Thanks a lot! (But I am sorry to say I cannot understand what "context" really means, and I suppose I shall sometimes crash even after understanding it. Sweat ^^;)

Dear Updating Name,

Thank you very much for your kind advice, including your valuable experience. I am probably much older than you and have no experience (other than a little Mathematica) with symbolic computation, etc. I have a lot of experience with FORTRAN programming and a little with assembler.

These older programs are written to execute in the order of the code statements, and the code looks spaghetti-complex because of the "go to" statements or jump statements, but the order of the operations is easy to understand, and debugging can be successful if done carefully, even if it takes time.

On the other hand, Mathematica's language is convenient for simple examples, but nasty pitfalls appear when I try to do anything even remotely complex. For example, is there an example of a function that needs to perform a symbolic computation in NMaximize[ ]? If numerical computation is inevitably assumed, shouldn't the argument be a number? I am also still stumped by the difference between a list and a number: when I have multiple answers in a list to a function defined using Module[ ], NMaximize[ ] does not work without a message even when I use Part[ ] to extract one answer as a number.

Ever since I started using Mathematica, I have wondered about the initialization of Mathematica programs at the start of execution. I always put the line

Remove["Global`*"];

at the beginning of each program, which is a spell without understanding what ` and * mean.

In conclusion, Mathematica is very useful for computing simple problems such as those given as samples, but when users try to solve their own customized problems, they often end up in a swamp-like pit requiring unknown spells. Wolfram seems to have no intention at this time of removing such difficulties in its product specifications. I am like a pilot who is confused by the autopilot specifications of an airliner and crashes.

Dear Shinji,

There is an example in the current documentation for NMinimize[] (here) that shows the use f2[a_?NumericQ] := .... Note NumberQ[Sqrt[2]] is False but NumericQ[Sqrt[2]] is True. Therefore NumberQ[] is a more restrictive test; per the docs, "NumberQ[expr] returns False unless expr is manifestly a number (i.e. has head Complex, Integer, Rational, or Real)." The expression Sqrt[2] has head Power (another somewhat frequently encountered problem). NumberQ[] may be slightly faster than NumericQ[], but RepeatedTiming[] shows no discernible difference in my tests.

This example had been part of the "knowledge base" tech notes, but I believe it was incorporated into the documentation some years ago. The knowledge base note is still available here: https://support.wolfram.com/12502.

The issue is also discussed in this StackExchange answer. Other answers in that Q&A discuss things someone thinks are helpful for Mathematica users to know, some of which you may find helpful.

As for "the critical language design policies" in general, I'm not sure I can really help. I think I "grok" them. When I started, I read The Mathematica Book by Stephen Wolfram back around 1990. It is (or was for the early versions) a good, intermediate-level guide, with a good introductory Part 1. I came away with a pretty good understanding of "everything-is-an-expression"; that "evaluation" meant rewriting expressions according to pattern-rule operations, and it was not strictly limited to number and string crunching as in C but was more like Lisp, two languages I knew; and that there was "standard evaluation" and "non-standard evaluation." Coming to grips with standard and non-standard evaluation, the significance of Hold* attributes, Hold[], Unevaluated[], Defer[], and up-/down-/sub-values associated to symbols took a bit longer. And now we have Inactive[]. It's complicated. That means it is easy to become confused. Some of it I gave up on and didn't understand for maybe fifteen years or more. Most work does not require an understanding of all the arcane aspects. Much of The Mathematica Book is available in the documentation, tutorial/VirtualBookOverview. It has necessarily been restructured and updated. "The Evaluation of Expressions" may be found under "Core Language."

I mention several aspects of evaluation, because that is at the heart of the problem in the OP, and an appropriate pattern (_?NumericQ) is needed to control the order of evaluation.

I might also mention that at the time I started learning Mathematica, I knew Postscript, which greatly helped with understanding graphics concepts in early Mathematica; and Macaulay and Maple, which greatly helped with understanding symbolic algebra concepts and symbolic computation in general.

I had the same problem as the original question posted above, so thank you for your answer. However, I would appreciate it if you could answer my question as to why Mathematica does not provide an answer to such a frequently encountered problem. I also wonder if there are too many random, no-regularity, cop-out patch options like "?NumberQ". Since the contents of Mathametica are a black box to users, I would like to know about the critical language design policies and syntax of this useful software.