Human and Machine Understanding of Mathematics
Edmund Furse
Mathematics provides a rich domain for discussion of deep questions about cognition in both humans and machines. Roger Penrose claims that it is impossible for a machine to understand mathematics, and therefore that a machine cannot be a mind. Research over several years into how university students learn and understand mathematics has led to a computational model of the task known as the Mathematics Understander (MU). MU can to some limited degree understand university level pure mathematics thus calling Penrose's argument into question. MU achieves its performance by a scruffy cognitive architecture which is able to learn a new branch of mathematics from scratch utilising its Contextual Memory System (CMS). To some degree, MU is able to "see" a relevant mathematics result at a glance demonstrating to some degree a sense of awareness that Penrose says is impossible in machines. The CMS provides an open ended hybrid cognitive architecture capable of modelling declarative learning.

Furse E., (1991), Human and Machine Understanding of Mathematics, Technical Report no. CS-97-1, Department of Computer Studies, University of Glamorgan.

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