Experimental mathematics, computers and the a priori

In recent decades, experimental mathematics has emerged as a new branch of mathematics. This new branch is defined less by its subject matter, and more by its use of computer assisted reasoning. Experimental mathematics uses a variety of computer assisted approaches to verify or prove mathematical hypotheses. For example, there is “number crunching” such as searching for very large Mersenne primes, and showing that the Goldbach conjecture holds for all even numbers less than 2 × 1018. There are “verifications” of hypotheses which, while not definitive proofs, provide strong support for those hypotheses, and there are proofs involving an enormous amount of computer hours, which cannot be surveyed by any one mathematician in a lifetime. There have been several attempts to argue that one or another aspect of experimental mathematics shows that mathematics now accepts empirical or inductive methods, and hence shows mathematical apriorism to be false. Assessing this argument is complicated by the fact that there is no agreed definition of what precisely experimental mathematics is. However, I argue that on any plausible account of ’experiment’ these arguments do not succeed.     

Mathematical forms and forms of mathematics: leaving the shores of extensional mathematics

In this paper, I introduce the idea that some important parts of contemporary pure mathematics are moving away from what I call the extensional point of view. More specifically, these fields are based on criteria of identity that are not extensional. After presenting a few cases, I concentrate on homotopy theory where the situation is particularly clear. Moreover, homotopy types are arguably fundamental entities of geometry, thus of a large portion of mathematics, and potentially to all mathematics, at least according to some speculative research programs.     

The need to explain

How do reasoners deal with inconsistencies? James (1907) believed that the rational solution is to revise your beliefs and to do so in a minimal way. We propose an alternative: You explain the origins of an inconsistency, which has the side effect of a revision to your beliefs. This hypothesis predicts that individuals should spontaneously create explanations of inconsistencies rather than refute one of the assertions and that they should rate explanations as more probable than refutations. A pilot study showed that participants spontaneously explain inconsistencies when they are asked what follows from inconsistent premises. In three subsequent experiments, participants were asked to compare explanations of inconsistencies against minimal refutations of the inconsistent premises. In Experiment 1, participants chose which conclusion was most probable; in Experiment 2 they rank ordered the conclusions based on their probability; and in Experiment 3 they estimated the mean probability of the conclusions' occurrence. In all three studies, participants rated explanations as more probable than refutations. The results imply that individuals create explanations to resolve an inconsistency and that these explanations lead to changes in belief. Changes in belief are therefore of secondary importance to the primary goal of explanation.     

The need to explain

How do reasoners deal with inconsistencies? James (1907) believed that the rational solution is to revise your beliefs and to do so in a minimal way. We propose an alternative: You explain the origins of an inconsistency, which has the side effect of a revision to your beliefs. This hypothesis predicts that individuals should spontaneously create explanations of inconsistencies rather than refute one of the assertions and that they should rate explanations as more probable than refutations. A pilot study showed that participants spontaneously explain inconsistencies when they are asked what follows from inconsistent premises. In three subsequent experiments, participants were asked to compare explanations of inconsistencies against minimal refutations of the inconsistent premises. In Experiment 1, participants chose which conclusion was most probable; in Experiment 2 they rank ordered the conclusions based on their probability; and in Experiment 3 they estimated the mean probability of the conclusions' occurrence. In all three studies, participants rated explanations as more probable than refutations. The results imply that individuals create explanations to resolve an inconsistency and that these explanations lead to changes in belief. Changes in belief are therefore of secondary importance to the primary goal of explanation.     

The influence of Platonism on mathematics research and theological beliefs

An age-old debate in the philosophy of mathematics is whether mathematics is discovered or invented. There are four popular viewpoints in this debate, namely Platonism, formalism, intuitionism, and logicism. A natural question that arises is whether belief in one of these viewpoints affects the mathematician’s research? In particular, does subscribing to a Platonist or a formalist viewpoint influence how a mathematician conducts research? Does the area of research influence a mathematician’s beliefs on the nature of mathematics? How are the beliefs regarding the nature of mathematics connected to theological beliefs? In order to investigate these questions, five professional research mathematicians were interviewed. The mathematicians worked in diverse areas within analysis, algebra, and within applied mathematics, and had a combined 160 years of research experience. Although none of the mathematicians wanted to be pigeonholed into any one category of beliefs, the study revealed that four of the mathematicians leaned towards Platonism, which runs contrary to the popular notion that Platonism is an exception today. This study revealed that beliefs regarding the nature of mathematics influenced how mathematicians’ conducted research and were deeply connected to their theological beliefs. The findings are presented in the form of vignettes that give an insight into the mathematical and theological belief structures of the mathematicians.     

HOW SHOULD WE BEHAVE? A REVIEW OF REASONS AND PERSONS BY DEREK PARFIT

In this book Parfit attempts to develop a rational, non-religious ethics. Instead of asking, “What does God tell us to do?” he asks, “What does reason tell us to do?” Given a set of simple assumptions, he considers whether it is possible to be consistently selfish or consistently good. Analyses of personal dilemmas (problems of self-control) and moral dilemmas (problems of social cooperation) show that neither consistent selfishness nor consistent goodness is logically possible. Instead, a fine balance must be maintained between, on the one hand, our immediate versus long-term good and, on the other, our own good versus that of other people. Ultimately Parfit fails to develop a formula by which such a balance may be struck. Parfit''s analysis is consistent with behavioral analysis in its reductionistic view of the self and the parallel it draws between relations with other people and relations with oneself at other times. Parfit''s analysis is inconsistent with behaviorism in its view of the mind as internal, available to introspection, and able to cause behavior. His nonfunctional mentalism leads Parfit to inconsistencies and blocks the path to a consistent ethics. Teleological behaviorism''s view of the mind in terms of patterns of overt behavior is not hampered by these inconsistencies and may lead to a functional rather than purely rational ethics.     

Is Intuition Necessary for Defending Platonism?

G?del asserts that his philosophy falls under the category of conceptual realism. This paper gives a general picture of G?del’s conceptual realism’s basic doctrines, and gives a way to understand conceptual realism in the background of Leibniz’s and Kant’s philosophies. Among philosophers of mathematics, there is a widespread view that Platonism encounters an epistemological difficulty because we do not have sensations of abstract objects. In his writings, G?del asserts that we have mathematical intuitions of mathematical objects. Some philosophers do not think it is necessary to resort to intuition to defend Platonism, and other philosophers think that the arguments resorting to intuition are too na?ve to be convincing. I argue that the epistemic difficulty is not particular to Platonism; when faced with skepticism, physicalists also need to give an answer concerning the relationship between our experience and reality. G?del and Kant both think that sensations or combinations of sensations are not ideas of physical objects, but that, to form ideas of physical objects, concepts must be added. However, unlike Kant, G?del thinks that concepts are not subjective but independent of our minds. Based on my analysis of G?del’s conceptual realism, I give an answer to the question in the title and show that arguments resorting to intuition are far from na?ve, despite what some philosophers have claimed.     

理念主义与形式主义的巴别塔

圣经里讲了一个关于语言的故事,早期的先民们使用相同的语言,尝试建立通天的巴别塔,这个计划因为语言变乱而失败。这个故事或许隐喻着:人类是不可能通过语言来达到真理的。然而,人类自从吃了知识果,走出了伊甸园,就担当起自己的运命。从理念主义(Rationalism)和形式系统(Formalism)这两条线索,人类尝试着重建新的巴别塔。我们从理念主义的线索,沿着柏拉图的理念世界,到康德的先天准则,再到胡塞尔的本质直观,评论了理念主义的不自明性,提出了使用处于构造之中的"共相"来代替"理念"或者"本质"。维特根斯坦提出"家族相似",反对理念主义和本质主义,但"家族相似"可能过于宽泛。另一方面,从形式主义的线索,沿着莱布尼兹的通用计算,到弗雷格和罗素的逻辑主义,到希尔伯特的形式主义纲领,再到哥德尔不完备性定理,评论了逻辑主义和形式主义的失败。形式主义失败之后,人们转向形式主义的能力范围研究,从而提出了图灵机,为计算机科学的出现奠定了理论基础。基于以上的思考,文中最后提出一些有待研究的问题,比如基于关系的数学,基于谓词的逻辑,矛盾但完备的形式系统。     

MAKING UP THE TRUTH

A recent account of the meaning of 'real' leads to a view of what anti-realism should be that resembles fictionalism, while not being committed to fictionalism as such or being subject to some of the more obvious objections to that view. This account of anti-realism explains how we might 'make up' what is true in areas such as mathematics or ethics, and yet these made-up truths are resistant to alterations, even by our collective decisions. Finally it is argued that the sort of anti-realism suggested explains the appearance that the ethical domain supervenes on the naturalistic.     

Towards a postmodern ethics: Sir Isaiah Berlin and John Caputo

Conclusions It may seem to their opponents that they are trying to have their cake and eat it too. Postmodernists admit that their own paradigm must be and will be placed into question by future thinkers. But if they can anticipate an eventual reaffirmation of their paradoxical stand in an ongoing oscillating debate, then cannot it be said that they have arrived at a truth that transcends their time and place in history? And, if so, is not their fallibilist stance in fact self-referentially inconsistent? The response of postmodernists is the claim that each reaffirmation of a fallibilist epistemology and ethics throughout history is in fact sui generis. And this is the case because each reaffirmation has its own unique context within which it is made. Modernists will of course argue that these contextual differences are nonessential and irrelevant. And the debate over the problem of the One and the Many is once again launched in a new context. Thus far from running away from the paradoxical position that what they assert is both true and false, postmodernists revel in such inconsistencies.But does not such an ethical stance resemble the Sisyphean nightmare of being condemned to roll a heavy stone up a cliff only to have it keep falling back to the bottom ad infinitum? If no decision is innocent of doing some harm in the world, why should we bother to play the moral game at all? Indeed, what possible help is a postmodernist ethics when it comes to making some of the complex and crucial decisions we face today if it refuses to say anything substantial beyond the recommendations that we be cautious and balanced?And the postmodernist can only reply that we are letting our neurotic need for solid foundations frighten us. For ethics is an art not a science. There are no absolute rules. If we do not like the way the game is set up, then we are simply revealing our ultimate hubris in the face of a mystery requiring deep humility.     

From metasemantics to analyticity

In this paper, I argue from a metasemantic principle to the existence of analytic sentences. According to the metasemantic principle, an external feature is relevant to determining which concept one expresses with an expression only if one is disposed to treat this feature as relevant. This entails that if one isn’t disposed to treat external features as relevant to determining which concept one expresses, and one still expresses a given concept, then something other than external features must determine that one does. I argue that, in such cases, what determines that one expresses the concept also puts one in a position to know that certain sentences are true—these sentences are thus analytic relative to this determination basis. Finally, I argue that there are such cases: some sentences are analytic relative to what determines that we express certain key concepts, and these sentences include ones that have always been thought to be the best candidates for being analytic, namely, stipulative truths, and first principles of mathematics.     

Applying Principles to Cases and the Problem of Judgment

We sometimes decide what to do by applying moral principles to cases, but this is harder than it looks. Principles are more general than cases, and sometimes it is hard to tell whether and how a principle applies to a given case. Sometimes two conflicting principles seem to apply to the same case. To handle these problems, we use a kind of judgment to ascertain whether and how a principle applies to a given case, or which principle to follow when two principles seem to conflict. But what do we discern when we make such judgments—that is, what makes such judgments correct? The obvious answer is that they are made correct by whatever makes other moral judgments correct. However, that cannot be right, for a principle can be inconsistent with morality yet still apply in a particular way to a given case. If the principle is inconsistent with morality, then morality cannot be what we discern when we judge whether and how that principle applies to a given case. I offer an alternative account of what makes such judgments correct.     

Are Young Adults' Suicides Psychologically Different from Those of Other Adults? (The Shneidman Lecture)

ABSTRACT: This Shneidman Lecture addresses the question: Are young adults' suicides psychologically different from those of other adults? Eight clusters (or patterns) as possible predictors of suicide notes and, by implication, suicide are discussed to address this question. The eight clusters are as follows: unbearable psychological pain, interpersonal relations, rejection-aggression, inability to adjust, indirect expressions, identification-egression, ego, and cognitive constriction. Comparisons between young adults (aged 18–25) and other adults indicate that young adults' suicides do differ psychologically in a number of patterns, but that considerable commonalities exist across the adult life span. The presentation emphasizes that a life span developmental perspective is essential when one is attempting to understand the suicides of young adults. A number of genuine suicide notes of young adults are presented to illustrate, from an idiographic view, what it might have been like for these individuals in the last moments of their lives.     

ON HELPING ONE'S NEIGHBOR

Few people doubt that severe poverty is a pressing moral issue. But what sorts of obligations, if any, do affluent people have toward the severely poor? If one accepts the idea that one has some obligations to the severely poor there still remains disagreement about the magnitude of this obligation and when it obtains. I consider Peter Singer's influential “shallow pond” argument, which holds that affluent people have greater obligations toward the severely poor than ordinary moral judgments suggest. Critics hold that Singer's view is excessively demanding and therefore untenable. I thus turn to the parable of the Good Samaritan and Christian accounts of neighbor‐love to help attenuate this criticism. Drawing from Christian conversations on neighbor‐love, I attempt to demonstrate that accepting an obligation to assist does not necessarily result in one abandoning one's special relations, abnegating self‐regard, or no longer pursuing other non‐moral strivings.     

Ideas and processes in mathematics: A course on history and philosophy of mathematics

This paper describes an attempt to develop a program for teaching history and philosophy of mathematics to inservice mathematics teachers. I argue briefly for the view that philosophical positions and epistemological accounts related to mathematics have a significant influence and a powerful impact on the way mathematics is taught. But since philosophy of mathematics without history of mathematics does not exist, both philosophy and history of mathematics are necessary components of programs for the training of preservice as well as inservice mathematics teachers.     

The ontological commitments of inconsistent theories

In this paper I present an argument for belief in inconsistent objects. The argument relies on a particular, plausible version of scientific realism, and the fact that often our best scientific theories are inconsistent. It is not clear what to make of this argument. Is it a reductio of the version of scientific realism under consideration? If it is, what are the alternatives? Should we just accept the conclusion? I will argue (rather tentatively and suitably qualified) for a positive answer to the last question: there are times when it is legitimate to believe in inconsistent objects.     

Pragmatic Inconsistency and Credibility

A critic may attack an arguer personally by pointing out that the arguer’s position is pragmatically inconsistent: the arguer does not practice what he preaches. A number of authors hold that such attacks can be part of a good argumentative discussion. However, there is a difficulty in accepting this kind of contribution as potentially legitimate, for the reason that there is nothing wrong for a protagonist to have an inconsistent position, in the sense of committing himself to mutually inconsistent propositions. If so, any such charge seems to be irrelevant. The questions to be answered in this essay are: what, if any, is the dialectical rationale for this type of criticism, and in what situations, if any, is this kind of charge dialectically legitimate? It will be shown that these attacks can be dialectically legitimate, in special circumstances, and that they can be seen as strategic?manoeuvres where a party attempts to reconcile his dialectical and his rhetorical objectives.     

Happiness in a Flux? The Instability Problem

In the recent discussion of happiness it has become popular to claim that being happy means having a certain positive attitude towards your life. This attitude involves both a judgement that your life measures up to your standards and a feeling of satisfaction with your life. In this paper, I am going to discuss a serious problem inherent in this account that has important ramifications for the normative question of how we ought to pursue happiness. If happiness is in part determined by your standards, how shall we determine whether you are happier in one life than in another when your standards change across these lives? Perhaps you will judge a life as a parent as better than a childless life, if you were to become a parent, but judge a childless life as better than a parenting life, if you were to remain childless. Which standard should determine the comparative happiness of the two lives? In this paper, I shall argue that some innocent-looking answers to this question will generate inconsistencies. To find an acceptable resolution, we need to make a difficult choice between what on the face of it look like two equally valid principles of happiness.     

Merleau‐Ponty on abstract thought in mathematics and natural science

In this paper, I argue that in spite of suggestions to the contrary, Merleau‐Ponty defends a positive account of the kind of abstract thought involved in mathematics and natural science. More specifically, drawing on both the Phenomenology of Perception and his later writings, I show that, for Merleau‐Ponty, abstract thought and perception stand in the two‐way relation of “foundation,” according to which abstract thought makes what we perceive explicit and determinate, and what we perceive is made to appear by abstract thought. I claim that, on Merleau‐Ponty's view, although this process can sometimes lead to falsification, it can also be carried out in such a manner that allows mathematics and natural science to articulate what we perceive in a way that is non‐distortive and in keeping with the demands of perception itself.