> m + n = n + m, m * n = n * m.
>
> When I read this what immediately popped into my mind was the following J
> statement:
>
> v~ -: v
Can we say
v is invariant wrt ~
Conjunctions do not take adverbs as parameters.
But for verbs, we can have
invarWrt=: 2 : 'u -: [EMAIL PROTECTED]'
load'trig'
sin invarWrt- _1 0 1
0
cos invarWrt- _1 0 1
1
^&2 invarWrt- _2 0 2
1
^&3 invarWrt- _2 0 2
0
> From: Tracy Harms <[EMAIL PROTECTED]>
>
> In the first chapter of _The Language of Mathematics_, Keith Devlin writes
> about the emergence of mathematics as a field of study and how that depended
> on the development of more and more abstract concepts.
>
> Here I quote from the section entitled "Symbolic progress":
>
> As an illustration of the distinction between the *use* of a mathematical
> device and the explicit recognition of the entities involved in that device,
> take the familiar observation that order is not important when a pair of
> counting numbers are added or multiplied. Using modern algebraic
> terminology, this principle can be compressed in a simple, readable fashion
> by the two commutative laws:
>
> m + n = n + m, m * n = n * m.
>
> In each of these two identities, the symbols m and n are intended to denote
> *any* two natural numbers. Using these symbols is quite different from
> writing down a particular instance of these laws, for example:
>
> 3 + 8 = 8 + 3, 3 * 8 = 8 * 3.
>
> The second case is an observation about the addition and multiplication of
> two particular numbers. It requires our having the ability to handle
> individual abstract numbers, at the very least the abstract numbers 3 and 8,
> and is typical of the kind of observation that was made by the early
> Egyptians and Babylonians. But it does not require a well-developed
> *concept* of abstract numbers, as do the commutative laws.
>
> (end quotation)
>
> When I read this what immediately popped into my mind was the following J
> statement:
>
> v~ -: v
>
> Here we find a natural extension to the trend of abstraction Devlin was
> emphasizing. At the point where one can read this simple bit of J, the
> prevailing ways of stating commutativity (m + n = n + m) seem about as
> clumsy as the use of examples with particular numbers. After all, the
> particular functions are no more relevant to the point of commutativity than
> are the values that fall within their domains. What makes a dyadic function
> commutative is an insensitivity of argument order, which is precisely what
> the additional abstraction of the J phrasing brings to the fore.
>
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