Non fungible numbers
Multiplication and addition that is no longer commutative
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Imagine a world where added numbers are identical but not equivalent.
So 4 apples + 3 oranges = 4 apples then 3 oranges
This is identical to 7 but not equivalent.
I am thinking of potential use in government accounting and algebraic programming with numbers.
Rather than all taxes going into a single taxable account and then spread out to different funding initiatives, you track the income of each area and track the costs of each taxraising and each funding. So roads should be self funding.
When I have a configuration for a web server I want to add to it a particular behaviour and have it slot into the right place automatically. So the data structure is a polynomial and I add a polynomial confifuration that "fits" the shape of the polynomial that is already there.
Usually in Kubernetes configuration there is few places that can accept my configuration fragment. I should be capable of "adding" to a particular YAML key. There is only small subset where it is valid. The polynomials aren't compatible except for certain areas.
好吧，您可以在乘法或加法或兩者兼而有之的情況下找到或提出許多非交換的數字環。而且，爲什麼只是乘法和加法？我們有很多 運營商。矩陣環在乘法等情況下是不可交換的。
所以，我想這不是什麼新鮮事，除了命名。爲什麼使用“可替代”一詞來表示“可交換”？實際上，這就是你如何定義等號“=”，在比特幣中，在計算複雜性或挖掘它所需的能量（或構成 1 個 BTC 的每個組成塊）下，1 BTC 不應該等於另一個 1 BTC，但是比特幣代碼定義的“平等”人爲地使它們“平等”。事實上，在基本現實中，也許沒有什麼是真正相同的，並且等式被構建爲過程循環（即，反轉其他程序的程序，形成狀態循環，或等價類 下的操作)。
我的意思是，流程通常是不可交換的，因此您必須通過在狀態空間中搜索路徑來處理它們（假設機器或雲基礎設施的狀態空間並不總是可以通過相同的操作可逆，但應用了不同的操作或操作對他們（=程序）需要應用以向後與向前達到相同的狀態）才能回到原始狀態。
除了自然事件之外，您還可以通過公理地選擇環屬性來設置加法的含義。例如，如果我們將“加法”定義爲數字相加，而不是數字相加，我們會得到乘法和加法不可交換：
4 3 = "43"，與 3 4 = "34" 不同
4 * 3 = "3333"，不等於 3 * 4 = "444"
歸根結底，數字是計算工具，而工具本質上是實用的——我會說是的——繼續前進，在方便的地方定義新的數字系統。
Well, you can find or come up with many rings of numbers, that are non-commutative, either under multiplication, or addition, or both. And, why just multiplication and addition? We have lots of operators. Ring of matrices are non-commutative under multiplication, etc.
So, I guess this is not a new thing, except for naming. Why use the word "fungible" for "commutative"? In reality, it's how you define the equality sign "=", in bitcoin, 1 BTC should not be equal to another 1 BTC under computational complexity or energy needed to mine it (or each constituent block making up that 1 BTC), but the "equality" defined by bitcoin code artificially makes them be "equal". In fact, at the base reality, perhaps nothing is really identical, and equalities get constructed as process loops (i.e., programs, that reverse other programs, forming state loops, or equivalence classes under operations).
I mean, processes are often non-commutative, so you have to deal with them by searching paths in state spaces (let's say a state space of a machine or cloud infrastructure can not always be reversible by the same operation but different operations or operators applied to them (=programs) need to be applied to reach the same state backward vs forward) to go and come back to the original state.
Apart from natural occurrences, you can also set the meaning of addition by choosing ring properties axiomatically. For example, if we define "addition" as adding up numerals, rather than numbers, we get that multiplication and addition are not commutative:
4 + 3 = "43", which is not the same as 3 + 4 = "34"
4 * 3 = "3333", which is not equal 3 * 4 = "444"
Ultimately, numbers are tools of computation, and tools are pragmatic in nature -- I'd say yes -- go forward, define new number systems, where it is convenient.