Tracking visually a target that moves in the physical world involves two types of eye movement which recruit parallel and largely distinct neuronal streams. The first movement is a very rapid movement (called saccade) which brings the target image within the central visual field, whereas the second movement, much slower, maintains the visual centering. Most of the time, the eyes do not rotate as fast as the visual target. The target image slips on the retina and the tracking is punctuated by catch-up saccades. The performance during which gaze moves as if it were attached to the target is not a spontaneous oculomotor behavior; it is the outcome of a repeated training. This artifact is important to consider when eye movements are used as a probe for detecting neurological or even psychiatric disorders. During the last five decades, multiple studies supposed equivalence between the measured orientations of the eyeballs and the neuronal processes driving their changes. Done with high resolution, the measurements consist of temporal series of angular deviation of the eyes from which magnitudes such as movement amplitude, duration and their ratio (speed) were calculated. Then, models were proposed where processes occurring within the brain would reduce a difference between internal estimates of gaze and target directions (for guiding the saccade), reduce a difference between internal estimates of eye and the target speeds (for accelerating the slow eye movement), and maintain the eye movement despite the stabilization of the target within the central visual field. This parallelism between the relative motions of rigid bodies (eyeball and target) and putative quasi-teleological processes also led electrophysiologists to investigate the correlation between the activity of some neurons and kinematic parameters (position, speed and acceleration) and to propose that the linear combination of these parameters would account for their firing rate. When the correlations were weak, various kinds of neural noise were then imagined and embedded within the brain activity. We have explained why the "neuronalization" of measured behavioral parameters is questionable (Goffart et al. 2018; Goffart 2019). Our critique does not lead the neurophysiology of eye movements, and more generally the neurophysiology of visuomotor functions, toward a dead end. Instead, it calls for the need to pursue the build-up of its foundations upon principles which are intrinsic to the brain functioning instead of embedding within the brain medium, notions which in fact belong to the science of the motion of rigid bodies or to the human culture. Once the change of viewpoint is made, slow and saccadic eye movements appear as restoring equilibria between neuronal assemblies which exert mutually opposing tendencies, and their synergy becomes the outcome of a learning-dependent synaptic plasticity (Bourrelly et al. 2016). Saccadic eye movements are very rapid, not because they would serve primary cost-benefit function or urgency to grasp information. They are rapid because the flows of visuomotor activity ultimately recruit premotor and motor neurons whose firing rate can emit burst of action potentials and because of the contractile properties of extraocular muscle fibers. It is the plasticity of the underlying neural network which then enables them to acquire secondary properties which fit with environmental constraints.