In primary school, it was said that all the adults in the family called mathematics "arithmetic", but the mathematics textbook in primary school did write "mathematics". Yes, my primary school mathematics is indeed a branch of mathematics, and it is a very basic branch, called arithmetic (the reason why it is not the most basic is because there are many theories that logic is the basis of mathematics, and I personally think that the definition of the four operations is also realized by sets). At that time, the mathematical symbol library was very simple, only
0 \ text { } 1 \ text { } 2 \ text { } 3 \ text { } 4 \ text { } 5 \ text { } 6 \ text { } 7 \ text { } 8 \ text { } 9 \ text { }+\ text { }-\ text { } \ times \ text { } \ div \ text { } = \ text { } \ text { }(\ text { })
Later, when I studied algebra, I added a symbol system, that is, 26 groups of Latin letters and 24 groups of Greek letters (including uppercase and lowercase letters), making a total of 100 letters. Then I began to study physics. Physical requirements can't set unknowns at will like mathematics, so I have to use the specified physical quantities. However, since then, I have realized the irrationality of the symbolic system of mathematics and physics. Then, in chemistry, the sudden appearance of the expression "let the quality ... be x" puzzled all the students who studied mathematics and physics at the same time. If you ask the teacher, the teacher's explanation usually includes "convention", "textbook regulations", "requirements of education bureau" and so on.
Then, the further you go, the more you will find that many branches of disciplines are taking care of their own scope, but never consider that once the disciplines cross, their symbolic meaning will lead to serious conflicts.
As mentioned in the title, \ sin {2} x and \ sin {- 1} x, under the so-called "convention", the latter actually represents an inverse trigonometric function, so that some teachers ask students to write \arcsin{x} in order to prevent confusion of meaning, and the function name uses multiple letters to represent the specific function, and negative exponential power cannot be written on it. This is a way to avoid ambiguity.
Then learn later, and there will be more and more slots. For example, e is used to represent eccentricity of conic curve, and \text{e} is used to represent Napier constant. But the problem is that there are many kinds of handwriting, and printing is divided into block letters and italics. How can people who read handwriting distinguish these two meanings? The general evasion method is to use \frac{c}{a} to represent eccentricity, and to abandon the representation of e. In addition, the differential symbol \text{d} and distance d are also silly and unclear when writing.
Of course, there is the matrix ~ haha, a_{ 1 1}. If this writing is not in the matrix environment, it is easy to be misunderstood as 1 1 of A _ n series. Will writing a comma kill the publisher?
I haven't learned the latter, but it is said that in order to expand the symbol library, the letters are written in various ways, flowery and scripted. ...
Oh, by the way, will you feel tired editing formulas with LaTeX?
The symbolic system of mathematics is flawed, and the flaws are really serious. This is related to the law of cognition, which can be said to be determined by the law of cognition.
Marxist philosophical viewpoint:
1, the world is infinite, and people's understanding is infinite. Cognition is repetitive and infinite.
There are only unknowns in the world, and there are no unknowns.
Man can't know the whole world like the back of his hand.
4. With the continuous development of knowledge, there will be more and more questions known by human beings that have no answers for the time being.
5. Knowledge comes from practice and is divided into two processes-perceptual knowledge and rational knowledge.
In other words, with the deepening of people's understanding, more knowledge will appear, and the mathematical symbol system must also be able to express this knowledge. In this case, it is difficult for human beings to cover so much knowledge with a limited number of words. In the development of mathematics, the creation of symbols has transformed the mathematical principle expressed by lines into a more acceptable formula. Obviously, "299,792,758", "99 million at two ends, 92,300 at seven ends and 58 at four ends" are just not as concise as "299,792,458". "The integral of sine function of this variable from zero to half pi" is not as happy as "\ int _ {0} {\ frac {\ pi} {2}} \ sin x \ text {d} x". Scientific counting provides a very good way to say numbers more easily. If a number exceeds 1000 billion, what is the higher "level"? (Some people may say "trillion"), there are words like "trillion", "billion" and "terrible" in English, but what about the bigger number? At this time, with the scientific counting method, how big a number is, as long as it is converted into a decimal, it becomes very simple at once. This is the advantage of mathematical symbols. But the disadvantage of mathematical symbols is that when human beings have not yet achieved global communication, the symbol system goes its own way. When human beings are communicating globally, the symbolic meaning is "establishment". Under the repetitive and infinite laws of cognition, under the current scientific research situation, each branch of the discipline system has its own way, which is miserable for students studying these disciplines. In addition, because of the finiteness of symbols, the same symbol means different meanings everywhere, and once the disciplines cross, it will cause very serious ambiguity. Indeed, it is very important to improve the symbol system of mathematics and physics.
Let me talk about my personal changes to the symbol system first.
1, appropriately reserved: for example, Arabic numerals, fractions, four operation symbols, absolute value symbols, simplex omitted multiplication symbols, specific function names, etc.
2. Symbol improvement of arithmetic, algebra and logic (set theory) 1: When changing the priority with brackets, use the length of brackets to distinguish between inside and outside. Parentheses indicate "off" and "on" respectively from the inside out.
3. Arithmetic symbol improvement 2: The greatest common divisor and the least common multiple are not expressed by brackets and brackets, but by gcd and lcm in brackets.
4. Improvement of algebraic symbols 2: Multiplication symbols shall not be omitted under the following circumstances: ① Numbers must be multiplied by numbers, and "×" must be used; ② When multiplying letters with brackets, you must add "\cdot" (to avoid confusion with functions); (3) When multiplying letters with fractions, "CDOT" must be added; (4) Functions that omit parameter brackets must be multiplied by a multiplication symbol. For example, the product of sine of x and cosine of y must be written as \ sin x \ cdot \ cos y.
5. Geometric symbol improvement 1: In order to prevent geometric points from being confused with numbers, the italicized letters of the original marking points were changed to straight bodies. The symbols of two points together represent a straight line, a long line on the top of the head represents a line segment, and an arrow represents a ray and a vector (meaning is repeated, but there is really no better solution). The symbols of a straight line and a segment are regarded as a set, and the length can only be calculated by adding an absolute value symbol. The symbols of three points that are not * * lines together represent a plane, and there is no need to add the word "plane" in front of it.
6. Improvement 2 of geometric symbols: the graphic symbols represent the points that form the graphic boundary, s is the length, a is the area, v is the volume, and d is the distance, where d_{\text{A-BC}} represents the distance from point A to BC line, and d_\text{AB-CD} represents the distance between parallel/non-planar AB line and CD line, using angle brackets (open).
7. Geometric symbol improvement 3: the symbol representing the graph (such as a single letter representing a straight line or a curve) is followed by a colon to represent the equation of the graph. For example, \text{AB}: 4x+5y+7=0 means that the equation of straight line AB is 4x+5y+7=0, and so is the curve. In addition, whether printing or not, a single letter represents a vector, and an arrow symbol must be added.
8. logical symbol improvement 2: full name and proper name propositions must indicate the set.
9. statistical symbol improvement 1: permutation and Combination, using Arr and com, with brackets (consider whether to synchronize them to represent the set composed of permutation and combination in the set).