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Proofs in Mathematics
Continue reading the Original PF Insights Post.
Proofs in Mathematics
Continue reading the Original PF Insights Post.
Proof by induction
A proof by induction is often used when we have to proof something for all natural numbers. The idea behind the proof by contradiction is that of falling dominoes.
Any problem that has a real-valued numeric answer can, in principle, be answered by posing an infinite sequence of yes or no questions. This is one of the ways that the computability of a real number is phrased within the mathematics of computing.doesn't searching for problems where it is possible to obtain a binary true or false answer severely limit the problems you can look at?
Since the number of finitely expressible problems is countably infinite, the notion of a "majority" is not clear cut. But yes, not all problems can be solved.it seems as though the vast majority of problems will be "unprovable."
So what? You're missing the point of this article, which is a short description of some types of proofs in mathematics. Each proof justifies a given statement in mathematics, which is either true or false. A proof gives us confidence that the statement is true.doesn't searching for problems where it is possible to obtain a binary true or false answer severely limit the problems you can look at?
Every question that has a non-binary answer, such as 'What is the value of ##e^{i\pi}##?', when given an answer, has a supplementary question: 'How do you know?', to which the answer is a proof.doesn't searching for problems where it is possible to obtain a binary true or false answer severely limit the problems you can look at?
I personally feel that is right, because of some vague, unformed connection to Godel's First Incompleteness Theorem. But it's just a feeling and, since the set of possible propositions is infinite, and the subsets that are provable and unprovable given any given logical language and set of non-logical axioms, both have cardinality ##\aleph_0##, it's not clear what we could mean by 'the vast majority'.it seems as though the vast majority of problems will be "unprovable."
I agree. @micromass, @bcrowell, or @mathwonk, one of you might want to take care of this.I believe it is "Bernoulli" instead of "Bernouilly".
Go ahead and start one, if you like. The first time that I recall computers being used for a proof was back in the mid-70s, in the Four Color Problem.Mark44 got mad at me (reasonably, I think) for making the thread somewhat off topic. If somebody wants to create a new thread which discusses the merits of proof in the 21st century world of big data and supercomputers, I would love to have that discussion.
1) Math isn't really considered a science anymore, and that is an old fashioned language popular during the time of Gauss. The scientific methods have diverged substantially from the mathematical methods although really both are quite severely intertwined in practice. Proof is used by science and logic and even law and argumentation, but this article is just about math proof, so no need to get all technical on the divergence of the term 'proof' itself.A concise and clear description of what a proof is.
1) One might add that proof is used in all scientific theories. The difference in mathematics is that proof gives certainty while in other sciences it does not. All theories deduce conclusions from axioms. Just as the Pythagorean theorem may be deduced from the axioms of Euclidean geometry so can elliptical orbits of a two body system be deduced from Newton's Laws. Proof is not unique to mathematics.
2) I think this sentence is badly stated.
"This reasoning goes against the heart of mathematics. In mathematics, we don’t just want the statement to hold for “most cases”, we want to make the statement work for “all cases”. Mathematics tries to provide results that are 100% true or 100% false. A result that holds for “most cases” is uninteresting (unless one can rigorously define what “most cases” means)."
Any theory wants to define conditions in which certain principles hold always.
I'm pretty sure that computerized proofs go back to mechanical computers computing Pi. If you'd like... I have a simple but informative source and could find this if you'd like.Go ahead and start one, if you like. The first time that I recall computers being used for a proof was back in the mid-70s, in the Four Color Problem.
The first time that I recall computers being used for a proof was back in the mid-70s, in the Four Color Problem.
I was talking about using computers in proofs of theorems, not for calculating numbers, such as the digits of ##\pi##.aikismos said:I'm pretty sure that computerized proofs go back to mechanical computers computing Pi.
Technically speaking, the approximation of irrational values to rational ones (such as those historically used as constants in calculations involving Pi) ARE theorems.I was talking about using computers in proofs of theorems, not for calculating numbers, such as the digits of ##\pi##.
That's really a stretch, IMO. There is a huge difference between a Univac-era computer cranking out the decimal digits of ##\pi##, as compared to a computer tabulating all possible ways that a map could be colored using four colors.Technically speaking, the approximation of irrational values to rational ones (such as those historically used as constants in calculations involving Pi) ARE theorems.
:) Not a stretch at all, and in fact, for two-thousand years, arguments over the nature of proof of fractions to approximate constants have raged between some math minds greater than you or I. As proof that rational approximations of irrational constants requires proof, cite me any irrational approximation, and I will show you that the value was derived from a mathematical technique PROVEN to be true. Now granted we don't teach them in school (and hence our lack of familiarity with the thousands of proofs used to find values for pi), but our ignorance doesn't imply that those proofs don't exist (they clearly do and are argued like any other) or that they are irrelevant (science relies heavily on math constants for their calculations). When anyone says that pi is about 3.1415, it's not an assumption, but a proven approximation. Both Karl Gauss and Leonard Euler devised proofs for approximations of pi. Of course, if you're suggesting your views on proof bear more weight, than I can respect your self-confidence. ;) Perhaps you're just biased towards proofs in discrete mathematics over real analysis?That's really a stretch, IMO. There is a huge difference between a Univac-era computer cranking out the decimal digits of ##\pi##, as compared to a computer tabulating all possible ways that a map could be colored using four colors.
My complaint about this is that the proofs were done beforehand -- for example, that ##\pi = 4\sum_{n = 0}^{\infty} \frac{(-1)^n}{2n + 1}## (Gregory-Liebniz series), to name just one formulation. There are many more here - https://en.wikipedia.org/wiki/Approximations_of_π. The role of the computer was to to the arithmetic, not the actual proof.:) Not a stretch at all, and in fact, for two-thousand years, arguments over the nature of proof of fractions to approximate constants have raged between some math minds greater than you or I. As proof that rational approximations of irrational constants requires proof, cite me any irrational approximation, and I will show you that the value was derived from a mathematical technique PROVEN to be true.