Insights Proofs in Mathematics - Comments

andrewkirk

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Nice article. I like the list of tips in the linked text by Knuth et al too. I'll read more of it at my leisure to try to improve my writing. I've a couple of comments:

For me the worst flaw in proof writing is when the writer does not explain their steps. For example the justification for getting from one line to the next may be a result that was observed briefly in passing 20 pages ago and has not been mentioned since, but the author doesn't reference it so the reader is left feeling stupid because they don't remember the result and can't understand how the step is justified. A number of widely-used physics texts do this frequently and I think it's very poor. This is one area where reading symbolic logic proofs can actually be easier than reading some more wordy mathematical ones, because it is a requirement of a symbolic logic proof that the justification of every line be formally stated.

The reason that many authors omit such justification is that it's a lot of work to insert all the correct references. But for every two minutes that an author saves herself by not locating and inserting the justification ref, she has cost her readers collectively dozens, maybe hundreds of hours (depending on how many readers there are!) racking their brains and leafing through the book trying to work out the justification.

Numbering all non-trivial equations is one way to make it much easier to include such refs.

Knuth et al suggest varying one's words in order to avoid monotony. For instance one might alternate 'so', 'hence', 'therefore', 'it follows that'. My practice in the past has been to do that but now I am questioning it. One doesn't read maths proofs for the beauty of their prose (that's what poetry and fiction are for) but to gain understanding, and unnecessarily varying the terms used seems to me to be more likely to detract from understanding than to support it. A problem that sometimes arises is that, in the search for synonyms, the writer ends up using a word that has alternative possible meanings, and hence introduces ambiguity into the text. This is a matter for judgement and I imagine that there's a sweet spot somewhere between the extremes of always using the same word and avoiding using the same key word twice in a paragraph. Currently I find I'm steering away from variety towards greater consistency and clarity in word choice, relative to where I was.

jbriggs444

Science Advisor
There appears to be a typo in...

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.

glaucousNoise

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?

it seems as though the vast majority of problems will be "unprovable."

jbriggs444

Science Advisor
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?
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.

it seems as though the vast majority of problems will be "unprovable."
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.

Mark44

Mentor
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?
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.

S.G. Janssens

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I believe it is "Bernoulli" instead of "Bernouilly".

andrewkirk

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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.

Indeed, in a sense a proof is an answer to the question 'How do you know that P?' where P is some proposition. That is a non-binary question. Yes or No doesn't cut it as a proof.
it seems as though the vast majority of problems will be "unprovable."
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'.

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glaucousNoise

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.

Mark44

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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.
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.

lavinia

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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.

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aikismos

1) You say "There are actually two separate skills that one must master: finding the proof and communicating the proof." and that you focus on the former, but really, you're focusing on the later. Devising mathematical proofs (which usually involves abduction and often empirical calculation these days) isn't really covered in the article at all.

2) You might also want to revise "A proof is a convincing statement" to say collection of statements to which logic is applied to arrive at a true conclusion. I've never seen a proof that is a single statement.

3) "By its very nature, a proof uses deductive reasoning and not empirical arguments." Proof is not strictly deductive in nature (see your article where you cite example and description of proof by induction). I think you might want to revise the article to contain Proof by Exhaustion as it is often used in education to show simple cases or is a part of larger proofs. It would be prudent to include the word counter-example in your example of the equation 412−41+41=412. Euclid didn't even do most of his research deductively. It's more of a communicational and didactic tool to use deduction than actual research math (which is way more creative and less constrictive than deduction).

4) You might want to compare examples of a paragraph proof, two-column proof, and flow proof if you're trying to help math newbies understand the nature of proof. Most newbies are sloshed around different classes which may ask them to do each.

Overall, a good start on a useful FAQ!

aikismos

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.
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.

4) "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)." - I think that given the target audience, this sentence is fair with the caveat that we obviously redefine our domains to make our statement (entirely) true or false. Once you start saying 100%, then we're moving into fuzzy sets!

aikismos

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.
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

Mentor
The first time that I recall computers being used for a proof was back in the mid-70s, in the Four Color Problem.
aikismos said:
I'm pretty sure that computerized proofs go back to mechanical computers computing Pi.
I was talking about using computers in proofs of theorems, not for calculating numbers, such as the digits of $\pi$.

aikismos

I was talking about using computers in proofs of theorems, not for calculating numbers, such as the digits of $\pi$.
Technically speaking, the approximation of irrational values to rational ones (such as those historically used as constants in calculations involving Pi) ARE theorems.

Mark44

Mentor
Technically speaking, the approximation of irrational values to rational ones (such as those historically used as constants in calculations involving Pi) ARE theorems.
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

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.
:) 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?

Oh, and to show you that the approximation of pi relies on a mathematical proof, here's an interesting case where a crank almost got a wrong value of pi passed as state law in the US. (To be fair the legislators were in Indiana). You may have never thought about it, and it may not give you the warm fuzzies, but fractions to approximate irrational values aren't picked out of a hat. They're proven with algebraic axioms to be true. :D https://en.wikipedia.org/wiki/Indiana_Pi_Bill

Mark44

Mentor
:) 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.
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.

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