This is a relatively new push that’s based on the idea that we shouldn’t talk about identity, but rather only about isomorphism. Equivalence is not Equality. Confusing the two leads to decategorification, and most of mathematics is decategorified.
The natural numbers were “invented” to simplify (decategorify) the category of finite sets. To be exactly accurate here, I’m talking about cardinal numbers, since we identify sets with their cardinalities. Two sets with the same number of elements have the same cardinality. Given sets A and B with cardinalities a and b, the natural-numbers statement “a=b” really means “there exists a bijection between A and B”. Addition is disjoint union of sets, while multiplication is Cartesian product.
So, within this context: prove that the disjoint union of two sets with one element each is in bijective correspondance with any set of two elements.
The limits of both of those sequences are symbolized by the number “one”, which may be written in two different ways in our number system, as either “1”, or as “.999…”.
So for you math philosophers - recently I heard a creationist talking, and he said that the laws of logic are evidence of a creator. This got me to thinking that the laws of logic are the same kind of thing as the laws of mathematics. These creationist guys say that our beautiful and “unreasonably effective” mathematics are the way they are because they were defined by God. Which leads to the observation that if God wanted to make them some other way, I suppose like a universe where 1+1=3, he could damn well do that.
Other thoughtful people, when asked the silly question about whether God could make a rock so heavy that he could not lift it, point out that an omnipotent God could do anything that wasn’t logically contradictory, such as a square circle is a contradiction. So I think these people are saying that the laws of logic (and math) would predate even God and he himself would be bound by them.
So is there any good observations about which would be right for the theist? I’d like a convenient way of refuting this argument for the existence of God, and this one seems pretty hollow to me, but I had not come across it before and was at a loss. What do you think?
Yes, the laws of logic are the same as those of mathematics – or perhaps better would be to say that mathematics is built on the laws of logic. However, you can build structures other than the natural numbers on the laws of logic: there are all kinds of algebra and geometry out there.
I’m not a professional philosopher, but my take on it is that the laws of logic and of mathematics are true for any possible universe, and they are true regardless of whether there is a creator. (At least in part that’s because a creator would have to follow the laws of logic).
In any case, evidence of a creator is not the same as evidence of the Jewish/Christian God in the first few chapters of Genesis.
My take is that the theorems and logical structures would still have the same logical truth values, but might be rendered meaningless. For instance, what if there were a universe where addition had no meaning? The logical system called mathematics would still be sound but not applicable, much like most random logical systems we can come up with don’t have much practical meaning.
No it doesn’t because category theory can be created independent of set theory. It isn’t usually, but it can be in the same way as set theory can be (has to be) created without set theory. So a category consists of objects and arrows in some pre-theoretical universe with the well-known properties. And a model for set theory is a particular kind of category that I could define but won’t since it won’t mean much to most of the people reading it.
All this doesn’t really get to the original question. Is math based on faith, just like religion. My answer is tentatively yes, but with a major difference. Math is tested all the time and passes all tests. Religious faith makes no testable predictions and so remains purely a matter of faith. When tests are attempted, such as the recent test of prayer, religious leaders insist that religion makes no such promises. I do think that some religions do and some don’t. At any rate, may god strike me dead on the spot if he wishes to demonstrate his power.
Incidentally, a colleague of mine claims that 2 is defined as the cardinality of a pair of platinum-iridium balls held in a termparature controlled vault in Sevres, France.
Ultimately, as many have said, give me your definition of 1, 2, +, and = and, i they are at all familiar ones, I can give you a proof. For a category theorist, 1 is a terminal object and 2 an object of truth values in a certain topos model of set theory and that fact that 1 + 1 = 2 is implicit in the model. Regardless, it makes the prediction that one ball taken with another ball gives two balls and that is verified regularly in every-day life, so it works.
I think it’s a joke on the Metre des Archives and the Kilogramme des Archives, which are platinum objects which define (or used to define) the length of a meter and the weight of a kilogram.
I have recently thought of the same thing, and came to a different conclusion: Math is not based on faith because any axiomatic system can and will be abandoned if it can be proven inconsistent. If an axiomatic system were inconsistent—if it could be used to prove both a statement and that statement’s negation—it would be abandoned instantly because all proofs derived from it would be immediately suspect. Religion simply doesn’t work that way.
Sounds about right. In case anybody’s interested, the metre is no longer defined this way – it is defined as the distance light travels in an absolute vacuum in 1/299,792,458 of a second. The kilogram is still defined this way, because nobody has come up with a definition for it that can be replicated in labs with a reasonable degree of accuracy. Wikipedia has a picture of The Kilogram.
Well, in theory that is correct, although don’t hold your breath. When Russell’s paradox, a genuine inconsistency, was first announced, a lot of people spent a lot of time trying to find flaws in Russell’s argument. Only when some time had passed and it hadn’t gone away did they finally decide they needed to look seriously at foundations. This eventually led to Zermelo-Frankel set theory, to Hilbert’s formalist program, to Goedel’s incompleteness theorem and many other things.
A couple decades ago, there was a case that both a theorem and a counter-example were claimed. It was in algebraic topology where proofs often lack real rigor and everyone, absolutely everyone, assumed that one of the two was in error. It must have been found for that was the last I heard of it.
But yes, eventually it would be abandoned. And that just fleshes out what I said originally: it is tested all the time and has not been found wanting. The claims of religion are not tested and it seems a mantra that they are not testable.
There’s a slight problem here: these days logic is rather variable. Just like there are all kinds of algebra and geometry out there, there are all kinds of logic. As an example, consider the usual relation between “standard” logic and set theory. The logical connectives correspond to union, intersection, and complement (in a suitable total set) of subsets.
Now, instead consider the open sets in the unit interval. Union and intersection can still correspond to OR and AND, but now the complement of an open set isn’t generally open. So what do we do? Take the interior of the complement – the largest open subset contained in the complement. That’s our new NOT.
Okay, so for an application, let X be the set [0,1]{0.5}. That is, all numbers but 1/2. The complement is just {0.5}, and its interior is empty, so NOT X is {}, the empty set. Now, NOT NOT X is [0,1], the whole interval. In this logic, NOT NOT X is not the same as X.
There’s a whole field of putting various logics on the same basis: topos theory.
Um, you’re talking to a category theorist here. I know this. It comes down to set theory because we’re talking about the category of finite sets.
The applicability of mathematics is “inductively” shown (in the philosophy of science sense, not the mathematics sense) but must be ultimately taken on faith in the same way that the sun coming up tomorrow must be. It’s happened millions of times before and never hasn’t, but that’s no reason it will tomorrow. Mathematics qua mathematics, however, is beyond such mundane real-world concerns.
Yes, that does raise interesting philosophical questions. One answer might be that what you described looks a lot like logic, but isn’t really, because it fails some tests. For example, in that example, X AND NOT X would equal X, not the universe, and in the usual logic X AND NOT X should be true for the universe.
O RLY? You’re assuming the Law of the Excluded Middle now? That’s something else that goes, and which a large number of very good mathematicians don’t fully accept.
Further, show me a proof that the LEM obtains in the real world. That is, the mathematics built in the “standard” topos is the one which most accurately describes our physical universe, and not the mathematics built in an “intuitionistic” topos.
OK, I’m having real trouble understanding the issue here, and, for that matter, most of the maths being discussed here, as I collapsed into a tiny whimpering ball at set theory in college and promptly entered the world of applications programming.
But I’m having trouble because I don’t know how you can ask for “proof” of one of the fundamentals of a completely abstract system. Math doesn’t exist *anywhere * except in the minds of sentients. You can prove lots and lots of things from the basic rules, but they’re all predicated on the definitions of the terms. We define addition as that operation which renders 2 when it is applied to 1 an 1. Or perhaps a better way to say that would be that 2 is defined as the result of the operation of addition between 1 and 1 (that is, after all, how little kids are taught using a number line, isn’t it?).
How can you possibly prove this? To me, addition constitutes one of the absolute fundamentals, but then I am a higher math idiot, and apparently there are more basic fundamentals than that. Apparently you can start with these other fundamentals and derive addition, as several far better educated posters than I have already done here. But sooner or later, it comes down to thought-based definitions. You can’t possibly prove them in a scientific sense, because *nothing * in math actually tangibly exists. We can say that addition as a concept has never in known history failed to accurately predict corresponding reality, but that can’t prove the concept - you know that nothing can ever be empiracally proven 100%; it can only be disproven.
This is *not * comparable to religion. Religion makes real claims about reality outside of the human mind. While we tend to use a verbal shorthand in speaking that suggests that math does too, the fact is that anyone with any education realizes that it is an abstract set of rules/concepts and has no *direct * correspondence with reality. You can not view addition in the same way that you can view, say, walking. Walking can be detected by the senses, both of the walker and the observer. Addition can only be demonstrated (never proven) to be a consistently viable model for reality, never reality itself. But religion is *not * a purely abstract concept; it purports to be a description of actual, physical reality. It’s simply not the same kind of thing at all.
Addition is comparable to walking in one way. Prove that someone who is walking is actually walking. You can’t, because the word “walking” is a definition, not a physical reality. It’s a symbol, as all words are. The difference is that walking is a word that defines an action that can be detected by the senses, and the rules that apply (i.e. the physical laws of the universe) are disprovable. But math is purely a self-contained system in and of the mind; you *can’t * disprove math; at most you could demonstrate its imperfection as a model for reality.
So, sverresverre, can you at last try to make the nature of your problem clearer to this admitted higher math idiot? Because this discussion is mostly too esoteric for me, but I’d like to understand what you’re asking better.
That’s pretty much exactly what I said in my first post where I mentioned structuralism. You can define some collection of things and rules where 1+1 doesn’t equal 2, but then that collection isn’t a model of the natural numbers.
This is my point about the “unreasonable effectiveness” of mathematics. There’s no justification beyond “it’s worked all the other times we’ve tried it” for the rules of mathematics being modelled in the physical world.
mathochist, that’s why I want sverresverre what he means by “proof.” Several people here have demonstrated that addition can be derived from lower level rules (or at least I think that’s what they’re saying - it goes over my head), but what can be proven about a system which is at its base totally imaginary? If we could find a counter example (i.e. a situation in which 1 + 1 <> 2) among the natural numbers, we could disprove that it’s a viable model. But since we have never found such a thing, we operate on the assumption that it is a good model. We can never absolutely prove that 1 + 1 = 2 is a perfect model, because you can’t prove that things are true, just that they’re untrue. And within mathematics itself, if you start at my level, you can’t prove that 1 + 1 = 2, because that’s the definition we’ve all agreed on for addition (or two). You can’t prove a definition.
