you're misidentifying two quite different things: how people actually learn arithmetic, and what are the truth conditions of arithmetical statements. That 2 plus 2 equals 4 is something with in no way deopends on us or the existenbce of what you call the physical reality. 2 dinosaurs plus 2 dinosaurs equals 4 dinasaurs althought there are not dinosaurs, so ther should be no constitutive lconnection between arithmetical truths and the alleged physical reality.

@@matiasguirado5899 l agree that it is a truth condition of a mathematical statement and does not require the physical. But to put it in words in demonstrations is the start of explaining, teaching and learning the underlying mathematical concept to others. Without the physical reality to use, how else can we define it? I may be slightly lacking in my ability to communicate it, but I am working with great effort to improve that. I think a lot of the confusion is caused by the fact that two in the mathematical is an adjective, but it is also often used as a noun. In two plus two equals four, it is being used as a noun meaning two of something. In learning, we start by demonstrating the concept of one first. Thank you for your words of wisdom in helping me with this. If we don't get the start correct, chaos will likely follow.

​@@richardhill3405 Numerals are proper names referring to numbers, this is part of the standard semantics for the language of arithmetic. For instance, '2' is the name of the number 2. And the number 2 is the object that numeral refers to. That's it. In thinking about two of something your are stepping out of arithmetic and countenancing some of aplications of arithmetic to the physical reality. That's another question. Right, we learn artihmetic by means of applications, but the truth conditions of the relevant statements are quite another story. Likewise physicists learn about electrons by means of Bohr model, but they know that this model does in no way reflect what electrons are actually like. If you want to know that you have to understand the principles of quantum mechanics. So my point is: don't let the learning process mess up the semantical analysis.

@@matiasguirado5899 The difference between nous and adjectives.

Thanks, William! Glad you enjoyed it, and thanks for writing. (I don't suppose you're by any chance the William Bristow who works at UW Milwaukee?)