Although construct measurement is critical to explanatory research and intervention efforts, rigorous measure development remains a notable challenge. For example, though the primary theoretical model for understanding health disparities among sexual minority (e.g., lesbian, gay, bisexual) adolescents is minority stress theory, nearly all published studies of this population rely on minority stress measures with poor psychometric properties and development procedures. In response, we developed the Sexual Minority Adolescent Stress Inventory (SMASI) with N = 346 diverse adolescents ages 14–17, using a (...) comprehensive approach to de novo measure development designed to produce a measure with desirable psychometric properties. After exploratory factor analysis on 102 candidate items informed by a modified Delphi process, we applied item response theory techniques to the remaining 72 items. Discrimination and difficulty parameters and item characteristic curves were estimated overall, within each of 12 initially derived factors, and across demographic subgroups. Two items were removed for excessive discrimination and three were removed following reliability analysis. The measure demonstrated configural and scalar invariance for gender and age; a three-item factor was excluded for demonstrating substantial differences by sexual identity and race/ethnicity. The final 64-item measure comprised 11 subscales and demonstrated excellent overall (α = 0.98), subscale (α range 0.75–0.96), and test–retest (scale r > 0.99; subscale r range 0.89–0.99) reliabilities. Subscales represented a mix of proximal and distal stressors, including domains of internalized homonegativity, identity management, intersectionality, and negative expectancies (proximal) and social marginalization, family rejection, homonegative climate, homonegative communication, negative disclosure experiences, religion, and work domains (distal). Thus, the SMASI development process illustrates a method to incorporate information from multiple sources, including item response theory models, to guide item selection in building a psychometrically sound measure. We posit that similar methods can be used to improve construct measurement across all areas of psychological research, particularly in areas where a strong theoretical framework exists but existing measures are limited. (shrink)

Goldbach’s conjecture, if not read in number theory, but in a precise foundation theory of mathematics, that refers to the metaphysical ‘theory of the participation’ of Thomas Aquinas, poses a surprising analogy between the category of the quantity, within which the same arithmetic conjecture is formulated, and the transcendental/formal dimension. It says: every even number is ‘like’ a two, that is: it has the form-of-two. And that means: it is the composition of two units; not two equal arithmetic units, but (...) two different formal-transcendental units, which are, in arithmetic, two prime numbers. (shrink)

Alexander George; Discussions: ‘Goldbach's Conjecture Can Be Decided in One Minute’: On an Alleged Problem for Intuitionism, Proceedings of the Aristotelian Soc.

So-called Neo-Russellians, such as Salmon, Braun, Crimmins, and Perry, hold that the semantic content of ‘ n is F ’ in a context c is the singular proposition ⟨ o, P ⟩, where o is the referent of the name n in c, and P is the property expressed by the predicate F in c. This is also known as the Neo-Russellian theory. Using truth ascriptions with names designating propositions, such as ‘Goldbach's conjecture’, in this paper, I will argue that, (...) together with highly plausible principles regarding a priori knowledge, the Neo-Russellian theory leads to unacceptable consequences. I will call this ‘the Goldbach puzzle’. Since the solution to the Goldbach puzzle cannot be to reject the discussed principles regarding a priori knowledge, the puzzle will undermine the Neo-Russellian theory. (shrink)

In this short note many conjectures on partitions of integers as summations of prime numbers are presented, which are extension of Goldbach conjecture.

i. a poetic solution of the Goldbach Conjecture; ii. several responses to the Epimenides Paradox; iii. the volitional solution to Russell's Paradox.

Except Goldbacher and Dee, most critics have held the latter part of the last book of the Metamorphoses of Apuleius to be almost entirely autobiographical; and many have taken the same view of the first chapter of the first book. Rohde, in particular, has championed this theory, and incidents of the last book are freely included in biographies of Apuleius.

By “epistemic modals,” I mean epistemic uses of modal words: adverbs like “necessarily,” “possibly,” and “probably,” adjectives like “necessary,” “possible,” and “probable,” and auxiliaries like “might,” “may,” “must,” and “could.” It is hard to say exactly what makes a word modal, or what makes a use of a modal epistemic, without begging the questions that will be our concern below, but some examples should get the idea across. If I say “Goldbach’s conjecture might be true, and it might be false,” (...) I am not endorsing the Cartesian view that God could have made the truths of arithmetic come out differently. I make the claim not because I believe in the metaphysical contingency of mathematics, but because I know that Goldbach’s conjecture has not yet been proved or refuted. Similarly, if I say “Joe can’t be running,” I am not saying that Joe’s constitution prohibits him from running, or that Joe is essentially a non-runner, or that Joe isn’t allowed to run. My basis for making the claim may be nothing more than that I see Joe’s running shoes hanging on a hook. (shrink)

I want to discuss a puzzle about the semantics of epistemic modals, like “It might be the case that” as it occurs in “It might be the case that Goldbach’s conjecture is false.”1 I’ll argue that the puzzle cannot be adequately explained on standard accounts of the semantics of epistemic modals, and that a proper solution requires relativizing utterance truth to a context of assessment, a semantic device whose utility and coherence I have defended elsewhere for future contingents (MacFarlane..

Number theory abounds with conjectures asserting that every natural number has some arithmetic property. An example is Goldbach’s Conjecture, which states that every even number greater than 2 is the sum of two primes. Enumerative inductive evidence for such conjectures usually consists of small cases. In the absence of supporting reasons, mathematicians mistrust such evidence for arithmetical generalisations, more so than most other forms of non-deductive evidence. Some philosophers have also expressed scepticism about the value of enumerative inductive evidence in (...) arithmetic. But why? Perhaps the best argument is that known instances of an arithmetical conjecture are almost always small: they appear at the start of the natural number sequence. Evidence of this kind consequently suffers from size bias. My essay shows that this sort of scepticism comes in many different flavours, raises some challenges for them all, and explores their respective responses. (shrink)

Just as we can be more or less certain that there is extraterrestrial life or that Goldbach’s conjecture is correct, we can be more or less certain about normative matters, such as whether euthanasia is permissible or whether utilitarianism is true. However, accommodating the phenomenon of degrees of normative certitude is a difficult challenge for non-cognitivist and expressivist views, according to which normative judgements are desire-like attitudes rather than beliefs. Several attempts have been made on behalf of non-cognitivism and expressivism (...) to meet the challenge Studies in metaethics, Oxford University Press, Oxford, 2007; Sepielli in Philos Stud 160: 191–207, 2012; Eriksson and Francén Olinder in Aust J Philos 94: 719–735, 2016). These attempts have all been found wanting. Michael Ridge has recently offered a quasi-realist solution, according to which expressivists can say exactly what cognitivists say about certitude, including normative certitude. In this paper, we explain the basic problem and Ridge’s quasi-realist solution. We then argue that the quasi-realist account of normative certitude faces severe difficulties that do not arise for cognitivist accounts, according to which normative judgements are beliefs. (shrink)

In recent decades, experimental mathematics has emerged as a new branch of mathematics. This new branch is defined less by its subject matter, and more by its use of computer assisted reasoning. Experimental mathematics uses a variety of computer assisted approaches to verify or prove mathematical hypotheses. For example, there is “number crunching” such as searching for very large Mersenne primes, and showing that the Goldbach conjecture holds for all even numbers less than 2 × 1018. There are “verifications” of (...) hypotheses which, while not definitive proofs, provide strong support for those hypotheses, and there are proofs involving an enormous amount of computer hours, which cannot be surveyed by any one mathematician in a lifetime. There have been several attempts to argue that one or another aspect of experimental mathematics shows that mathematics now accepts empirical or inductive methods, and hence shows mathematical apriorism to be false. Assessing this argument is complicated by the fact that there is no agreed definition of what precisely experimental mathematics is. However, I argue that on any plausible account of ’experiment’ these arguments do not succeed. (shrink)

SummaryThe platonist, in affirming the principle of bivalence for sentences for which there is no decision procedure, disconnects their truth‐conditions from conditions that would enable us to prove them ‐ as if Goldbach's conjecture, say, might just happen to be true. According to Dummett, what has gone wrong here is that the meaning of the relevant sentences has been conceived so as to go beyond what could be learned in learning to use them, or displayed in using them competently. Dummett (...) draws the general conclusion that accounts of meaning must traffic only in decidable circumstances. I suggest that Dummett can be right about platonism but wrong in this general conclusion: the centrality of decidable circumstances in competent use of language is a special feature of mathematical language. This means that someone who recoils from the anti‐realism constituted by Dummett's generalized anti‐platonism, in the case of, say, statements about other minds, need not be recoiling into a close analogue of platonism, as Dummett suggests. We can reinstate the intuitive idea that platonism goes wrong by inappropriately modelling the epistemology and metaphysics of mathematics on the epistemology and metaphysics of the natural world. And we make room for the suggestion that anti‐realism makes a converse mistake; in this vein, I propose a picture of Dummettian anti‐realism as a novel expression of familiar and suspect epistemological and metaphysical thoughts. (shrink)

In 1932, von Neumann proposed classifying the statistical behavior of differentiable systems. Joint work of B. Weiss and the author proved that the classification problem is complete analytic. Based on techniques in that proof, one is able to show that the collection of recursive diffeomorphisms of the 2-torus that are isomorphic to their inverses is $\Pi ^0_1$-hard via a computable 1-1 reduction. As a corollary there is a diffeomorphism that is isomorphic to its inverse if and only if the Riemann (...) Hypothesis holds, a different one that is isomorphic to its inverse if and only if Goldbach’s conjecture holds and so forth. Applying the reduction to the $\Pi ^0_1$-sentence expressing “ZFC is consistent” gives a diffeomorphism T of the 2-torus such that the question of whether $T\cong T^{-1}$ is independent of ZFC. (shrink)

The purpose of this article is to defend Hume's claim that whatever is conceivable is possible from a criticism by William Kneale. Kneale argues that although a mathematician can conceive of the falsehood of the Goldbach conjecture, he does not conclude that it is not necessarily true. The author suggests that by taking into account Hume's distinction between intuitive and demonstrative knowledge, a revised version of his claim can be offered which is not open to Kneale's criticism.

This book is written so as to be ‘accessible to philosophers without a mathematical background’. The reviewer can assure the reader that this aim is achieved, even if only by focusing throughout on just one example of an arithmetical truth, namely ‘7+5=12’. This example’s familiarity will be reassuring; but its loneliness in this regard will not. Quantified propositions — even propositions of Goldbach type — are below the author’s radar.The author offers ‘a new kind of arithmetical epistemology’, one which ‘respects (...) certain important intuitions’ 1 : apriorism, realism, and empiricism. The book contains some clarification of these ‘isms’, and some thoughtful critiques of major positions regarding them, as espoused by such representative figures as Boghossian, Bealer, Peacocke, Field, Bostock, Maddy, Locke, Kant, C.I. Lewis, Ayer, Quine, Fodor, and McDowell. The philosophical reader will find some interest and value in these wider-ranging discussions. Our concern in this review, however, is to examine closely the original positive proposal on offer.Arithmetical truths, the author maintains, are conceptual truths. Knowing truths like 7+5=12 involves no ‘epistemic reliance on any empirical evidence’; but that, she says, is not to claim ‘epistemic independence of the senses altogether’. She wants to show that "experience grounds our concepts … and then mere conceptual examination enables us to learn arithmetical truths ." Concepts that are ‘appropriately sensitive’ to ‘the nature of [an independent] reality’ she calls grounded. Because of the role of grounded concepts, ‘arithmetical truths explain our arithmetical beliefs in the right sort of way for those beliefs to count as knowledge’ .In the context of her concentration on the special nature of arithmetical knowledge, the author offers what could strike some bystanders as an unnecessarily over-ambitious account of knowledge tout court. Knowledge, for the author, is "true belief which … ". (shrink)