Several comments independently propose incrementation as coming before addition, multiplication.
One can only make sense of progressions like this by finding relationships between the levels. There are at least two ways to see a connection between addition and multiplication: The logarithm, and polynomials. Polynomials are well-studied expressions combining addition and multiplication.
Algebraic geometry studies sets defined by polynomial equations, as linear algebra studies sets defined by linear equations. Tropical geometry [1] is a version of algebraic geometry that replaces addition by minimization, and multiplication by addition.
Pure mathematicians like to hear music in their choices; they want these choices to arise naturally and support deep theories. Tropical geometry passes this test. One wants an inevitability to one's choices, a belief that alien life would reach the same conclusions.
If there are two answers to "what's zeroth? What's fourth?" that itself is interesting. However, one wants extensive evidence to believe that there's branching. Otherwise, we decide that we've simply stepped onto the wrong trail and need to backtrack. Pure mathematicians live in profound fear of just playing house, making stuff up because it sounds good, even if it looks to everyone else like that's what we do.
One can only make sense of progressions like this by finding relationships between the levels. There are at least two ways to see a connection between addition and multiplication: The logarithm, and polynomials. Polynomials are well-studied expressions combining addition and multiplication.
Algebraic geometry studies sets defined by polynomial equations, as linear algebra studies sets defined by linear equations. Tropical geometry [1] is a version of algebraic geometry that replaces addition by minimization, and multiplication by addition.
Pure mathematicians like to hear music in their choices; they want these choices to arise naturally and support deep theories. Tropical geometry passes this test. One wants an inevitability to one's choices, a belief that alien life would reach the same conclusions.
If there are two answers to "what's zeroth? What's fourth?" that itself is interesting. However, one wants extensive evidence to believe that there's branching. Otherwise, we decide that we've simply stepped onto the wrong trail and need to backtrack. Pure mathematicians live in profound fear of just playing house, making stuff up because it sounds good, even if it looks to everyone else like that's what we do.
[1] https://en.wikipedia.org/wiki/Tropical_geometry