Proofs and Refutations: The Logic of Mathematical Discovery by Imre LakatosDespite playing such a major role in philosophys formal genesis, the dialogue has often presented a challenge to contemporary philosophers. Many are apt to shy away from it due to its apparent levity and lack of rigor. However, the dialogue possesses significant didactic and autotelic advantages. At its best, it can reveal without effort the dialectic manner in which knowledge and disciplines develop. This way, the reader has a chance to experience the process.
Proofs and Refutations is a paragon of dialogical philosophy. Using just a few historical case studies, the book presents a powerful rebuttal of the formalist characterization of mathematics as an additive process in which absolute truth is gradually arrived at through infallible deductions. The logic of discovery, he claims, is a much messier affair. Theorems begin as mere conjectures, whose proofs are informal and whose terms are vaguely defined. It is only through a dialectical process, which Lakatos dubs the method of proofs and refutations, that mathematicians finally arrive at the subtle definitions and absolute theorems that they later end up taking so much for granted.
Lakatos contrasts the formalist method of approaching mathematical history against his own, consciously heuristic approach. Instead of treating definitions as if they have been conjured up by divine insight to allow the mathematician to deduce theorems from the bottom up, the heuristic approach recognizes the very top down aspect of performing mathematics, by which definitions develop as a consequence of the refinement of proofs and their related concepts. Ultimately, the naive conjecture (the top) is where the mathematician begins, and it is only after the process of proofs and refutations has finalized that we are even prepared to present mathematics as beginning from first principles and flourishing therefrom.
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Imagine that we place several points on the circumference of a circle and connect every point with each other. This divides the circle into many different regions, and we can count the number of regions in each case. The diagrams below show how many regions there are for several different numbers of points on the circumference. We have to make sure that only two lines meet at every intersection inside the circle, not three or more. We can immediately see a pattern: the number of regions is always twice the previous one, so that we get the sequence 1, 2, 4, 8, 16, … This means that with 6 points on the circumference there would be 32 regions, and with 7 points there would be 64 regions. We might decide that we are happy with this result.
A mathematical proof is an inferential argument for a mathematical statement , showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems ; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms ,    along with accepted rules of inference. Proofs are examples of exhaustive deductive reasoning or exhaustive inductive reasoning which establish logical certainty, and are distinguished from empirical arguments or non-exhaustive inductive reasoning which establish "reasonable expectation". Enumerating many confirmatory cases is not enough for a proof, which must demonstrate that the statement is always true occasionally by listing all possible cases and showing that it holds in each. An unproven proposition that is believed to be true is known as a conjecture. Proofs employ logic expressed in mathematical symbols, along with some amount of natural language which usually admits some ambiguity.
Mathematical proofs can be difficult, but can be conquered with the proper background knowledge of both mathematics and the format of a proof.
poems to describe her beauty
Pure mathematics consists entirely of such asseverations as that, if such and such a proposition is true of anything, then such and such another proposition is true of that thing It's essential not to discuss whether the proposition is really true, and not to mention what the anything is of which it is supposed to be true If our hypothesis is about anything and not about some one or more particular things, then our deductions constitute mathematics. Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true. Although the ubiquity of people who neither know what they're talking about nor know whether what they're saying is true may incorrectly suggest that mathematical genius is rampant, the quote does give a succinct, albeit overstated, summary of the formal axiomatic approach to mathematics. Both opinions are enjoyable and thought provoking.