The title of my dissertation is *Seeing* *What* *We* *Think:* *Frege* *and* *Kant* *on* *Mathematical* *Writing,* *Demonstrating,* *and* *Understanding*. What is striking about mathematics as a field of study is that its practitioners paradigmatically can, do, and must make their discoveries, demonstrate their findings, and convince their peers *simply* *by* *writing* *things* *out* and *visually* *attending* *to* *what* *is* *written* *out*. Some of what is recognizably mathematics may be carried out in a natural language, but robust participation in the discipline requires learning one or more symbolic languages. The burden of this essay is to explain what it is, or what it takes, for a sy…

The title of my dissertation is *Seeing* *What* *We* *Think:* *Frege* *and* *Kant* *on* *Mathematical* *Writing,* *Demonstrating,* *and* *Understanding*. What is striking about mathematics as a field of study is that its practitioners paradigmatically can, do, and must make their discoveries, demonstrate their findings, and convince their peers *simply* *by* *writing* *things* *out* and *visually* *attending* *to* *what* *is* *written* *out*. Some of what is recognizably mathematics may be carried out in a natural language, but robust participation in the discipline requires learning one or more symbolic languages. The burden of this essay is to explain what it is, or what it takes, for a symbolic language to be the typically eloquent vehicle it is. The big idea is this: having framed a concept of some mathematically interesting structure, we marshal certain sign designs into inscriptional arrays whose two-dimensional physiognomy we take to symbolize the character of the structure we have conceived. I clarify, explore, and defend this idea by tracing the ways it finds expression throughout the corpora of Frege and Kant. In their eyes (but my words), participating in mathematical inquiry requires mastering a symbolic language not so much because it lets us *describe* the character of what we conceive with a clarity unmatched by natural language but because it lets us *see* the character of what we conceive with a clarity unmatched by describing it. This puts every practitioner in a position to become a direct witness in the discovery and demonstration of truths, which, I suggest, following Frege and Kant, enables the kind of understanding we are apt to count as distinctively mathematical.