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What is the relationship between the concept of an equation (a statement) and the concept of a function (and the concept of morphisms in category theory)?

I'm going to use equations as the most important subclasses of statements. To take a seemingly trivial example, the equation (E1) 2*x = 5. This equation which could be rewritten as (E2) f(x) => 2*x -5, but E1 and E2 are technically not equivalent. With f(x)=> I mean, the statement is a function of x. For every x f(x) returns true or false, depending on the contents of f(x). The equation E1 can be true for values depending on x. The truth value is dependent on the value of x, i.e. E1 is a function which returns a bool, given a value of x. E2 returns a value. I haven't defined domain and co-domain. Say it is N. E2 is always true. So if I write (E3) f(x) = 2*x -5 and f(x)=0, E3 and E1 become equivalent. In other words, everything is a function, because equations are simply boolean functions. In (E2), f(x) could be seen as a placeholder for (E1). E2 says: there is an expression, which is an equation, which has the contents 2*x=5. In a programming language this could be all said much clearer. In python E1 would be: 2*x ==5 and E3: def f(x): return 2*x-5==0.

More generally, as I understand it, the approach of Russell/Wittgenstein was to reduce everything to statements. In category everything is reduced to functions (or morphisms in it's own parlance).

Edit: some further notes to clarify.

  • 5
    You say that the equation $2x=5$ ‘could be rewritten as’ $f(x)=2x-5$; this is absolutely false. $(E1)$ is a statement, something with a truth value, while $(E2)$ is an incomplete definition $-$ specifically, the definition of the function $f$ $-$ and as such has no truth value; they are completely different species of animal.2012-11-30
  • 0
    I have rewritten the statement. However I don't see anything wrong with writing f(x) = 2x -5. This can be interpreted as saying: 2x -5 depends on x. Rather the traditional statement f(x) = 2x-5, should be written as f(x) := 2x-5, because it is a definition of a function, and not an equation. "=" acts as an assignment operator not as a comparison operator ("=="). This works obviously in most cases, because mathematicians can tell the difference between "=" and "==". This doesn't mean it's completely rigorous.2012-11-30
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    I saw that you’d rewritten it; my objection still holds. I don’t agree that ‘$f(x)=2x-5$’ can be interpreted as saying that $2x-5$ depends on $x$. It has two possible interpretations: (1) an incomplete definition of $f$, and (2) an assertion that some previously defined function $f$ is the same as the function $2x-5$. The latter does make it a statement, but one that is unrelated to the statement $2x=5$.2012-11-30
  • 0
    The relationship is that f(x) takes the value of 0 when the equation is true. In fact equations can be rewritten as A=B <=> A-B=0. define f(x)=A-B and let f(x)=0. There you go.2012-12-01
  • 2
    No. Under interpretation (1) of my previous comment it’s meaningless to talk about the truth or falsity of $f(x)=2x-5$. Under interpretation (2) the statement ‘$f(x)=2x-5$ is true’ is meaningful but has nothing at all to do with the statement ‘$2x=5$’; rather, it’s equivalent to the statement ‘$f(x)-(2x-5)=0$’.2012-12-01

3 Answers 3