Logic and Foundations of Mathematics Commons^™

Three Essays In Intuitionistic Epistemology, Tudor Protopopescu

Dissertations, Theses, and Capstone Projects

We present three papers studying knowledge and its logic from an intuitionistic viewpoint.

An Arithmetic Interpretation of Intuitionistic Verification

Intuitionistic epistemic logic introduces an epistemic operator to intuitionistic logic which reflects the intended BHK semantics of intuitionism. The fundamental assumption concerning intuitionistic knowledge and belief is that it is the product of verification. The BHK interpretation of intuitionistic logic has a precise formulation in the Logic of Proofs and its arithmetical semantics. We show here that this interpretation can be extended to the notion of verification upon which intuitionistic knowledge is based. This provides the systems of intuitionistic epistemic logic …

Keats, Truth, And Empathy, Peter Shum

Sophia and Philosophia

At one level, Keats’s sonnet entitled On Peace (1814) is full of philosophical certainties. The speaker believes, for example, that a nation’s people have a right to live in freedom under the rule of law, and that the rule of law should be applicable to everybody. Political and philosophical commitments of this kind do not seem to be called into question in this poem, or made the subject of an enquiry. On the contrary, it is as though we are confronted with somebody who, in certain central thematic respects at least, appears to know his own mind.

The Rhetoric Of Tyranny: Callicles The Rhetor And Nietzsche's Zarathustra, Greg Whitlock

Sophia and Philosophia

Here I will work through the rhetoric of tyranny as practiced by Callicles and as reflected in Nietzsche’s Zarathustra, in particular. In Part 2 it will be shown that Nietzsche’s account of Plato as the complex figure with a Socratic exterior but a latent alternative ego of the tyrant, arrived at an image consistent with E.R. Dodds’ later thesis. Callicles the rhetor, featured as a student of Gorgias, embodies this alter-ego. In Part 3 we find Callicles and Zarathustra shared very similar beliefs once they overcame shame and gained honesty. Indeed, Callicles expounded a number of propositions foundational to …

The Problem Of Obviousness, Benjamin Goldberg

Sophia and Philosophia

1. The Problem of Obviousness
There’s no such thing as obviousness.

This isn’t, of course, itself obvious; nor is it clear why it should be a problem. So let me start elsewhere, with the anti-vaccine movement. A friend of mine laid out the ‘obvious’ position: there are facts and rationality on one side, unenlightened ignorance and bigotry on the other. Scientists versus fools, and the fools don’t even know what game is being played.

Poetic Justice: Apology Overdue, Stephen Faison

Sophia and Philosophia

Jurors of our republic, I do not know whether you were persuaded by the case made against me, but I certainly hope that you were not. Some of what the prosecutor told you is accurate, though much of it is untrue. To put it another way, some of his facts are correct, yet the conclusions he presented were usually misleading distortions and in some cases simply incorrect. If the indictment is clarified to its essentials, I am accused of corrupting the young and not believing in the gods in whom the city believes. I intend to show that these charges …

Cindy, Ken Clatterbaugh

Sophia and Philosophia

Cindy Norton sang loud in slightly accented Spanish. The song was “Perfidia” (“Wicked Woman”). She remembered the melody, and the Spanish language radio reminded her of the lyrics, although she could hardly hear the radio or herself above the grinding, popping, and rattling of her Volkswagen Rabbit. Her radio picked up only one station.

‘To Warm Our Hands’, Emmanuel O. Angulo

Sophia and Philosophia

Lovers often die shortly one after the other. Romeo and Juliet. June Carter and Johnny Cash. My grandfather and my grandmother. Leonard Cohen and Marianne Ihlen.

Marianne was the inspiration, most famously, of Cohen’s song “So long, Marianne”, but also of “Bird on the wire” and poems from the collection Flowers for Hitler. Cohen’s last words for her reached her just two days before her death—and a few months before his own. They said: ‘you know that I’ve always loved you for your beauty and your wisdom, but I don't need to say anything more about that because you know …

The Descent Of Winter: William Carlos Williams Under The Influence Of Paris, Phillip Barron

Sophia and Philosophia

The Descent of Winter, published in Ezra Pound’s magazine The Exile in 1928, is an uneven experiment in unclassifiable writing. Williams began writing the diaristic entries aboard the SS Pennland “in the fall of 1927. He was returning from Europe, where he had left Florence Williams with their two sons, who were to attend school in Geneva.” (Imaginations, 231) The slim book alternates between spare, image driven poems composed of short lines and diaristic prose. Most entries are titled only with the dates of their composition. In aesthetics, Williams always concerned himself with form. The variety of …

"Part Of That Force That Always Wills The Evil And Always Produces The Good" On A Devilish Incoherence, Peter Baumann

Sophia and Philosophia

When Mephisto was asked by Faust, "Well now, who are you then?" (“Nun gut, wer bist Du denn?”), he gave the well-known answer, "Part of that force that always wills the evil and always produces the good" (“Ein Teil von jener Kraft, die stets das Böse will und stets das Gute schafft”: Goethe, Faust, 1334-1336). This answer raises some important questions which in turn lead to interesting answers. Let us start by looking at the content of Mephisto's utterance.

Vol 7 No 2 Contents Page

Comparative Philosophy

No abstract provided.

Vol 7 No 2 Information Page

Comparative Philosophy

No abstract provided.

Vol 7 No 2 Cover Page

Comparative Philosophy

No abstract provided.

Clearing Up Calculus With Adaptive Learning Habits, Jennifer Patterson

Honors Theses

No abstract provided.

The C3 Conditional: A Variably Strict Ordinary-Language Conditional, Monique L. Whitaker

Dissertations, Theses, and Capstone Projects

In this dissertation I provide a novel logic of the ordinary-language conditional. First, however, I endeavor to make clearer and more precise just what the objects of the study of the conditional are, as a lack of clarity as to what counts as an instance of a given category of conditional has resulted in deep and significant confusions in subsequent analysis. I motivate for a factual/counterfactual distinction, though not at the level of particular instances of the conditional. Instead, I argue that each individual instance of the conditional may be interpreted either factually or counterfactually, rather than these instances dividing …

The Logic Of Concessive Statements, Aharon Grenadir

School for Lifelong Education Publications

Concessive statements appear frequently in everyday reasoning. They are one of the eleven types of statement mentioned in the Ramchal‟s Sefer Derech Tevunos. In addition, in Talmudic discussions, every statement has a presupposition (hava-amina) and a conclusion (ka mashma lan). This goal of this research is to organize the categorizations of concessive statements that are stated in the technical literature. Using the distintinction in lomdus between dechiyah (overriding a law) and hutrah (removal of a law), a novel categorization can be added, according to the type of denial of the expectation by the main clause. That is the subject of …

Gronwall’S Inequality And Systems Of Integral Equations, Bradford Garcia

Honors Theses

No abstract provided.

Avatars Of Oneself, Patrick L. Miller

Sophia and Philosophia

Zoe is an American woman who has found that “drawing the line and standing firm has always made me feel like a bitch, and, actually, I feel that people saw me as one too.”[3] For two years, however, she played an online role-playing game using a male character where “as a man I was liberated from all that.” She made mistakes in her unfamiliar role, but learned from them. “I got better at being firm but not rigid,” she says; “I practiced, safe from criticism.” Case is an American man, who plays a similar game but always appears as a …

On The Relationship Of Alcibiades’ Speech To The Rest Of The Speeches In Plato’S Symposium[1], Andy Davis

Sophia and Philosophia

To get to the point immediately concerning how I think about the relationship between the first five speeches and Socrates’ speech: it seems to me the claim that Plato has only brought together inadequate perspectives on Eros in order to present Socrates’ speech over and against them as the only correct one is completely in error. Socrates himself does not deny these speeches their accolades, he comes back to many things in them as he assigns each single perspective its own due place. Much more, I believe that from the first speech to the last a decisive progress takes place, …

Jane Austen Meets The Gps: Place And Space, David Kolb

Sophia and Philosophia

When one reads Jane Austen’s novels, one finds that her heroines’ lives center around a beloved and comfortable home, in a local region including a small town, some neighboring estates, and local hills and valleys. It is a detailed and textured home area of nearby places reachable on foot or horse. One to three miles are walkable to a friend’s house or a favorite scenic hill. Beyond this region is no longer “home.” Fifteen or twenty-five miles can be distant.

Kirkos, Prabhu Venkatarama

Sophia and Philosophia

Yesterday morning I received a newspaper clipping that changed the course of my life. It arrived in a small grey envelope, my mother’s Cyrillic handwriting on the front instantly recognizable by its beautiful loops and arches. I was expecting it; my mother had told me it was in the mail. The clipping was of an obituary, that of Robert Mascas, a younger cousin of mine who’d lived in my former hometown of Kirkos. He had turned eighteen three months ago, precisely seven years to the day after I had. The obituary didn’t mention what my mother had on the phone, …

Xanthippe To Her Mother, Ginger Osborn

Sophia and Philosophia

The following is a translation of an ancient manuscript, presumably a late-Hellenistic school exercise, recovered from the so-called Villa of the Papyri in Herculaneum, which was entombed by the eruption of Vesuvius in 79 BCE. The library was well-stocked with philosophical works, mostly of an Epicurean bent, but with a variety of other traditions represented as well. The text below is the result of the editorial work and translation of the Italo-Brtitish philosophical eccentric Michael Tommasi, completed presumably in Cambridge in the 1940s, but never published; his literary executors discovered the manuscript among his posthumous papers. Several revisions to Tommasi's …

Nietzsche's Views On Plato Pre-Basel, Daniel Blue

Sophia and Philosophia

In an essay published in 2004[1] Thomas Brobjer surveyed Nietzsche’s attitudes toward Plato and argued that, far from entering into a dedicated agon with that philosopher, he had little personal engagement with Plato’s views at all. Certainly, he did not grapple so immediately and fruitfully with him as he did with Emerson, Schopenhauer, Lange, and even Socrates. Instead, he merely “set up a caricature of Plato as a representative of the metaphysical tradition … to which he opposed his own.”[2] This hardly reflects the view of Nietzsche scholarship in general, but Brobjer argued his case vigorously by ranging broadly over …

Empty Souls: Confession And Forgiveness In Hegel And Dostoevsky, Ryan Johnson

Sophia and Philosophia

“Towards the end of a sultry afternoon early in July a young man came out of his little room in Stolyarny Lane and turned and in the direction of Kameny Bridge in central St. Petersburg.”[1] Right then, this young man, a former law student named Rodion Raskolnikov, is caught in an agonizing conversation with himself over whether or not to commit the ultimate crime: to murder an innocent person. Exasperated, wondering what to do with such a weighty decision, he cried aloud, “that’s why I don’t act, because I am always talking. Or perhaps I talk so much just because …

Review Of Swimming Home, By Vincent Katz, Phillip Barron

Sophia and Philosophia

After a reading in San Francisco, I asked Katz about the significance of using the term “poetry” repeatedly throughout “Sidewalk Poem.” He answered, “poetry is about making, as much as any hand skill like carpentry or welding, but the craft of poetry is not respected.” For Charles Reznikoff and Federico Garcia Lorca, walking the streets of New York was integral to the act of writing poetry. The craft of poetry contains not only those moments when pen smears ink on paper or keyboard presses pixel on screen. Making poetry is more sensuous and active than the still, quiet moments that …

Nietzsche And Heraclitus: Notes On Stars Without An Atmosphere, Niketas Siniossoglou

Sophia and Philosophia

I awake estranged from everyone. Words have lost their meaning; they sound indifferent and homonymous. The word No appears to mean Yes, or rather: Yes and No are malleable, ephemeral, and transparent. A decades-old or perhaps centuries-old movement of miry clay has resulted in a miscarriage of words. Iinquire whether anyone still holds the resources needed for a direct, sincere affirmation of life—a Yes that is definitively and essentially affirmative—or a No that is definitively and essentially negative—words bursting forth splendour like a crystal. I am told that formulations of this sort are incomprehensible; they are too metaphorical and, …

Toward A Kripkean Concept Of Number, Oliver R. Marshall

Dissertations, Theses, and Capstone Projects

Saul Kripke once remarked to me that natural numbers cannot be posits inferred from their indispensability to science, since we’ve always had them. This left me wondering whether numbers are objects of Russellian acquaintance, or accessible by analysis, being implied by known general principles about how to reason correctly, or both. To answer this question, I discuss some recent (and not so recent) work on our concepts of number and of particular numbers, by leading psychologists and philosophers. Special attention is paid to Kripke’s theory that numbers possess structural features of the numerical systems that stand for them, and to …

Wabi-Sabi Mathematics, Jean-Francois Maheux

Journal of Humanistic Mathematics

Mathematics and aesthetics have a long history in common. In this relation however, the aesthetic dimension of mathematics largely refers to concepts such as purity, absoluteness, symmetry, and so on. In stark contrast to such a nexus of ideas, the Japanese aesthetic of wabi-sabi values imperfections, temporality, incompleteness, earthly crudeness, and even contradiction. In this paper, I discuss the possibilities of “wabi-sabi mathematics” by showing (1) how wabi-sabi mathematics is conceivable; (2) how wabi-sabi mathematics is observable; and (3) why we should bother about wabi-sabi mathematics

Explanatory Proofs And Beautiful Proofs, Marc Lange

Journal of Humanistic Mathematics

This paper concerns the relation between a proof’s beauty and its explanatory power – that is, its capacity to go beyond proving a given theorem to explaining why that theorem holds. Explanatory power and beauty are among the many virtues that mathematicians value and seek in various proofs, and it is important to come to a better understanding of the relations among these virtues. Mathematical practice has long recognized that certain proofs but not others have explanatory power, and this paper offers an account of what makes a proof explanatory. This account is motivated by a wide range of examples …

Abstraction And Epistemic Economy, Marco Panza

MPP Published Research

Most of the arguments usually appealed to in order to support the view that some abstraction principles are analytic depend on ascribing to them some sort of existential parsimony or ontological neutrality, whereas the opposite arguments, aiming to deny this view, contend this ascription. As a result, other virtues that these principles might have are often overlooked. Among them, there is an epistemic virtue which I take these principles to have, when regarded in the appropriate settings, and which I suggest to call ‘epistemic economy’. My purpose is to isolate and clarify this notion by appealing to some examples concerning …

Restall's Proof-Theoretic Pluralism And Relevance Logic, Teresa Kouri

Philosophy Faculty Publications

Restall (Erkenntnis 79(2):279–291, 2014) proposes a new, proof-theoretic, logical pluralism. This is in contrast to the model-theoretic pluralism he and Beall proposed in Beall and Restall (Aust J Philos 78(4):475–493, 2000) and in Beall and Restall (Logical pluralism, Oxford University Press, Oxford, 2006). What I will show is that Restall has not described the conditions on being admissible to the proof-theoretic logical pluralism in such a way that relevance logic is one of the admissible logics. Though relevance logic is not hard to add formally, one critical component of Restall’s pluralism is that the relevance logic that gets added must …