Cynosure对《Trick or Truth?》的笔记(2)

笔记梳理文章的逻辑结构，记录了我认为重要的一些信息。对于本文，我的理解是，作者的4个分论点并没有充分地回答“数学有效性”。好在作者也考虑到了这个不足，所以引用了Hamming和Putnam的话。这两个人的话很赞！所以在我看来，本文最大的意义在于搜集了前人对于“数学有效性”的优秀的观点，并呈现给我们。

Children of the Cosmos

Presenting a Toy Model of Science with a SupportingCast of Infinitesimals，by Sylvia Wenmackers

In 1960,Wigner wrote an essay ponderingsuch correspondences entitled, “The Unreasonable Effectiveness of Mathematics inthe Natural Sciences”. 他问了个问题：数学为什么能如此有效？比如，甚至能预测新的基本粒子。本书是fqxi发起的讨论，试图回答Wigner的“数学有效性”这个问题。本文是这次讨论的一等奖，代表了不少人认可的一种观点。

本文的论点：Mathematics may seem unreasonably effective in the natural sciences, inparticular in physics. In this essay, I argue that this judgment can be attributed, at leastin part, to selection effects.

（作者对自然科学和数学的看法是：To me, mathematics is a long-lasting and collective attempt at thinking systematically about hypothetical structures）

四个分论点：

分论点1。we are creatures that evolved within this Universe, and that our pattern finding abilities are selected by this very environment. (we are selected (A Natural History of Mathematicians))

（我对pattern的理解是：当我们从纷繁复杂的自然现象（或数据）中找出规律时，就可以认为我们找到了pattern；当我们面对复杂系统而不知该如何理解它时，我们选择观察该复杂系统有没有什么规律（pattern），然后通过研究这些规律去理解该复杂系统。

我们的物理规律可以说是 我们认为自然界呈现给我们的pattern(s) ）

把论证该分论点转化为回答两个问题：

问题1。What enables us to do mathematics at all?

问题2。how is it that we cannot simply describe real-world phenomena with mathematics, but even predict later observations with it?

首先，人类（和动物）天生有计数的能力。这种能力源于the biological evolution of our species and its predecessors.

其次，Our current abilities are advanced, yet limited. 。。。We are prone to patternicity, which is a bias that makes us see patterns in accidental correlations。这可能源于In our evolutionary past, appropriately identifying many patterns yielded a largeradvantage than the disadvantage due to false positives.（以识别老虎为例）

总结该分论点：mathematics is a form of human reasoning—the most sophisticated of its kind. When this reasoning is combined with empirical facts, we should not be perplexed that—on occasions—this allows us to effectively describeand even predict features of the natural world. The fact that our reasoning can be applied successfully to this aim is precisely why the traits that enable us to achievethis were selected in our biological evolution.

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分论点2。our mathematics—although not fully constrained by the natural world—is strongly inspired by our perception of it. (our mathematics is selected (Mathematics as Constrained Imagination);)

I argue that there are additional factors at play that can explain this success（success指数学有效性）—makingthese unintentional applications of mathematics more likely after all.

In my view, mathematics is about exploring hypothetical structures; some call it thescience of patterns. Where do these structures or patterns come from? Well, they maybe direct abstractions of objects or processes in reality, but they may also be inspiredby reality in a more indirect fashion.

The hypothetical structures of mathematics arenot concocted in a physical or conceptual vacuum. Even in pure mathematics, thisphysical selection bias acts very closely to the source of innovation and creativity。

数学发展也是一个有选择的进化过程。The selection process is mainly driven by cultural factors, which are internal to mathematics (favoring theories that exhibit epistemic virtues such asbeauty and simplicity). But, 。。。empirical factors come into play as well, mediated by external interactions with science. Although mathematics is often described as an a priori activity, unstained by any empiricalinput, this description itself involves an idealization. In reality, there is no a priori.

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分论点3。（it）finds fault with the usual assessment of the efficiency of mathematics: our focus on the rare successes leaves us blind to the ubiquitous failures (selection bias).(effective applications of mathematics remain the exception rather thanthe rule (Mathematics Fails Science More Often Than Not).)

即，赞成“数学有效性”的人没有考虑到“有很多数学理论失败了”。

我的理解是：比如，首先，数学家常常失败了很多次之后才建立了一个逻辑自洽的理论（比如四元数）。其次，数学家（理论物理学家）曾发展新的数学工具把广义相对论推广为很多不同的版本。后来其中的某些版本被实验否定了，那么与之相关的数学工具也被认为是错误的（无效的）。

所以，research, even in pure mathematics,is biased towards the themes of the natural sciences.

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分论点4。the act of applying mathematics provides many more degrees offreedom than those internal to mathematics. (the application of mathematics has degrees of freedom beyond those internal to-mathematics(Abundant Degrees of Freedom in Applying Mathematics: The Caseof Infinitesimals);)

论证：微积分里Infinitesimals这个概念一直不够让人信服。为此，后来数学家发明了hyperreal numbers即‘non-standard analysis’(NSA)。

NSA的优点1: It has been suggested that NSA can provide a post hoc justification for how infinitesimals are used inphysics 。

NSA的优点2: NSA can be employed to make sense of classical limits in physics: classical mechanics can be modelled as quantum mechanics with an infinitesimal Planckconstant [22].

NSA的优点3: When an apparatus with betterresolving power is developed, some quantities that used to be unobservably smallbecome observable [26, 27]. This shift in the observable-unobservable distinctioncan be modelled by a form of NSA, called relative analysis, as a move to a finercontext level [28].

NSA的优点4: The interpretation of (relative) infinitesimals as (currently) unobservable quantities is suggestive of why the calculus is so applicable to the natural sciences: it appearsthat infinitesimals provide scientists with the flexibility they need to fit mathematicaltheories to the empirically accessible world.

以上四个分论点论述完毕。我的理解，作者的逻辑是：人类进化得越来越聪明，数学也进化得越来越复杂、描述了越来越多的自然现象。但我觉得还有一个问题没有回答：

即使这样，为什么数学“就能”描述之？（类似“为什么宇宙是可以理解的？”）或者说，有没有这么一个可能：终于有一天，数学家无论如何也无法发明出新的数学工具（以辅助物理学家）去解释新的自然现象？

我觉得文章接下来就是在回答这个问题。

Could our cosmos have been different—so different that a mathematical descriptionof it would have been fundamentally impossible？作者的回答是“No”，即无论如何都是可以理解的。（the answer to the speculative question at the start of this section is ‘no’ and trivially so, for otherwiseit would not be a cosmos.）

首先，明确概念：cosmos。。。referred to the order of the Universe (not the Universe itself).It is closely related to the search for archai or fundamental ordering principles.。。。Since these archai had to be understandable to humans, without divine intervention or mystical revelation, they had to be limited in number and sufficiently simple.。。。

然后，进一步明确论点：On my view of mathematics, the further step amounts to claiming that nature itself is—at leastin principle—understandable by humans. I think that all we understand about nature are our mathematical representations of it. Ultimately, reality is not something to be understood, merely to be. （更简洁的表述：with mathematics, we can think the unthinkable.）

最后，开始论证：

第一，作者以随机现象为例：When we try to imagine a world that would defy our mathematical prowess, it is tempting to think of a world that is totally random. 论证逻辑是：起初人们无法理解随机现象，后来发明了概率统计来研究之。最后人们认为自己理解了这些随机现象。

第二，作者以奇幻绘画作品为例：Many people are able to recognize a Dalí painting instantly as his work,even if they have not seen this particular painting before. Since we started from human works of art, unsurprisingly, the strategy fails to outpace our own constrained imagination. 我所理解的作者的逻辑是：虽然人们没有看过Dalí的某个画作A，但人们曾经看过他的其他画作B。所以当看到画作A时，人们一眼就认出A是Dalí的作品。这说明A具有Dalí的绘画风格。这说明A和B有相同的pattern。

（有一幅画作C。鉴赏师X从某些细节推断C符合Dalí的绘画风格（pattern），认为C是真迹；鉴赏师Y从另一些细节推断C不符合Dalí的绘画风格，认为C是赝品。）

第三，作者以“尝试理解一个无论如何也无法理解的世界”为例。I can imagine a world in which processes cannot be summarized orapproximated in a meaningful way. Our form of intelligence is aimed at finding the gist in information streams, so it would not help us in this world (in which it wouldnot arise spontaneously by biological evolution either).

对于“无法理解的世界”，作者先是反驳了Max Tegmark对该问题的解释：From myview of mathematics as constrained imagination, however, the idea of a mathematicalmultiverse is still restricted by what is thinkable by us, humans.（Max Tegmark的multiverse是已经被理解的（用数学描述的）。但现在面对的是一个无法理解的世界，所以Max Tegmark的回答不令人信服。）

作者然后通过思考Wigner和Einstein来继续回答“如何去理解一个无法理解的世界”。

对于Wigner，作者引用了几个前人的回答：

Hamming的回答：

if mathematicians would have started out with a system in which those crucial results would not hold, then。。。they would have changed their postulates until they did.

Putnam的回答：

mathematical knowledge resembles empirical knowledge in many respects: “the criterion of success in mathematics is the success of its ideas in practice”

（我对Hamming和Putnam的话的理解是：

假设物理学家从现象Pa、Pb里分别总结出规律La、Lb。

数学家把La和Lb的共同之处归纳为基本假设A：

“La和Lb的共同之处是X”

之后数学家基于A发展和完善新的数学工具。

后来，随着实验精度的提高，现象Pb里有了新的发现Pb’，对此物理学家总结出新的规律Lb’。但数学家发现La和Lb’没有任何共同之处。如何归纳出新的共同之处呢？苦苦思索后，数学家决定把假设A更新为B：

“La和Lb的共同之处是“它们相互之间没有共同之处””

之后数学家基于B发展和完善新的数学工具。）

Abbott的回答：

“all physical laws and mathematical expressions of those laws are […] necessarily compressed due to the limitations of the human mind”[35, p. 2150]. He explains that the associated loss of information does not precludeusefulness “provided the effects we have neglected are small”,。。。“the class of successful mathematical models is preselected”, whichhe described as a “Darwinian selection process”。。。he is well aware that “when analytical methods become too complex,we simply resort to empirical models and simulations”

对于Einstein，作者引用他的话：geometry stems from empirical “earth-measuring”, but modern axiomatic geometry, which allows us to consider multiple axiomatisations (including non-Euclidean ones), remains silent onwhether any of these axiom schemes applies to reality。。。axiomatic geometry can be supplemented with a proposition to relate mathematical concepts to objects of experience: “Geometry thus completed is evidently a natural science”

我的理解是（可能不太准确），欧氏几何的5条公设是和测量相关的（比如，有一道平面几何的选择题，求两点间距离。虽然我不知道该如何计算，但我可以“做实验”：用尺规把题目的各个条件画出来，然后用尺子测量一下待求的两点间的距离。最后选择与此最接近的那个答案。）

。但现代几何是基于群论的，其中不包含测量这个概念。所以无法用现代几何的思路“做实验”。那么为了能让其“做实验（成为自然科学）”，就需要给现代几何加上额外的proposition。

CH7 The Deeper Roles of Mathematicsin Physical Laws，by Kevin H. Knuth

这篇文章可以说是在讨论“加法的本质是什么”——这也太酷炫了吧！——借此说明“数学有效性”这个问题。

p84 用inclusion-exclusion principle（Sum Rule）来解释：

χ = V − E + F （Euler characteristic）

E = (A + B + C) − π （Spherical excess）（这个是怎么推导的？因为是ABC三个数，而不是两个）

太酷炫了！

Knuth K.H., Skilling J. 2012. Foundations of Inference. Axioms 1:38–73, doi:10.3390/axioms1010038, arXiv:1008.4831 [math.PR] ，似乎提到了乘法(从product rule of probability的角度去理解乘法）

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How Not to Factor a Miracle, by Derek K. Wise

区分了两种reductionism。

举例quotient （ group）的例子非常易于理解。

对于“宇宙为什么对于我们是可知的”这个问题的回答是：the shocking thing is that we have the mentalcapacity to do this in practice. The real miracle is the complexity of the worldrelative to our own intelligence. This is indeed cause to marvel. It is one thing forthe universe to be sensible in some precise way, and quite another for some entitywithin the universe to make sense of it to the extent we have.The desire to ‘factor out’ the human element no doubt stems from reductionisttendencies, where in this case the component we attempt to ignore is ourselves. I seeno sensible way to do this.

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Demystifying the Applicabilityof Mathematics，by Nicolas Fillion

物理规律多是表述为方程。能得到方程，表示我们理解了规律。但如果要能应用规律，我们还需要得到方程的解。有些方程有解析解，有些方程没有解析解，有些方程我们目前还不知道解是什么。对于后两种情况，我们使用计算机模拟的方式去理解方程的解。随之而来的问题是模拟的精度。

作者从认识论的角度来看到这个问题。epistemology。。。envisages a better scenario in which the claims, hypotheses,models, theories, and methods are accounted for not by fortunate mistakes, idiosyncrasies, etc., but by a rationally compelling presentation they ought to have. To usethe term introduced by Carnap [3], the object of scientific epistemology is a rationalreconstruction of science.

The successes of applied mathematics crucially depend on the methods of perturbation theory.

Ian Morris在《The Measure of Civilization》说过：量化不能使你表达观点更客观，但量化可以使你更准确地表达你的观点。

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Beyond Math，by Sophía Magnúsdóttir

In this essay I reflect on the use and usefulness of mathematics from theperspective of a pragmatic physicist.。。。how we can do physics without using math

区分：observations are math和observations can be described by math。

M：the entirety of all mathematics

O：all observations.

T： a map that maps the models to observations.

o： the observations that humans already have so far。

【我在想，是否可以把宗教定义为映射S：

S: O —> religion concepts，

考虑到宗教无法作用于自然现象，所以S只能是单向的，即单射。】

文章在“Models. Behaving. Gladly”一节里说：Think of any scientific explanation of any observable phenomenon. Take, say, thecomet on its orbit around the sun. The process of science is not, as Practical Physicisthad been taught, finding and using a map from mathematics to observation. No, themodels that we use are subsystems of our universe, just like the system we want todescribe (Fig. 2).The model is you doing a calculation with pen on paper. It’s a computer doingthe calculation for you. It’s a computer running a Monte Carlo algorithm. It’s your student plotting a graph. In each of these cases we map one observation to another.We take the results of a calculation—a plot, a table, a number—and match them toanother observation.

对此，我的理解是：以西方的星历表为例。以水星为例，我由水星的星历表可知水星在某时间的位置，也可以依行李表做预测。第一，在这种情况下，我不需要知道水星的轨迹是否是某圆锥曲线，也不需要知道它满足开普勒定律，更不需要知道宇宙里存在万有引力定律。其星历表的数据甚至已经包含了牛顿定律无法解释的“进动”现象。在此过程中，我不需要任何物理知识。第二，要想预测今年的水星位置，如果不需要太精确的话，可以拿一百年前的水星星历表来用。第三，假设发现了宇宙某恒星A及其邻近的某行星B，观察发现B的星历表和水星星历表很近似，所以若精度要求不高的话，就用水星星历表去代替B的星历表。结论是，整个过程不需要高深的数学知识。

《Questioning Foundations of Physics》里Chapter 18 Rethinking the Scientific Enterprise:In Defense of Reductionism，by Ian T. Durham，作者把科学探索的过程分为三部分：measurement, description, and predictive explanation.（传统上，科学探索的过程分为是两部分：theory and experiment）制作星历表的过程属于measurement。

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http://www.bilibili.com/video/av4036429，2015 Math Panel with Donaldson, Kontsevich, Lurie, Tao, Tayl_演讲•公开课。17分开始简单讨论了数学有效性这个话题