You will interesting commentary here:
The "received views" generally ignore such possibilities because of history and folklore. Category theory and its child homotopy type theory are changing that situation. But, they do not do so by addressing the classical criticisms with respect to "second order" or how "verificationalism" does not properly distinguish between truth and provability.
Among the traditional foundational authors, Russell acknowledged how group theory and projective geometry could impact the significance of logicistic sentiment embodied by the analysts arithmetizing the study of real numbers. In fact, prior to his logicist period, Russell wrote a book on geometry proposing how projective geometry might correspond with Kant's "pure geometry" associated with external spatial intuition. (The common disparagement of Kant in which this is assumed to refer to Euclidean geometry has been shown to be misinformed by a translation by Ewald in which Kant actually calls for the development of alternative geometries long before such alternatives had been realized in the nineteenth century.).
Laughably, as logicians began to "represent" truth values with 0 and 1, other avenues of mathematical research began representing proper mathematical objects (such as finite geometries obtained from finite groups) using the two-element Galois field. The sixteen basic Boolean functions that ground classical propositional semantics correspond with the 4-dimensional vector space over GF(2). This vector space has the form of a finite affine plane. If you perform negations and de Morgan conjugations and their composition on some fixed representation of truth tables, you will be performing collineations in this plane.
Additionally, this vector space corresponds with a group of order 16 which carries a finite configuration called a Kummer configuration. String theorists are interested in the continuous correlate to this configuration, and, Melmendier has specifically correlated the theta functions of interest to these 16 vector space elements.
I guarantee that you can use devise a compositional system over 16 constants in which truth tables are recovered and with which you can implement propositional logic.
Arithmetization? Logicism? Formalism?
No one can can prove the "truth" of their presuppositions. And, as you learn about the famous distinction between "syntax" and "semantics" look up "pragmatics" as it had been associated with Rudolph Carnap's work. People trying to sit on a three-legged stool with only two legs look ridiculous.