日日深杯酒满，朝朝小圃花开。自歌自舞自开怀，无拘无束无碍。



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

“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 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.

… great mathematical expositor.

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

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.”

下面是数学公式测试片段:

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

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