In Chapter 8, I deal with the threat that the present account strips arithmetical knowledge of all the important characteristics traditionally associated with it: apriority, objectivity, necessity and universality. I argue that apriority can be saved in the strong sense of arithmetical knowledge being contextually a priori in the context set by our cognitive and physical capacities. Objectivity can be saved in the sense of maximal inter-subjectivity, while necessity can be saved in the sense of arithmetical theorems being true in all possible worlds where cognitive agents with proto-arithmetical abilities have developed. Finally, universality of arithmetical truths is saved through arithmetic being universally applicable and shared by all members of cultures that develop arithmetic based on proto-arithmetical abilities.

In Chapter 7, I focus on the ‘conventionalist threat’ against the account presented in this book. I present Warren’s conventionalist account of mathematics and discuss Wittgenstein’s philosophy of mathematics and its debated conventionalism. I identify strict conventionalism according to which mathematical truths are fundamentally arbitrary as the main threat against the present account. I then show that, due to arithmetic’s foundations in the evolutionarily developed proto-arithmetical abilities, my account survives the conventionalist threat. Arithmetical knowledge, under the present understanding, is maximally inter-subjective, which is enough to deal with the threat of arbitrariness. I then discuss the nature of mathematical conventions further, pointing out their importance without succumbing to the view that arithmetic is fundamentally based only on conventions.