Modernizing the philosophy of mathematics

The distinction between analytic and synthetic propositions, and with that the distinction between a priori and a posteriori truth, is being abandoned in much of analytic philosophy and the philosophy of most of the sciences. These distinctions should also be abandoned in the philosophy of mathematics. In particular, we must recognize the strong empirical component in our mathematical knowledge. The traditional distinction between logic and mathematics, on the one hand, and the natural sciences, on the other, should be dropped. Abstract mathematical objects, like transcendental numbers or Hilbert spaces, are theoretical entities on a par with electromagnetic fields or quarks. Mathematical theories are not primarily logical deductions from axioms obtained by reflection on concepts but, rather, are constructions chosen to solve some collection of problems while fitting smoothly into the other theoretical commitments of the mathematician who formulates them. In other words, a mathematical theory is a scientific theory like any other, no more certain but also no more devoid of content.     

The Development of Proportional Reasoning: Effect of Continuous Versus Discrete Quantities

This study examines the development of children's ability to reason about proportions that involve either discrete entities or continuous amounts. Six-, 8- and 10-year olds were presented with a proportional reasoning task in the context of a game involving probability. Although all age groups failed when proportions involved discrete quantities, even the youngest age group showed some success when proportions involved continuous quantities. These findings indicate that quantity type strongly affects children's ability to make judgments of proportion. Children's greater success in judging proportions involving continuous quantities appears to be related to their use of different strategies in the presence of countable versus noncountable entities. In two discrete conditions, children—particularly 8- and 10-year-olds—adopted an erroneous counting strategy, considering the number of target elements but not the relation between target and nontarget elements, either in terms of number or amount. In contrast, in the continuous condition, when it was not possible to count, children may have relied on an early developing ability to code the relative amounts of target and nontarget regions.     

The Irony of Measurement by Subjective Estimations

In 1948 S. S. Stevens, in his famous Science article, proposed a theory of measurement that radically differed from the dominate theory of the time. The dominate theory held that all strong forms of scientific measurement— for example, those that yielded ratio scales—had to be based on an observable ordering and an observable commutative and associative operation. Stevens proposed different criteria and introduced his method of magnitude estimation. Stevens as well as measurement theorists considered his method to be radically different from those based on commutative and associative operations. Although his method was controversial, it became a standard tool in the behavioral sciences. This article argues that Stevens' method, together with implicit assumptions he made about the scales of measurement it generated, is from a mathematical perspective the same as the measurement process based on commutative and associative operations. The article also provides a theory of qualitative numbers and shows an interesting relationship between qualitative numbers and Stevens' method.     

Photographic Evidence and the Problem of Theory-Ladenness

Scientists use visualisations of different kinds in a variety of ways in their scientific work. In the following article, we will take a closer look at the use of photographic pictures as scientific evidence. In accordance with Patrick Maynard’s thesis, photography will be regarded as a family of technologies serving different purposes in divergent contexts. One of these is its ability to detect certain phenomena. Nonetheless, with regard to the philosophical thesis of theory-ladenness of observation, we encounter certain reservations concerning the status of photography and that of photographic pictures in the process of measurement in science. Accordingly, the aim of this paper is twofold: We will discuss suggested solutions both for the technological and for the psychological part of the problem of theory-ladenness appearing in the context of the use of photography in scientific observations. The essential proposal will be to follow Christian Suhm in his advice to make a distinction between theory-relativity and theory-ladenness.     

Germs and angels: the role of testimony in young children's ontology

In three experiments, children's reliance on other people's testimony as compared to their own, first-hand experience was assessed in the domain of ontology. Children ranging from 4 to 8 years were asked to judge whether five different types of entity exist: real entities (e.g. cats, trees) whose existence is evident to everyone; scientific entities (e.g. germs, oxygen) that are normally invisible but whose existence is generally presupposed in everyday discourse; endorsed beings (e.g. the Tooth Fairy, Santa Claus) whose existence is typically endorsed in discourse with young children; equivocal beings (e.g. monsters, witches) whose existence is not typically endorsed in discourse with young children; and impossible entities (e.g. flying pigs, barking cats) that nobody believes in. Children make a broad dichotomy between entities and beings that they claim to exist (real entities; scientific entities; and endorsed beings) and those whose existence they deny (equivocal beings and impossible entities). They also make a more fine-grained distinction among the invisible entities that they claim to exist. Thus, they assert the existence of scientific entities such as germs with more confidence than that of endorsed beings such as Santa Claus. The findings confirm that children's ontological claims extend beyond their first-hand encounters with instances of a given category. Children readily believe in entities that they cannot see for themselves but have been told about. Their confidence in the existence of those entities appears to vary with the pattern of testimony that they receive.     

Living with the abstract: realism and models

A natural way to think of models is as abstract entities. If theories employ models to represent the world, theories traffic in abstract entities much more widely than is often assumed. This kind of thought seems to create a problem for a scientific realist approach to theories. Scientific realists claim theories should be understood literally. Do they then imply (and are they committed to) the reality of abstract entities? Or are theories simply—and incurably—false (if there are no abstract entities)? Or has the very idea of literal understanding to be abandoned? Is then fictionalism towards scientific theories inevitable? This paper argues that scientific realism can happily co-exist with models qua abstracta.     

ERKENNTNISTHEORETISCHE UND ONTOLOGISCHE PROBLEME DER THEORETISCHEN BEGRIFFE

Operationalism and theoretical entities. The thesis of the“theory ladenness” of observation leads to an antinomy. In order to solve this antinomy a technical operationalism is sketched, according to which theories should in principle not contain anything that cannot be reduced to technical procedures. This implies the rejection of Quine's underdeterminacy thesis and of many views about the theoretical-observational distinction, e.g. neopositivistic views, van Fraassen's view, Sneed-Stegmüller's view. Then I argue for the following theses: 1. All scientific concepts are theory laden in the sense that they allow us to anticipate possible experiences, but they have to be in principle fully observable, i.e. integrally convertible into operational-technical applications. 2. The observation/theory distinction can be maintained as a historical one: what is observable depends on the instruments that are available at any stage of the development of science. 3.In principle theoretical entities are empirically real in Hacking's sense. However, some aspects of Hacking's realism are to be criticized. Theoretical entities are to be resolved into the totality of the interrelated properties accessible to us by means of theoretical points of view embodied in scientific instruments.     

Relating numeric cognition and language processing: Do numbers and words share a common representational platform?

Numerical processing and language processing are both grounded in space. In the present study we investigated whether these are fully independent phenomena, or whether they share a common basis. If number processing activates spatial dimensions that are also relevant for understanding words, then we can expect that processing numbers may influence subsequent lexical access to words. Specifically, if high numbers relate to upper space, then they can be expected to facilitate understanding of words such as bird that are having referents typically found in the upper vertical space. The opposite should hold for low numbers. These should facilitate the understanding of words such as ground referring to entities with referents in the lower vertical space. Indeed, in two experiments we found evidence for such an interaction between number and word processing. By eliminating a contribution of linguistic factors gained from additional investigations on large text corpora, this strongly suggests that understanding numbers and language is based on similar modal representations in the brain. The implications of these findings for a broader perspective on grounded cognition will be discussed.     

Function and Argument in Begriffsschrift

It is well known that the formal system developed by Frege in Begriffsschrift is based upon the distinction between function and argument—as opposed to the traditional distinction between subject and predicate. Almost all of the modern commentaries on Frege's work suggest a semantic interpretation of this distinction, and identify it with the ontological structure of function and object, upon which Grundgesetze is based. Those commentaries agree that the system proposed by Frege in Begriffsschrift has some gaps, but it is taken as an essentially correct formal system for second-order logic: the first one in the history of logic. However, there is strong textual evidence that such an interpretation should be rejected. This evidence shows that the nature of the distinction between function and argument is stated by Frege in a significantly different way: it applies only to expressions and not to entities. The formal system based on this distinction is tremendously flexible and is suitable for making explicit the logical structure of contents as well as of deductive chains. We put forward a new reconstruction of the function-argument scheme and the quantification theory in Begriffsschrift. After that, we discuss the usual semantic interpretation of Begriffsschrift and show its inconsistencies with a rigorous reading of the text.     

Two Kinds of Mental Realism

I argue that there is a distinction to be drawn between two kinds of mental realism, and I draw some lessons for the realism-antirealism debate. Although it is already at hand, the distinction has not yet been drawn clearly. The difference to be shown consists in what realism is about: it may be either about the interpretation of folk psychology, or the ontology of mental entities. I specify the commitment to the fact-stating character of the discourse as the central component of realism about folk psychology, and from this I separate realism about mental entities as an ontological commitment towards them. I point out that the two views are mutually independent, which provides the possibility of considering folk psychology as not being in cognitive competition with scientific psychology. At the end I make a tentative suggestion as to how to interpret the former in order to avoid this conflict.     

Reasoning strategies with rational numbers revealed by eye tracking

Recent research has begun to investigate the impact of different formats for rational numbers on the processes by which people make relational judgments about quantitative relations. DeWolf, Bassok, and Holyoak (Journal of Experimental Psychology: General, 144(1), 127–150, 2015) found that accuracy on a relation identification task was highest when fractions were presented with countable sets, whereas accuracy was relatively low for all conditions where decimals were presented. However, it is unclear what processing strategies underlie these disparities in accuracy. We report an experiment that used eye-tracking methods to externalize the strategies that are evoked by different types of rational numbers for different types of quantities (discrete vs. continuous). Results showed that eye-movement behavior during the task was jointly determined by image and number format. Discrete images elicited a counting strategy for both fractions and decimals, but this strategy led to higher accuracy only for fractions. Continuous images encouraged magnitude estimation and comparison, but to a greater degree for decimals than fractions. This strategy led to decreased accuracy for both number formats. By analyzing participants’ eye movements when they viewed a relational context and made decisions, we were able to obtain an externalized representation of the strategic choices evoked by different ontological types of entities and different types of rational numbers. Our findings using eye-tracking measures enable us to go beyond previous studies based on accuracy data alone, demonstrating that quantitative properties of images and the different formats for rational numbers jointly influence strategies that generate eye-movement behavior.     

Transient Things and Permanent Stuff

A view of individuals as constituted of quantities of matter, both understood as continuants enduring over time, is elaborated in some detail. Constitution is a three-place relation which can't be collapsed to identity because of the place-holder for a time and because individuals and quantities of matter have such a radically different character. Individuals are transient entities with limited lifetimes, whereas quantities are permanent existents undergoing change in physical and chemical properties from time to time. Coincidence, considered as a matter of occupying the same place, is developed, alongside sameness of constitutive matter, as a criterion of identity for individuals. Quantities satisfy the mereological criterion of identity, applicable to entities subject to mereological relations and operations such as regions of space and intervals of time. A time-dependent analogue of mereological parthood is defined for individuals, in terms of which analogues of the other mereological relations can be defined. But it is argued that there is no analogue of the mereological operation of summation for individuals.     

Foxhole atheism,revisited: The effects of mortality salience on explicit and implicit religious belief

Although fear of death features prominently in many historical and contemporary theories as a major motivational factor in religious belief, the empirical evidence available is ambivalent, and limited, we argue, by imprecise measures of belief and insufficient attention to the distinction between implicit and explicit aspects of cognition. The present research used both explicit (questionnaire) and implicit (single-target implicit association test; property verification) measurement techniques to examine how thoughts of death influence, specifically, belief in religious supernatural agents. When primed with death, participants explicitly defended their own religious worldview, such that self-described Christians were more confident that supernatural religious entities exist, while non-religious participants were more confident that they do not. However, when belief was measured implicitly, death priming increased all participants' beliefs in religious supernatural entities, regardless of their prior religious commitments. The results are interpreted in terms of a dual-process model of religious cognition, which can be used to resolve conflicting prior data, as well as to help explain the perplexing durability of religious belief.     

The correlation-based law of effect

It is commonly understood that the interactions between an organism and its environment constitute a feedback system. This implies that instrumental behavior should be viewed as a continuous exchange between the organism and the environment. It follows that orderly relations between behavior and environment should emerge at the level of aggregate flow in time, rather than momentary events. These notions require a simple, but fundamental, change in the law of effect: from a law based on contiguity of events to a law based on correlation between events. Much recent research and argument favors such a change. If the correlation-based law of effect is accepted, it favors measures and units of analysis that transcend momentary events, extending through time. One can measure all consequences on a common scale, called value. One can define a unit of analysis called the behavioral situation, which circumscribes a set of values. These concepts allow redefinition of reinforcement and punishment, and clarification of their relation to discriminative stimuli.     

Cleaner fish <Emphasis Type="Italic">Labroides dimidiatus</Emphasis> discriminate numbers but fail a mental number line test

Several species of primates, including humans, possess a spontaneous spatial mental arrangement (i.e. mental number line MNL) of increasing numbers or continuous quantities from left to right. This cognitive process has recently been documented in domestic chicken in a spatial–numerical task, opening the possibility that MNL is a cognitive capacity that has been conserved across vertebrate taxa. In this scenario, fish might possess the MNL as well. Here we investigated whether cleaner fish Labroides dimidiatus show evidence for MNL in two experiments. In Experiment I, we tested fish’s abilities in number discrimination, presenting simultaneously either small (2 vs 5) or large (5 vs 8) continuous quantities where one quantity was systematically rewarded. Experiment II used a protocol of an MNL task similar to the study on chickens. We trained cleaners with a target number (i.e. 5 elements), then we presented them with an identical pair of panels depicting either 2 elements or 8 elements, and we recorded their spontaneous choice for the left or right panel on each presentation. Cleaner fish showed high abilities in discriminating small and large numbers in Experiment I. Importantly, cleaners achieved this discrimination using numerical cues instead of non-numerical cues such as the cumulative surface area, density, and overall space. In contrast, cleaners did not allocate continuous quantities to space in Experiment II. Our findings suggest that cleaner fish possess numbering skills but they do not have an MNL. While similar studies on animals from various clades are needed to trace the evolution of MNL within vertebrates, our results suggest that this cognitive process might not be a capacity conserved across all vertebrate taxa.     

A Dogma of Naturalism

One of the major historical effects of Quine's attacks upon the analytic‐ synthetic distinction has been to popularise the belief that philosophy is continuous with science. Currently, most philosophers believe that such continuity is an inevitable consequence of naturalism. This article argues that though Quine's semantic holism does imply that there is no sharp distinction between truths discoverable by scientific investigation and truths discoverable by philosophical investigation, it also implies that there is a perfectly sharp and natural distinction between natural science and naturalistic philosophy.     

Semantic alignment across whole-number arithmetic and rational numbers: evidence from a Russian perspective

ABSTRACT

Solutions to word problems are moderated by the semantic alignment of real-world relations with mathematical operations. Categorical relations between entities (tulips, roses) are aligned with addition, whereas certain functional relations between entities (tulips, vases) are aligned with division. Similarly, discreteness vs. continuity of quantities (marbles, water) is aligned with different formats for rational numbers (fractions and decimals, respectively). These alignments have been found both in textbooks and in the performance of college students in the USA and in South Korea. The current study examined evidence for alignments in Russia. Textbook analyses revealed semantic alignments for arithmetic word problems, but not for rational numbers. Nonetheless, Russian college students showed semantic alignments both for arithmetic operations and for rational numbers. Since Russian students exhibit semantic alignments for rational numbers in the absence of exposure to examples in school, such alignments likely reflect intuitive understanding of mathematical representations of real-world situations.     

Mathematical Knowledge for Teaching Fraction Multiplication

The present study contrasts mathematical knowledge that two sixth-grade teachers apparently used when teaching fraction multiplication with the Connected Mathematics Project materials. The analysis concentrated on those tasks from the materials that use drawings to represent fractions as length or area quantities. Examining the two teachers' explanations and responses to their students' reasoning over extended sequences of lessons led to a theoretical frame that emphasizes relationships between teachers' unit structures and pedagogical purposes for using drawings. In particular, the present study builds on the distinction made in past research between reasoning with two and with three levels of quantitative units and demonstrates that reasoning with three levels of units is necessary but insufficient if teachers are to use students' reasoning with units as the basis for constructing generalized numeric methods for fraction arithmetic. Teachers need also to assemble three-level unit structures with flexibility supported by drawn versions of the distributive property.