We extend prior results of Cody-Eskew, showing the consistency of GCH with the statement that for all regular cardinals κ≤λ, where κ is the successor of a regular cardinal, there is a rigid saturated ideal on Pκλ. We also show the consistency of some instances of rigid saturated ideals on Pκλ where κ is the successor of a singular cardinal.

For each substance-like quantity, a theorem about its conservation or non-conservation can be formulated. For the electric charge e.g. it reads: Electric charge can neither be created nor destroyed. Such a statement is short and easy to understand. For some quantities, however, the proposition about conservation or non-conservation is usually formulated in an unnecessarily complicated way. Sometimes the formulation is not generally valid; in other cases only a consequence of the conservation or non-conservation is pronounced. A clear and unified formulation could improve the comprehensibility and simplify teaching

El debate actual en la filosofía de la ciencia se plantea en términos del conflicto entre realismo e instrumentalismo. Tanto el realismo científico como el instrumentalismo científico se nos presentan, en principio, como dos concepciones de la ciencia bien perfiladas pero antagónicas.

This is a discussion of Delia Fara’s theory of vagueness, and of its solution to the Sorites paradox, criticizing some of the details of the account, but agreeing that its central insight will be a part of any solution to the problem. I also consider a wider range of philosophical puzzles that involve arguments that are structurally similar to the argument of the Sorites paradox, and argue that the main ideas of her account of vagueness helps to respond to some of those puzles.

From large cardinals we show the consistency of normal, fine, κ-complete λ-dense ideals on Pκ(λ) for successor κ. We explore the interplay between dense ideals, cardinal arithmetic, and squares, answering some open questions of Foreman.

We show that every homeomorphism between closed measure zero subsets extends to a measure preserving auto-homeomorphism, whenever the Cantor set is endowed with a suitable probability measure. This is valid both for the standard product measure, as well as for the universal homogeneous rational measure.

We present the theory of special relativity here through the lens of differential geometry. In particular, we explicitly avoid any reference to hypotheses of the form “The laws of physics take the same form in all inertial reference frames” and “The speed of light is constant in all inertial reference frames”, or to any other electrodynamic phenomenon. For the author, the clearest understanding of relativity comes about when developing the theory out of just the primitive concept of time (which is also a concept inherent in any standard exposition) and the basic tenets of differential geometry. Perhaps surprisingly, once the theory is framed in this way, one can predict existence of a “universal velocity” which stays the same in all “inertial reference frames”. This prediction can be made by performing much more basic time measurement physical experiments that we outline in these notes, rather than experiments of an electrodynamic nature. Thus, had these physical experiments been performed prior to Michelson-Morley type experiments (which, in principle, could have been done in any period with precise enough time keeping instruments), the Michelson-Morley experiments would simply give us an example of a physical entity, i.e., light, which enjoys this special “universal” status.

This paper offers a partial defense of the group-theoretical reading of Helmholtz along the lines of Ernst Cassirer in the fourth volume of The Problem of Knowledge of 1940. In order to avoid the problem raised by Schlick, Cassirer relied on a Kantian view of space not so much as an object of geometry, but as a precondition for the possibility of measurement. Although the concept of group does not provide a description of space, the modern way to articulate the concept of space in terms of transformation groups reveals something about the structure and the transformation of spatial concepts in mathematical and natural sciences.

The paper investigates Ernst Cassirer’s structuralist account of geometrical knowledge developed in hisSubstanzbegriff und Funktionsbegriff (1910). The aim here is twofold. First, to give a closer study of several developments in projective geometry that form the direct background for Cassirer’s philosophical remarks on geometrical concept formation. Specifically, the paper will survey different attempts to justify the principle of duality in projective geometry as well as Felix Klein’s generalization of the use of geometrical transformations in his Erlangen program. The second aim is to analyze the specific character of Cassirer’s geometrical structuralism formulated in 1910 as well as in subsequent writings. As will be argued, his account of modern geometry is best described as a “methodological structuralism”, that is, as a view mainly concerned with the role of structural methods in modern mathematical practice.

El objetivo del presente artículo es someter a escrutinio la afirmación de Hintikka según la cual la verdadera lógica elemental no es la clásica sino la lógica IF y, en consecuencia, el marco en que ordinariamente son pensadas las relaciones entre lógica y matemáticas es por completo inadecuado. Para ello, primero se exponen las funciones o características que una lógica debe poseer y, en segundo lugar, se presentan las ideas constitutivas de la lógica IF. Más adelante se demuestran algunas de las propiedades matemáticas de la lógica IF y se analizan las complejidades a que da lugar la negación en este sistema. Por último, se ofrecen algunas razones para matizar o poner en duda las conclusiones que Hintikka extrae de su propuesta para la filosofía de las matemáticas.