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EuclideanSpace
- hurryer 3y ago> We now know, through the theorems of Kirt Gödel, that there is no definitive way to classify mathematics Is this true?
- jeffparsons 3y agoNot quite; I think it was actually Kurt Gödel. Jokes aside, I assume they are referring to Gödel's incompleteness theorems, which might well reasonably be characterised in that way.
- eru 3y agoThough it only applies to some mathematical systems. Some are simple enough to be decidable.
- throwoutway 3y agoWhich? Are they consistent? Are they useful? From wikipedia: > The incompleteness theorems apply to formal systems that are of sufficient complexity to express the basic arithmetic of the natural numbers and which are consistent and effectively axiomatized.
- eigenket 3y agoTarski's axioms for geometry are the standard example of a system which is both complete and consistent. It is even decidable - there is an algorithm which you can feed a statement and it outputs whether it is true or not.
- throwoutway 3y agoOk, how relevant is Tarski's though? Seems academically, they are weak > We partially prove the theorem of [7] that Tarski’s (extremely weak!) plane geometry axioms imply Hilbert’s axioms. https://sciendo.com/article/10.2478/forma-2014-0017 https://sciendo.com/article/10.2478/forma-2014-0017 Also, thats for geometric space but I dont see how that's particularly relevant to this. Elsewhere I find that it doesn't contradict Gödel
- eigenket 3y agoI don't understand your questions. They are relevant for geometry, they aren't really relevant to anything else but they're axioms for geometry so being relevant for geometry is what you'd expect.
- throwoutway 3y agoIIRC, incompleteness theorems applies to arithmetic not geometry, the difference being subtle right?
- eigenket 3y agoThe original incompleteness theorems apply to any logical system which can formulate and prove a certain chunk of arithmetic, yes. It turns out that the amount of arithmetic you actually need is incredibly small, so these results apply to many systems including some formulations of geometry.
- eigenket 3y agoOh, by the way, it matters here that "weak" in the source you quote had a very specific technical meaning. If axioms are weak it means they apply to many things. Stronger axioms make more restrictions and apply to more specific things. So (arguably) axioms being weak is a good thing. Generally you want to work with axioms that are as weak as you can get away with.
- throwoutway 3y ago
- eru 3y agoYes, they are consistent. Yes, there are useful systems. See https://en.wikipedia.org/wiki/Agda_(programming_language) https://en.wikipedia.org/wiki/Agda_(programming_language) for a reasonably practical example: > Agda is a total language, i.e., each program in it must terminate and all possible patterns must be matched.
- nateburke 3y agoCan the theorems of Kirt Gödel be classified, though, is my question. And if so, would their domains of applicability within the universe of formal systems actually be a classification system of that universe in and of itself?
- bigdict 3y agoWho is Kirt Gödel?
- xeonmc 3y agoActually, I believe it was rather his distant cousin, Kirk Girdle, with his Incompetence Theorem.
- abetusk 3y agoGodel's incompleteness theorem say that a system of sufficient complexity can not be bother complete and consistent [0]. Consistency ensures that statements can not be both true and false (under a given set of axioms). Completeness says that everything that can be written down can be proved either true or false (under a given set of axioms). You must sacrifice one or the other. As we mostly care more about consistency (we don't want to make nonsense statements), completeness is sacrificed. Meaning, there will be things we can write down that we won't ever be able to prove true or false (relative to a set of axioms). There are many other implications and subtleties, like potentially not being able to show a set of axioms is complete or consistent, potentially not being able to show the equivalence of axioms, not knowing whether adding an axiom is still consistent, etc. Turing understood Godel's incompleteness theorem and applied it to computation. The computer science way of saying the above is that one can never make a program that can determine whether other programs halt or not (in finite time). Practically, this means that there will never be a compiler that will tell you whether you program has a bug. This is called the Halting Problem [1] and directly relates to Godel's incompleteness theorem as one can cast mathematics through the lens of computation and running Turing machines. Presumably the "classification" here is some combination of whether models are equivalent or consistent and complete. [0] https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_... [1] https://en.wikipedia.org/wiki/Turing_machine https://en.wikipedia.org/wiki/Turing_machine
- altruios 3y agoone small detail to clarify. We can't build a general 'halting determiner' for any arbitrary program. But to say that for any particular program a halting machine can be developed is a different proposition. A simple example is the fact we don't have to run this code for ever to determine if it halts or not: ``` while(true){} ``` but we won't ever get a function() named X that outputs halt/not where x(turning machine / computer program). we able to, however, make a X(TM/C) where it outputs trivially halts or undetermined (past y steps).
- abetusk 3y agoYes you're right. I should have said "it's impossible to create a program that, in general, for all input programs, determines whether it halts or not (in finite time)." The whole of mathematics and computer science is, in some sense, analyzing those programs which we can prove to halt.
- nh23423fefe 3y agoI've become obsessed with Geometric Algebra recently. I like this resource https://euclideanspace.com/maths/algebra/clifford/index.htm https://euclideanspace.com/maths/algebra/clifford/index.htm