Hot answers tagged

32

More fun than equations are patterns that seem to hold. Put a dot on a circle, connect it to all the other dots (none yet), there is 1 region. Second dot connects to first, two regions. Third dot connects to first two, there are now 4 regions. It sure looks like the regions are doubling. In fact, they are not. It's a fun problem, and takes you by surprise. ...


22

From day one. In my experience in Germany, proofs are taken seriously from day one, or even before that. We had a voluntary prep course before the first semester that was half a repetition of calculus (which is a part of the high school curriculum here) and half an introduction to proofs. And the very first homework assignments in analysis and linear algebra ...


21

Welcome Kostya! The mapping view is definitely important, but I don't think it's supreme. For me here's how I think about it. There are three ways to think about (basic) linear algebra: As a theory of matrices, linear maps and systems of linear equations. It is a fascinating result that when you formalize these three views the mathematical objects are ...


17

(1) Here is a $3$-case proof from Larry Cusick's webpages: Theorem. There are no rational number solutions to the equation $x^3 + x + 1 = 0$. Proof. (Proof by Contradiction.) Assume to the contrary there is a rational number $p/q$, in reduced form, with $p$ not equal to zero, that satisfies the equation. Then, we have $p^3/q^3 + p/q+ 1 = 0$. After ...


15

Most answers describe proof as an improved correctness checking tool. But, I don't think this gets at the core of the issue. The goal of math isn't to check off theorems, but to understand them and consequently nature (broadest possible meaning of the word). Proofs are a way to describe insights into a subject in a clear manner. Impossibility proofs tell ...


14

Is there any better alternative to the three-dot notation? The usual general advice is to use words instead of symbols. The best notation is no notation; whenever it is possible to avoid the use of a complicated alphabetic apparatus, avoid it. A good attitude to the preparation of written mathematical exposition is to pretend that it is spoken. Pretend that ...


14

One thing that can't be shown without proof techniques is the impossibility of something. One reasonably concrete example of this is the impossibility of constructing a square and a circle with the same area using a straight edge and a compass. Individual cases can be shown to not work and to demonstrate that it's a hard problem, but there's a world of ...


13

Theorem: Suppose you start with the number 1, and you're allowed to multiply it by 2, 3, and/or 5, as many times as you like. In this way, it's possible to get any whole number. Proof: Check this for 1, 2, 3, 4, 5, and 6. Also true for other random examples I think of, such as 10, 100, and 96. It's clearly true.


12

In my experience (U.S.), that's on the boundary between 2nd and 3rd year -- either the end of sophomore year or the start of junior year. Two years ago I did a survey of Associate in Science (2-year) Mathematics degree programs. It's not common to have a dedicated only-proofs course, but I think many use a Discrete Mathematics course as a vehicle where proof-...


12

Maybe you can use some non-mathematical examples... Black swans. Europeans assumed that all swans were white, because all swans in Europe are white, to the extent that a black swan was considered impossible--like a "flying pig". When they finally got to Australia and discovered that there are black swans, it was a big shock. Suppose someone tells ...


12

I really like the idea of a "discovery fiction" -- it gives a name to something I often try to use when teaching. Here is one suggestion. I will try to come back and write a more elaborated version of this answer later, with diagrams and proper notation, but briefly: (a) Don't let on that you are going to prove the Pythagorean Theorem -- don't ...


9

You might know (or not) enough computer science to know there are such things as functional programming languages. These are programming languages (the most popular are probably Scheme, ML, and Haskell) where loops and variables are avoided, computation is largely done by recursion, and, most importantly, functions are treated like data and can be passed as ...


9

A great start to doing proofs is working through Daniel Velleman's How to Prove it: A Structured Approach, 2nd Edition.. I've used it many times in teaching, usually as a supplementary text.


9

Because mathematicians want to be really, really certain. There are lots of ways you can be convinced that something is true, and be wrong. By requiring proofs before you are convinced, you can make that amazingly less likely. A proof requires that someone lay bare all of the weaknesses in their claim, and show how those weaknesses are not a problem. ...


7

Aside from any particular book, I'd say that you need a human being reviewing and giving feedback on your proofs. This is a type of writing for consumption by other people. One of the main things is that a proof should be clear, explanatory, and insightful. Partly this criteria depends on the level of expertise of the expected audience. Now, I'm not entirely ...


7

An example that directly contradicts the idea of "seeing that an equation works after testing it for a few values:" Theorem (not): The polynomial $x^2-x+41$ is prime for every integer value of $x$. This works after testing it for every integer $x = 0, 1, 2, \dots, 40$. So "obviously" (according to your friend!) its value is a prime number ...


6

From my experience in French Classes Préparatoires, we learn proofs during year one without a specific course about it, just while we learn calculus and linear algebra, starting on day one (we actually start a bit in high school). Maths in these classes are very rigorous, and everything that is taught gets proven (with few exceptions), even requiring to re-...


6

I can start with how it was/is done in Russia. Logic was taught in Russian gymnasiums as a separate subject in late 19th century. When Bolsheviks came to power they pulled logic out of the curriculum. Logic was reinstated in Soviet schools in late 1940s only to be abolished again by the end of 1950s when Khruschev took over. AFAIK, there is no separate logic ...


6

I am someone who had a lot of trouble with this, having no formal education in higher mathematics until very recently, and thus having to figure out what is meant by a "correct" proof all on my own. This is a topic that is close to my heart. Specifically your quote here But I would say it happened organically. It would be the same way a native ...


6

Put 100 red marbles in a box and 1 white marble. When you blindly pick a few marbles and they are all red, it does not mean all marbles in the box are red.


6

Most of the answers assume, as does the OP, that the purpose of proof is verification. In fact "establishing the truth of a proposition" is only one of the reasons that mathematicians prove things. The most-cited paper on this subject is probably Michael de Villiers' "The Role and Function of Proof in Mathematics" (Pythagoras, 1990, pp. ...


5

As a general introduction, the book Theorems, Corollaries, Lemmas, and Methods of Proof by Richard J. Rossi can be useful. For example, how to prove that a sequence converges? A detailed explanation is given on pages 168-170. As a general rule, to get better we have to do a lot of proofs. A lot of sentences to be proved can be found in problem books on the ...


5

Daniel Solow, How to Read and Do Proofs. Wiley, 6th Edition, 2013.


5

This answer is meant to supplement Daniel R. Collins' answer, which is excellent—my goal is to draw a little more distinction between the "levels" of education and training in the US. Short Answer In the United States, "proofs based" courses (and formal proofs in general) are typically regarded as topics in "higher mathematics", ...


5

In the UK, students usually learn proofs in the first year of a mathematics degree. My experience is similar to Sumyrda's answer. They also gain some exposure to proof techniques before university in A-Level Mathematics and Further Mathematics, which include proof by contradiction, trig proofs, elementary algebraic proof and proof by induction.


5

I don't think fighting for the definition of a word makes most sense. After all, we could consider Robbie mastery versus Kahn Academy mastery (as different attributes). For what it's worth, the Kahn Academy sense of mastery is a rather normal one. The idea comes from the theory of building block automaticity. That you have skills down to the sense that ...


5

Sharing the impressions of a person who earned 2 IMO bronze medals in his youth, but whose dreams of a successful research career were never truly fulfilled :-) Mathematics is, indeed, not only about problem solving, but it isn't only about building theory either. Individual mathematicians may place themselves near one of the end points of this "...


5

From a practical point of view, your friend is right. One just has to be careful to test enough numbers. Engineers don't believe calculus because it has proofs based on analysis which is based on set theory; engineers believe calculus because lots of engineers before them used calculus to build bridges and very few of the bridges fell down. We have ...


5

This is a great question. Love it love it love it!! The following is just what occurs to me off the top of my head. Show a graph paper grid with a dot at the origin and a dot at (3,4). Say we want to find the distance between the dots. We could guess that it would be 3+4=7. Well, that would be right if these were city blocks, but it's not the right answer as ...


5

How about this: Let $k$ be an infinite field, and let $f \in k[x]$. Assume $f(t) = 0$ for all $t \in k$. Assume to the contrary that $f$ is not the zero polynomial. Then $f$ is a polynomial of degree $n \geq 1$. Choose distinct elements $z_1, z_2, z_3, \dots, z_{n+1}$ of $k$. By repeated use of the linear factor theorem, we know that $f(x) = (x-z_1)(x-z_2)(...


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