It is important to note that the meaning of propositions is not defined within the language itself, but rather is a matter of interpretation: the statement P could mean anything I want the statement P ∧ Q could mean any two things I want. For example, ~A ∨ B is false if A is true and B is false, and true otherwise. The truth-value of a statement depends on the truth-values of its constituent propositions and which connectives are used. The simplest form of logic is called propositional logic, which lets us form statements out of propositions (which can be either true or false, and are represented by uppercase letters) joined by the connective symbols ∧ (AND), ∨ (OR), ~ (NOT) and → (IMPLIES). If a valid proof finishes in a statement, then it proves that statement. The only statements which do not need to be generated by those rules are the axioms. Importantly, each link in the chain must be generated by one of a few strict rules, which are chosen to be as obvious as possible (for example, from the statement A ∧ B, meaning that A and B are both true, there is a rule allowing us to derive the statement A, meaning that A is true). The key things to know are that logical statements are purely symbolic, and logical proofs consist of branching chains of statements. The answer is that we encode each statement and proof in a logical language we can only trust the edifice of mathematics which we are about to build up as much as we trust the rules of that language. Firstly, how do we know, rigorously, which proofs are correct and which aren't? Secondly, what do we mean by statements - can they include words, numbers, symbols? Thirdly, how can we interpret statements without any ambiguity? Would the 3rd system be decidable as well? We can't answer this yet, because the explanations so far have skimmed over a few important things. A system which only had one axiom, reading "There is a man named Socrates", would be consistent and true, but definitely not complete. ![]() A system which is complete, and for which there is an algorithm which enumerates every correct proof, must also be decidable: simply run the proof-enumerating algorithm, and stop when it reaches either the statement you want to prove or its negation, which must happen eventually.A system which included as axioms every possible statement would be complete and decidable, but inconsistent and false. ![]() It's important to thoroughly understand the properties above, and how they relate to each other in general. We now know, thanks to Einstein's relativity, that space is not flat and therefore we need to use a different version of geometry to model it properly. Fourthly, it is false in the real world (in an intuitive sense describing rigorously what that means is incredibly difficult). Thirdly, it is complete: every geometric statement can either be proved or disproved from the axioms. Secondly, it is decidable: there is an algorithm which, for any geometrical statement, decides whether it has a correct proof or not. Firstly, it is consistent: there is no way to derive a contradiction from the axioms. Euclidean geometry (as formalised more rigorously by Tarski) has a few interesting properties. ![]() ![]() Greek maths was focused on geometry, which was formalised by Euclid into 5 axioms - foundational assertions - from which he proved many theorems. Although the Babylonians were able to do some interesting calculations (particularly astronomical), mathematics in the sense of deductive arguments from premises to conclusions began with Thales, Pythagoras and their disciples. Let's start with some historical background. Any discussion or errata are of course very welcome. If you're philosophically inclined, definitely look at the last section on Truth and Induction. I hope everyone will get something out of it (including my tutors!) but if you're new to mathematics, that might not be much. Be warned that this essay is rather long it starts relatively easy and gets significantly harder. To do that, I'll explore logic, set theory, computability, information theory and philosophy. I want to investigate the nature of and limits of mathematics.
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