What are mathematics helpful for ? Mathematics are helpful for physics. Physics helps us make fridges. Fridges are made to contain spiny lobsters, and spiny lobsters help mathematicians who eat them and have hence better abilities to do mathematics, which are helpful for physics, which helps us make fridges which… ——Laurent Schwartz

The mathematical sciences particularly exhibit order symmetry and limitations; and these are the greatest forms of the beautiful. –Aristotle


Nulla poena sine lege



No stress, friend


“A novice was trying to fix a broken Lisp machine by turning the power off and on. Knight, seeing what the student was doing, spoke sternly: “You cannot fix a machine by just power-cycling it with no understanding of what is going wrong.” Knight turned the machine off and on. The machine worked.”


Neque enim ingenium sine disciplina aut disciplina sine ingenio perfectum artificem potest efficere.——Marcus Vitruvius Pollio


“We must know, we shall know,” and concludes his response with “And with this thought in mind, I will happily continue to keep hammering pitons into the sides of the infinite mountain of mathematical truth, as we all slowly inch our way up its irresistible slopes.” ——Ron Graham



什么是内心的声音?权威一无是处吗? 不积跬步,何以至千里.

…We shall not cease from exploration
And the end of all our exploring
Will be to arrive where we started
And know the place for the first time


I was raised by a pack of wild mathematicians. We roamed the great planes proving theorems and conjecturing.


You either die a hero, or you live long enough to see yourself become the villain.

As Jean-Pierre Serre reportedly quipped to his mathematician colleague Raoul Bott, “While the other sciences search for the rules that God has chosen for this Universe, we mathematicians search for the rules that even God has to obey.”

… great mathematical expositor.

… 深解义趣,涕泪悲泣.

Point n’est besoin d’espérer pour entreprendre, ni de réussir pour perséverer.——Willem van Oranje Nassau


I write an average of five new programs every week. Poets have to write poems. I have to write computer programs.——Donald Knuth


But still a large part of mathematics which became useful developed with absolutely no desire to be useful, and in situations where nobody could possibly know in what area it would become useful; and there were no general indications that it ever would be so. By and large it is uniformly true in mathematics that there is a time lapse between a mathematical discovery and the moment when it is useful; and that this lapse of time can be anything (rom thirty to a hundred years, in some cases even more; and that the whole system seems to function without any direction, without any reference to usefulness. . . This is true for all of science. Successes were largely due to forgetting completely about what one ultimately wanted, or whether one wanted anything ulti- mately; in refusing to investigate things which profit, and in relying solely on guidance by criteria of intellectual elegance; it was by follow- ing this rule that one actually got ahead in the long run, much better than any strictly utilitarian course would have permitted.


Allez en avant, et la foi vous viendra

Grothendieck comparing two approaches, with the metaphor of opening a nut: the hammer and chisel approach, striking repeatedly until the nut opens, or just letting the nut open naturally by immersing it in some soft liquid and let time pass: “I can illustrate the second approach with the same image of a nut to be opened. The first analogy that came to my mind is of immersing the nut in some softening liquid, and why not simply water? From time to time you rub so the liquid penetrates better, and otherwise you let time pass. The shell becomes more flexible through weeks and months—when the time is ripe, hand pressure is enough, the shell opens like a perfectly ripened avocado! A different image came to me a few weeks ago. The unknown thing to be known appeared to me as some stretch of earth or hard marl, resisting penetration… the sea advances insensibly in silence, nothing seems to happen, nothing moves, the water is so far off you hardly hear it… yet it finally surrounds the resistant substance.”



Wir mussen wissen. Wir werden wissen.


You can lead a horse to water but you can’t make him drink


  • $h_\theta(x) = \Large\frac{1}{1 + \mathcal{e}^{(-\theta^\top x)}}$ ;
  • $a^2 + b^2 = c^2$ ;
  • $\sum_{i=1}^m y^{(i)}$;
  • $\frac{1}{\pi} \int_{-\pi}^{\pi}|f(t)|^{2} d t=\frac{1}{2} a_{0}^{2}+\sum_{n=1}^{\infty} a_{n}^{2}+b_{n}^{2}$ ;
  • $f(x)\sim \frac{a_0}{2}+\sum_{n=1}^{\infty}a_n\cos(nx)+b_n\sin(nx) $ ;



  1. $ a^{2}+b^{2}=c^{2} $

  2. $ \log x y=\log x+\log y $

  3. $ \frac{\mathrm{d} f}{\mathrm{d} t}=\lim _{h \rightarrow 0} \frac{f(t+h)-f(t)}{h} $

  4. $ F=G \frac{m_{1} m_{2}}{r^{2}} $

  5. $ i^{2}=-1 $

  6. $ V-E+F=2 $

  7. $ \Phi(x)=\frac{1}{\sqrt{2 \pi \rho}} e^{\frac{(x-\mu)^{2}}{2 \rho^{2}}} $

  8. $ \frac{\partial^{2} u}{\partial t^{2}}=c^{2} \frac{\partial^{2} u}{\partial x^{2}} $

  9. $ f(\omega)=\int_{\infty}^{\infty} f(x) e^{-2 \pi i x \omega} d x $

  10. $ \rho\left(\frac{\partial \mathbf{v}}{\partial t}+\mathbf{v} \cdot \nabla \mathbf{v}\right)=-\nabla p+\nabla \cdot \mathbf{T}+\mathbf{f} $

  11. $ \nabla \cdot \mathbf{E}=\frac{\rho}{\varepsilon_{\mathrm{e}}} $ $ \nabla \cdot \mathbf{H}=0 $ $ \nabla \times \mathbf{E}=-\frac{1}{c} \frac{\partial \mathbf{H}}{\partial t} $ $ \nabla \times \mathbf{H}=\frac{1}{c} \frac{\partial E}{\partial t} $

  12. $ \mathrm{d} S \geq 0 $

  13. $ E=m c^{2} $

  14. $ i h \frac{\partial}{\partial t} \Psi=H \Psi $

  15. $ H=-\sum p(x) \log p(x) $

  16. $ x_{t+1}=k x_{t}\left(1-x_{t}\right) $

  17. $ \frac{1}{2} \sigma^{2} S^{2} \frac{\partial^{2} V}{\partial S^{2}}+r S \frac{\partial V}{\partial S}+\frac{\partial V}{\partial t}-r V=0 $


Theory of everything(so far):
$Z=\int \mathcal{D}($ Fields $) \exp \left(i \int d^{4} x \sqrt{-g}\left(R-F_{\mu \nu} F^{\mu \nu}-G_{\mu \nu} G^{\mu \nu}-W_{\mu \nu} W^{\mu \nu}\right.\right.$ $+\sum_{i} \overline{\psi}_{i} \not D \psi_{i}+\mathcal{D}_{\mu} H^{\dagger} \mathcal{D}^{\mu} H-V(H)-\lambda_{i j} \overline{\psi}_{i} H \psi_{j}))$

Fads are as common in mathematics as in any other human activity, and it is always difficult to separate the enduring from the ephemeral in the achievements of one’s own time. At present there is a strong current of abstraction flowing through our graduate schools of mathematics. This current has scoured away many of the individual features of the landscape and replaced them with the smooth, rounded boulders of general theories. When taken in moderation, these general theories are both useful and satisfying; but one unfortunate effect of their predominance is that if a student doesn’t learn a little while he is an undergraduate about such color- ful and worthwhile topics as the wave equation, Gauss’s hypergeometric function, the gamma function, and the basic problems of the calculus of variations—among many others—then he is unlikely to do so later. The natural place for an informal acquaintance with such ideas is a leisurely introductory course on differential equations. Some of our current books on this subject remind me of a sightseeing bus whose driver is so obsessed with speeding along to meet a schedule that his passengers have little or no opportunity to enjoy the scenery. Let us be late occasionally, and take greater pleasure in the journey. –George F.Simmons

What is the use of these results? The answer is that I don’t know. They will almost certainly produce some theorems in the theory of partial differential equations, and some of them may find application in imaging with MRI or ultrasound, but that is by no means certain. It is also beside the point. Quinto and I are studying these topics because they are interesting in their own right as mathematical problems, and that is what science is all about. —Cormack