Interestingly, I just read a piece in Martin Gardner’s book Are Universe’s Thicker than Blackberries? that touched on this. His point – which I totally agree with – is that math clearly exists outside of human experience. His off the cuff example (paraphrasing): If there are two dinosaurs in a field, and two more dinosaurs come into the field, there will be four dinosaurs in the field. Doesn’t matter whether there are any humans around to see it, or to use the words “two” and “four”, the math of “2 + 2 = 4” exists in and of itself.
To the OP: Ask your friend if the same logic can be applied to religion.
Much as I admire Martin Gardner I think he is wrong in this case. There are only 2+2=4 dinosaurs if someone, or something, is around to count them. And, of course in another number base there aren,t 4. There might be 10+10=100, or even 2+2=11 dinosaurs.
Right, but you see, you’re confusing the human notation with the independent concepts. There are an infinite number of ways to denote the concept of “2”, and the concept of “2 + 2 = 4”. But the concept itself exists independently of any notation. The symbol “2” is nothing more than how we in the western world denote the concept of “two”. The symbols “10” denotes the exact same concept in the binary number base. And the expression “2+2=11” would denote the same concept as “2+2=4” if the symbols “11” were agreed to denote the concept of “four”. You have to separate the concept from the symbols. The concept exists, whatever symbols are used.
Those dinosaurs in the field exist, without any humans around to count them. I mean, please, unless you subscribe to the notion that all of existence springs only from human minds, then you have to agree with that, right? And two of them joining another two of them means that there are four of them. That is just true, in English, in French, in Japanese, in binary, in hexadecimal, on Earth, on any other planet in the Universe, using any notation or descriptional language.
Find me another system in which the concept of " two plus two equals four" does not hold.
Not to speak for David, but I don’t think that’s what he meant. Addition is itself defined arbitrarily. It has to be; it cannot even be proven that every natural number has a successor. And successor is undefined. You might as well call expressions like “2 + 2 = 4” Peanoisms.
No I’m not. There is no concept of a definite number of items or quantity of material without someone or something to form the concept. In my view, mathematics is the human game of finding all the consequences of a few assumptions about numbers.
The more I read from you, the more I like. I agree with what you said, and I’ll go further. There is no concept of one thing being distinct from another — so they can be counted — without someone or something to form the concept of difference. I like thinking of a snake crawling down a tree. When he reaches the ground, he does not differentiate tree and ground. It’s nothing to him but a different slope (if even that), not a different entity. All he knows is prey and not-prey.
You mean wrt Peano’s axioms, right? wrt ZF set theory and the standard model, it’s a short proof.
Anyway, no one is denying that two dinosaurs plus two dinosaurs gives you four dinosaurs. The question is this: if we kill every being that has a concept of the number 2, does the number 2 still exist?
Some things are there. Suppose a human were there observing them. (Never mind that dinosaurs are extinct.) Can they be counted, like rocks (what’s a rock?), or can they not be counted, like water? A human sees a couple of similar-looking collections of matter that look like things called “dinosaurs”, which he has seen before. Someone has seen these things and decided that such collections of matter are around enough that they deserve a name, “dinosaur”. Once they’ve been named, some characteristics may be associated with them - they are big, slow, living things of a certain shape, color and texture that tend to do this and that. What if one of the dinosaurs is dead? Is it still a dinosaur? What if there was a dead dinosaur, subsequently eaten by ants, and all the ants are present in the field? What if there’s a dinosaur fetus, or just a blastocyst, inside one of the dinosaurs - does that count? If these things are going to be countable, you have to make certain judgments. Let’s say you are only counting living, fully grown dinosaurs, and you have come up with consistent and workable definitions of all those terms. To your own satisfaction.
Now maybe you can count the dinosaurs. But wait, how do you decide whether a dinosaur is “in the field” and thus to be counted, or “not in the field”, and thus not to be counted, or even if the concept of “the field” permits a reasonable definition of “in” and “out”. “The field” starts where the trees end. But there are a couple of saplings in the middle of the field. Maybe if you define “the field” as “out of the trees, in this area I see here.” No, that’s no good, because you see a different area depending on where you’re standing, where you’re looking, and the time of day. Also, if you walk to the other side of the field and look back, are you looking at the same field? Once, you looked at a pencil sticking out of a glass of water and it looked like two pencils, but it turned out to be a single pencil. Could the converse be going on with the field?
Mathematicians are people who never quite get comfortable with all these arbitrary judgments required to actually describe real, physical phenomena.
I don’t have time for more than a drive by (and I haven’t finished reading it yet), but Where Mathematics Comes From seems to be exactly on point for this discussion, and it’s a great read (thanks for the recommendation, SM). Perhaps more after I’ve finished reading and digesting it
So, are you saying that if something cannot be “proven” within some system of logic that humans created, then it has no absolute existence? Ok, then you don’t exist, because you can’t prove that you do. The Earth doesn’t exist, because you can’t prove that it does.
That’s just nonsense. Physicality and addition have an existence that is independent of humans or any intelligence.
I’m sorry, but you’re wrong. You are entitled to your view, but it’s not correct. Things have a physical exstence, independent of any intelligent observation. Four dinosaurs in a field are there, they exist, whether any intelligence observes them or not.
The problem isn’t with the “exist” part. The problem is with the “four dinosaurs in a field” part. The dinosaurs, or at least the stuff that makes up what you define to be the dinosaurs, exists. But the “fourness” of the dinosaurs requires abstraction and human reasoning.
