Mathematical Realists Research Articles

Coincidence Avoidance and Formulating the Access Problem

AbstractIn this article, I discuss a trivialization worry for Hartry Field’s official formulation of the access problem for mathematical realists, which was pointed out by Øystein Linnebo (and has recently been made much of by Justin Clarke-Doane). I argue that various attempted reformulations of the Benacerraf problem fail to block trivialization, but that access worriers can better defend themselves by sticking closer to Hartry Field’s initial informal characterization of the access problem in terms of (something like) general epistemic norms of coincidence avoidance.

J M Keynes’s Mathematical Platonism(Realism) in the A Treatise on Probability, Like Kurt Godel’s Mathematical Platonism, Held That All Mathematical Entities Were Independent of the Human Mind, but They Could Be Perceived or Intuited Directly From The Abstract Universe in Which They Existed: Keynes Was Both a Platonist (Logicist) and Aristotelian (Empiricist)

J M Keynes was never simply a Platonist or Realist like G E Moore was alleged to be. Keynes was a Mathematical Realist (Mathematical Platonist) similar to Kurt Godel, much like 95 % of all practicing mathematicians are today. Weak to Strong Mathematical Platonism has dominated the field of mathematics for the last 2,400 years, dating back to Pythagoras and Plato. Economists, who have written on Keynes, have confused and conflated Platonism with Keynes’s Mathematical Platonism, which is very close to the Mathematical Platonism of Kurt Godel, the greatest mathematical logician of the 20th century. Keynes never accepted Plato’s theory of Forms. Keynes‘s logical relations, P(a/h) =α,and V(a/h) =w, in his A treatise on Probability are mathematical entities.Therefore, since they are mathematical entities, they must, like all other mathematical entities, exist independently from the human mind in an abstract, metaphysical universe that can be perceived or intuited directly by the human mind. Heterodox economists, Post Keynesians and Fundamentalist Keynesians simply lack the necessary philosophical background, knowledge and training that is needed in order to grasp Keynes’s philosophy of mathematics.They have categorized Keynes as a Platonist without realizing that, as a mathematician, Keynes would naturally categorize all mathematical entities, just like Bertrand Russell and William Ernest Johnson, as existing independently from the human mind. Russell later moved from a Strong to Weak perspective on mathematical Platonism. The other two major philosophies of mathematics, Hilbert’s formalism, which regards mathematics as being mere scribble written on a piece of paper that the practicing mathematician hopes to make some sense out of later, and Brouwer’s Intuitionism, which views mathematical entities as properties of each individual human mind’s intuition, make up the other 5 % of practicing mathematicians views of the nature of mathematical entities. Brouwer’s approach is problematic once one asks the question,”Is 2 +2 still equal to 4 if the earth is completely destroyed, so that there are no more human minds in existence? ”A “yes” answer means that the mathematical entities were never internal to the human mind. This paper examines the flawed philosophical understanding of mathematical Platonism exhibited by O’Donnell (1989, 1990, 1992), Bateman (1996), Backhouse (2010) and Davis (1991, 1994,1997, 2003), who have confused and/or conflated Platonism with Mathematical Platonism while ignoring Keynes’s Aristotelianism.These misconceptions have appeared again in 2017 in papers published in the Journal of Markets and Morality. Keynes was both a Platonist, mathematical Platonist, and Arietotelian. A dangerous, pseudo-scientific approach that relies on a dualism based on only binary choices has arisen since the late 1930’s in the work of G L S Shackle and Joan Robinson, and especially since the late 1970’s among Keynesian Fundamentalists. Supposedly, Keynes’s work is presented as being only one of two conflicting choices: it is either in historical time or logical time, atomic or organic, open or closed, ergodic or non ergodic, formalist or anti-formalist, mathematical or anti-mathematical, based on uncertain or certain knowledge, knowledge or unknowledge, rationalist or empiricist, Platonic or Aristotelian, all or nothing. Aristotle himself referred to himself explicitly as a Platonist in his Metaphysics. Like Keynes after him, he maintained many aspects of Platonism (Virtue ethics), but emphasized other aspects ignored by Plato, such as physics and biology (empirical). The same holds for Keynes.

On Saving the Astronomical Phenomena: Physical Realism in Struggle with Mathematical Realism in Francis Bacon, al-Bitruji, and Averroës

When we examine the history of astronomy up to the end of the seventeenth century by considering the relation between mathematical astronomy and natural philosophy, it has been argued that there were two groups of philosophers and astronomers: instrumentalists and realists. However, this classification is deficient when we consider attitudes toward the explanatory power of mathematics in determining astronomical theories. I offer the solution of dividing realists into two subcategories—mathematical realists and physical realists. Mathematical realists include those who thought mathematics could provide the real motions of celestial bodies. The physical realists, who include Ibn Rushd (Averroës), believed mathematics could not provide us with the real structure of the heavens. To avoid contradictions between the physical and mathematical models, the physical realists held that a mathematical model of the heavens should be designed by considering physics. I argue, in this article, that Francis Bacon can be considered a physical realist.

Importance and Explanatory Relevance: The Case of Mathematical Explanations

A way to argue that something (e.g. mathematics, idealizations, moral properties, etc.) plays an explanatory role in science is by linking explanatory relevance with importance in the context of an explanation. The idea is deceptively simple: a part of an explanation is an explanatorily relevant part of that explanation if removing it affects the explanation either by destroying it or by diminishing its explanatory power, i.e. an important part (one that if removed affects the explanation) is an explanatorily relevant part. This can be very useful in many ontological debates. My aim in this paper is twofold. First of all, I will try to assess how this view on explanatory relevance can affect the recent ontological debate in the philosophy of mathematics—as I will argue, contrary to how it may appear at first glance, it does not help very much the mathematical realists. Second of all, I will show that there are big problems with it.

Indispensability, causation and explanation

When considering mathematical realism, some scientific realists reject it, and express sympathy for the opposite view, mathematical nominalism; moreover, many justify this option by invoking the causal inertness of mathematical objects. The main aim of this note is to show that the scientific realists’ endorsement of this causal mathematical nominalism is in tension with another position some (many?) of them also accept, the doctrine of methodological naturalism. By highlighting this conflict, I intend to tip the balance in favor of a rival of mathematical nominalism, the mathematical realist position supported by the ‘Indispensability Argument’ – but I do this indirectly, by showing that the road toward it is not blocked by considerations from causation.

Can mathematical objects be causally efficacious?

Callard argues that it is metaphysically possible that a mathematical object, although abstract, causally affects the brain. I raise the following objections. First, a successful defence of mathematical realism requires not merely the metaphysical possibility but rather the actuality that a mathematical object affects the brain. Second, mathematical realists need to confront a set of three pertinent issues: Why a mathematical object does not affect other concrete objects and other mathematical objects, what counts as a mathematical object and how we can have knowledge about an unchanging object.

Pendampingan Relawan Matematika Asyik (Rematika) Terhadap Siswa di Sekolah dan Anak-Anak Nelayan Pinggir Pantai Kenjeran Surabaya

Selama ini matematika menjadi matapelajaran yang sulit bagi siswa. Oleh karena itu sebagian orang tua memilih jalan mendatangkan guru privat untuk membantu anaknya memahami matematika. Pengabdian ini adalah pendampingan pembelajaran bagi siswa di Sekolah dasar Muhammadiyah 18 Surabaya dan anak-anak nelayan yang masih sekolah di daerah kenjeran surabaya. Pengabdian ini dilakukan oleh sejumlah Realwan Matematika Asyik (Rematika) Universitas Muhammadiyah Surabaya. Para relawan membuat media inovasi pembelajaran sejumlah 7 media untuk membantu siswa memhami matematika dengan cara asyik dan menyenangkan. Diantaranya Pembelajaran himpunan dengan media Boneka Pintar (Bopin) . Smart Turtle. Kotamat (Kotak Matematika ). Merangka (Menara Perang Matematika). Kurung Sakti. Pirates of Mathematics I dan II

Can the Cumulative Hierarchy Be Categorically Characterized

Mathematical realists have long invoked the categoricity of axiomatizations of arithmetic and analysis to explain how we manage to fix the intended meaning of their respective vocabulary. Can this strategy be extended to set theory? Although traditional wisdom recommends a negative answer to this question, Vann McGee (1997) has offered a proof that purports to show otherwise. I argue that one of the two key assumptions on which the proof rests deprives McGee's result of the significance he and the realist want to attribute to it. I consider two strategies to deal with the problem --- one of which is outlined by McGee himself (2000) --- and argue that both of them fail. I end with some remarks on the prospects for mathematical realism in the light of my discussion.

Mathematical objects and the object of theology

AbstractThis article brings mathematical realism and theological realism into conversation. It outlines a realist ontology that characterizes abstract mathematical objects as inaccessible to the senses, non-spatiotemporal, and acausal. Mathematical realists are challenged to explain how we can know such objects. The article reviews some promising responses to this challenge before considering the view that the object of theology also possesses the three characteristic features of abstract objects, and consequently may be known through the same methods that yield knowledge of mathematical objects.

Non-Representational Mathematical Realism

This paper is an attempt to convince anti-realists that their correct intuitions against the metaphysical inflationism derived from some versions of mathematical realism do not force them to embrace non-standard, epistemic approaches to truth and existence. It is also an attempt to convince mathematical realists that they do not need to implement their perfectly sound and judicious intuitions with the anti-intuitive developments that render full-blown mathematical realism into a view which even Gödel considered objectionable (Gödel 1995, p. 150).</p><p>I will argue for the following two theses: (i) that realism, in its standard characterization, is our default position, a position in agreement with our pre-theoretical intuitions and with the results of our best semantic theories, and (ii) that most of the metaphysical qualms usually related to it depends on a poor understanding of truth and existence as higher-order concepts.

Mathematical Explanations Of Empirical Facts, And Mathematical Realism

A main thread of the debate over mathematical realism has come down to whether mathematics does explanatory work of its own in some of our best scientific explanations of empirical facts. Realists argue that it does; anti-realists argue that it doesn't. Part of this debate depends on how mathematics might be able to do explanatory work in an explanation. Everyone agrees that it's not enough that there merely be some mathematics in the explanation. Anti-realists claim there is nothing mathematics can do to make an explanation mathematical; realists think something can be done, but they are not clear about what that something is. I argue that many of the examples of mathematical explanations of empirical facts in the literature can be accounted for in terms of Jackson and Pettit's [1990] notion of program explanation, and that mathematical realists can use the notion of program explanation to support their realism. This is exactly what has happened in a recent thread of the debate over moral realism (in this journal). I explain how the two debates are analogous and how moves that have been made in the moral realism debate can be made in the mathematical realism debate. However, I conclude that one can be a mathematical realist without having to be a moral realist.

Morality and Mathematics: The Evolutionary Challenge

The Evolutionary Challenge for moral realism is, roughly, the challenge to explain our having many true moral beliefs, given that those beliefs are the products of evolutionary forces that would be indifferent to the moral truth. This challenge is widely thought not to apply to mathematical realism. In this article, I argue that it does. Along the way, I substantially clarify the Evolutionary Challenge, discuss its relation to more familiar epistemological challenges, and broach the problem of moral disagreement. I conclude that there may be no epistemological ground on which to be a moral antirealist and a mathematical realist.

Magicicada, Mathematical Explanation and Mathematical Realism

Baker (2005) claims to provide an example of mathematical explanation of an empirical phenomenon which leads to ontological commitment to mathematical objects. This is meant to show that the positing of mathematical entities is necessary for satisfactory scientific explanations and thus that the application of mathematics to science can be used, at least in some cases, to support mathematical realism. In this paper I show that the example of explanation Baker considers can actually be given without postulating mathematical objects and thus cannot be used by the mathematical realist. I also show that, despite this, mathematics keeps playing an important methodological role in the explanation and does not reduce to a merely computational or descriptive framework.

Naturalism, Truth and Beauty in Mathematics

Can a scientific naturalist be a mathematical realist? I review some arguments, derived largely from the writings of Penelope Maddy, for a negative answer. The rejoinder from the realist side is that the irrealist cannot explain, as well as the realist can, why a naturalist should grant the mathematician the degree of methodological autonomy that the irrealist's own arguments require. Thus a naturalist, as such, has at least as much reason to embrace mathematical realism as to embrace irrealism.

Holistic Realism: A Response to Katz On Holism and Intuition

In his book, Realistic Rationalism (henceforth RR), and his article, “Mathematics and Metaphilosophy”(henceforth MM), Jerrold Katz develops and defends a philosophy of mathematics that is realist in ontology and rationalist in epistemology. On his view, mathematics contains a body of necessary truths about certain abstract entities that are known a priori though the exercise of reason. Katz believes that his epistemology answers the familiar challenge, so well articulated by Paul Benacerraf, to mathematical realists that they explain how beings like us can acquire knowledge about causally inert, nonspatial, nontemporal, mathematical objects. Katz thinks the key to responding successfully to this challenge is to develop a “no contact epistemology,” that is, one that posits no physical process connecting us to mathematical objects. By basing his account of mathematical knowledge upon reason and intuition, Katz believes that he has avoided mysticism, such as that associated with Plato’s epistemology of recollection, and has given a deeper account than, say, Frege’s view that we grasp or apprehend abstract objects or Godel’s brief comparison of mathematical intuition with ordinary sense perception. Katz thinks that the only other no contact epistemology worthy of consideration is W. V. Quine’s confirmational holism. But he argues that this falls prey to his own Revisability Paradox and is inconsistent (RR, 72–4; MM, 375).