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 and understanding mindsets' structure (in Latin forma mentis) from textual data. Combining network science, psycholinguistics and Big Data, TFMNs successfully identified relevant concepts in benchmark texts, without supervision. Once validated, TFMNs were applied to the case study of distorted mindsets about the gender gap in science. Focusing on social media, this work analysed 10,000 tweets mostly representing individuals' opinions at the beginning of posts. "Gender" and "gap" elicited a mostly positive, trustful and joyous perception, with semantic associates that: celebrated successful female scientists, related gender gap to wage differences, and hoped for a future resolution. The perception of "woman" highlighted jargon of sexual harassment and stereotype threat (a form of implicit cognitive bias) about women in science "sacrificing personal skills for success". The semantic frame of "man" highlighted awareness of the myth of male superiority in science. No anger was detected around "person", suggesting that tweets got less tense around genderless terms. No stereotypical perception of "scientist" was identified online, differently from real-world surveys. This analysis thus identified that Twitter discourse mostly starting conversations promoted a majorly stereotype-free, positive/trustful perception of gender disparity, aimed at closing the gap. Hence, future monitoring against discriminating language should focus on other parts of conversations like users' replies. TFMNs enable new ways for monitoring collective online mindsets, offering data-informed ground for policy making.