Platonic Number in the Parmenides and Metaphysics XIII

I argue here that a properly Platonic theory of the nature of number is still viable today. By properly Platonic, I mean one consistent with Plato's own theory, with appropriate extensions to take into account subsequent developments in mathematics. At Parmenides 143a-4a the existence of numbers is proven from our capacity to count, whereby I establish as Plato's the theory that numbers are originally ordinal, a sequence of forms differentiated by position. I defend and interpret Aristotle's report of a Platonic distinction between form and mathematical numbers, arguing that mathematical numbers alone are cardinals, by reference to certain non-technical features of a set-theoretical approach and other considerations in philosophy of mathematics. Finally I respond to the objections that such a conception of number was unavailable in antiquity and that this theory is contradicted by Aristotle's report in Metaph . XIII that Platonic numbers are collections of units. I argue that Aristotle reveals his own misinterpretation of the terms in which Plato's theory was expressed.     

Non‐Classical Knowledge

The Knower paradox purports to place surprising a priori limitations on what we can know. According to orthodoxy, it shows that we need to abandon one of three plausible and widely‐held ideas: that knowledge is factive, that we can know that knowledge is factive, and that we can use logical/mathematical reasoning to extend our knowledge via very weak single‐premise closure principles. I argue that classical logic, not any of these epistemic principles, is the culprit. I develop a consistent theory validating all these principles by combining Hartry Field's theory of truth with a modal enrichment developed for a different purpose by Michael Caie. The only casualty is classical logic: the theory avoids paradox by using a weaker‐than‐classical K₃ logic. I then assess the philosophical merits of this approach. I argue that, unlike the traditional semantic paradoxes involving extensional notions like truth, its plausibility depends on the way in which sentences are referred to—whether in natural languages via direct sentential reference, or in mathematical theories via indirect sentential reference by Gödel coding. In particular, I argue that from the perspective of natural language, my non‐classical treatment of knowledge as a predicate is plausible, while from the perspective of mathematical theories, its plausibility depends on unresolved questions about the limits of our idealized deductive capacities.     

Establishing Connections between Aristotle's Natural Deduction and First-Order Logic

This article studies the mathematical properties of two systems that model Aristotle's original syllogistic and the relationship obtaining between them. These systems are Corcoran's natural deduction syllogistic and ?ukasiewicz's axiomatization of the syllogistic. We show that by translating the former into a first-order theory, which we call T _RD, we can establish a precise relationship between the two systems. We prove within the framework of first-order logic a number of logical properties about T _RD that bear upon the same properties of the natural deduction counterpart – that is, Corcoran's system. Moreover, the first-order logic framework that we work with allows us to understand how complicated the semantics of the syllogistic is in providing us with examples of bizarre, unexpected interpretations of the syllogistic rules. Finally, we provide a first attempt at finding the structure of that semantics, reducing the search to the characterization of the class of models of T _RD.     

What are numbers?

A number is the number of a class which is an objective, nonactual, mathematical object. The concept of class is analyzed and it is concluded that a number is the number of a pure founded class. A tempting strategy of explaining numbers away is rejected. Some well‐known definitions of numbers are analyzed and it is concluded that this analysis purports the thesis that the unique notion of number does not exist. Numbers are conventional. Nevertheless, an argument is offered purporting the thesis that von Neumann's ordinal numbers are the ordinal numbers. Accordingly, the corresponding von Neumann's cardinal numbers are the numbers.     

Theories and inter‐theory relations in Bošković

During 1745–1755 Bo?kovi? explicitly used the concept of scientific theory in three cases: the theory of forces existing in nature, the theory of transformations of geometric loci, and the theory of infinitesimals. The theory first mentioned became the famous theory of natural philosophy in 1758, the second was published in the third volume of his mathematical textbook Elementorum Universae Matheseos (1754), and the third theory was never completed, though Bo?kovi? repeatedly announced it from 1741 on. The treatment of continuity and infinity in natural philosophy, geometry and infinitesimal analysis brought about inter‐theory relations in Bo?kovi?'s work during his Roman period. The two constructed theories of Bo?kovi?, the theory of forces and the theory of geometric transformations, directly influenced the idea for the construction of his third theory. These written theories refer to understanding and effective application of continuity and infinity in natural philosophy and geometry, and this task, according to Bo?kovi?, requires methodological support from the theory of infinitesimals.     

Numerical predictors of arithmetic success in grades 1–6

Math relies on mastery and integration of a wide range of simpler numerical processes and concepts. Recent work has identified several numerical competencies that predict variation in math ability. We examined the unique relations between eight basic numerical skills and early arithmetic ability in a large sample (N = 1391) of children across grades 1–6. In grades 1–2, children's ability to judge the relative magnitude of numerical symbols was most predictive of early arithmetic skills. The unique contribution of children's ability to assess ordinality in numerical symbols steadily increased across grades, overtaking all other predictors by grade 6. We found no evidence that children's ability to judge the relative magnitude of approximate, nonsymbolic numbers was uniquely predictive of arithmetic ability at any grade. Overall, symbolic number processing was more predictive of arithmetic ability than nonsymbolic number processing, though the relative importance of symbolic number ability appears to shift from cardinal to ordinal processing.     

Language and the use of language

Antony Flew's ‘A Strong Programme for the Sociology of Belief (Inquiry 25 {1982], 365–78) critically assesses the strong programme in the sociology of knowledge defended in David Bloor's Knowledge and Social Imagery. I argue that Flew's rejection of the epistemological relativism evident in Bloor's work begs the question against the relativist and ignores Bloor's focus on the social relativity of mathematical knowledge. Bloor attempts to establish such relativity via a sociological analysis of Frege's theory of number. But this analysis only succeeds if the rejection of an explanatory theory entails that there are reasonable grounds for the rejection of the set of propositions which that theory was intended to explain. I argue against such an entailment, and thus against Bloor's attempt to relativize mathematical knowledge.     

How representations of number and numeracy predict decision paradoxes: A fuzzy-trace theory approach

Higher numeracy has been associated with decision biases in some numerical judgment-and-decision problems. According to fuzzy-trace theory, understanding such paradoxes involves broadening the concept of numeracy to include processing the gist of numbers—their categorical and ordinal relations—in addition to objective (verbatim) knowledge about numbers. We assess multiple representations of gist, as well as numeracy, and use them to better understand and predict systematic paradoxes in judgment and decision-making. In two samples (Ns = 978 and 957), we assessed categorical (some vs. none) and ordinal gist representations of numbers (higher vs. lower, as in relative magnitude judgment, estimation, approximation, and simple ratio comparison), objective numeracy, and a nonverbal, nonnumeric measure of fluid intelligence in predicting: (a) decision preferences exhibiting the Allais paradox and (b) attractiveness ratings of bets with and without a small loss in which the loss bet is rated higher than the objectively superior no-loss bet. Categorical and ordinal gist tasks predicted unique variance in paradoxical decisions and judgments, beyond objective numeracy and intelligence. Whereas objective numeracy predicted choosing or rating according to literal numerical superiority, appreciating the categorical and ordinal gist of numbers was pivotal in predicting paradoxes. These results bring important paradoxes under the same explanatory umbrella, which assumes three types of representations of numbers—categorical gist, ordinal gist, and objective (verbatim)—that vary in their strength across individuals.     

Frege's Cardinals Do Not Always Obey Hume's Principle

Hume's Principle, dear to neo-Logicists, maintains that equinumerosity is both necessary and sufficient for sameness of cardinal number. All the same, Whitehead demonstrated in Principia Mathematica's logic of relations (where non-homogeneous relations are allowed) that Cantor's power-class theorem entails that Hume's Principle admits of exceptions. Of course, Hume's Principle concerns cardinals and in Principia's ‘no-classes’ theory cardinals are not objects in Frege's sense. But this paper shows that the result applies as well to the theory of cardinal numbers as objects set out in Frege's Grundgesetze. Though Frege did not realize it, Cantor's power-theorem entails that Frege's cardinals as objects do not always obey Hume's Principle.     

KORSGAARD'S REJECTION OF CONSEQUENTIALISM

Abstract: In her recent book Self‐Constitution: Agency, Identity, and Integrity, Christine Korsgaard does a wonderful job developing her Kantian account of normativity and the rational necessity of morality. Korsgaard's account of normativity, however, has received its fair share of attention. In this discussion, the focus is on the resulting moral theory and, in particular, on Korsgaard's reason for rejecting consequentialist moral theories. The article suggests that we assume that Korsgaard's vindication of Kantian rationalism is successful and ask whether, nonetheless, her account is consistent with consequentialism. It suggests further that we grant that moral reasons are not based on substantive principles, and that they must instead emerge from the purely formal principles of practical reason. Can consequentialist principles nonetheless emerge from the formal constraints of practical reason? Why can't a consequentialist embrace Korsgaard's account of self‐constitution and normativity?     

Are the Natural Numbers Fundamentally Ordinals?

There are two ways of thinking about the natural numbers: as ordinal numbers or as cardinal numbers. It is, moreover, well‐known that the cardinal numbers can be defined in terms of the ordinal numbers. Some philosophies of mathematics have taken this as a reason to hold the ordinal numbers as (metaphysically) fundamental. By discussing structuralism and neo‐logicism we argue that one can empirically distinguish between accounts that endorse this fundamentality claim and those that do not. In particular, we argue that if the ordinal numbers are metaphysically fundamental then it follows that one cannot acquire cardinal number concepts without appeal to ordinal notions. On the other hand, without this fundamentality thesis that would be possible. This allows for an empirical test to see which account best describes our actual mathematical practices. We then, finally, discuss some empirical data that suggests that we can acquire cardinal number concepts without using ordinal notions. However, there are some important gaps left open by this data that we point to as areas for future empirical research.     

A Transdisciplinary Perspective Concerning the Origin of the Species: The Migratory Theory of Genetic Fitness

Although the Neo-Darwin Theory of Evolution is one of the most celebrated theories in science, nonetheless it has received many criticisms. These criticisms are documented and a new transdisciplinary theory of origin is introduced. Darwin's original argument was that natural selection, through heritable changes, changed simple organisms over time. These heritable changes are responsible for the complex plethora of life seen around us today. Darwin's original theory, however, was deconstructed after the fact into a mutation-based theory. This mutation-based theory in its current form is an insufficient and indeed unnecessary transdisciplinary explanation. A subsequent statistical comparison between the six extant scientifically based primary theories of origin was undertaken and based on current biological knowledge a statistically significantly (p < .05) best fit phenotypic model emerged.     

Identifying the Source of Misfit in Item Response Theory Models

When an item response theory model fails to fit adequately, the items for which the model provides a good fit and those for which it does not must be determined. To this end, we compare the performance of several fit statistics for item pairs with known asymptotic distributions under maximum likelihood estimation of the item parameters: (a) a mean and variance adjustment to bivariate Pearson's X², (b) a bivariate subtable analog to Reiser's (1996) overall goodness-of-fit test, (c) a z statistic for the bivariate residual cross product, and (d) Maydeu-Olivares and Joe's (2006) M₂ statistic applied to bivariate subtables. The unadjusted Pearson's X² with heuristically determined degrees of freedom is also included in the comparison. For binary and ordinal data, our simulation results suggest that the z statistic has the best Type I error and power behavior among all the statistics under investigation when the observed information matrix is used in its computation. However, if one has to use the cross-product information, the mean and variance adjusted X² is recommended. We illustrate the use of pairwise fit statistics in 2 real-data examples and discuss possible extensions of the current research in various directions.     

Assessing Approximate Fit in Categorical Data Analysis

A family of Root Mean Square Error of Approximation (RMSEA) statistics is proposed for assessing the goodness of approximation in discrete multivariate analysis with applications to item response theory (IRT) models. The family includes RMSEAs to assess the approximation up to any level of association of the discrete variables. Two members of this family are RMSEA₂, which uses up to bivariate moments, and the full information RMSEA_n. The RMSEA₂ is estimated using the M₂ statistic of Maydeu-Olivares and Joe (2005, 2006), whereas for maximum likelihood estimation, RMSEA_n is estimated using Pearson's X² statistic. Using IRT models, we provide cutoff criteria of adequate, good, and excellent fit using the RMSEA₂. When the data are ordinal, we find a strong linear relationship between the RMSEA₂ and the Standardized Root Mean Squared Residual goodness-of-fit index. We are unable to offer cutoff criteria for the RMSEA_n as its population values decrease as the number of variables and categories increase.     

ADOLF GRÜNBAUM ON THE STEADY‐STATE THEORY AND CREATIO CONTINUA OF MATTER OUT OF NOTHING

Abstract The ideas of creatio ex nihilo of the universe and creatio continua of new matter out of nothing entered the arena of natural science with the advent of the Big Bang and the steady‐state theories in the mid‐twentieth century. Adolf Grünbaum has tried to interpret the steady‐state theory in such a way, to show that the continuous formation of new matter out of nothing in this theory can be explained purely physically. In this paper, however, it will be shown that Grünbaum's interpretation encounters at least three problems: not distinguishing between material and efficient causes, inconsistency, and misconceiving the law of density conservation.     

Mathematical Proof Theory in the Light of Ordinal Analysis

We give an overview of recent results in ordinal analysis. Therefore,we discuss the different frameworks used in mathematical proof-theory, namely subsystem of analysis including reversemathematics, Kripke–Platek set theory, explicitmathematics, theories of inductive definitions,constructive set theory, and Martin-Löfs typetheory.     

Identity in Fiction

Anthony Everett ( 2005 ) argues that those who embrace the reality of fictional entities run into trouble when it comes to specifying criteria of character identity. More specifically, he argues that realists must reject natural principles governing the identity and distinctness of fictional characters due to the existence of fictions which leave it indeterminate whether certain characters are identical and the existence of fictions which say inconsistent things about the identities of their characters. Everett's critique has deservedly drawn much attention and a number of defensive moves have been made by, or on the behalf of, fictional realists. My goal in this paper is to move this debate on a further step. I have three goals: (i) to clarify the importance of Everett's discussion of identity criteria within the context of fictional realism, (ii) to reassess Everett's objections to realism in light of the resultant literature, and (iii) to develop a novel strategy for responding to Everett's concerns. On the approach to be developed, the problems emerge due to an indeterminacy inherent in the concept of a fictional character itself.     

The metaphysics of quantity

A formal theory of quantity T _Q is presented which is realist, Platonist, and syntactically second-order (while logically elementary), in contrast with the existing formal theories of quantity developed within the theory of measurement, which are empiricist, nominalist, and syntactically first-order (while logically non-elementary). T _Q is shown to be formally and empirically adequate as a theory of quantity, and is argued to be scientifically superior to the existing first-order theories of quantity in that it does not depend upon empirically unsupported assumptions concerning existence of physical objects (e.g. that any two actual objects have an actual sum). The theory T _Q supports and illustrates a form of naturalistic Platonism, for which claims concerning the existence and properties of universals form part of natural science, and the distinction between accidental generalizations and laws of nature has a basis in the second-order structure of the world.     

Hubert's multi-rater kappa revisited

There is a frequent need to measure the degree of agreement among R observers who independently classify n subjects within K nominal or ordinal categories. The most popular methods are usually kappa-type measurements. When R = 2, Cohen's kappa coefficient (weighted or not) is well known. When defined in the ordinal case while assuming quadratic weights, Cohen's kappa has the advantage of coinciding with the intraclass and concordance correlation coefficients. When R > 2, there are more discrepancies because the definition of the kappa coefficient depends on how the phrase ‘an agreement has occurred’ is interpreted. In this paper, Hubert's interpretation, that ‘an agreement occurs if and only if all raters agree on the categorization of an object’, is used, which leads to Hubert's (nominal) and Schuster and Smith's (ordinal) kappa coefficients. Formulae for the large-sample variances for the estimators of all these coefficients are given, allowing the latter to illustrate the different ways of carrying out inference and, with the use of simulation, to select the optimal procedure. In addition, it is shown that Schuster and Smith's kappa coefficient coincides with the intraclass and concordance correlation coefficients if the first coefficient is also defined assuming quadratic weights.     

The Role of Structural Reasoning in the Genesis of Graph Theory

The concept of quantity (Größe) plays a key role in Frege's theory of real numbers. Typically enough, he refers to this theory as ‘theory of quantity’ (‘Größenlehre’) in the second volume of his opus magnum Grundgesetze der Arithmetik (Frege 1903). In this essay, I deal, in a critical way, with Frege's treatment of the concept of quantity and his approach to analysis from the beginning of his academic career until Frege 1903. I begin with a few introductory remarks. In Section 2, I first analyze Frege's use of the term ‘source of knowledge’ (‘Erkenntnisquelle’) with particular emphasis on the logical source of knowledge. The analysis includes a brief comparison between Frege and Kant's conceptions of logic and the logical source of knowledge. In a second step, I examine Frege's theory of quantity in Rechnungsmethoden, die sich auf eine Erweiterung des Größenbegriffes gründen (Frege 1874). Section 3 contains a couple of critical observations on Frege's comments on Hankel's theory of real numbers in Die Grundlagen der Arithmetik (Frege 1884). In Section 4, I consider Frege's discussion of the concept of quantity in Frege 1903. Section 5 is devoted to Cantor's theory of irrational numbers and the critique deployed by Frege. In Section 6, I return to Frege's own constructive treatment of analysis in Frege 1903 and succinctly describe what I take to be the quintessence of his account.