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A 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) a
by chongli 3y ago
A 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.