On expandability of models of Peano arithmetic. III

Already after sending the first two parts of this paper ([5], [6]) to the editor, two new results on the subject have appeared — namely the results of G. Wilmers and Z. Ratajczyk. So for the sake of completeness let us review them here.Allatum est die 2 Septembris 1976     

On the autonomy of mental processes: a case study of arithmetic

We define a process as autonomous if it can begin without intention, and if it can run on to completion without intention. We develop empirical criteria for determining whether a process can begin without intention, for determining whether it begins in the same way without intention as it does with intention, and for determining whether it can run on to completion without intention once it begins. We apply these criteria to assess the autonomy of the processes underlying simple mental arithmetic--the addition and multiplication of single digits--and find evidence that simple arithmetic may be only partially autonomous: It can begin without intention, but does not begin in the same way without intention as with intention and does not run on to completion without intention. This conclusion suggests there may be a continuum of autonomy, ranging from completely autonomous to completely nonautonomous.     

On the status of statistical inferences

Can the axioms of probability theory and the classical patterns of statistical inference ever be falsified by observation? Various possible answers to this question are examined in a set theoretical context and in relation to the findings of microphysics.     

Cognitive arithmetic: Evidence for obligatory activation of arithmetic facts

In two experiments, obligatory activation of arithmetic facts resulted in interference on a simple number-matching task. Subjects were required to verify the presence of a probe number (e.g., 5) in a previously presented pair (e.g., 5+1). Items for which the probe was the sum of the initial pair (e.g., 5+1 and 6) were rejected more slowly than items for which the probe was not the sum (e.g., 5+1 and 3), and this effect was largest at stimulus onset asynchronies of less than 180 msec between the number pair and the probe. The results are consistent with the notion that arithmetic knowledge is represented in an associative network and accessed by means of spreading activation.     

On the form of the forgetting function: the effects of arithmetic and logarithmic distributions of delays

Forgetting functions with 18 delay intervals were generated for delayed matching-to-sample performance in pigeons. Delay interval variation was achieved by arranging five different sets of five delays across daily sessions. In different conditions, the delays were distributed in arithmetic or logarithmic series. There was no convincing evidence for different effects on discriminability of the distributions of different delays. The mean data were better fitted by some mathematical functions than by others, but the best-fitting functions depended on the distribution of delays. In further conditions with a fixed set of five delays, discriminability was higher with a logarithmic distribution of delays than with an arithmetic distribution. This result is consistent with the treatment of the forgetting function in terms of generalization decrement.     

The logic of arithmetic

Conclusion The arithmetic developed along the lines indicated by Cantor has up until now found no application in the real, as opposed to the mathematical, world. It is to be hoped that the changes suggested in this paper bring mathematics closer to human thought and allow it to be of increasing benefit in its service to mankind.     

Deflationism beyond arithmetic

The conservativeness argument poses a dilemma to deflationism about truth, according to which a deflationist theory of truth must be conservative but no adequate theory of truth is conservative. The debate on the conservativeness argument has so far been framed in a specific formal setting, where theories of truth are formulated over arithmetical base theories. I will argue that the appropriate formal setting for evaluating the conservativeness argument is provided not by theories of truth over arithmetic but by those over subject matters ‘richer’ than arithmetic, such as set theory. The move to this new formal setting provides deflationists with better defence and brings a broader perspective to the debate.

     

Some quick arithmetic

Mathematical abilities for the four simple arithmetic operations were studied on a sample of 53 female right-handers. True and false statements were presented auditorily and a manual response was required as to the trueness of the statements. Response times, accuracy, and laterality index showed no significant ear advantage, but the responses to true statements were significantly faster than to the false ones. The shortest response times were found for addition problems, and the longest for subtractions. The correlation between the size of the difference between the two operands and response time was not conclusive but the trend was unexpectedly positive.     

Parental socioeconomic status and the neural basis of arithmetic: differential relations to verbal and visuo‐spatial representations

We examined the relation of parental socioeconomic status (SES) to the neural bases of subtraction in school‐age children (9‐ to 12‐year‐olds). We independently localized brain regions subserving verbal versus visuo‐spatial representations to determine whether the parental SES‐related differences in children's reliance on these neural representations vary as a function of math skill. At higher SES levels, higher skill was associated with greater recruitment of the left temporal cortex, identified by the verbal localizer. At lower SES levels, higher skill was associated with greater recruitment of right parietal cortex, identified by the visuo‐spatial localizer. This suggests that depending on parental SES, children engage different neural systems to solve subtraction problems. Furthermore, SES was related to the activation in the left temporal and frontal cortex during the independent verbal localizer task, but it was not related to activation during the independent visuo‐spatial localizer task. Differences in activation during the verbal localizer task in turn were related to differences in activation during the subtraction task in right parietal cortex. The relation was stronger at lower SES levels. This result suggests that SES‐related differences in the visuo‐spatial regions during subtraction might be based in SES‐related verbal differences.     

On the status of location in visual attention

The authors propose a distinction between four issues underlying the debate on the status of location in visual selective attention. Three of them concern the representation within which attention operates. The grouping question focuses on whether or not this representation segments the visual field into perceptual groups. The space-invariance question focuses on whether it describes objects in spatio-topic or in space-invariant coordinates. Finally, the feature-coding question concerns whether or not it contains information about objects' non-spatial features. The last issue focuses on whether or not attention can be guided preattentively towards items possessing certain pre-specified physical properties other than location, and is referred to as the attentional-guidance question. A critical survey of the literature within the proposed framework is presented. Based on its conclusions, the status of location in current research is outlined, and avenues for further research are suggested.