Proof and Canonical Proof

Certain anti-realisms about mathematics are distinguished by their taking proof rather than truth as the central concept in the account of the meaning of mathematical statements. This notion of proof which is meaning determining or canonical must be distinguished from a notion of demonstration as more generally conceived. This paper raises a set of objections to Dummett's characterisation of the notion via the notion of a normalised natural deduction proof. The main complaint is that Dummett's use of normalised natural deduction proofs relies on formalisation playing a role for which it is unfit. Instead I offer an alternative account which does not rely on formalisation and go on to examine the relation of proof to canonical proof, arguing that rather than requiring an explicit characterisation of canonical proofs we need to be more aware of the complexities of that relation.     

Creativity and schizotypy from the neuroscience perspective

Behavioral research has revealed that some cognitive features may be similar between creative and psychotic/schizophrenic-like thoughts. In this study, we addressed the potential link between creativity and schizotypy at the level of the brain by investigating functional patterns of brain activity (using functional magnetic resonance imaging) during creative cognition in preselected groups with low versus high psychometrically determined schizotypy. Our findings revealed an association between the originality component of creativity and reduced deactivation of right parietal brain regions and the precuneus during creative cognition, congruent with the idea that more-creative people may include many more events/stimuli in their mental processes than do less-creative people. Similarly, the high-schizotypy group showed weaker deactivation of the right precuneus during creative cognition. The fact that originality and schizotypy show similar functional brain activity patterns during creative ideation (i.e., reduced deactivation of the right precuneus) strongly supports the contention that similar mental processes may be implicated in creativity and in psychosis proneness.     

Generalization peak shift in rats under conditions of positive reinforcement and avoidance

Rats were trained to discriminate between two click frequencies. One frequency was associated with either variable-interval food reinforcement (Experiment 1) or free-operant avoidance (Experiment 2). The other frequency was associated with the absence of food in Experiment 1 and the absence of shock in Experiment 2. On a click frequency generalization test, the rats in both experiments showed positive peak shift with the shape of the relative gradients being very similar. This is the first reported instance of peak shift in rats when responding was maintained by an avoidance contingency. Nondifferentially trained controls showed that this shift was due exclusively to associative processes, with nonassociative stimulus factors in themselves apparently making no contribution to increased rates at particular stimulus values. These results show the comparability of appetitive and aversive control and support the position that gradient differences do not result from approach versus avoidance per se.     

The effect of reinforcement differences on choice and response distribution during stimulus compounding

In Experiments I and II, rats were trained to respond on one lever during light and another during tone. The absence of tone and light controlled response cessation. In the multiple schedule of Experiment I, all reinforcements were received for responding in tone or light; in the chain schedule of Experiment II, all reinforcements were received in no tone + no light for not responding. Experiment I subjects, for which tone and light were associated with response and reinforcement increase, responded significantly more to tone-plus-light than to tone or light alone (additive summation). Experiment II subjects, for which tone and light were associated with response increase and reinforcement decrease, responded comparably to tone, light, and tone + light. Thus, additive summation was observed when stimulus-response and stimulus-reinforcer associations in tone and light were both positive, but not when they were conflicting. All subjects in both experiments responded predominantly on the light-correlated lever during tone + light, even when light intensity was reduced in testing. Furthermore, when a light was presented to a subject engaged in tone-associated responding, all subjects immediately switched the locus of responding to the light-correlated lever. No change in locus occurred when a tone was presented to a subject engaged in light-associated responding, irrespective of the stimulus-reinforcer association conditioned to tone. The light-lever preference in tone + light indicates that the heightened responding observed in Experiment I was not the summation of tone-associated behavior with light-associated behavior. Rather, it appears to be the result of a facilitation of one operant (light-associated responding) by the reinforcement-associated cue for the other.