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142

Theoretical computer science could certainly be considered a branch of mathematics. This branch of computer science deals with computers and computer programs as mathematical objects. Theoretical computer scientists could be described as computer scientists who know little about computers. However, when people say "computer science" they usually include ...


102

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 ...


67

An example is Pythagorians discovery of irrationality of $\sqrt{2}$. They kept it as a secret for a while because of their special philosophical point of view about the rationality of all numbers in the world. In fact their cosmology were based on a presumption that everything in the nature is made of numbers and their ratios. Some stories say that finally a ...


62

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 ...


54

Let $C$ and $M$ be the set of all things which are considered "computer science" and "mathematics", respectively. If I understand you correctly, your question is: Is $C\subset M$? If this is the case, your question is not well posed because neither $C$ nor $M$ are well defined. How do you draw the line between what is math and what is not math without ...


52

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 ...


48

Actually, I think the best example is of a mathematician who is alive nowadays and proved many results which were unpublished for many years just because he wanted to present them all together in order to solve a famous problem: Fermat's last theorem.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


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 ...


39

Generally speaking, mathematical logic investigates the nature and limitations of logical systems. For example, consider the question: "can every true statement be proved?" This is a problem about logic. When we study geometry, algebra and other branches of mathematics we are interested in using logical thinking, but the purpose is to understand something ...


36

The Laplacian $\Delta f (p)$ is the lowest-order measurement of how $f$ deviates from $f(p)$ "on average" - you can interpret this either probabilistically (expected change in $f$ as you take a random walk) or geometrically (the change in the average of $f$ over balls centred at $p$). To make this second interpretation precise, write the Taylor series $$ ...


36

The diagrams are a way of describing a group generated by reflections. Any collection of reflections (in Euclidean space, say) will generate a group. To know what this group is like, you need to know more than just how many generators there are: you need to know the relationships between the generators. The Coxeter diagram tells you that information. There ...


34

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 ...


28

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 ...


26

Elementary functions are finite sums, differences, products, quotients, compositions, and $n$th roots of constants, polynomials, exponentials, logarithms, trig functions, and all of their inverse functions. The reason they are defined this way is because someone, somewhere thought they were useful. And other people believed him. Why, for example, don't we ...


24

Are there known results in mathematics that are not based on logic? Well, there's always Ramanujan's approximation of $\pi$ as $\sqrt[4]{\dfrac{2143}{22}}$ , based on a dream which he had one night about a certain Hindu goddess that his family worshipped. :-)


22

I would say that computer science is a branch of mathematics. Donald Knuth is a famous computer scientists and is also considered a great mathematician. He wrote a series of books called "The Art of Computer Programming" which is extremely rigorous and mathematical. Edit: To make my position more clear since it is apparently controversial. Almost all ...


22

Rudin, Rudin, Rudin. Baby analysis, Real and Complex Analysis, and Functional Analysis. His terse style is like no other as he makes you work for your understanding. Moreover, the exercises are fun, hard, and instructional.


21

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 ...


21

I imagine one could write books on the subject, but probably the first moment one encounters homological algebra in algebraic geometry is the following. If $f: X\to Y$ is a proper map of varieties, then given a coherent sheaf $\mathcal{F}$ on $Y$ one can form the pull-back $f^{*} \mathcal{F}$. Now, the pull-back functor $f^{*}: \mathcal{C}oh(Y) \rightarrow ...


19

Here's what I wrote in the preface to the second edition of Introduction to Smooth Manifolds: I have deliberately not provided written solutions to any of the problems, either in the back of the book or on the Internet. In my experience, if written solutions to problems are available, even the most conscientious students find it very hard to resist ...


19

See Intrinsic weak derivatives and Sobolev spaces between manifolds by Alexandra Convent and Jean Van Schaftingen Gromov's compactness theorem for pseudo holomorphic curves by Rugang Ye


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 ...


17

It is not uncommon to hear ideas along the lines that computer science is computer programming without practical constraints theoretical computer science is computer science without physical constraints mathematics is computer science without finiteness constraints Each subject in the chain is seen as a limiting case of the one before, where some ...


17

I would approach the question this way. We can think of our "library" of functions being built up recursively: Start with a few basic functions (polynomials, exponentials, logarithms, trig functions, etc.), and start composing, concatenating, integrating, etc. At each stage you have a collection of functions that have been defined "so far". What ...


17

Euler apparently had some trouble deriving the Jacobian used in change of variables for double integrals. He began by considering congruent transformations consisting of (affine) linear functions, and got something like $$\mathrm{d}x\,\mathrm{d}y=m\sqrt{1-m^2}\,\mathrm{d}t^2+(1-2m^2)\,\mathrm{d}t\,\mathrm{d}v-m\sqrt{1-m^2}\,\mathrm{d}v^2$$ which he ...


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 ...


17

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. ...


16

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 ...



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