multiply mysteries beyond necessity. (iv) To say that mathematical causation is unintelligible and inexplicable is not (...) to complain about causation in general. (shrink)

Models are indispensable tools of scientific inquiry, and one of their main uses is to improve our understanding of the phenomena they represent. How do models accomplish this? And what does this tell us about the nature of understanding? While much recent work has aimed at answering these questions, philosophers' focus has been squarely on models in empirical science. I aim to show that pure mathematics also deserves a seat at the table. I begin by presenting two cases: Cramér’s (...) random model of the prime numbers and the function field model of the integers. These cases show that mathematicians, like empirical scientists, rely on unrealistic models to gain understanding of complex phenomena. They also have important implications for some much-discussed theses about scientific understanding. First, modeling practices in mathematics confirm that one can gain understanding without obtaining an explanation. Second, these cases undermine the popular thesis that unrealistic models confer understanding by imparting counterfactual knowledge. (shrink)

In his influential book, The Nature of Morality, Gilbert Harman writes: “In explaining the observations that support a physical theory, scientists typically appeal to mathematical principles. On the other hand, one never seems to need to appeal in this way to moral principles.” What is the epistemological relevance of this contrast, if genuine? This chapter argues that ethicists and philosophers of mathematics have misunderstood it. They have confused what the chapter calls the justificatory challenge for realism about an area, (...) D—the challenge to justify our D-beliefs—with the reliability challenge for D-realism—the challenge to explain the reliability of our D-beliefs. Harman’s contrast is relevant to the first, but not, evidently, to the second. One upshot of the discussion is that genealogical debunking arguments are fallacious. Another is that indispensability considerations cannot answer the Benacerraf–Field challenge for mathematical realism. (shrink)