Math by Pure Thinking: R First and the Divergence of Measures in Hegel's Philosophy of Mathematics

We attribute three major insights to Hegel: first, an understanding of the real numbers as the paradigmatic kind of number (which also accords with their role in physical measurement); second, a recognition that a quantitative relation has three elements (the two things being related and the relation itself), which is embedded in his conception of measure; and third, a recognition of the phenomenon of divergence of measures such as in second‐order or continuous phase transitions in which correlation length diverges (e.g., the critical point of water at which the reciprocal size of the droplets diverges). For ease of exposition, we will refer to these three insights as the R First Theory, Tripartite Relations, and Divergence of Measures. Given the constraints of space, we emphasize the first and the third in this paper.     

Distinct subsystems for the parafoveal processing of spatial and linguistic information during eye fixations in reading

Two experiments examined readers' use of parafoveally obtained word length information for word recognition. Both experiments manipulated the length (number of constituent characters) of a parafoveally previewed target word so that it was either accurately or inaccurately specified. In Experiment 1, previews also either revealed or denied useful orthographic information. In Experiment 2, parafoveal targets were either high- or low-frequency words. Eye movement contingent display changes were used to show the intact target upon its fixation. Examination of target viewing duration showed completely additive effects of word length previews and of ortho-graphic previews in Experiment 1, viewing duration being shorter in the accurate-length and the orthographic preview conditions. Experiment 2 showed completely additive effects of word length and word frequency, target viewing being shorter in the accurate-length and the high-frequency conditions. Together these results indicate that functionally distinct subsystems control the use of parafoveally visible spatial and linguistic information in reading. Parafoveally visible spatial information appears to be used for two distinct extralinguistic computations: visual object selection and saccade specification.     

Blocking as a retrieval failure: Reactivation of associations to a blocked stimulus

Conditioned lick suppression in rats was used to examine the effectiveness of three different “reminder” treatments in reactivating associations to a blocked stimulus in a Kamin blocking paradigm. Experiment I indicated that with our parameters prior tone-footshock pairings could block manifestation of a light-footshock association that would otherwise be evident following pairings of a light-tone compound stimulus with footshock. In Experiment II, exposure to either the US, the blocked stimulus (light), or the apparatus cues between the compound conditioning trials and testing decreased blocking. Experiments III(a) and III(b) replicated the unblocking effects seen in Experiment II and included control groups that received the identical training and reminder treatments except for the omission of the light from the compound stimulus. These latter animals failed to display behaviour comparable to the blocked and reminded subjects, thereby establishing the associative basis of suppression to the light in the animals reminded following treatment known to produce blocking. Experiment IV also replicated the results of Experiment II and included control groups that received identical light-tone compound trials and reminder treatments without prior conditioning to the tone alone. In these control groups, reminder treatments tended to disrupt rather than increase evidence of conditioning to the light. The results suggest that associations are formed to the added element of a compound despite prior conditioning to the initial element, and that failure on the test trial to retrieve these associations to the blocked CS, rather than a failure to attend to or learn about the added element, is at least in part responsible the Kamin blocking effect.