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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.glaucousNoise said: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.glaucousNoise said: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.glaucousNoise said: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.Krylov said: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.glaucousNoise said: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.
lavinia said: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.Mark44 said: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 said: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.Mark44 said: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.aikismos said:Technically speaking, the approximation of irrational values to rational ones (such as those historically used as constants in calculations involving Pi) ARE theorems.
Mark44 said: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.aikismos said::) 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.
Mark44 said: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.
aikismos said: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.
2) Theories aren't arrived at from deduction of axioms. I think you meant theorems. The former is scientific nomenclature for assertions, while the latter is mathematical.
3) While Pythagorean's Theorem can be proven from algebraic or geometric theorems, actual elliptical orbits are never deduced strictly from Newton's laws. Strictly speaking, specific ellipses can be even in mathematical models, but elliptical orbits are physical phenomena which require initial states obtained through astronomical measurements and are actually subject to complicated gravitational fields (the two-body problem is an ideal and simple model). Then after modeling, generally physical orbits have to be checked against more measurements.
glaucousNoise said:definitely seems as though aikismos, lavinia, and myself would enjoy from a new insights post on the philosophy of mathematical proof if we can find some qualified people!
perhaps you should stop debating the philosophy of mathematical proofs thenlavinia said:These Forums are not for Philosophy (which is why I always object to the term "real world") and this should be taken off line.
You're the one who keeps wanted to steer the thread in this direction. Much of this thread has been a discussion of the Insights article in the first post. Later, the discussion went off on somewhat of a tangent about computers being used to prove mathematical statements.glaucousNoise said:perhaps you should stop debating the philosophy of mathematical proofs then
In logic, we often work with “formal proofs”. These are proofs with a very rigid structure and contain only symbols and not words. They are often quite difficult to read. In theory all proofs should be “formal proofs”, but this would be unmanageable. Instead, mathematicians write informal proofs that are easy to read, but still convincing enough.
The purpose of comments in mathematical proofs is to provide additional explanation or clarification for the steps or reasoning used in the proof. They help to make the proof more understandable and accessible to readers.
Comments in a mathematical proof should be written in a clear and concise manner, using proper mathematical notation and terminology. They should also be placed strategically throughout the proof to correspond with the relevant steps or concepts being explained.
No, comments are not necessary in all mathematical proofs. They are typically used in more complex or lengthy proofs to provide additional insight or to explain particularly challenging steps. In simpler proofs, comments may not be needed as the steps and reasoning are more straightforward.
No, comments should not be used as a substitute for a proper proof. While they can provide helpful explanations, comments alone do not constitute a valid proof. The proof should still follow logical and mathematical principles and be able to stand on its own without the comments.
Yes, there are some general guidelines for using comments in mathematical proofs. These include keeping the comments concise and relevant, using proper mathematical notation and terminology, and placing the comments strategically throughout the proof. It is also important to avoid using comments as a crutch and to ensure that the proof itself is clear and logical without the comments.