追Penrose的旧文，追到一本50年前的文集，Cécile M. DeWitt和John A. Wheeler编的Battelle Rencontres: 1967 Lectures in Mathematics and Physics. W. A. Benjamin, Inc., 1968. 编者的前言提出了现实的问题，在今天更现实。

   他们引冯诺依曼1947年的话说，数学分支众多，千姿百态，一个数学家能懂四分之一的领域就不错了；而物理学高度集中，好的理论物理学家能懂一半的领域。

Mathematics falls into a great number of subdivisions, differing from one another widely in character, style, aims, and influence. It shows the very opposite of the extreme  concentration of theoretical physics. A good theoretical physicist may today still have a working knowledge of more than half of his subject. I doubt that any mathematician now living has much of a relationship to more than a quarter.（这段话出自John von Neumann的一篇“散文”The Mathematician，原发表于Works of the Mind Vol. I no. 1 (University of Chicago Press, Chicago, 1947), 180-196.）

   老冯说的是30-40年代的数学和物理学，到60年代，情形不同了，数学和物理学都发生了巨变，不同科目之间的距离，甚至比Snow在50年代所说的“两种文化”（自然科学与社会科学）之间的距离还遥远，数理圈内都呈现着“百样文化”的态势（farther than ever from being Two Cultures, in the sense of Snow, they sometimes seem closer to being a hundred cultures），犹如新时期的巴别塔。于是，著名的Battelle纪念研究院（Battelle Memorial Institute）请了一群数学家和物理学家来讲各自的东西，这就是第一次“巴特尔之约”（Battelle Rencontres in Mathematics and Physics）（后来还有其他领域的）。Battelle成立于1929年，是应用和技术科学的研究机构，其创始者Gordon Battelle在遗嘱中写的目标是，for the purpose of education in connection with and the encouragement of creative and research work and the making of discoveries and inventions in connection with the metall urgy of coal, iron, steel, zinc and their allied industries.

   这第一次数理约会的东西虽然距离老Gordon的应用还遥远得很，但它影响了后来的物理学风尚——今天的物理学家不缺数学了，甚至弦理论家自豪的一点就是反过来影响了数学。    

I  Lie Groups and Symmetric Spaces / SigurdurHelgason

II  Special Functions and Representations of LieGroups / Leon Ehrenpreis

III Commutativité del'algèbre des opérateurs différentiels invariants / André Lichnerowicz

IV  Hyperbolic Partial Differential Equations on a Manifold / Yvonne Choquet-Bruhat

V  Topics on Space-Time / André Lichnerowicz

VI  Relativistic Fluids in Cosmology / Charles W.Mistier

VII  Structure of Space-Time / Roger Penrose

VIII  The Structure of Singularities / Robert Geroch

IX  Superspaceand the Nature of Quantum Geometrodynamics / John Archibald Wheeler

X  The Topology of Wheeler's Superspace / BryceS. De Witt

XI  Boundary Conditions for the State Functionalin Quantum Theory of Gravity / H.  Leutwyler

XII  The Everett-Wheeler Interpretation of QuantumMechanics / Bryce S. De Witt

XIII  Progress and Goals in Renormalization Theory /Klaus Hepp

XIV  Perturbation Theory in Quantum Field Theoryand Homology / Jean Lascoux

XV  Landau Singularities in the Physical Region / FredericPham

XVI  Algebraic Topology Methods in the Theory ofFeynman Relativistic Amplitudes / Tullio Regge XVII  The Use of Padé Approximations in ParticlePhysics / Marcel Froissart

XVIII Topics in Topology and DifferentialGeometry / Raoul Bott and John Mather

XIX  Continuous Solutions of Linear Equations—SomeExceptional Dimensions in Topology / Beno Eckmann

XX  Differentiable Dynamical Systems / StephenSmale

XXI  Characterization of Stable Mappings / John N.Mather

XXII  One-Parameter Subgroups Do Not Fill a Neighborhood of the Identity in an Infinite-Dimensional Lie (Pseudo-) Group / CharlesFreifeld

XXIII  A Dynamical Theory for Morphogenesis:Elementary Catastrophes on R / René  Thorn

XXIV  How to Turn a Sphere Inside Out / StephenSmale

XXV  Eversion of the 2-sphere / Bryce S. De Witt    

   以这个标准来看今天的数学课，多个数学文化之间的鸿沟在日益扩张，犹如东非大裂谷，也许会成为新的大洋。