James Franklin

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.

The logical interpretation of probability, or "objective Bayesianism'' – the theory that (some) probabilities are strictly logical degrees of partial implication – is defended. The main argument against it is that it requires the assignment of prior probabilities, and that any attempt to determine them by symmetry via a "principle of insufficient reason" inevitably leads to paradox. Three replies are advanced: that priors are imprecise or of little weight, so that disagreement about them does not matter, within limits; that it is possible to distinguish reasonable from unreasonable priors on logical grounds; and that in real cases disagreement about priors can usually be explained by differences in the background information. It is argued also that proponents of alternative conceptions of probability, such as frequentists, Bayesians and Popperians, are unable to avoid committing themselves to the basic principles of logical probability.

Review of Jaynes, Probability Theory: The Logic of Science; Marrison, The Fundamentals of Risk Management; and Hastie, Tibshirani and Friedman, The Elements of Statistical Learning. A standard view of probability and statistics centers on distributions and hypothesis testing. To solve a real problem, say in the spread of disease, one chooses a “model”, a distribution or process that is believed from tradition or intuition to be appropriate to the class of problems in question. One uses data to estimate the parameters of the model, and then delivers the resulting exactly specified model to the customer for use in prediction and classification. As a gateway to these mysteries, the combinatorics of dice and coins are recommended; the energetic youth who invest heavily in the calculation of relative frequencies will be inclined to protect their investment through faith in the frequentist philosophy that probabilities are all really relative frequencies. Those with a taste for foundational questions are referred to measure theory, an excursion from which few return. That picture, standardised by Fisher and Neyman in the 1930s, has proved in many ways remarkably serviceable. It is especially reasonable where it is known that the data are generated by a physical process that conforms to the model. It is not so useful where the data is a large and little understood mess, as is typical in, for example, insurance data being investigated for fraud. Nor is it suitable where one has several speculations about possible models and wishes to compare them, or..

Aristotle divided arguments that persuade into the rhetorical (which happen to persuade), the dialectical (which are strong so ought to persuade to some degree) and the demonstrative (which must persuade if rightly understood). Dialectical arguments were long neglected, partly because Aristotle did not write a book about them. But in the sixteenth and seventeenth century late scholastic authors such as Medina, Cano and Soto developed a sound theory of probable arguments, those that have logical and not merely psychological force but fall short of demonstration. Informed by late medieval treatments of the law of evidence and problems in moral theology and aleatory contracts, they considered the reasons that could render legal, moral, theological, commercial and historical arguments strong though not demonstrative. At the same time, demonstrative arguments became better understood as Galileo and other figures of the Scientific Revolution used mathematical proof in arguments in physics. Galileo moved both dialectical and demonstrative arguments into mathematical territory.

While the consequences of mathematically-based software, algorithms and strategies have become ever wider and better appreciated, ethical reflection on mathematics has remained primitive. We review the somewhat disconnected suggestions of commentators in recent decades with a view to piecing together a coherent approach to ethics in mathematics. Calls for a Hippocratic Oath for mathematicians are examined and it is concluded that while lessons can be learned from the medical profession, the relation of mathematicians to those affected by their work is significantly different. There is something to be learned also from the codes of conduct of cognate but professionalised quantitative disciplines such as engineering and accountancy, as well as from legal principles bearing on professional work. We conclude with recommendations that professional societies in mathematics should sponsor an (international) code of ethics, institutional mission statements for mathematicians and syllabuses of ethics courses for incorporation into mathematics degrees.

The death of a person is a tragedy while the explosion of a lifeless galaxy is a mere firework. The moral difference is grounded in the nature of humans: humans have intrinsic worth, a worth that makes their fate really matter. This is the worth proposed as the foundation of ethics. Ethics in the usual sense of right and wrong actions, rights and virtues, and how to live a good life, is founded on something more basic that is not itself about actions, namely the worth of persons. Human moral worth arises from certain properties that distinguish humans from the rest of creation (though some animals share a lesser degree of those properties): rationality, consciousness, the ability to act for reasons, emotional structure and love, individuality. This complex package makes humans the "piece of work" of which Hamlet says, “How noble in reason, how infinite in faculty." The Worth of Persons establishes a foundation for ethics in the equal worth of persons, which makes ethics absolutely objective and immune to relativist attacks because it is based on the metaphysical truth about humans. Endless debate about ethical dilemmas, rules, and principles fails to connect with what is really important ethically, that is, what makes humans matter.

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.

In order to fulfil their essential roles as the bearers of truth and the relata of logical relations, propositions must be public and shareable. That requirement has favoured Platonist and other nonmental views of them, despite the well-known problems of Platonism in general. Views that propositions are mental entities have correspondingly fallen out of favour, as they have difficulty in explaining how propositions could have shareable, objective properties. We revive a mentalist view of propositions, inspired by Artificial Intelligence work on perceptual algorithms, which shows how perception causes persistent mental entities with shareable properties that allow them to fulfil the traditional roles of (one core kind of) propositions. The clustering algorithms implemented in perception produce outputs which are (implicit) atomic propositions in different minds. Coordination of them across minds proceeds by game-theoretic processes of communication. The account does not rely on any unexplained notions such as mental content, representation, or correspondence (although those notions are applicable in philosophical analysis of the result).

There is a wide range of realist but non-Platonist philosophies of mathematics—naturalist or Aristotelian realisms. Held by Aristotle and Mill, they played little part in twentieth century philosophy of mathematics but have been revived recently. They assimilate mathematics to the rest of science. They hold that mathematics is the science of X, where X is some observable feature of the (physical or other non-abstract) world. Choices for X include quantity, structure, pattern, complexity, relations. The article lays out and compares these options, including their accounts of what X is, the examples supporting each theory, and the reasons for identifying the science of X with (most or all of) mathematics. Some comparison of the options is undertaken, but the main aim is to display the spectrum of viable alternatives to Platonism and nominalism. It is explained how these views answer Frege’s widely accepted argument that arithmetic cannot be about real features of the physical world, and arguments that such mathematical objects as large infinities and perfect geometrical figures cannot be physically realized.

Antitheodicy objects to all attempts to solve the problem of evil. Its objections are almost all on moral grounds—it argues that the whole project of theodicy is morally offensive. Trying to excuse God’s permission of evil is said to deny the reality of evil, to exhibit gross insensitivity to suffering, and to insult the victims of grave evils. Since antitheodicists urge the avoidance of theodicies for moral reasons, it is desirable to evaluate the moral reasons against theodicies in abstraction from the intellectual reasons for and against them. It is argued that the best known theodicies such as those based on soul-making and free will are guilty of moral faults as alleged. But Leibniz’s best of all possible worlds theory, often thought to be the most morally offensive ‘Panglossian’ theodicy, is morally blameless because it excuses God by the absolute impossibility of his choosing any world better than the present one. Theodicy should not be conceived of as a search for greater goods which may excuse God’s permitting evils. From the divine point of view, creation is an upfront choice between scenarios—in modern parlance, a Trolley problem rather than a Transplant problem. In cases of forced choice among scenarios, it is morally improper to criticize one who chooses the best.

Among worldviews, in addition to the options of materialist atheism, pantheism and personal theism, there exists a fourth, “local emergentism”. It holds that there are no gods, nor does the universe overall have divine aspects or any purpose. But locally, in our region of space and time, the properties of matter have given rise to entities which are completely different from matter in kind and to a degree god-like: consciousnesses with rational powers and intrinsic worth. The emergentist option is compared with the standard alternatives and the arguments for and against it are laid out. It is argued that, among options in the philosophy of religion, it involves the minimal reworking of the manifest image of common sense. Hence it deserves a place at the table in arguments as to the overall nature of the universe.

Fifty years of effort in artificial intelligence (AI) and the formalization of legal reasoning have produced both successes and failures. Considerable success in organizing and displaying evidence and its interrelationships has been accompanied by failure to achieve the original ambition of AI as applied to law: fully automated legal decision-making. The obstacles to formalizing legal reasoning have proved to be the same ones that make the formalization of commonsense reasoning so difficult, and are most evident where legal reasoning has to meld with the vast web of ordinary human knowledge of the world. Underlying many of the problems is the mismatch between the discreteness of symbol manipulation and the continuous nature of imprecise natural language, of degrees of similarity and analogy, and of probabilities.

Let me tell you what philosophy is about, then about how Sydney does it in its own special way. Does life have a meaning, and if so what is it? What can I be certain of, and how should I act when I am not certain? Why are the established truths of my tribe better than the primitive superstitions of your tribe? Why should I do as I’m told? Those are questions it’s easy to avoid, in the rush to acquire goods and prestige. Even for many of a more serious outlook, they are questions easy to dismiss with excuses like “it’s all a matter of opinion” or “let’s get on with practical matters” or “they’re too hard”. They are questions that may be ignored, but they do not go away. They’re philosophical questions. There’s a right way to approach them – you read the writings of the classical and recent philosophers and consider carefully their arguments back and forth. There are many wrong ways to approach them, such as choosing at random among the ideas your parents or friends or gurus have, or ideas that feel good. Or you can just not bother. Sydney has a certain reputation for superficiality in this regard. A character in David Williamson’s Emerald City says “No-one in Sydney ever wastes time debating the meaning of life — it’s getting yourself a water frontage”, says If you have a Writers Festival or a conference on Happiness in Sydney, you don’t normally expect philosophers to be invited. Caroline Jones’ radio series, ‘The Search for Meaning’.

Probabilistic inference from frequencies, such as "Most Quakers are pacifists; Nixon is a Quaker, so probably Nixon is a pacifist" suffer from the problem that an individual is typically a member of many "reference classes" (such as Quakers, Republicans, Californians, etc) in which the frequency of the target attribute varies. How to choose the best class or combine the information? The article argues that the problem can be solved by the feature selection methods used in contemporary Big Data science: the correct reference class is that determined by the features relevant to the target, and relevance is measured by correlation (that is, a feature is relevant if it makes a difference to the frequency of the target).

Argues that information, in the animal behaviour or evolutionary context, is correlation/covariation. The alternation of red and green traffic lights is information because it is (quite strictly) correlated with the times when it is safe to drive through the intersection; thus driving in accordance with the lights is adaptive (causative of survival). Daylength is usefully, though less strictly, correlated with the optimal time to breed. Information in the sense of covariance implies what is adaptive; if an animal can infer what the information implies, it increases its chances of survival.

Defends the cosmological argument for the existence of God against Hume's criticisms. Hume objects that since a cause is before its effect, an eternal succession has no cause; but that would rule of by fiat the possibility of God's creating the world from eternity. Hume argues that once a cause is given for each of a collection of objects, there is not need to posit a cause of the whole collection; but that is to assume the universe to be a heap of things arbitrarily grouped rather than a whole arbitrarily divided.

Some courses achieve existence, some have to create Professional Issues and Ethics in existence thrust upon them. It is normally Mathematics; but if you don’t do it, we will a struggle to create a course on the ethical be.” I accepted. or social aspects of science or mathematics. The gift of a greenfield site and a bull- This is the story of one that was forced to dozer is a happy occasion, undoubtedly. But exist by an unusual confluence of outside cirwhat to do next? It seemed to me I should cumstances. ensure the course satisfied these require- In the mid 1990s, the University of New ments: South Wales instituted a policy that all its • It should look good to students, to staff.

Brains, unlike artificial neural nets, use symbols to summarise and reason about perceptual input. But unlike symbolic AI, they “ground” the symbols in the data: the symbols have meaning in terms of data, not just meaning imposed by the outside user. If neural nets could be made to grow their own symbols in the way that brains do, there would be a good prospect of combining neural networks and symbolic AI, in such a way as to combine the good features of each. The article argues the cluster analysis provides algorithms to perform the task, and that any solution to the task must be a form of cluster analysis