Developmental and Academic Learning differences for Pre-School Children in North Jeddah, Saudi Arabia

The aim of the research was to determine the size of the problem of learning differentials for preschool children for the target group during the period from 20/4/1436 to 3/5/1436 AH, while applied to the sample studied during the period from 5/5/1436 to 13/5 / 1436 e. The researcher used the descriptive method of "survey study". The researcher applied some scientific tests and measurements to measure some learning differences on a sample of (100) children and their child (53 males and 47 females) in pre-school in Jeddah. The results of the study showed that the differences in visual memory were the most common learning differences (51%) and the auditory differences (25%), followed by learning differences (24%) and mathematical inference (24%) respectively. The results also showed significant differences between males and females in the tests of hyperactivity, dizziness, attention, social interaction, writing, and auditory perception for females, and males and females are equal in tests of social maturity, pre-reading, mathematical inference and visual memory. Among students born in 2010 and students born in 2011 in social maturity tests, social interactions, writing, auditory perception, visual memory, pre-reading for age 5, attention tests, impulse and hyperactivity. For (4) years, while there were no differences according to the variables (gender - age) in other measures. The researcher recommends taking into account the child's access to the level of maturity appropriate for learning to read and write and account, and not forced to do so in the pre-school, and interest in building programs of motor education for children and the need to apply to students outside the classroom. As well as the importance of focusing teachers on activities that are concerned with the development of the child's senses in general, and on audio-visual activities in particular, and assigning the task of pre-school teaching to qualified teachers in the field of child education.

After Non-Euclidean Geometry: Intuition, Truth and the Autonomy of Mathematics

The mathematical developments of the 19th century seemed to undermine Kant’s philosophy. Non-Euclidean geometries challenged Kant’s view that there is a spatial intuition rich enough to yield the truth of Euclidean geometry. Similarly, advancements in algebra challenged the view that temporal intuition provides a foundation for both it and arithmetic. Mathematics seemed increasingly detached from experience as well as its form; moreover, with advances in symbolic logic, mathematical inference also seemed independent of intuition. This paper considers various philosophical responses to these changes, focusing on the idea of modifying Kant’s conception of intuition in order to accommodate the increasing abstractness of mathematics. It is argued that far from clinging to an outdated paradigm, programs based on new conceptions of intuition should be seen as motivated by important philosophical desiderata, such as the truth, apriority, distinctiveness and autonomy of mathematics.

MATHEMATICAL INFERENCE AND LOGICAL INFERENCE

The deviation of mathematical proof—proof in mathematical practice—from the ideal of formal proof—proof in formal logic—has led many philosophers of mathematics to reconsider the commonly accepted view according to which the notion of formal proof provides an accurate descriptive account of mathematical proof. This, in turn, has motivated a search for alternative accounts of mathematical proof purporting to be more faithful to the reality of mathematical practice. Yet, in order to develop and evaluate such alternative accounts, it appears as a necessary prerequisite to first possess a clear picture of what the deviation of mathematical proof from formal proof consists in. The present work aims to contribute building such a picture by investigating the relation between the elementary steps of deduction constituting the two types of proofs—mathematical inference and logical inference. Many claims have been made in the literature regarding the relation between mathematical inference and logical inference, most of them stating that the former is lacking properties that are constitutive of the latter. Such differentiating claims are, however, usually put forward without a clear conception of the properties occurring in them, and are generally considered to be immediately justified by our direct acquaintance, or phenomenological experience, with the two types of inferences. The present study purports to advance our understanding of the relation between mathematical inference and logical inference by developing a detailed philosophical analysis of the differentiating claims, that is, an analysis of the meaning of the differentiating claims—through the properties that occur in them—as well as the reasons that support them. To this end, we provide at the outset a representative list of the different properties of logical inference that have occurred in the differentiating claims, and we notice that they all boil down to the three properties of formality, generality, and mechanicality. For each one of these properties, our analysis proceeds in two steps: we first provide precise conceptual characterizations of the different ways logical inference has been said to be formal, general, and mechanical, in the philosophical and logical literature on formal proof; we then examine why mathematical inference does not appear to be formal, general, and mechanical, for the different variations of these notions identified. Our study results in a precise conceptual apparatus for expressing and discussing the properties differentiating mathematical inference from logical inference, and provides a first inventory of the various reasons supporting the observations of those differences. The differentiating claims constitute thus a set of data that any philosophical account of mathematical inference and proof purporting to be more faithful to mathematical practice ought to be able to accommodate and explain.

Logical and Mathematical Inference: Syntax and Semantics of Proof

The paper treats the relation between mathematical and logical inferences in mathematics and analyses an ontological approach for explaining the indispensability of the semantic content from mathematical proof. It was shown that such an approach entails serious metaphysical commitments, that is why it is concluded that the epistemological approach is preferable in explaining the nature of the difference between the formal and mathematical inferences.