Mindset reconstruction maps how individuals structure and perceive knowledge, a map unfolded here by investigating language and its cognitive reflection in the human mind, i.e. the mental lexicon. Textual forma mentis networks (TFMN) are glass boxes introduced for extracting, representing and understanding mindsets' structure, in Latin "forma mentis", from textual data. Combining network science, psycholinguistics and Big Data, TFMNs successfully identified relevant concepts, without supervision, in benchmark texts. Once validated, TFMNs were applied to the case study of the gender gap in science, which was strongly linked to distorted mindsets by recent studies. Focusing over social media perception and online discourse, this work analysed 10,000 relevant tweets. "Gender" and "gap" elicited a mostly positive perception, with a trustful/joyous emotional profile and semantic associates that: celebrated successful female scientists, related gender gap to wage differences, and hoped for a future resolution. The perception of "woman" highlighted discussion about sexual harassment and stereotype threat (a form of implicit cognitive bias) relative to women in science "sacrificing personal skills for success". The reconstructed perception of "man" highlighted social users' awareness of the myth of male superiority in science. No anger was detected around "person", suggesting that gap-focused discourse got less tense around genderless terms. No stereotypical perception of "scientist" was identified online, differently from real-world surveys. The overall analysis identified the online discourse as promoting a mostly stereotype-free, positive/trustful perception of gender disparity, aware of implicit/explicit biases and projected to closing the gap. TFMNs opened new ways for investigating perceptions in different groups, offering detailed data-informed grounding for policy making.