Weakly electric fish generate electric current and use hundreds of voltage sensors on the surface of their body to navigate and locate food. Experiments (von der Emde and Fetz 2007 J. Exp. Biol. 210 3082-95) show that they can discriminate between differently shaped conducting or insulating objects by using electrosensing. One approach to electrically identify and characterize the object with a lower computational cost rather than full shape reconstruction is to use the first order polarization tensor (PT) of the object. In this paper, by considering experimental work on Peters' elephantnose fish Gnathonemus petersii, we investigate the possible role of the first order PT in the ability of the fish to discriminate between objects of different shapes. We also suggest some experiments that might be performed to further investigate the role of the first order PT in electrosensing fish. Finally, we speculate on the possibility of electrical cloaking or camouflage in prey of electrosensing fish and what might be learnt from the fish in human remote sensing.