modify chapter 9 and 6

Author: yimaeecs

chapters/chapter6/consistent-representations.tex

@@ -91,7 +91,10 @@ \chapter{Consistent and Self-Consistent Representations}
 \begin{equation}
     d(\X, \hat \X).
 \end{equation}
-This leads to the concept of {\em consistent representation}. As we
+This leads to the concept of {\em consistent representation}.\footnote{Note that here the notion
+of consistency is empirical or inductive in nature. It is different from the notion of 
+logic and deductive consistency in mathematics,
+also known as the Hilbert's Principle, which we will have more discussions in Chapter \ref{ch:future}. } As we
 have briefly alluded to in
 \Cref{ch:intro} (see \Cref{fig:autoencoder}), {\em autoencoding},
 which integrates the

chapters/chapter6/consistent-representations_zh.tex

@@ -47,7 +47,9 @@ \chapter{一致与自一致表征}
 \begin{equation}
     d(\X, \hat \X).
 \end{equation}
-这引出了{\em 一致表征}（consistent representation）的概念。正如我们在\Cref{ch:intro}中简要提到的（见\Cref{fig:autoencoder}），将编码和解码过程结合在一起的{\em 自编码}（autoencoding）是学习这种表征的自然框架。我们在\Cref{ch:classic-models}中研究了一些重要的特例，其中分布的支撑是分段线性的。在
+这引出了{\em 一致表征}（consistent representation）的概念。\footnote{请注意，这里“自洽”是经验或者归纳意义上的。有别于数学里的逻辑”自洽“的希尔伯特原理。
+我们在第\ref{ch:future}会有更多的讨论。}
+正如我们在\Cref{ch:intro}中简要提到的（见\Cref{fig:autoencoder}），将编码和解码过程结合在一起的{\em 自编码}（autoencoding）是学习这种表征的自然框架。我们在\Cref{ch:classic-models}中研究了一些重要的特例，其中分布的支撑是分段线性的。在
 本章的\Cref{sec:consistent-representation}中，我们将研究如何将自编码扩展到支撑为非线性的更一般的分布类别（如图 \ref{fig:manifold-autoencoding} 所示），同时强制保证 $\X$ 和 $\hat \X$ 之间的一致性。
 \begin{figure}
     \centering

chapters/chapter9/future.tex

@@ -380,9 +380,14 @@ \subsection{From Inductive to Deductive Intelligence}
 understood an abstract concept such as:
 \begin{enumerate}
     \item the notion of equality between two quantities,
-    \item the notion of numbers (natural, rational, or real),
+    \item the notion of numbers (natural, rational, real, or imaginary),
     \item the notion of infinity,
-    \item and the notion of mathematical induction,
+    \item the notion of mathematical induction,
+    \item the notion of logic consistency\footnote{In mathematics, if a set of axioms and their deduced results are consistent and 
+    do not contradict one another, then they exist. This is known as Hilber's Principle. This is different from the 
+    notion of ``consistency'' in Chapter \ref{ch:consistent-representations}, which requires the learned
+    representations to be consistent with the empirical data distributions. One consistency is inductive 
+    and the other is deductive.} and proof by contradition,
 \end{enumerate}
 just to name a few? Or has it simply memorized a massive number of
 examples of such notions? Note that state-of-the-art large language

chapters/chapter9/future_zh.tex

@@ -229,7 +229,9 @@ \subsection{从归纳智能到演绎智能}
     \item 两个量之间绝对相等的概念，
     \item 数字（自然数、有理数或实数）的概念，
     \item 无穷的概念，
-    \item 以及数学归纳法的概念。
+    \item 以及数学归纳法的概念，
+    \item 逻辑自洽\footnote{在数学里，一个公理系统只要逻辑上是自洽的，它就是可以合理存在的。这也称为希尔伯特原理。这与我们在第\ref{ch:consistent-representations}
+    章里提到的学到的表征与观测到的数据分布要一致不一样。一个是经验的，一个是抽象逻辑上的。}和反证法。
 \end{enumerate}
 还是它仅仅记住了大量关于这些概念的例子？学到这些例子的一些经验规律。请注意，最先进的大语言模型仍然在诸如“3.11比3.9大还是小？”这样简单的数学问题上挣扎。
 \footnote{一些模型通过有针对性的工程设计纠正了它们对这类问题的回答，或者结合了额外的验证机制，