A_Seagull wrote: ↑Sun Sep 17, 2017 12:08 am
I don't question that 5=5, I only question what some people seem to think it means and what its foundations are.
Ok. Well for the sake of discussion, this is an interesting discussion!
I was put on to the law of identity on this very forum, perhaps a year or more ago. Someone posed the following subtle question.
We know that in formal set theory, we have the
axiom of extensionality. This axiom tells us when two sets are equal. Two sets are equal when they have exactly the same elements.
So the set {1,2,3} is the same as the set {3, 2, 1} because they have the same elements. Order doesn't matter. But they are both different than the set {1, 2, 3, 4} because that set contains an element, namely 4, that's not present in {1, 2, 3}. So those are not the same set.
Now the question is ... before I have formally defined equality of sets ...
how do I know when two elements are the "same"? That is, if I claim that {5} = {5} because both sets contain exactly the same elements, namely the single element 5 ...
how do I know the 5 on the left is the same as the 5 on the right?
I confess this question gave me quite a bit of trouble. Especially because it was posed by a poster who was, let's say, a bit cranky in his presentation, yet seemed to have a lot of insight into things. One often finds that among alternative viewpoints. Being crazy and being interesting are not mutually exclusive. On the contrary. It's sanity that's boring.
So I mulled this question over, and asked myself how we know that 5 = 5 before we have defined 5 as a particular set, as is done in the development of the natural numbers within set theory. In fact 5 is nothing more than the set {0 ,1, 2, 3, 4}. And therefore 5 = 5 by the axiom of extensionality.
But what is 5 BEFORE we have so modeled it within set theory?
The only answer I can come up with is that this is an application of the law of identity from logic. This is what I meant earlier when I said that the law of identity is logically prior to math. You objected to that. What I meant was that the law of identity is logically prioer to set theory. Whether you take set theory as a reasonable proxy for math is a question of philosophy of course!
So that's where I'm coming from. In math we start by assuming a thing is equal to itself. Then we stipulate the axiom of extensionality, define 5 as a set, and prove that 5 = 5 within set theory.
But before set theory, before any formal development of math, is the principle of identity. A thing is equal to itself. And yes I fully agree that this is something we must be explicit about once in a while. The law of identity is assumed. It's hard to imagine what kind of formal reasoning we could do if we were not allowed to assume that 5 = 5.
A_Seagull wrote: ↑Sun Sep 17, 2017 12:08 am
For me what "5=5" is is a string of symbols which is theorem of mathematics which can be useful when mapped to real objects.
I perfectly agree. I'm one who often takes the formalist stance. If someone says, "well how can .999... = 1 or pi have infinitely many digits with no repeating block or how can irrational numbers exist when they don't exist in nature," I just say: I don't care! You don't ask is chess is real. Math is a formal game. Stop asking me for it to mean anything. It's fun to play and keeps me out of trouble. Go harass the chess players and ask them if the knight "really" moves that way. You see the nonsense of expecting math to "mean" anything. Math is a formal game and Hilbert agrees!
But then on the OTHER hand, of course math is derived from our experience in the world. Our experience of space and time and quantity and measure. And the study of formal math is inextricably bound up with our knowledge of the world. No physicist thinks it's all a formal game, they're studying the real world and they use math. Physicists mock mathematicians for their abstract formalism. Math is about the world. Ask any physicist!
So I see both sides of this.
What I DON'T see is trying to claim that 5 = 5 is questionable because you don't believe the law of identity. That is not a sensible thing to be arguing in my opinion.
But if all you want to do is get me to agree that the law of identity is an assumption, and you can't step in the same river twice, of course I agree with you.
The assumption that "a thing is equal to itself" is itself a particular worldview. Western rationality and all that. A perfectly sensible alternative to model the real world is:
A thing is never equal to itself.
I have no problem with that. I rather think it's probably true about the world we live in. Western rationality is a point of view. It's useful. It's logical. It's fun. But is it necessarily at the heart of the universe? No, I rather doubt that.
I just don't see how we could do math that way. In math you have to start with the law of identity.
What do you think?