Quantum Mechanical Unbounded Operators and Constructive Mathematics – a Rejoinder to Bridges

As argued in Hellman (1993), the theorem of Pour-El and Richards (1983) can be seen by the classicist as limiting constructivist efforts to recover the mathematics for quantum mechanics. Although Bridges (1995) may be right that the constructivist would work with a different definition of closed operator, this does not affect my point that neither the classical unbounded operators standardly recognized in quantum mechanics nor their restrictions to constructive arguments are recognizable as objects by the constructivist. Constructive substitutes that may still be possible necessarily involve additional incompleteness in the mathematical representation of quantum phenomena. Concerning a second line of reasoning in Hellman (1993), its import is that constructivist practice is consistent with a liberal stance but not with a radical, verificationist philosophical position. Whether such a position is actually espoused by certain leading constructivists, they are invited to clarify.     

CULTURE,COUNTING AND NUMBER CONSERVATION

Children from 3 remote village populations in Papua New Guinea were administered notational counting and number conservation tasks. The results replicate previous research conducted in the United States that shows children develop the use of counting to mediate the comparison and reproduction of sets prior to understanding number conservation. The significance of these findings is discussed with respect to current models of number development.     

Orientation effects on contour interaction in the Zöllner illusion

The magnitude of the Zöllner illusion was measured as a joint function of the angle of intersection between inducing and test contours and the orientation of the complete display. The intersect angle at which judgmental error was maximal varied as the display was rotated from 0 to 90 deg. An explanation of the Zöllner illusion in terms of selective adaptation of neural orientation specific detectors accounts for the interactive effects of display orientation and intersect angle if it is assumed that different orientation detectors have different tuning characteristics.     

Moray vs. the rest: The effects of extended shadowing practice

The processing requirements of the shadowing response have led to criticisms of its use in selective attention experiments. The present experiment tested a single highly skilled shadower (N. Moray) using a task requiring the detection of digits embedded in dichotic lists of letters, with attention being directed to one list by use of the shadow technique. This subject was able to detect considerably more digits in both the attended and unattended messages than were groups of unskilled shadowers previously tested. Although performance to the unattended message was high, it remained below performance to the attended message.     

Approximate methods in calculating discriminant functions

Approximate methods of solving for discriminant functions have been tried on three sets of data. The principal illustration is the problem of finding a weighted sum of scores, on four psychological tests, so that men and women may be distinguished most clearly. The work starts from the complete solution, due to R. A. Fisher, where it is necessary to solve as many simultaneous equations, dependent on the standard deviations of the tests and their mutual correlations, as there are tests. It is proposed, by way of numerical simplification, that a set of equations be substituted where some one quantity replaces all the correlations. A solution is obtained where the weights to be assigned the tests are very simply expressed in terms of differences between the mean values of tests, the standard deviations of tests, and the said quantity. The difficulty remains of finding an estimate of the arbitrary constant that will give good discrimination. If an optimal solution is made a result is obtained which, in the three sets of data considered, is almost indistinguishable from that yielded by the complete solution. The calculation of this optimal common quantity is, however, itself so considerable that another estimate, previously suggested by R. W. B. Jackson, appears more profitable. This estimate is derived simply from the variability between the total scores for each subject and the variability of each test. Using this estimate, the discriminant functions can be rapidly calculated; the results compare very favorably, in the case of the data considered, with those from the complete solution.The present work was done while the writer was employed by the Ontario Department of Health.     

Johannes Kepler’s computer simulation of the universe: Some remarks about theory in psychology

Computer simulation may be an overly seductive way of formulating theory in the social sciences. The capability of constructing complex computer simulation models may remove some of the incentive for finding simple laws. This point is illustrated via a fantasy in which the sixteenth century astronomical theorist, Johannes Kepler, uses Ms computer instead of his mind to account for extant data on stellar and planetary motion. Similarities between sixteenth century cosmology and twentieth century psychology are discussed.