Seven and a Half Billion Pure Mathematicians

January 12, 2021

Vienna, January 12, 2021 - Most extraordinary achievements in mathematics are products of the abstract branch of the discipline, called pure mathematics, a field that common people would not even try to comprehend let alone dare to venture into. The best scholars in pure mathematics have been those remarkable individuals, who could easily think in terms of complex mathematical concepts, called “mathematical objects,” such as sets, groups, graphs, manifolds or vector spaces independently of any application outside mathematics. These mathematical objects are defined and, therefore, can be understood by their relationship to and interaction with other objects. Nothing could be further from our everyday experience and interest, where objects mean concrete visible physical things, such as a hammer on a flower, with boundaries, parts, and colors. Abstraction is nowhere to be found in the process of perceiving a hammer as an object. Or is it?
A new study in Nature Communication ( shows that automatic implicit learning of statistically co-occurring features of the visual environment lead observers to consider the set of those features as “objects” even if the feature set has no visible boundary contours or parts with similar visual properties as true objects do, and the observers have no explicit knowledge about the cohering features. Nevertheless, those “consciously invisible” objects evoke the same attentional processes in the brain as boundary-defined visible objects do.
The team behind the behavioral study led by Dr Jozsef Fiser (CEU, Austria) exposed humans to two types of alternating blocks in a visual statistical learning paradigm. In the first block, participants watched, without any task, a number of multi-element visual scenes composed of a set of abstract shapes with some shape pairs appearing more often together than others. In the second block, they performed an object-based attention (OBA) task while looking at scenes comprising shapes used in the observational blocks. In the final phase of the experiments, observers performed a test assessing how well they learned implicitly, without any dedicated training which shape pairs appeared together consistently in the blocks without any task.
Earlier studies of OBA with true objects demonstrated that observers perform the task faster and more precisely, when the attentional cue and/or the targets appear within the same object compared to when they appear in different objects. The present results showed that the very same OBA effect emerged with the “objects” defined purely by co-occurrence as with true boundary-defined objects. Moreover, there was a direct correlation between how well the observers learned implicitly such a co-occurrence-based object and how strong OBA effect was that object evoked.
“These results suggest that the visual routines traditionally attributed to the initial levels of object recognition, such as close boundary detection or figure-ground segregation can be interpreted as specific examples of a more general principle applied by our brain that leads to object representations: identifying consistent statistical properties co-occurring in the input. This view provides a flexible framework to incorporate a number of hitherto unexplained and peculiar observations of object perception,” Gabor Lengyel CEU PhD student and the lead author stated.
“Our finding has a wider implication by linking object representations to implicit statistical learning,” Fiser noted. “Objectness in our brain seems to be a graded concept instead of being all-or-none, and also a concept changing dynamically and with experience. Two visually distinct static parts clearly separated in space are strongly viewed as two objects until suddenly they start to move together, when they get reinterpreted as a moderately likely single object. This indicates that an enormous amount of implicit computation is going on in our brain all the time based on relational information and also the existence of a correspondingly complex emerging internal representation of objectness due to perpetual learning.”
It is hard to miss the similarity between this approach to the concept of everyday objects and mathematicians’ technique of defining and using mathematical objects for interpreting complex rules and abstract phenomena. After all, each of us might just do pure mathematics all the time without knowing it.