121–130 of 206 posts
Earlier quoted context omitted.
> It's because formulas have a convention, where the LHS is a single term naming the value you want, and the RHS contains the terms for values you have. This is simply not true.
I agree this is not entirely true, I have often heard physicists express similar formulas in the way the parent described (I.e. 2K = mv^2)
f(x) = 2x + 3
For all x in the domain of f, f(x) = 2x + 3
f = { (x, y) in Cartesian product of domain and codomain | y = 2x + 3 }
The articles other point in this example is that we might way “when x = 2, f(x) = 7.” Claiming that x is used both as an indeterminate value and a concrete value. Again, I think the ambiguity is resolved when translate using the correct quantifies, something like “for all x in the domain of f, if x = 2, then f(x) = 7.”
Or perhaps you might claim, “there exists an x in the domain of f such that f(x) = 7.” The important point being that the function f is formally NOT the formula f(x) = 2x + 3, but a particular set of ordered pairs, of which you can make formal statements about in first order logic.
Another example used was
A = {n^2 | n = 1, 2, ... 100}
A = { n^2 | n in {1, 2, ..., 100}}Computer languages make no exception: they are nothing else than formalisms to express computations. As for every other formalism, the meaning of signs is chosen to be what appears most comfortable in that context by the formalism designer. The statement "x = x+1" has very different interpretations depending on whether you consider it written in C or in standard polynomial equation theory; but in both cases there is a well known meaning for it. In exactly the same way the word "case" has different meaning depending on whether your are reading in English or in Italian.
- John Baez ( http://math.ucr.edu/home/baez/week147.html )
Earlier quoted context omitted.
I agree this is not entirely true, I have often heard physicists express similar formulas in the way the parent described (I.e. 2K = mv^2)
PV=nRT
(I assume this was inspired by https://news.ycombinator.com/item?id=16803874 ) The use of ‘=’ for assignment in programming languages comes, not directly from mathematics, but indirectly from the use of mathematics in science and engineering. As an example, consider the formula for kinetic energy, commonly written 𝑚𝑣² 𝐾 = ─── 2 Why isn't it written 2 K = m v ², which expresses the same mathematical equality in a s…
But there is a good reason to write it as K=mv^2/2 which has nothing to do with specifying a computation. It is the result of symbolic integration of p=mv with respect to v.
The author itself admits in the postscript that he has embellished a bit the article, but allow me to take it at its face value: to me, it seems that the article confuses mathematics with its notation (and the same for computer science, but at this level CS is just a branch of mathematics). All the funny stuff he goes on describing follow from this confusion. When a mathematician does mathematics, they have very well…
Earlier quoted context omitted.
Piling on with a bit more pedantry, my experience is a bit different. In my current ring theory course, we have indeed written things like 3 * 3 = 1 when working in |F_5 (not sure that notation is going to work as well as I hope, looks alright in the app I use), but it's not the equality symbol is overloaded, but the numbers themselves. Rather than using = to mean numeric equality and equality w.r.t. equivalence clas…
That seems odd to me. I don't think I've read any ring/algebra/module theory text that doesn't explicitly denote equivalence classes with, for example, square brackets.
This is not limited to rings of the form Z/nZ.
Geometry distinguishes between equivalence and value. An "angle" isn't its degrees, but the geometric figure (two rays or segments meeting at an end-point of each). It's the measure of the angle that is the degrees. You don't say "angles are equal" - you say they are congruent . It's their measures that are "equal". Although congruency implies measure equality, it doesn't really mean that, but that the shapes are the…