History of Mathematics

Set Theory: the foundation of modern mathematics

Modern mathematics has countless branches, but they all have a common foundation-set theory-because of it, the family of mathematics has a common language. Set theory has some basic concepts: Set
(SET), relation, function, and other (equivalence) are almost inevitable in the languages of other mathematical branches. For these
The understanding of a single concept is the foundation for further learning other mathematics. I believe that college students in science and engineering are not unfamiliar with this.

However, a very important thing is not so well-known-that is, axiom
Choice ). This theory means that "any group of non-empty sets can take out one element from each set ." -- It seems that it is obviously impossible to make a clear proposition. However, this seems to be common
However, the principle can deduce some strange conclusions, such as the pinball theorem of Guanghe-taski-"A ball can be divided into five parts to perform a series of rigid transformations (translation rotation) on them) can be combined into two
The ball of the sample size ". It is precisely because of these conclusions that are totally against common sense that the mathematics field once had a heated debate over whether to accept it for a long time. Now, mainstream mathematicians should basically accept it, because
The important theorem of many mathematical branches depends on it. In the subjects we will discuss later, the following theorem relies on the choice principle:

1. Topology: baire
CATEGORY Theorem
2. Real Analysis (measurement theory): the existence of the unmeasurable set of Lebesgue
3.
Four major theorems of functional analysis: Hahn-Banach extension theorem, and steinhaus Theorem
(Uniform Boundedness Principle), Open Mapping Theorem, closed graph
Theorem

Based on the set theory, modern mathematics has two families: Analysis and algebra ). For others, such as ry and Probability
In the classical mathematical age, they are tied with algebra, but their modern versions are basically based on analysis or algebra. Therefore, in the modern sense, they are not parallel to analysis and algebra.
System.

Analysis: the magnificent building calculus built on the Limit: the classical age of analysis-from Newton

Let's talk about analysis. It is developed from calculus (caculus ).
That is why some calculus textbooks are called "Mathematical Analysis. However, the scope of analysis is far more than that. The calculus we study in the first year of college can only be an entry to classical analysis. Analysis and Research
There are many objects, including the derivative (derivatives), integral (Integral), differential equation (differential
Equation), and the level (infinite
Series) -- these basic concepts are introduced in elementary calculus. If there is an idea that runs through it, it is the limit-this is the soul of the entire analysis (not just calculus.

I
The stories that many people have heard of are Newton and lavenitz.
Leibniz. In fact, in their time, many calculus tools began to be applied in science and engineering, but the foundation of calculus was not really built. That
The ghost of "an infinitely small number" that has been unclear for a long time has plagued the mathematics field for more than one hundred years-this is the Second Mathematical Crisis ". It was not until kherk re-established the basic of calculus from the perspective of the limit of the series.
Concept, this discipline began to have a solid foundation. Until today, the entire analysis building is built on the foundation of the limit.

This is the development of analysis.
He provided a rigorous language, but he did not solve all the problems of calculus. In the 19th century, the world of analysis still had some lingering dark clouds. The most important one is the "letter
Whether the number can be accumulated ". The point we learned in the current calculus textbook that uses "infinitely segmented intervals, matrix area and limit" is about 1850 by Riann)
The point is called the riman integral. However, what function has a Cartesian integral )? Mathematicians have long proved that the continuous function defined in the closed interval is product-free by the family. However, this result is not
Satisfactory. Engineers need to integrate functions of piecewise continuous functions.

Real analysis: a modern analysis is established in the theory of real number and measurement.

In the middle and late 19th century, the problem of non-continuous functions is always an important issue of analysis. For closed
The Study of the range-based Cartesian points found that the key to the accumulation is that "the number of Discontinuous Points is small enough ". Only finite discontinuous functions are product-able, but many mathematicians have constructed many non-sequential functions in the infinite.
Product function. Obviously, when measuring the size of a point set, limitations and infinity are not an appropriate standard. During the discussion of the "point set size" issue, mathematicians discovered the real number axis, which they once thought
Something that is fully understood-has many features that they do not think. With the support of the limit idea, the real number theory is established at this time. It marks several equivalent theorems that characterize the real number completeness.
(Definite theory, interval set theorem, cosine convergence theorem, Bolzano-Weierstrass Theorem and Heine-Borel
Theorem and so on) -- these theorems clearly express the fundamental difference between real numbers and rational numbers: completeness (not strictly speaking, it is closed to limit operations ). With the deep understanding of real numbers, how to measure "points"
The problem of large and small sets has also made breakthroughs. leberger creatively considers the algebra of sets and outer
The concept of content (a prototype of "external measure") is combined to establish the measurement theory (Measure
Theory), and further establishes the points based on measure-Lebesgue (
Integral ). With the support of this new point concept, the problem of accumulation becomes clear at a glance.

The real number theory, measure theory, and leberger points mentioned above constitute our current
Real Analysis
Analysis). Some books are also called the theory of real variable functions. For application science, real analysis does not seem as "practical" as classical calculus-it is difficult to obtain any algorithm directly based on it. And,
It is unrealistic for engineers to solve some "difficulties", such as non-continuous functions everywhere or continuous and non-micro functions everywhere. However, I think it is not a pure mathematical concept.
The practical significance of games is to provide a solid foundation for many modern branches of applied mathematics. Below, I will only list a few of its usefulness:

1.
The functional space of the product is incomplete, but the functional space of the product is complete. To put it simply, the function that the column of the product of a column of the product converges to is not necessarily a product of the product.
The function columns must converge to a leberger product function. In functional analysis and approximation theory, we often need to discuss "the limit of a function" or "the series of a function ".
The theory is almost unimaginable. We sometimes look at the LP function space mentioned in some paper, which is based on leberger points.
2.
Leberger points are the basis of Fourier Transformation (which is everywhere in engineering. Many elementary teaching materials on signal processing may bypass leberger points and talk about practical things instead of their mathematical bases.
However, it is not always possible for in-depth research issues-especially those who want to do some work in theory-to bypass.
3.
As we can see below, the measurement theory is the basis of modern probability theory. Topology: analysis is extended from the real number axis to the general space -- Abstract basis of Modern Analysis

With the establishment of the real number theory, we began to extend the limit and continuous score to a more general place.
Analysis. In fact, many concepts and theorems based on real numbers are not unique to real numbers. Many features can be abstracted and promoted to a more general space. The promotion of the real number axis facilitates the Point Set Topology
(Point-Set
Topology. Many concepts that originally existed only in real numbers are extracted for general discussion. In topology, four C components constitute its core:

1. Closed
Set (closed set ). In modern topology, open set and closed set are the most basic concepts. Everything comes from this. These two concepts are the promotion of open and closed intervals. Their fundamental position is not
It was first recognized. After a long period of time, people realized that the concept of open set is the foundation of continuity, and closed set is closed to limit operation-and the limit is the foundation of analysis.
2.
Continuous Function
(Continuous function ). In calculus, a continuous function is defined in the epsilon-delta language. In topology, it is defined as "opening a set as a function of opening a set ". The second definition and
One is equivalent, but is rewritten in a more abstract language. I personally think that its third (equivalent) Definition fundamentally reveals the nature of a continuous function-"a continuous function is a function that maintains the limit operation"
-- For example, Y is a series of x1, x2, X3 ,... F (y) is F (X1), F (X2), F (X3 ),
... . The importance of continuous functions can be compared from other branches. For example, in group theory, the basic operation is "multiplication". For a group, the most important ing is "homomorphic ing"-to maintain "multiplication ".
Ing. In the analysis, the basic operation is "Limit", so the position of continuous functions in the analysis is equivalent to that of homomorphic ing in algebra.
3. connected set
(Connected set ). A concept that is a little narrower than it is
Connected), that is, any two points in the set are connected in a continuous path. This may be a concept that most people understand. In general, the concept of connectivity is slightly abstract. In my opinion, the connectivity has two duplicates.
Use Case: one is used to prove the general mean value theorem (Intermediate Value ).
Theorem), there is also the order of the fundamental group in the algebraic topology, topological group theory and Li Qun theory.
4. Compact
Set ). Compactness does not seem to appear specifically in elementary calculus, but there are several real-number theorems related to it. For example, "a bounded series must have a consortium.
Column "-- in the compactness language --" the real number space has a bounded closed set ". In topology, it is generally defined as something that sounds abstract-"arbitrary tightly set
There is a limited subcoverage for open coverage ". This definition is very convenient when discussing the topology theorem. It can often help to implement the transition from infinite to finite. For analysis, it is more of a form.
-- "The number of closely concentrated columns must have a subcolumn for convergence" -- it reflects the most important "Limit" in analysis ". Compactness is widely used in modern analysis and cannot be fully described. Two important parameters in Calculus
Theory: Extreme Value Theory, and uniform convergence
Theorem.

In a sense, the point set topology can be viewed as a general theory about the "Limit", which is abstracted from the real number theory.
The concept has become the general language of almost all modern analysis disciplines, and is also the foundation of the entire modern analysis.

Differential ry: Analysis on a manifold -- introducing a differential structure in a Topological Space

Topology promotes the concept of limit to a general topology, but this is not the end of the story, but just
Start. In calculus, We have differentiation, derivation, and points after the limit. These things can also be extended to the topological space and established on the basis of topology-this is the differential ry. In terms of teaching, differentiation
There are two different types of Teaching Materials. One is the "classical differential ry" based on the classical microcomputer ", it mainly involves the calculation of some geometric quantities in 2D and 3D spaces, such as curvature. There is also
Built on the basis of modern topology, this is also called "Modern Differential ry"-its core concept is "manifold" (manifold)-that is, in the Topology Space
Added a set
Structures that can be used for Differential Operations. Modern Differential ry is a very rich discipline. For example, generally, the definition of differentiation on a manifold is richer than that of a traditional one. I have seen three equivalent definitions from different angles.
-- This makes things more complicated, but it also gives different understandings of the same concept, which often leads to different ideas when solving problems. In addition to promoting the concept of calculus
Many new concepts: tangent space, cotangent space, push forward, pull back, fiber
Bundle, flow, immersion, submersion, and so on. Differential ry: Analysis on a manifold -- introducing a differential structure in a Topological Space

Topology promotes the concept of limit to the general topology space, but this is not the end of the story, but only
Yes. In calculus, We have differentiation, derivation, and points after the limit. These things can also be extended to the topological space and established on the basis of topology-this is the differential ry. In terms of teaching, differentiation
There are two different types of geometric textbooks. One is the "classical differential ry" based on the classical microcomputer ", it mainly involves the calculation of some geometric quantities in 2D and 3D spaces, such as curvature. There is also
It is built on the basis of modern topology. It is called "Modern Differential ry" here. Its core concept is "manifold", which is in the topological space.
Add
Sets the structure that can be used for Differential Operations. Modern Differential ry is a very rich discipline. For example, generally, the definition of the differential on a manifold is richer than that of the traditional differential. I have seen three equivalent definitions from different angles.
Meaning-this aspect makes things more complicated, but it also gives different understandings of the same concept, which often leads to different ideas when solving problems. In addition to promoting the concept of calculus, it also introduces
Has many new concepts:

Tangent space, cotangent space, push forward, pull back, fiber
Bundle, flow,
Immersion, submersion, and so on.

In recent years
Learning seems quite fashionable. However, frankly speaking, to understand some basic manifold algorithms, or even "CREATE" Some manifold algorithms, there is no need for the basis of differential ry. For my research, micro
The most important application of sub- ry is another branch built on it: Li Qun and Lie algebra-a beautiful marriage between the two family analyses and algebra in mathematics. Another important combination of analysis and Algebra
It is a functional analysis and harmonic analysis based on it.

Algebra: an abstract world

Abstract Algebra

Let's look back at another big family-algebra.

If classical calculus is an introduction to analysis, then modern Algebra
The entry point is two parts: Linear Algebra (linear algebra) and basic abstract algebra (Abstract
Algebra)-it is said that some textbooks in China are called modern algebra.

Algebra -- the name seems to be studying numbers. In my opinion, the main research is computational rules. An algebra, in fact
They all abstract some basic rules from a specific computing system, establish a system of justice, and then conduct research on this basis. Adding a set of operation rules to a set constitutes an algebraic structure. In the main
In the algebraic structure, the simplest is group, which has only one reversible operation that conforms to the combination rate, usually called multiplication ". If this operation also meets the exchange rate, it is called the Abel group.
(Abelian
Group ). If there are two kinds of operations, one is addition, which satisfies the exchange rate and combination rate, and the other is multiplication, which satisfies the combination rate and satisfies the allocation rate. This richer structure is called ring ),
If the multiplication on the ring satisfies the exchange rate, it is called the commutative
Ring ). If the addition and multiplication of a ring have all the good properties, it becomes a field ). Based on the domain, we can establish a new structure that can be used for addition and multiplication.
A linear algebra (linear algebra ).

The advantage of algebra is that it only cares about the deduction of calculation rules, regardless of the objects involved in calculation. Just define
Then, let a cat take a dog and get a pig :-). All the theorems obtained based on abstract calculation rules can be applied to the CAT/dog multiplication mentioned above. Of course, we still want to use it in practical use.
Do something meaningful. All those who have learned abstract algebra know that many important conclusions can be derived based on the simplest rules, such as the combination law. These conclusions can be applied to all places that satisfy these simple rules.
Fang -- this is the power of algebra. We no longer need to re-establish so many Theorems for every specific field.

Abstract Algebra is based on some basic theorem.
There are two schools: Studying finite discrete algebra structures (such as finite groups and finite fields), which are usually used in number theory, encoding, and integer equations; another school is to study continuous algebra, usually
It is associated with topology and analysis (such as topology group and Li Qun ). In my learning, the focus is mainly the latter.
Linear Algebra: basic position of "Linearity"

For
Learning, vision,
For optimization or statistics, linear algebra is the most commonly used, which we started learning in the lower grades of college. Linear Algebra, including built on it
Based on various disciplines, the two core concepts are vector space and linear transformation. The position of linear transformation in linear algebra is the same as that of Continuous Functions in analysis or homomorphic ing in group theory.
-- It maintains the ing of basic operations (addition and multiplication.

In learning, there is a tendency to despise linear algorithms and advertise non-linearity. Maybe in many cases
We need non-linearity to describe the complex real world, but at any time, linearity is fundamental. Without linear foundation, it is impossible to have the so-called non-linear extension. Our commonly used non-linear
The methods include manifold and kernelization, both of which require regression linearity at a certain stage. A manifold needs to establish a ing with a linear space in each region, by connecting many local Linear Spaces
While kernerlization maps the original linear space to another linear space by replacing the inner product structure, and then performs operations in the linear space.
. In the field of analysis, linear operations are everywhere, such as differentiation, integral, Fourier transformation, and Rapp.
Also, the mean values in the statistics are linear.

Functional Analysis: moving from finite dimension to infinite dimension

Linear Algebra learned in college is simple mainly because it is carried out in finite dimension space, because it is limited, we do not
Too many analysis methods are required. However, a finite dimension space cannot effectively express our world. Most importantly, a function forms a linear space, but it is an infinite dimension space. All of the most important operations on a function
In an infinite dimension space, such as Fourier transformation and wavelet analysis. This shows that in order to study functions (or continuous signals), we need to break the limitations of finite-dimensional space and enter the infinite-dimensional function space-Here
The first step is functional analysis.

Functional Analysis (Functional
Analysis) is a general linear space, including finite dimension and infinite dimension. However, many things are very trivial under finite dimension, and the real difficulties often occur in the infinite dimension. In
In functional analysis, elements in space are still called vectors, but linear transformations are usually called operators ). In addition to addition and multiplication, some operations are further added here, such as adding the norm
Expression of "vector length" or "element distance". Such space is called "normed linear space "(
Space). Further, you can add an inner product operation. This space is called the inner product space ).

When you enter
When there is an infinite dimension of time, many old ideas no longer apply, and everything needs to be re-examined.

1.
All finite-dimensional spaces are complete (convergence of the sequence of the keys), while many infinite-dimensional spaces are incomplete (such as continuous functions on closed intervals ). Here, a complete space has a special name: a complete Norm Space
It is called the he space, and the complete inner product space is called the Hilbert space ).
2.
In a finite dimension space, space and its dual space are completely homogeneous. In an infinite dimension space, they have subtle differences.
3.
In a finite dimension space, all linear transformations (matrices) are bounded transformations, while in an infinite dimension, many operators are unbounded (unbounded ), the most important example is to evaluate the function.

4.
In a finite dimension space, all bounded closed sets are tight, such as the unit ball. In all infinite dimension spaces, the unit ball is not tight-that is to say, an infinite point can be sprinkled in the unit ball without
Vertex.
5. In a finite dimension space, the spectrum of a linear transformation (matrix) is equivalent to all the feature values in an infinite dimension space.
The operator's spectral structure is much more complex than this. Apart from the Point Spectrum composed of feature values, there are also approximate point
Spectrum and residual spectrum. Although complicated, it is more interesting. This forms a very rich branch-Spectrum
Theory ).
6.
In a finite dimension space, any point always has a projection on any sub-space. In an infinite dimension space, this is not necessarily the case, the sub-spaces with such good characteristics have a special name for the chbihov space.
(Chebyshev space ). This concept is the basis of modern approximation theory (Approximation
Theory ). Function space approximation theory should play a very important role in learning. However, there are not many articles that use modern approximation theory.

Continue
Anterior: he algebra, harmonic analysis, and Lie Algebra

There are two important directions for basic functional analysis. The first is the he algebra (
Algebra), which introduces multiplication (different from multiplication) on the basis of the real-time space (a complete inner product space ). For example, in addition to addition and multiplication, a matrix can also be used for multiplication. This constitutes
He algebra. In addition, the fully-qualified bounded operators and square product functions of the value domain can constitute the he algebra. The he algebra is the abstraction of functional analysis, and many conclusions on bounded operators are derived, as well as operator spectra.
In theory, many theorems are not only applicable to operators, but can be obtained from the general ing algebra and applied outside the operators. The he algebra allows you to look at functional analysis at a higher level.
But I still need to think about what it can bring more in practical problems than functional analysis.

Another important direction that can best combine functional analysis with practical problems is harmonic analysis.
(Harmonic
Analysis ). I will list its two sub-fields, Fourier analysis and wavelet analysis. I think this shows its practical value. The core issue of its research is how to use base functions to approach
And construct a function. It studies the problems of function space and must be based on functional analysis. In addition to Fourier and wavelet, harmonic analysis also studies some useful function spaces, such
Space, Sobolev
Space, which has many excellent properties and has important applications in engineering and physics. For vision, harmonic analysis in signal expression, Image
Constructor, both are non-
Commonly used tools.

When analysis and linear algebra walk together, functional analysis and harmonic analysis are generated. When analysis and group theory walk in, we have
Group) and lie
Algebra ). They give Algebra to elements in a continuous group. I always think this is a very beautiful mathematics: in a single system, topology, differentiation, and algebra come together. Under certain conditions,
Through the relationship between Lie groups and Lie algebra, it converts the combination of geometric transformations into linear operations and converts subgroups into linear subspaces, this will introduce many important Models and Algorithms in learning to geometric motion.
Modeling creates necessary conditions. Therefore, we believe that Li Qun and Lie algebra are of great significance to vision, but learning
Its path may be very difficult, and you need to learn a lot before it
Mathematics.

Modern Probability Theory: Regeneration Based on Modern Analysis

Finally, let's talk about the branch of mathematics that many learning researchers are particularly concerned with: probability theory. From
Since Kolmogorov introduced measure into probability theory in the 1930s S, the theory of measure has become the basis of modern probability theory. Here, probability is defined as measurement, random variables are defined as measurable functions, and conditions
A random variable is defined as the projection of a measurable function in a function space. The mean value is the integral of a measurable function for probability measurement. It is worth noting that many modern ideas start to look at the basis of probability theory based on the concept of functional analysis.
Concept: a random variable forms a vector space, while a signed probability measure forms its dual space, where one party applies to the other party to form the mean. Although the angle is different, the two methods share the same path,
The basis is equivalent.

On the basis of modern probability theory, many traditional branches have been greatly enriched and the most representative include the theory of Yang.
(Martingale)-the theory triggered by research on gambling is now mainly used in Finance (here we can see the theoretical connection between gambling and finance:-P), Brownian
Motion)-the basis of the Continuous Random Process and the stochastic
Calculus), including random points (points on the path of the random process, which is representative of Ito
And Random differential equations. This knowledge is indispensable for the Study of Applying continuous ry to establish probability models and transformation of distribution.