CORRECTED TYPOS, IMPROVED CLARITY AND STRUCTURE

The matter is quite complex. He is wrong, yet I believe he is on something, and Qiaochu Yuan even more.

The pivotal point is: it all depends on what "trait d'union" between sum and product you chose to use to extend and generalize.

Here some possibilities:

1) (linear math pov) While it is true that multiplication is not limited to be a geometric scaling, it is, at the same time, captured somehow by abstract scaling as an action of a field over a vector space. This is one possible way to extend the meaning of summing and multipling/scaling and it is a point of view strong of the argument that rings are universally a good way to think abstractly of + and x.

This is the point of view that the real deal here is distributivity, and this is the engine of most of the modern mathematics: linearity.

I believe Yuan comes from another place: he notice the fact that exponentiation can be extended in many, often orthogonal, directions. This further weakens the confidence in claim that it, the exponentiation, IS the next step of a sequence contianing sum and product.

2) (Logic/foundational pov) Product and sum, their arithmetic, can be seen as a lower reflection, incarnation, of the algebra of cardinalities of sets. Sums and multiplications have to do with operations on object of some category (like that of sets) so + and x are really to be tought as shadows of what happens one level up, at the level of sets, and this, in turn, is a shadow of what happens at the level of logic itself. This pov places the roots of + and x and their essence at the level of logic itself (namely cartesian product and coproduct, logic AND and logic OR). This point is even stronger, and leaves apparently less room to the higer operations.

3) (Peano/Ackermann/Goodstein recurisive pov) Then, at the end, or in principle, there was Peano's conception of arithmetic and it's axiomatic theory. Peano puts the essence of arithmeitcal operations, their "true traid d'union" in their recursive relationship: from successor to sum, from sum to product and from product to exponentiation. In this view, higher operations, in the sense of Goodstein, are the real deal and assume some feeling of naturality. This point of view can be generalized to categories themselves and, it turns out, some weakened form of Goodstein contruction inside categories, applied to coproduct yelds some properties that are, surprisingly, linked to categorical products. I can not explain this yet to myself, but this seems to deserve more inquiry.

3,5) From the layman, common sense, point of view. Recursion has its purpose and meaning in making expressions shorter, at the cost of increasing complexity. So multiplication is a short notation for repeted addition. Everything has a practical explaination and going up in the ranks is only a funny exercise in googology.

(09/15/2024, 12:43 AM)marracco Wrote: Devlin is wrong. He claims that multiplication is scaling, but scaling only works as multiplication in $1D

[...]

Therefore, the product should be understood as encompassing different concepts: multiplication, scaling, and rotations. Maybe each rank adds more meanings for the product, which we do not notice due to symmetries, that higher hyperperations break.

The matter is quite complex. He is wrong, yet I believe he is on something, and Qiaochu Yuan even more.

The pivotal point is: it all depends on what "trait d'union" between sum and product you chose to use to extend and generalize.

Here some possibilities:

1) (linear math pov) While it is true that multiplication is not limited to be a geometric scaling, it is, at the same time, captured somehow by abstract scaling as an action of a field over a vector space. This is one possible way to extend the meaning of summing and multipling/scaling and it is a point of view strong of the argument that rings are universally a good way to think abstractly of + and x.

This is the point of view that the real deal here is distributivity, and this is the engine of most of the modern mathematics: linearity.

I believe Yuan comes from another place: he notice the fact that exponentiation can be extended in many, often orthogonal, directions. This further weakens the confidence in claim that it, the exponentiation, IS the next step of a sequence contianing sum and product.

2) (Logic/foundational pov) Product and sum, their arithmetic, can be seen as a lower reflection, incarnation, of the algebra of cardinalities of sets. Sums and multiplications have to do with operations on object of some category (like that of sets) so + and x are really to be tought as shadows of what happens one level up, at the level of sets, and this, in turn, is a shadow of what happens at the level of logic itself. This pov places the roots of + and x and their essence at the level of logic itself (namely cartesian product and coproduct, logic AND and logic OR). This point is even stronger, and leaves apparently less room to the higer operations.

3) (Peano/Ackermann/Goodstein recurisive pov) Then, at the end, or in principle, there was Peano's conception of arithmetic and it's axiomatic theory. Peano puts the essence of arithmeitcal operations, their "true traid d'union" in their recursive relationship: from successor to sum, from sum to product and from product to exponentiation. In this view, higher operations, in the sense of Goodstein, are the real deal and assume some feeling of naturality. This point of view can be generalized to categories themselves and, it turns out, some weakened form of Goodstein contruction inside categories, applied to coproduct yelds some properties that are, surprisingly, linked to categorical products. I can not explain this yet to myself, but this seems to deserve more inquiry.

3,5) From the layman, common sense, point of view. Recursion has its purpose and meaning in making expressions shorter, at the cost of increasing complexity. So multiplication is a short notation for repeted addition. Everything has a practical explaination and going up in the ranks is only a funny exercise in googology.

MSE MphLee

Mother Law \((\sigma+1)0=\sigma (\sigma+1)\)

S Law \(\bigcirc_f^{\lambda}\square_f^{\lambda^+}(g)=\square_g^{\lambda}\bigcirc_g^{\lambda^+}(f)\)