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Does mathematics contain a lot of theorems that rely on unproven assumptions?
by Jeff_Brown 3y ago
Does mathematics contain a lot of theorems that rely on unproven assumptions?
- LanceH 3y agoAbsolutely.
- roywiggins 3y agoTechnically, all of them do, that's what axioms are.
- agensaequivocum 3y agoAll knowledge ultimately relies on some self-evident first principals which are not demonstrable.
- alasdair_ 3y agoALL knowledge does? Can you prove that? :)
- ducttapecrown 3y agoWell, all knowledge capable of modeling the natural numbers is incapable of proving every true statement, so in a sense there can be no universal axiom set.
- kdmccormick 3y agoSure, I'll take a swing! If we define "knowing" some fact F as "we have proven that F follows from prerequisite facts A1..AN", then we can imagine all knowledge forms a graph where prerequsities point to the facts that they prove. Either this graph is cyclic, or there are nodes with no inbound edges. Therefore, there are facts which are either non-demonstrable, or only demonstrable via cyclic logic. I think this works, unless of course you have another definiton of "knowing" :)
- alasdair_ 3y agoI see no reason why knowledge needs to come from prerequisite facts. Showing my ignorance here, but isn't this the whole point of "cogito ergo sum"? The "fact" that we are thinking is a first principle that comes from nothing else without a prerequisite axiom.
- prox 3y agoThat is still a presupposition you are taking, you are making a statement “that thinking is the first principle that comes from nothing.” There are host of ways to debate that philosophically.
- T-A 3y agohttps://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_...
- agensaequivocum 3y agoIf first principals were demonstrated by definition they rely on some other first principles (see syllogisms). Hence you have an infinite regress which is unknowable. Thus no knowledge could be acquired. Therefore the first principals in syllogisms must be self-evident.
- zmgsabst 3y ago> Dubito ergo cogito ergo sum. Technically, we can derive one fact — there is something rather than nothing: Asking the question is itself proof.
- TheRealPomax 3y agoNot just unproven, there are a decent number that are known to be unprovable.
- alanbernstein 3y agoProbably, but we call them conjectures to indicate that we know that, and only call them theorems when we have tricked ourselves into believing otherwise
- chongli 3y agoA theorem is just a statement which has been proved. Proofs are logical deductions that begin with a set of assumptions (called the hypothesis of the theorem) and follow a sequence of valid steps to reach a result (called the conclusion of the theorem). Without any assumptions at all, you have nowhere to go. There’s nothing you can conclude if you begin by assuming nothing.
- rcme 3y agoHowever, in order to be a theorem, it's generally required that the hypothesis be consistent with the axiomatic system, which is not possible if the hypothesis can be proven false.
- chongli 3y agoAxioms can just be thought of as baseline assumptions that are included by the terms in your hypothesis. If your theorem is about some vector space V, then you’re assuming that vector spaces exist, and also assuming all of the building blocks do as well (fields all the way down to natural numbers and set theory).
- dragonwriter 3y agoYou aren't assuming existence (math is abstract, none of it relies on things existing), you are assuming certain definitions of vector spaces, etc.
- chongli 3y agoExistence is a very common term in mathematics. It doesn't mean existence like a table or a piece of paper. It means that a mathematical object could (in principle) be constructed without leading to a contradiction.
- anon291 3y agoGodel's incompleteness theorems say that all mathematics that is complicated enough to encode basic arithmetic must rely on unproven assumptions. Unproven assumptions are the basis of all mathematics. For example, even numbers rely on very unproven assumptions. We assume, for example, that there is always a number following another number, but it is not at all obvious what that means.
- denton-scratch 3y ago> We assume, for example, that there is always a number following another number Assuming you're referring to natural numbers or integers, that's not an assumption: https://proofwiki.org/wiki/Natural_Numbers_are_Infinite https://proofwiki.org/wiki/Natural_Numbers_are_Infinite
- ducttapecrown 3y agoThe existence of a set with infinite cardinality is one of the axioms of ZFC. https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory#7._Axiom_of_infinity https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_t...
- anon291 3y agoThe statement 'there are infinitely many natural numbers' is not the same as 'there's a natural number following every other natural number'. In particular, the real numbers are infinite, but there is no unique real number following another real. Moreover, the natural numbers are typically defined axiomatically, either directly or via a set-based representation. Either way, you run into the same issue which is that eventually you arrive at a set that, due to known physical constraints of the universe, is impossible to identify and you must reasonably ask yourself if such a number exists.
- denton-scratch 3y agoReal numbers != Natural numbers. Every natural number has a 'next'; there is no 'next' for a real number.