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112

Niccolò “Tartaglia” Fontana invented the first general method to find the roots of an arbitrary cubic equation (based on earlier work by himself and others on how to solve cubics of particular forms), but kept his method secret so as to preserve his advantage in problem-solving competitions with other mathematicians. He divulged the secret to his student ...


70

William Sealy Gosset, while working at Guinness, developed a way of gauging the quality of raw materials with very few samples ($\implies$ less lab work $\implies$ cheaper!). As the story goes, company policy at Guinness forbade its chemists from publishing their findings. Thus Gosset had to publish under the pseudonym "Student". The results of his work are ...


58

Cryptography is often a good source of such instances: At GCHQ, Cocks was told about James H. Ellis' "non-secret encryption" and further that since it had been suggested in the late 1960s, no one had been able to find a way to actually implement the concept. Cocks was intrigued, and developed, in 1973, what has become known as the RSA encryption ...


55

Actually, I think the best example is of a mathematician who is alive nowadays and proved many results which were unpublished for many years just because he wanted to present them all together in order to solve a famous problem: Fermat's last theorem. His name is Andrew Wiles.


48

I think the answer to this question is, unfortunately, a little difficult. As many will point out, Wolfram is beyond egotistical and that fact definitely colors the reception of the book. There is a long list of (mostly negative) reviews here. The negativity reaches its apex in the review by Cosma Shalizi. There are some positive reviews as well, though, ...


43

6accdae13eff7i3l9n4o4qrr4s8t12ux Added later: I posted the "aenrsw" above in cryptic form as a bit of secretive levity, but in case the link gets broken, the character string is an anagram used by Newton in a 1677 letter to Leibniz to lay claim to the fundamental theorem of calculus without revealing it. A second, famous example, from an 1829 letter ...


38

The writing of Stephen Wolfram's book A New Kind of Science involved Wolfram Research employees. Wolfram considers proofs done by these employees to be covered by their NDAs, which is exemplified by Matthew Cook's proof that the cellular automata Rule 110 is Turing complete. That proof was first written in 1994, but Matthew Cook was dissuaded from trying to ...


37

There are now quite a few excellent ones, but most of these are pitched at fairly sophisticated readers-graduate students or professional mathematicans. The thinking according to such textbooks, of course,is that the readers are very far along in thier mathematical training and are ready to use that mathematics to learn physics at a very high level. Whether ...


32

Let $A$, $B$ be two closed subsets of $[0, 1]$, both with measure $\frac{1}{2}$, and suppose $A\cap B = \emptyset$. Then $$m([0, 1]\setminus(A\cup B)) = m([0, 1]) - (m(A) + m(B)) = 1 - \left(\tfrac{1}{2} + \tfrac{1}{2}\right) = 0.$$ But $[0, 1]\setminus(A\cup B)$ is open and the only open set with measure zero is the empty set, so $[0, 1]\setminus(A\cup ...


26

In reality, an exact side length of one meter does not exist, either. Nor does an exact square shape. Also note that the digit sequences as such are irrelevant as they depend on the units involved - with a suitable unit, the diagonal is maybe one kellicap long and the side length is irrational.


24

It's not the number $\sqrt{2}$ that's non-terminating; it's the decimal expansion of the number that's non-terminating. If you try to write down the entire decimal expansion of the number, you'll be writing forever, but the number itself is just a small number between $1.4$ and $1.5$.


24

Just to get a data point, using Maple I took $2000$ random quintics with coefficients pseudo-random numbers from -100 to 100 (but the coefficient of $x^5$ nonzero). $1981$ of these were irreducible (of course the reducible ones are solvable). All $1981$ irreducible quintics were not solvable. EDIT: Quintics with a rational root are solvable, and these are ...


23

One sparkling gem at the intersection of number theory and geometry is Aubry's reflective generation of primitive Pythagorean triples, i.e. coprime naturals $\,(x,y,z)\,$with $\,x^2 + y^2 = z^2.\,$ Dividing by $z^2$ yields $\,(x/z)^2\!+(y/z)^2 = 1,\,$ so each triple corresponds to a rational point $(x/z,\,y/z)$ on the unit circle. Aubry showed that we can ...


22

There are influential posthumous papers by authors who did not publish them. Galois had an immense influence on algebra because of the publication of something he wrote the night before he died in a duel. He gave us the word "group". Thomas Bayes had an influential posthumous paper in which he found the conditional probability distribution of a random ...


21

The words "never" and "anything" are somewhat restrictive (and we know that absolute statements are always wrong). Nowadays, you can't be recognized at all when you're not publishing. Additionally, the term "Genius" is hard to define... One person came to my mind is Grigori Yakovlevich Perelman. He hardly published anything, did not even defend his ...


20

We usually denote $n$-tuples in the form $(a_1,\ldots,a_n)$, so for example $(x,y,z)$ is a triple, $(x,y)$ is a pair, somewhat redundantly $(x)$ could be called a one-tuple and $()$ a (in fact: the) zero-tuple.


19

Onsager announced in 1948 that he and Kaufman had found a proof for the fact that the spontaneous magnetization of the Ising model on the square lattice with couplings $J_1$ and $J_2$ is given by $M = \left(1 - \left[\sinh 2\beta J_1 \sinh 2\beta J_2\right]^{-2}\right)^{\frac{1}{8}}$ But he kept the proof a secret as a challenge to the physics community. ...


18

I do not know whether this is appropriate for an answer: the work being secret, I cannot tell what it is (this is not a joke). And I would not be enough of a mathematician to describe it anyway. It concerns Alexander Grothendieck, 86 today, winner of the Fields Medal in 1966. My original information is from the French magazine La Recherche, n°486, April ...


18

Why do we stop at the exponentiation stage in the arithmetic of natural numbers? We don't actually stop there. Knuth's up-arrow notation generalizes the operations up to any level you want and Conway's chained arrow notation even further. Note though that all these generalizations are applicable only for $n\in\mathbb{N}$. Inability to generalize and ...


18

First of all: you shouldn't give up on problems after 30 minutes. Take a break, try a different problem, maybe wait a few days and try again -- you'll gain a lot more from the problem if you struggle and solve it yourself. Having access to solutions can be helpful, but you don't want to find yourself relying on them. (There's a phrase that gets thrown around ...


17

Descriptive Geometry was invented by Gaspard Monge in 1765 (he was eighteen) for military applications. For 15 years (possibly more) it was classified and kept as a military secret. It is a geometric technique to represent 3-dimensional objects through projections on planes. It allows 3D geometric constructions similar to what is usually done in 2D, through ...


16

I'm not sure I understand your question correctly. Are you asking for elementary results proved using much more complicated theorems? If so, here's my favourite example. We will prove that the $n^{th}$ root of $2$ is irrational, for $n>2$ Suppose $\sqrt[n]{2}=\frac{p}{q}$, then $2 = \frac{p^n}{q^n}$, which implies $p^n=q^n+q^n$ This contradicts ...


16

No, SGA VI it is neither necessary nor sufficient! Beware that whatever your prerequisites Fulton's book is incredibly difficult . I would advise you to concentrate on chapter one and to take your time for digesting the rich but concisely explained material there. A great strategy would be to simultaneously read Eisenbud-Harris's online treatise, ...


16

Since no one has mentioned it, I must add V.I Arnold's excellent "Mathematical Methods of Classical Mechanics". As the title suggests, the book focuses on classical mechanics, which is always a good start for a physics education. I'd suggest reading this or other good classical mechanics texts before moving on to various quantum theories, as it's difficult ...


16

Possibly Abbott, Understanding Analysis


16

I will give you the same answer I gave to a friend some years ago (I don't know if it's right... How can we know? Is this question about mathematics?): Irrational numbers are the result of calculations, not of measurements with rulers. These calculations are based on axioms that were extrapolated from experience and influenced by human intuition. We can ...


15

The paragraph you refer to is about probably 50th and 60th, and I am not well aware of the book from that period. However, I would like to point out that starting from 1980 and till 1992 a series of math and physics books was published under the title "Библиотечка Кванта" (Kvant's library). Some of these books are translations of very insightful books, but ...


14

Generally, no. But you could say: Let us denote something with (). ... and start using (), if you think this would convey your point. There is no need for academic reference validating this, it is the matter of basic author freedom. Let's say you want to wear a violet tie with blue dots. You don't ask if there is a law permitting you to do this. ...


14

$a^n+b^n=c^n$ has non-trivial integer solutions if and only if $n\le2$



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