Numbers are abstract constructs of humankind  initially used to count objects or properties they had and later developed to measure sizes of measurable things and so on. The development of numbers came along with the  development of human's abstraction capacity and the sheer demand needed for their usage. 

One peculiar point how indeed numbers emerged for sheer application purposes is the absence of the concept zero by early humans, the very reason it was conceived and created much later than the early created, known numbers called natural or counting numbers, 1,2,3,4,5,..... .  

But as science grows and humans expand their knowledge and wanted to have influence on their environment,  mathematical tools were becoming necessities and therefore the creation and expansion/augmentation of different kinds of numbers,  from counting numbers to Clifford numbers were imperative. Numbers continue to exist because we can not live, think and function without them. 

Kronecker said:  " God created the integers, all else is the work of man" to express his fascination, admiration and the indispensability of integers. 

No mathematics without symbols-------we have symbols of numbers, symbols of geometric objects, symbols of arithmetic, symbols of logics,…. But numbers as mathematical symbols have longest history among all.

From the number spectrum (the expansion of number family), we witness a truth: the creation of mathematical numbers expedites the growth of mathematics.

How big the number family is now and how does it expand?

Most (not all) cultures of the present and past know natural numbers (staring at 1). These numbers refer to quantities and are instrumental in counting (determining quantities). More sophisticated cultures added fractions for measurement, soon extended by algebraic numbers (like square root of 2). Technical demands required zero and negative numbers (leading to the integer number system, Z and the system of positive and negative fractions, Q).

For some time, the real number system, R,  seemed to capture the technical and practical needs of medieval Europe, but the problem of finding roots of polynomials lead to the complex number system, C. In algebraic terms, this is R[i] (reals with an additional imaginary unit "i" added. In the early 19th century, Gauss developed integer arithmetic modulo a prime p (number system Zp -- not to be confused with the p-adic number system Qp) and made good use of the number system Z[i] (Gaussian integers like 2+3i).  Hamilton developed the quaternion number system H, with numbers looking like a+i*b+j*c+k*d (i,j,k are different imaginary units). They are used nowadays in computing the visual effects of computer games.

Mathematicians experimented with algebraic systems like Z[square root of 2]. Such structures are technically known as rings (an axiomatic "number system" having addition and multiplication, subject to familiar rules). Some of these rings qualify as Euclidean rings having division-with-remainder , e.g., Z[square root of 2], Z[square root of -2], or Z[i], but also rings of polynomials with coefficients in a field (which is something like Q, Qp, R, C, or Zp).

Nowadays, one might say that  number systems are part of the discipline of algebra and that there is no clear limitation on what should be understood by "number".

We witness many "numbers" created in our history; but is there a law for us to operate, when and how can people create new "number"?

Kronecker's statement, about integers being created by the deity, all others being made by mankind, seems to apply. The way they are being used suggests that non-integer number systems are shaped according to needs and technical demands of science and of mathematics itself.

The only thing that various systems seem to have in common is an algebraic viewpoint on "arithmetic operations" expressed as axioms for domains and (skew) fields. Some of these (like polynomials or formal infinite power series over a field) may seem far away from what we have in mind with "number".

Kronecker's statement just shows his respects to our ancestors’ great creation for integers-------their popularity, beauty, usefulness, ….

As we know in fact, all number forms have been created by us human, it doesn’t matter what forms they are and how far away from what we may have in mind.

"...is there a law for us to operate, when and how can people create new "number?"

Judging on the existing number systems that are in use, the only rule seems to be a pragmatic one: does it serve a purpose?

For a nice illustration, there is some interest in p-adic numbers on behalf of physicists. Roughly, the idea is that all computations about physics are finite approximations to something that is supposed to exist, at least, in some "idealistic completion". The real numbers are the most common way to complete the number system of fractions but, in fact, a p-adic number system is another completion with entirely different properties. That makes p-adic numbers potentially interesting as a benchmark.

Do the creations of different number forms such as quaternion number, Fuzzy number, finite number, infinite number,…expedite the growth of mathematics? Why?

Each of the known number systems has its own use and range of applicability. You need specialists in each of these systems for extensive explanation.

Geng's quote of "infinite numbers" probably refers to ordinal and cardinal numbers, which do not fit into traditional algebraic classification. They are of great importance to the fundamentals of set theory but (to my knowledge) nowhere else.

Various systems of "complex" integers (of Gauss, of Eisenstein), as well as integers modulo a prime, have interesting applications to the traditional integers besides being of intrinsic interest.

As to quaternions and beyond, see the paper On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry by John H. Conway and Derek A. Smith.  Bull. Amer. Math. Soc. 42 (2005), 229-243. There is a brief article in Wikipedia on octonians. The latter fail not only to have commutative multiplication (like quaternions), but also fail associativity. There is nothing field-like beyond octonians in the chain of extensions

          reals ---> complex numbers ---> quaternions --->octonians.

If we understand the foundation and meaning of a certain mathematical numbers, we can use them well and develop them into a system such as the invention of “0”; but if we don’t understand the foundation and meaning of a certain mathematical numbers, we may meet troubles-------people have been troubling by the “non-number infinitesimal (variables)” related "tyranny of epsilons and deltas formal language" in “standard analysis” ever since.

Now, “non-standard analysis” is on its way with its “standard numbers vs. infinitesimals” instead of the "epsilons and deltas vs. variables" in standard analysis. A very important feature acknowledged for “non-standard analysis” is its deep structural equivalence with “standard analysis” and this is just because nonstandard structure is constructed within the confines of present classical infinite related theory system-----all the problems and defects disclosed by the suspended infinite paradoxes (such as the newly discovered Harmonious Series Paradox) will be shifted to “non-standard analysis” and people don’t understand the foundation and meaning of the newly invented “being-number infinitesimal” either. So, will the new “being-number infinitesimal” related “tyranny of infinitesimal operation” in non-standard analysis trouble us from now on?

Dear Dejenie，

Do you think infinite things in universe can be numbers in our mathematics? Can we category numbers according to our cognizing areas such as “finite relating area”, “infinite relating area”,…?

Dear Akira, thank you for such frank opinions.

Yes, we are living in a colorful world, it is natural that different people look at things from different angles and care for different things, that’s ok and everyone is happy. We understand what happen around us and have our own value of mathematics------we just do what we are called for according to our own instinct and intuition and we are very happy.

Dear Mr. Akira Kanda, would you please tell me what you mean by “intuition on the limits”?

People believe that the formal language of epsilon-delta was created to scientific our “intuition on the limits”------although this attempt fails and the theory of “non-standard analysis” was created instead with the same fate. The problem people have been fighting very hard is: how to scientific our “intuition on the limits?”

“How to scientific our behaviors in our science?” is really very necessary.  Do you have some good ways to “scientific our infinite related behaviors in our science?”

Dear Akira,

But, it is impossible for nonstandard real numbers which include infinitesimals to solve those defects in present classical infinite related theory disclosed by the family members of suspended Zeno’s Paradox such as Harmonious Series Paradox.

Akira,

(quote)1. Ordinal arithmetic is a form of algebra. Algebra is a very abstract concept.(unquote).

I fully agree on this.  Notice, however, my mentioning of traditional algebraic classification (emphasis added). We are talking about number systems in this thread, which are "core algebra". Ordinal arithmetic fails some of the core axioms.

(quote) 2. Ordinal concept is essential in many mathematical constructions as a natural extension of the so called inductive definition. (unquote)

Again, I fully agree with this and my reference to set theory is indeed too restricted. I should have noticed that arithmetic is basically about addition and multiplication of natural numbers, which fails some of the "core algebraic" axioms. I should indeed have paid attention to (first-order) arithmetic. Gentzen assigned an ordinal number to first-order arithmetic theories to prove their consistency and to measure their strength.

Akira, I'm treating (some of) your posts individually as I find your remarks most pertinent. As to your post on the pedagogical problem of mathematical education, you hit on a well-known and painful problem at secondary schools and even at the undergraduate level of the average university curriculum.

The concept of "closure" or "completion" of a number system deserves perhaps a bit more attention. From the natural numbers up to the complex numbers, all intermediate stages involve a closure-or-completion principle. The integers arise from the natural numbers (with zero) as a closure for solving the problem "a+x=b" in x. The extension from integers to fractions is based on a desire to solve equations "a*x=b" in x (for non-zero a). The extension of fractions to reals is based (originally) on the desire to have limits. With the introduction of "real closed fields", it can (partly) be explained alternatively by the desire to have square roots of positive numbers and roots for odd-degree polynomials. The extension to complex numbers, finally, is motivated by a desire to have roots for all polynomials of positive degree. As to the extension of reals to non-standard reals, I intend to prepare a separate post.

I have a few remarks concerning

(quote) (...)I learned what ax is for real x using infinite series. From this we can derive the laws of exponents for real exponents. (unquote)

It is unclear to me whether you are simply referring to the traditional series expansion (as in calculus), or to a method of formal power series, employed in discrete mathematics and in algebra. In this method, one simply gives names like "log(x)" or "ex" to certain formal power series (with no concern at all for convergence), and one proves certain formulas (motivating the use of names like "logarithm" and "exponent" by merely algebraic manipulations. To apply this to analysis, you have to consider real evaluation of power series and to face the problem of convergence (specifically, of convergence radius).

Whatever way you meant it, the problems of how to explain this to kids in secondary education are enormous. We can't blame the "front line" people for avoiding the whole matter or speculating on "physical intuition".

It is said that number is the language of mathematics and it is true that “number history” is a vivid picture of our mathematics history.

We human have been using infinite related number forms since the notation of “infinite” came into our science and really trying very hard to describe “infinite things in universe” with quantity. They are the products of mathematical theory systems, the defected theory system will have defected languages even hard to produce their related languages-----“non-number infinitesimal number (variables)” is a typical example and a helpless interlude.

Hi, a few hours ago I thought about answering. For me there were a few names and theories missing. In my opinion we should begin looking for an answer in the end of the 19th century by having a look on Dedekinds The Nature and Meaning of Numbers and Freges The Foundations of Arithmetic. I would say this also leads us to the question if logic is a mathematical discipline or mathematics could be reducible to logic, that is the point of logizism wherefore Frege and Dedekind are more or less representatives.

Sorry, if this is apart of your discussion, but i thought it should be mentioned under this topic. We find a definition of number in Freges work and his involvement of the former works on this topic. It might be interesting what he says about psychologism which I see also here in use to answer.

Dear Akira,

I believe if “Physicist version of infinitesimals are not the mathematical version of infinitesimals and there is no obvious connection between Robinson's infinitesimal and Physicists infinitesimal”, it is a tragedy of mathematics.

So, we actually come to the bottom: What is science, what is mathematics and what is physics?

Sincerely yours,

Geng

Dear Akira,

>It is my understanding that Zeno's paradox is a problem of kinematics not that of mathematics. Could you check if the Harmonious Series Paradox is the issue of physics or that of mathematics?

It is no doubt that in human science, the “problem of kinematics” in family members of suspended Zeno's paradox can be studied and described mathematically------ “potential infinite”.

And, it is the defects in present classical “potential infinite--actual infinite” theory system that have been troubling people and have been unable us to create right number forms ever since------“non-number infinite numbers (variables)” is a helpless interlude in our mathematics history.

Dear Akira，

Who desides wether or not there is  concept of taking limit to infinitesimals in infinite series? 

Geng

So, it is wrong not to  take limit to infinitesimals in infinite series? 

Akira,

(quote) It is my view that in this scheme of absolute elitism, it is possible to teach secondary school students who can handle the whole development of the theory of numbers and built correct analysis on it. (unquote)

This is certainly a bold ambition. Moreover, basing it on a scheme of absolute elitism (your description of it seems historically correct) goes against popular thinking of the present pragmatic era. A few years ago, I (loosely) participated in a discussion on the matter of mathematical education at secondary school on LinkedIn. Most participants were "front line" people, teaching classes of students aged 12-18. From what I remember, many were rather unsatisfied with the current math curriculum. They have no choice but to follow it and, at best, they invent tricks to motivate students. (Actually, it is not just a problem of mathematics; the current system of school education is questioned.)

The dilemma is a practical one and even the "good" teachers think like this: most of these kids won't encounter any math of significance in later life, so why bothering them with the discipline  of mathematics?

For some time, I thought that a solution could be to go even one step further towards pragmatism: to present mathematics as a kind of experimental (computer-assisted) physics:

Testing numbers on primality, doing computations modulo a prime, and verifying that it obeys the same rules as algebra with real numbers does. One discovers an exotic number system related to "clock reading"...

One may discover primitive elements in Zp for various primes p, and use their powers (for a large prime) to produce random looking sequences (imitating throwing dice or picking a card).

Computing modulo 2 relates with bit arithmetic on a computer. That 1+1=0 gives an amazing and simple method to garble a text with random bits; using the same sequence again un-garbles the mess. That's cryptography.

There are other topics that can be treated experimentally. For instance, integration as surface calculation and Archimedes' method of inscribed and circumscribed regular polygons to compute the circumference of the circle by approximation (determining the mysterious pi).

Good students may pick up the fact that the computer only does the dummy labor, and that all this beauty originates in the human mind and allows exact control. Most other students may at least have appreciated the fun of this. However, I can also think of objections that will be thrown (e.g., no practical skills are being learned, acquired knowledge is difficult to measure).

As a pure mathematician, I consider math to be more than a toolkit for science and engineering. It is also world-wide accepted art and culture. But can we impose its discipline and hardships on all kids between 12 and 18?

Akire, I understand your point of view. Some universities in the US (like Princeton, Berkeley, MIT) are "centers of excellence". In European countries, there is an ambition to start similar centers of excellence. It seems, however, that you aim for such centers at the level of secondary education. Europe has developed a strong tradition of "egalitarism" which is difficult to break and which makes this a highly sensitive topic when it concerns young people. Things may change when the pressure of running behind becomes larger.

The subject is perhaps a bit off topic in this thread.

Why we have failed to create a useful “number language” to tell what infinite is mathematically ever since.

Physicist (all kinds of practical sciences) versions of “infinite numbers” must be the mathematical version of “infinite numbers”-------“infinite numbers” have only one version in human science (physics is one of human science).

Another work relating to number is challenging us.

When we study ”the meaning of zero" and the location of zero in “number spectrum” in our mathematics, an unbalanced defect can be easily discovered: “zero" appears on one side of the “number spectrum” as a kind of mathematical language telling people a situation of “ nothing, not-being,…”; but on the other side of the “number spectrum” we lack of another kind of mathematical language telling people an opposite situation to “zero”------“ something, being,…”.

We need a new number symbol (“yan”) with opposite meaning to zero locating at the opposite side of zero in the “number spectrum” to make up the structural incompleteness of “number spectrum” and to complete the existence of “zero”.

Geng,

(quote)We need a new number symbol (“yan”) with opposite meaning to zero locating at the opposite side of zero in the “number spectrum”(unquote)

Let me suggest computation modulo 2: it has only two numbers, 0 and 1, with simple addition and multiplication tables suggesting truth values (no / yes). It is also the dominating number system of computers (fundamental memory unit) and can be used to describe all  natural numbers by writing them in binary (e.g. 1001 is "9").

I really don't see what makes you so dissatisfied about "numbers". Number systems exist for all tastes and purposes.

I can largely agree with Akira's last post (though I do not dramatize the gap between physics and mathematics: it seems to me that theoretical physics -- like theoretical computer science -- runs over fluently into pure mathematics).

Some people may not like the fact that infinitesimals were re-introduced by Robinson in an unusual way -- via logic. However, this route has a particular advantage. It is based on the Ultrapoduct Theorem of Jerzy Los (Polish "L"). Restricted to the first-order language of ordered fields, it states that the real line and any of its ultrapowers have exactly the same properties.

Knowing that ultrapowers of the real number system have infinitesimals and infinite numbers, this is even more remarkable: apparently, an inhabitant of the real line cannot tell it apart from a model with infinitesimals. Both are real closed fields: positive numbers are squares and odd degree polynomials have a root.

It needs a more extensive language to distinghuish them: the standard real line is Archimedian: the number sequence 1, 1+1, 1+1+1,... eventually outperforms every real number. The model with infinitesimals (and hence with "infinities") does not have this property. In fact, non-Archimedian ordered fields are exactly the ones with infinitesimals. 

This may perhaps explain why infinitesimals were once introduced with no physical evidence for their existence. In a precise context of logic, one could add them or not to the standard system without making a difference. Assuming infinitesimals is basically an alternative for assuming completeness of the real number system (which is instrumental for the concept of limit).

Dear Marcel,

I am very satisfied about "numbers", I just want to develop it and make it a better “mathematical language” serving us human better than before. Computation modulo 2 is also a kind of “mathematical language” doing its own business; these two “mathematical languages” help each other.

Sincerely yours,

Geng

Akira,

I am not in a position to discuss the methodology of physics with authority. I just notice that physics (like biology and chemistry) is a science of physical reality. As such, its truths are dictated primarily by what is observed, putting logic into a submissive role. Of course, the mathematical translation of observations or of models must be treated with due respect of logic. In my (perhaps naive) opinion, this mathematics is largely following computational paths, rather than following a path of reasoning.

As to (theoretical) computer science, its results sometimes contain build-in restrictions. My paper "Theories with the Independence Property" (available on RG) takes up a possible property of logic theories that was originally observed in Artificial Intelligence for special cases like theories of feature trees (with or without constraints).  They turned the property into quite effective decision procedures. I found that lots of quite common (mathematical) theories have the Independence Property: most theories about algebraic structures, about partial order, about functions, and even Pasch-Peano geometry with or without density and dimension. However, I was not able to extend the application to decision procedures as the AI people did. So, their restriction to special theories cannot be said to be unnecessary. They just are more pragmatic than I am.

Dear Marcel,

According to my studies, 0 means “non-existing” while “a new symbol (yan) with opposite meaning to zero in numeric system” means “existing” (may be understood as “no” and “yes” as well, but not exactly the same as computation modulo 2 with only two numbers, just simply 0 and 1) both number forms without “big” or “small” value of quantity meaning. Those numbers as 1, 2, 3 and so on are between two endpoints of "zero" and “yan” in “number spectrum”.

Sincerely yours,

Geng

Geng,

I am not particularly fond of mystical interpretations. Mathematics is a quite sober discipline where the "meaning" of a prime (= elementary, basic) object is reduced to its behavior as regulated in the assumed axioms. What you call "meaning" is just what you want to think about it, which has no mathematical relevance. Perhaps it may help to develop a feeling for a subject.

Your desire can easily be met with. In the literature one finds occasional extensions of the natural numbers (including 0) with an additional symbol "oo" for infinity. Usually, one agrees that n < oo (n a finite number), oo + n = oo = n + oo (n any number, including "oo"), oo  * n  = oo = n * oo (n nonzero), leaving oo * 0 and 0 * oo undefined. 

In many cases, it is just a temporary agreement to facilitate some recursive definition (e.g., dimension of a space, which can be infinite). It is not something to make a lot of fuzz about, but it is (probably) expressing what you want.

Dear Marcel, thank you.

Then what do you think zero is in numeric system of our mathematics, is it mystical? If zero is not mystical, why “a new symbol with opposite meaning to zero in numeric system” is mystical?

Sincerely yours,

Geng

Dear Akira, what do you think of following ideas?

From the cognition point of view:

As a kind of mathematical language, the roles zero plays are decided both mathematically and linguistically.

As a basic numerical (number) element, zero locates at the right position in the Table of the Numerical Elements (Number Spectrum) such as Table of the Chemical Elements in chemistry and Light Spectrum in physics.

Sincerely yours,

Geng

"0" is a mathematical object (and hence a potential subject of statements), whereas "nonexistence" is a type of statement (like "there does not exist a real root of x2 + 1" or "there is no field with 24 elements". That zero is a typical outcome of unsuccessful or vain processes (e.g., of  counting, of measuring,..) partially depends on conventions (see footnote); anyhow it doesn't give "0" a statement-like status! I cannot  understand how anyone can ever be confused about this.

I am equally surprised to hear that (some?) physicists think continuum is countably infinite. I thought that most physicist believe the universe to be essentially finite. Do they perhaps have a separate definition of "physical continuum"?

=====footnote: counting is a convention to run in succession through 1,2,3,... with each next object to be counted. It there is none, you should have agreed that 0 is named before the actual counting starts. Computer programs that count things take this provision explicitly.

Just for fun: the product of "no numbers at all" is... 1. This convention relates with a programming habit to set a multiplication register to 1 at the start and, with each number involved, to multiply the current register content with it.  At the end, the register contains the demanded result. As a bonus it fits even nicely into computation rules with exponents, with the convention "a to the power 0" equals 1 (usually retricted to positive a).

I am sorry to say that this particular topic has no deep grounds.

Dear Marcel,

Do you think numbers (such as "0") is a kind of mathematical language?

Sincerely yours,

Geng

Geng, numbers are (mathematical) objects. The symbolic names "0", "1", ... are part of the mathematical language, serving as names of potential subjects of statements.

Akira, the idea of "zero" seems to come from ancient India. It was adopted and preserved in the Muslim Culture, and later accepted in medieval Europe  after some dispute, which now seems ridiculous. Some found it a waste of effort to "give a name to nothing", others even feared that it might wake up evil spirits. It is indeed irrelevant for the process of counting at which number you start counting. The role of 0 and 1 has been fixed by proclaiming them as neutral elements for addition and multiplication, respectively. That makes them into true benchmarks in the landscape of numbers.

Disputes about the role and meaning of 0 and 1, or disputes on topics like "existence of a square root of -1", have been reduced to their essence by the mathematical revolution of the last century. It created a universe of discourse for mathematics which is essentially independent of physical reality by adopting axioms of set theory and rules of predicate logic as its basis.

Dear Akira, thank you for your frank opinion on zero.

> I said, There is a categorical difference between non-existent and exist and its value is zero. I notice that many scientists including mathematicians have difficulty in understanding this difference, which is very subtle but enormously important.

Dear Marcel,

I agree with you that numbers have been created in our science to cognize things in universe, so we say numbers is mathematical language. We have different number forms for different things such as natural number, fuzzy number, approximate number, ….

But because of some defected understanding of “what is number and how does it exist” ever since, there have been many disputes on the number forms such as “non-number infinite related numbers”, “zero”, ….

Sincerely yours,

Geng

Dear Akira,

I beg your pardon that I didn’t express clearly Akira, the 2nd and the 3rd questions are not for your post. I have corrected my mistake in my last post.

I agree with you. Actually, in many places zero serves as a kind of reference (generator, starter, beginner, origin, …) in a scientific system (a chosen system). Its “location” may be in the middle of left-right or up-low------it is us human that give “something” in universe a name of zero in a mathematical language system （we could even just give it another name like ” generator” or “middle” or… ）. As a kind of reference, this “zero” is without numerical value, not for numerical calculating.

Sincerely yours,

Geng

As a mathematical language, can numbers be illustrated by following “number tree”?

Ontology-number (absolute number, objective number, quality number, distinct number, …): such as 2 pieces of stone, 5 water molecules,… and natural number, number of absolutely being, objective 0 (zero), ….

Form-number (relative number, subjective number, numerical number, indistinct number, …): such as 2 kg of rice, 5 meters, … and approximate number, fuzzy number, infinite number, subjective 0 (zero),….

We have many different kinds of number forms: 0 (zero), natural number, fuzzy number, …, but at the time when they were born in our science, must they be proved mathematically or just came out as needed and staying there without proof?

I think originally the just came out as needed, though it may be different now! 

Dear Akira and Samuel, you hit the point.

So for long, we actually are unable to know what are numbers and what not in mathematics!

It is the very reason that no one can tell why those infinite related numbers are not numbers.

My best regards to you, Geng

Dear Akira,

Is it possible to have a definition of “what are numbers” in human science?

Yours, Geng

Dear Akira, good morning! (Is it morning in your city now?)

If a “defect” is discovered, why not try to do something?

Yours, Geng

Akira, I am really very happy to hear from you again but sorry to know that you have some healthy problems.

Dear Mr. Akira Kanda, health is the most important thing. I am sure you know much more than me that without health everything is 0.

My best, sincerely regards to you, Geng,

Akira,

"Dedekind [...] defined real numbers as the cut of rational numbers and developed mathematical analysis. This process was mathematically simple but lacks in intuitive explanation."

Until the nineteenth century real numbers were simply thought to be present in the real world, just like Euclid's points and lines. After that, mathematics shifted to definitions that made concepts independent from reality. If you look closely at set theory, you will see that natural numbers are available there as:

0 ={} (emptyset), 1={0}={{}}, 2={0,1}={{},{{}}}, 3={0,1,2}=....

The definition n={0,1,...,n-1} is nice, smart, economical, and (I dare say) rather intuitive. The technical workout {{},--} is terrible (and rarely used), but it creates a universe of numbers out of nothing.

While natural numbers can still be seen in nature (with a small mental effort), real numbers are far more tricky. One may think that pi is available in nature as (circle circumference)/(circle diameter). That's cheating. It requires a sophisticated assignment of exact length along a curved line and you can only have a result if you have the numbers to define length and express it.

Dedekind's definition of the reals (rationals + gaps, with gaps expressed as "Dedekind cuts") is a marvelous and adequate view and I find it intuitive (and close to practice). Remember how the ancient Greec struggled with square root of 2? It is somewhere in the gap between 1.414213562 and 1.414213563. If you don't like the technical workout of the "gap view", you may use the alternative viewpoint of completion which provides the meaning of an infinite sequence of decimals. (The idea of completion is widely used in mathematics.)

There is no royal road to mathematics (Menaechmus to Alexander the Great).

Akira, in mathematics, once a class of objects is created and its basic properties have been established, the class and its members become black boxes with the proven basic properties as assumed rules. It is no longer important, then, how they were constructed; they simply "go by the rules".

For instance, when you deal with fractions a/b (with a, b integer and b not 0) you are "allowed" to eliminate a common divisor d of a and b by dividing d out in the nominator and denominator. In (mathematical) reality, a fraction is an equivalence class of pairs (a,b) (with a, b as above) where two pairs (a,b) and (c,d) are declared equivalent if and only if a*d = b*c. The equivalence class of (a,b) is denoted a/b and the above "simplification rule" follows from this construction. The fraction a/b becomes a "black box" with the behavior that one would expect on intuitive grounds.

Dedekind's real numbers (gaps) are shown to be ordered and to obey certain rules of addition and multiplication which constitute the axioms of a complete ordered field. It has been shown that any two of such fields must be isomorphic, in other words: they can only differ by being produced in different ways.

Limits, derivatives and integration are derived from the knowledge module "complete ordered field". Pushing these concepts further back into the original construction is grossly inefficient and should never be done by the above remark on isomorphism.

It is a pedagogical rule that all learning must start with activating the intuition. In the case at hand, this means to rely on the available intuition about real numbers to collect enough basic properties. After that, one can introduce limits, derivatives, and integration. This holds for excellent students too. It is only when one realizes the relevance and usefulness of a subject that one may get motivated to learn its technical foundations.

The modular structure of mathematics is an invitation to such an approach.

It is natural and understandable that more people have been busily engaged in the field of surface mathematics (applied mathematics) than in fundamental mathematics (pure mathematics), because applied mathematics is really a very important cognizing tool with visible present importance in both our life and science, and this tool does directly bring to the users material properties-------the researchers working in this field may cash their work immediately (the work in this field is hard and challenging of course) while fundamental mathematics is for surface mathematics (applied mathematics) and many of it may be without visible present importance in both our life and science-------the researchers working in this field may be very difficult to cash their work immediately although their works are roots and basic stones for surface mathematics (applied mathematics).

Dear Akira and Marcel, I sincerely hope to have your criticisms on my following superficial ideas:

1, so far as I know many people believe that pedagogy operations is with its own characteristics not exactly step by step as academic facts go, but the most important thing is the teachers should be aware of these phenomenon. Now, the problem is there are really some defects in academic things in number theory.

2, people used to believe that rationals are continuous and than situation changed when “real numbers as the cut of rational numbers”.... Than according to the “cut (gap) theory”, what is by the cut of real numbers?

3, “Mathematics is now reduced to calculators. What can we do? Nobody understands what is infinite either.”

“In the university, they teach only Calculus. No analysis. Without understanding the fundamental theorem of calculus, students are demanded to calculate mindless differentiation and integration of functions which they will never ever use in their life.”

If we don’t really understand what infinite is either, limit theory is a big problem. So, not only students but for all people, differentiation and integration become the operations on “operation lines”-------fingers and hands moving up and down, pressing buttons,… ; just do, it is impossible and no need to have any analysis knowing what they are treating in front of them and why ”the infinite related very small numerical things (such as infinitesimals)” in and out of the formulas------whatever they are called: very small numbers, infinitesimals, variables, monads, …; products are out and paradoxes are still there, …. It is one of the reasons that some people working in the physics field even say “no infinite things in mathematics and science”.

4, “Any how, is there anybody who wants to discuss differentiation and integration in terms of cut limit?”    It is impossible to do this work well if we don’t really understand what infinite, limit, and infinite related numbers are.

Applied mathematics is really important in both our life and science but I sincerely hope more people pay attention to the defects being discovered in fundamental mathematics (pure mathematics) in the 21th century. It would be a tragedy for mathematics if we just pay attention to surface mathematics (applied mathematics) and keep ignoring those fundamental defects in our future mathematical work.

Thank you!

My best, sincerely regards to you, Geng,

Dear Akira,

I agree with you that “to learn physics one has to learn the history of it and the way how ideas are put together to produce a theory is important”. I think this is especially important for academic researchers, and I really benefit mathematical history a lot in my work.

Regards, Geng

Science evolves, and evolution tends to cover its traces and origins. This may explain why it is good to know history.

Speaking for mathematics only, I see that, on the one hand, it went through a "foundational" revolution in  the previous century and, on the other hand, that some of its most relevant branches (notably: geometry, analysis) largely remained their course. These branches are still responsible for much applied mathematics. For many workers in these areas, the presence of set-theoretic foundations doesn't make much difference to their mathematical activities.

There is a growing body of novel applications coming from algebra (including number theory) and discrete mathematics, which took profit of abstractions originating in the nineteenth and twentieth century. It is perhaps a turbulent flow of developments but, again, not a revolution. That term should be used only for mathematical logic, topology, category theory, and (of course) set theory itself.

So, my view on mathematical evolution is rather mixed, but the field doesn't seem to be in a state of crisis. Above all, there is a crisis in education, caused by the growing demands of a technological society and by the growing potentials of electronic devices in teaching. Mathematics at primary and secondary school is under fire: traditional reasons for teaching (basic) mathematics are loosing value. 

Mathematics is our human’s; we can develop it in a good way and make it evolve better along with our human evolution.

Akira,

I fully agree that, through the ages, mathematics followed a relatively stable path, independent of social changes and even of culture. The differences between Euclid's geometry and modern mathematics are largely technical. Both use accurate reasoning starting from elementary knowledge about an idealized world. Nowadays, methods of reasoning have been formalized, elementary knowledge is cast into axioms, and idealized worlds are made into an independent and unified mathematical reality of sets, structures, and number systems.

This gives good hope that mathematics will survive today's obstacles. Many years ago, I considered the opportunities for a book about "mathematical culture", which should contain and explain all mathematics needed to understand its current applications. Roger Penrose's The Road to Reality (BCA, 2004) goes in that direction, but it is not quite what I thought of. It should rather present an accurate development of mathematics, starting with formal logic, and presenting the main mathematical subjects as modules starting from axioms or building on earlier modules. It should end at a level where applications can be indicated.

Such a project is not of commercial interest and requires a team of specialists and much coordination. Today, I feel that a project of this size should be left to the young.

As most people agree that human science is a kind of human’s property (some other creatures in universe may also “enjoy” this property) composed by human’s mental work (subjective world) and universe (objective world) ------- no human science without human. So, the unbalanced things of subjective and objective compounds of our knowledge are not scientific. Sometimes the subjective factor or objective factor is overstated, each causes confusion.

Human science is human’s, so we must present all the things in our science with our human’s ways. Our science history proves that this is really not easy and is really very hard.

Dear Akira,

Number theories are the knowledge we human gained imprinted with our former experiences, some are “correct” and some are “incorrect” ontologically or formally. The “correct” one will last longer than the “incorrect” one. The human knowledge system is our human’s; so, as we human evolve, our knowledge system accompanying us evolves. Evolution is a nature law.

It is not easy to discover the mistakes in present knowledge system and it is even more difficult to do something to remove some “mistakes” in present knowledge system--------fighting with traditional experiences and ideas and pinioning a new field.

Regards, Geng

Five of my published papers have been up loaded onto RG to answer such questions:

1，On the Quantitative Cognitions to “Infinite Things” (I)

https://www.researchgate.net/publication/295912318_On_the_Quantitative_Cognitions_to_Infinite_Things_I

2，On the Quantitative Cognitions to “Infinite Things” (II)

https://www.researchgate.net/publication/305537578_On_the_Quantitative_Cognitions_to_Infinite_Things_II

3，On the Quantitative Cognitions to “Infinite Things” (III)

https://www.researchgate.net/publication/313121403_On_the_Quantitative_Cognitions_to_Infinite_Things_III

4 On the Quantitative Cognitions to “Infinite Things” (IV)

https://www.researchgate.net/publication/319135528_On_the_Quantitative_Cognitions_to_Infinite_Things_IV

5 On the Quantitative Cognitions to “Infinite Things” (V)

https://www.researchgate.net/publication/323994921_On_the_Quantitative_Cognitions_to_Infinite_Things_V