Chaos (SEP)

The big news about chaos is supposed to be that the smallest of changes in a system can result in very large differences in that system’s behavior. The so-called butterfly effect has become one of the most popular images of chaos. The idea is that the flapping of a butterfly’s wings in Argentina could cause a tornado in Texas three weeks later. By contrast, in an identical copy of the world sans the Argentinian butterfly, no such tornado would have occurred in Texas. The mathematical version of this property is known as sensitive dependence and such sensitivity has implications for predictability of future behavior. Clarifying sensitive dependence’s significance is important given there have always been limits on prediction. Chaos studies have highlighted these implications in fresh ways, enabled forms of mitigation as well as control of chaos, and have led to other implications for how we think about our world.

In addition to exhibiting sensitive dependence, chaotic systems are deterministic and nonlinear and exhibit aperiodic behavior (Lorenz 1963). This entry discusses systems exhibiting these properties and their philosophical implications. For those not familiar with the basic phenomenology of chaos, reading nontechnical treatments such as Smith (2007) or Bishop (2023) is highly recommended. Because of the distinctive nature of quantum chaos, it is treated separately in the Supplement: Quantum Chaos, needed for discussions of broader implications in §6.

Read the rest here: https://plato.stanford.edu/entries/chaos/

Chance versus Randomness

Certainty (SEP)

Certainty, or the attempt to obtain certainty, has played a central role in the history of philosophy. Some philosophers have taken the kind of certainty characteristic of mathematical knowledge to be the goal at which philosophy should aim. In the Republic, Plato says that geometry “draws the soul towards truth and produces philosophic thought by directing upwards what we now wrongly direct downwards” (527b). Descartes also thought that a philosophical method that proceeds in a mathematical way, enumerating and ordering everything exactly, “contains everything that gives certainty to the rules of mathematics” (Discourse on the Method, PW 1, p. 121). Other philosophers have adopted different models for how to best understand certainty. For example, Aristotle and Aquinas take scientific explanation to be essential to certainty (see Pasnau 2017, pp. 5–7), while Al-Ghazālī thought that certainty arose out of religious practice (see Albertini 2005). For many empiricists, certainty with respect to empirical matters is to be found in basic beliefs that are grounded in some fundamental aspect of perceptual experience (see, e.g., Lewis 1952).

Like knowledge, certainty is an epistemic property of beliefs. (In a derivative way, certainty is also an epistemic property of subjects: S is certain that p just in case S’s belief that p is certain.) Although some philosophers have thought that there is no difference between knowledge and certainty, it has become increasingly common to distinguish them. On this conception, then, certainty is either the highest form of knowledge or is the only epistemic property superior to knowledge. One of the primary motivations for allowing kinds of knowledge less than certainty is the widespread sense that skeptical arguments are successful in showing that we rarely or never have beliefs that are certain (see Unger 1975 for this kind of skeptical argument) but do not succeed in showing that our beliefs are altogether without epistemic worth (see, for example, Lehrer 1974, Williams 1999, and Feldman 2003; see Fumerton 1995 for an argument that skepticism undermines every epistemic status a belief might have; and see Klein 1981 for the argument that knowledge requires certainty, which we are capable of having).

As with knowledge, it is difficult to provide an uncontentious analysis of certainty. There are several reasons for this. One is that there are different kinds of certainty, which are easy to conflate. Another is that the full value of certainty is surprisingly hard to capture. A third reason is that there are two dimensions to certainty: a belief can be individually certain at a particular moment, or it can be certain over some greater length of time in a system of beliefs.

Read the rest here: https://plato.stanford.edu/entries/certainty/

Category Mistakes (SEP)

Category mistakes are sentences such as ‘The number two is blue’, ‘The theory of relativity is eating breakfast’, or ‘Green ideas sleep furiously’. Such sentences are striking in that they are highly odd or infelicitous, and moreover infelicitous in a distinctive sort of way. For example, they seem to be infelicitous in a different way to merely trivially false sentences such as ‘2+2=5
’ or obviously ungrammatical strings such as ‘The ran this’.

The majority of contemporary discussions of the topic are devoted to explaining what makes category mistakes infelicitous, and a wide variety of accounts have been proposed including syntactic, semantic, and pragmatic explanations. Indeed, this is part of what makes category mistakes a particularly important topic: a theory of what makes category mistakes unacceptable can potentially shape our theories of syntax, semantics, or pragmatics and the boundaries between them. As Camp (2016, 611–612) explains: “Category mistakes … are theoretically interesting precisely because they are marginal: as by-products of our linguistic and conceptual systems lacking any obvious function, they reveal the limits of, and interactions among, those systems. Do syntactic or semantic restrictions block ‘is green’ from taking ‘Two’ as a subject? Does the compositional machinery proceed smoothly, but fail to generate a coherent proposition or delimit a coherent possibility? Or is the proposition it produces simply one that our paltry minds cannot grasp, or that fails to arouse our interest? One’s answers to these questions depend on, and constrain, one’s conceptions of syntax, semantics, and pragmatics, of language and thought, and of the relations among them and between them and the world.”

Moreover, the question of how to account for the infelicity of category mistakes has implications for a variety of other philosophical questions. For example, in metaphysics, it is often argued that a statue must be distinct from the lump of clay from which it is made because ‘The statue is Romanesque’ is true, while ‘The lump of clay is Romanesque’ is not—indeed, the latter ascription arguably constitutes a category mistake. Correspondingly, an assessment of this argument depends on one’s account of category mistakes (see §4.3 below).

Read the rest here: https://plato.stanford.edu/entries/category-mistakes/

Medieval Theories of the Categories (SEP)

This entry is intended as a brief and general introduction to the development of category theory from the beginning of the Middle Ages, in the sixth century, to the Silver Age of Scholasticism, in the sixteenth. This development is fascinating but extraordinarily complex. Scholars are just beginning to take note of the major differences in the understanding of categories and of how these differences are related to the discussion of other major philosophical topics in the Middle Ages. Much work remains to be done, even regarding the views of towering figures, so necessarily we have had to restrict our discussion to only a few major figures and topics. Still, we hope that the discussion will serve as a good starting point for anyone interested in category theory and its history.

1. Issues

Philosophers speak about categories in many different ways. There is one initial, and rather substantial, difference between philosophers who allow a very large number of categories and those who allow only a very small number. The first include among categories such different things as human, green, animal, thought, and justice; the second speak only of very general things such as substance, quality, relation, and the like, as categories. Among twentieth-century authors who allow many categories is Gilbert Ryle (b. 1900, d. 1976). Roderick Chisholm (b. 1916, d. 1999) is an example of those who have only very few. Medieval authors follow Aristotle’s narrow understanding.

The disagreement concerning categories in the history of philosophy does not end there. Even if we restrict the discussion to a small number of items of the sort that Aristotle regards as categories, many issues remain to be settled about them, and philosophers frequently disagree about how to settle them. These issues may be gathered into roughly ten groups.

The first group comprises what may be described roughly as extensional issues; they have to do with the number of categories. The extension of a term is comprised by the things of which the term can be truthfully predicated. Thus the extension of ‘cat’ consists of all the animals of which it is true to say that they are cats. Philosophers in general frequently disagree on how many categories there are. For example, Aristotle lists up to ten, but gives the impression that the ultimate number is not settled at all. Plotinus (204/5–270) and Baruch Spinoza (1632–77) reduce the number radically, but their views do not by any means establish themselves as definitive. In the Middle Ages the number of categories is always small (ten or less) but it nonetheless varies.

Read the rest here: https://plato.stanford.edu/entries/medieval-categories/

Bounded Rationality (SEP)

Herbert Simon introduced the term ‘bounded rationality’ (Simon 1957b: 198; see also Klaes & Sent 2005) as shorthand for his proposal to replace the perfect rationality assumptions of homo economicus with a concept of rationality better suited to cognitively limited agents:

Broadly stated, the task is to replace the global rationality of economic man with the kind of rational behavior that is compatible with the access to information and the computational capacities that are actually possessed by organisms, including man, in the kinds of environments in which such organisms exist. (Simon 1955a: 99)

Bounded rationality now describes a wide range of descriptive, normative, and prescriptive accounts of effective behavior which depart from the assumptions of perfect rationality. This entry aims to highlight key contributions—from the decision sciences, economics, cognitive- and neuropsychology, biology, physics, computer science, and philosophy—to our current understanding of bounded rationality.

1. Homo Economicus and Expected Utility Theory

Bounded rationality has come to encompass models of effective behavior that weaken, or reject altogether, the idealized conditions of perfect rationality assumed by models of economic man. In this section we state what models of economic man are committed to and their relationship to expected utility theory. In later sections we review proposals for departing from expected utility theory.

The perfect rationality of homo economicus imagines a hypothetical agent who has complete information about the options available for choice, perfect foresight of the consequences from choosing those options, and the wherewithal to solve an optimization problem (typically of considerable complexity) that identifies an option which maximizes the agent’s personal utility.

Read the rest here: https://plato.stanford.edu/entries/bounded-rationality/

Boundary (SEP)

We think of a boundary whenever we think of an entity demarcated from its surroundings. There is a boundary (a line) separating Maryland and Pennsylvania. There is a boundary (a circle) isolating the interior of a disc from its exterior. There is a boundary (a surface) enclosing the bulk of this apple. Sometimes the exact location of a boundary is unclear or otherwise controversial (as when you try to trace out the borders of a desert, the edges of a mountain, or even the boundary of your own body). Sometimes the boundary lies skew to any physical discontinuity or qualitative differentiation (as with the border of Wyoming, or the boundary between the upper and the lower halves of a homogeneous sphere). But whether sharp or blurry, natural or artificial, for every object there appears to be a boundary that marks it off from the rest of the world. Events, too, have boundaries — at least temporal boundaries. Our lives are bounded by our births and by our deaths; the soccer game began at 3pm sharp and ended with the referee’s final whistle at 4:45pm. It is sometimes suggested that even abstract entities, such as concepts or sets, have boundaries of their own (witness the popular method for representing the latter by means of simple closed curves encompassing their contents, as in Euler circles and Venn diagrams), and Wittgenstein could emphatically proclaim that the boundaries of our language are the boundaries of our world (1921: prop. 5.6). Whether all this boundary talk is coherent, however, and whether it reflects the structure of the world or simply the organizing activity of our mind, or of our collective practices and conventions, are matters of deep philosophical controversy.

1. Issues

Euclid defined a boundary as “that which is an extremity of anything” (Elements, I, def. 13). Aristotle defined the extremity of a thing as “the first thing beyond which it is not possible to find any part [of the given thing], and the first within which every part is” (Metaphysics, V, 1022a4–5). Together, these two definitions deliver the classic account of boundaries, an account that is both intuitive and comprehensive and offers the natural starting point for any further investigation into the boundary concept. Indeed, although Aristotle’s definition concerned primarily the extremities of spatial entities, it applies equally well in the temporal domain. Just as the Mason-Dixon line marks the boundary between Maryland and Pennsylvania insofar as no part of Maryland can be found on the northern side of the line, and no part of Pennsylvania on its southern side, so “the now is an extremity of the past (no part of the future being on this side of it), and again of the future (no part of the past being on that side of it)” (Physics, VI, 233b35–234a2). Similarly for concrete objects and events: just as the surface of an apple marks its spatial boundary insofar as the apple extends only up to it, so the referee’s whistle marks the temporal boundary of the game insofar as the game protracts only up to it. In the case of abstract entities, such as concepts and sets, the account is perhaps adequate only figuratively. Still, it is telling that one of the Greek words for ‘boundary’, ὅρος, is also a word for ‘definition’: as John of Damascus nicely put it, “definition is the term for the setting of land boundaries taken in a metaphorical sense” (The Fount of Knowledge, I, 8). Likewise, it is telling that in point-set topology the standard definition of a set’s boundary (from Hausdorff 1914, §7.2) reflects essentially the same intuition: the boundary, or frontier, of a set A is the set of those points all of whose neighborhoods intersect both A and the complement of A (where a neighborhood of a point p is, intuitively, a set of points that entirely “surround” p). It is not an exaggeration, therefore, to say that the Euclidean-Aristotelian characterization captures a general intuition about boundaries that applies across the board. Nonetheless, precisely this intuitive characterization gives rise to several puzzles that justify philosophical concern.

Read the rest here: https://plato.stanford.edu/entries/boundary/