Dear James

I hope you don't mind repeating two answers I gave much earlier in a different thread:

"What is mathematics? It was only within the last thirty years or so that a definition of mathematics emerged on which most mathematicians now agree: mathematics is the science of patterns. What the mathematician does is examine abstract patterns-numerical patterns, patterns of shape, patterns of motion, patterns of behavior, voting patterns in a population, patterns of repeating chance events, and so on. These patterns can arise from the world around us or from the depths of space and time."

And a more detailed answer:

Earlier, I have asked the question "What is mathematics?" I have answered that mathematics is the science of patterns. Different kinds of patterns give rise to different branches of mathematics. For example:

-Arithmetic and number theory study patterns of numbers and counting.

-Geometry studies patterns of shape.

-Calculus allows us to handle patterns of motion.

-Logic studies patterns of reasoning.

Probability theory deals with paterns of chance.

Topology studies patterns of closeness and position.

In his 1940 book 'A Mathematician's Apology', the accomplished English mathematician G. H. Hardy wrote:

"The mathematician's patterns, like the painter's or the poet's, must be beautiful, the ideas, like the colors or the words, must fit together in a harmonious way. Beauty is the first test; there is no permanent place in the world for ugly mathematics...It may be very hard to define mathematical beauty, but that is just as true of beauty of any kind-we may not know quite what we mean by a beautiful poem, but that does not prevent us from recognizing one when we read it."

The beauty to which Hardy was referring to is a beauty that can be observed , and appreciated, only by those sufficiently well trained in the discipline. It is a beauty "cold and austre," according to Bertrand Russel, the famous English mathematician and philosopher, who wrote, in his 1918 book Mysticism and Logic:

"Mathematics, rightly viewed, possesses not only truth, but supreme beauty-a beauty cold and austre, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of stern perfection such as only the greatest art can show."

Mathematics' greatest success has undoubtedly been in the physical domain, where the subject is rightly referred to as both the queen of the (natural) sciences by Carl Friedrich Gauss (see my introduction to the main question of this thread) and as the queen and servant of the (natural) sciences( As noted by Eric Bell, a Scottish American mathematician).

In an age when the study of the heavens dominated scientific thought, Galileo said:

"The great book of nature can be read only by those who know the language it was written. And this language is mathematics."

In a much later era and striking a similar note, when the study of the inner workings of the atom had occupied the minds of many scientists for a generation, the Nobel Prizewinner Richard Feynman had this to say about mathematics:

"To those who do not know mathematics it is difficult to get across a real feeling as to the beauty, the deepest beauty, of nature ... If you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in".

I can hardly do better than Galileo’s excellent descriptions:

"The great book of nature can be read only by those who know the language in which it was written. And that language is mathematics.” - Galileo

Dear James

I hope you don't mind repeating two answers I gave much earlier in a different thread:

"What is mathematics? It was only within the last thirty years or so that a definition of mathematics emerged on which most mathematicians now agree: mathematics is the science of patterns. What the mathematician does is examine abstract patterns-numerical patterns, patterns of shape, patterns of motion, patterns of behavior, voting patterns in a population, patterns of repeating chance events, and so on. These patterns can arise from the world around us or from the depths of space and time."

And a more detailed answer:

Earlier, I have asked the question "What is mathematics?" I have answered that mathematics is the science of patterns. Different kinds of patterns give rise to different branches of mathematics. For example:

-Arithmetic and number theory study patterns of numbers and counting.

-Geometry studies patterns of shape.

-Calculus allows us to handle patterns of motion.

-Logic studies patterns of reasoning.

Probability theory deals with paterns of chance.

Topology studies patterns of closeness and position.

In his 1940 book 'A Mathematician's Apology', the accomplished English mathematician G. H. Hardy wrote:

"The mathematician's patterns, like the painter's or the poet's, must be beautiful, the ideas, like the colors or the words, must fit together in a harmonious way. Beauty is the first test; there is no permanent place in the world for ugly mathematics...It may be very hard to define mathematical beauty, but that is just as true of beauty of any kind-we may not know quite what we mean by a beautiful poem, but that does not prevent us from recognizing one when we read it."

The beauty to which Hardy was referring to is a beauty that can be observed , and appreciated, only by those sufficiently well trained in the discipline. It is a beauty "cold and austre," according to Bertrand Russel, the famous English mathematician and philosopher, who wrote, in his 1918 book Mysticism and Logic:

"Mathematics, rightly viewed, possesses not only truth, but supreme beauty-a beauty cold and austre, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of stern perfection such as only the greatest art can show."

Mathematics' greatest success has undoubtedly been in the physical domain, where the subject is rightly referred to as both the queen of the (natural) sciences by Carl Friedrich Gauss (see my introduction to the main question of this thread) and as the queen and servant of the (natural) sciences( As noted by Eric Bell, a Scottish American mathematician).

In an age when the study of the heavens dominated scientific thought, Galileo said:

"The great book of nature can be read only by those who know the language it was written. And this language is mathematics."

In a much later era and striking a similar note, when the study of the inner workings of the atom had occupied the minds of many scientists for a generation, the Nobel Prizewinner Richard Feynman had this to say about mathematics:

"To those who do not know mathematics it is difficult to get across a real feeling as to the beauty, the deepest beauty, of nature ... If you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in".

Dear Issam,

Many thanks for your contribution. Your points are excellent.

You write: Topology studies patterns of closeness and position. Tou might want to add that a topologist considers the nearness of points and sets.

There are abstract algebra (groupoids, semigroups, groups, rings and fields(, category theory (e.g., NEAR and SET), metric space theory, approach space theory, prixunuty soace theory and uniform space theory (and other research areas) to consider to obtain a more complete picture of mathematics.

I am inclinded to think that mathematics can be viewed in its own right as well as the language of nature. After all of the execellent points you have made, the answer to the question What is mathematics? is still open. I say this because there are many aspects of mathematics that are independent of nature. And that independence starts with sets of points in, for example, the Euclidean plane. Such points are not foiund in nature.

It does look like mathematics has its roots in nature, if we consider the notion of ananta (endless or unending) in the Indian or lines in the sand (Archimedes), and so on. But there has been, through the centuries, a transition to a much more accessible ground in mathematics than nature itself supplies.

According to me, Mathematics is a tool to explain the complexity of the universe.

Regards,

Nitish

Arno, you write

Cantor understood that the continuum is unlimited, however it should be realized that the corresponding physical continuum must be limited because its components and elements may physically be neither infinitely small nor infinitely large even though this would mathematically be thinkable, Mathematical Physics, p. 7.

Yes, precisely, This suggests that pure mathematics provides an approximate description of the physical continuum.

What is Mathematics?

math·e·mat·ics n. (used with a sing. verb). The study of the measurement, properties, and relationships of quantities, using numbers and symbols. [From Middle English mathematik, from Old French mathematique, from Latin mathematica, from Greek mathematike (tekhne)].

---The American Heritage Dictionary

math-e-mat-ics n. ... the science of numbers and their operations, interrelations, combinations, generalizations, and abstractions and of space configurations and their structure, measurement, transformations, and generalizations.

---Webster's New Collegiate Dictionary

Mathematics is a language.

---Gibbs, Josiah Willard. 1839-1903.

American mathematician and physicist

To those who do not know Mathematics it is difficult to get across a real feeling as to the beauty, the deepest beauty of nature. ... If you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in.

---Richard Feynman. 1918-1988.

American physicist.

The Character of Physical Law

Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.

---Bertrand Russell. 1872–1970

British philosopher, mathematician.

Mysticism and Logic

All science requires Mathematics. The knowledge of mathematical things is almost innate in us... This is the easiest of sciences, a fact which is obvious in that no one’s brain rejects it; for laymen and people who are utterly illiterate know how to count and reckon.

---Roger Bacon. 1214-1294

English philosopher, scientist.

Opus Maius

"What is mathematics? Most people would say it has something to do with numbers, but numbers are just one type of mathematical structure. Saying "math is the study of numbers" (or something similar) is like saying that "zoology is the study of giraffes". Math may be better thought of as the study of patterns, but this too falls short...

The more I study math, the more I wonder about what exactly math is. Actually nobody knows. It seems to be a product or our minds, and yet reflects the external universe with uncanny accuracy. A mathematician develops a mathematical theory for its aesthetic unworldly beauty and it's compelling evolution, with no thought of how it might be applied to the world. A century later a physicist finds this theory to be perfect to use as a framework to express his physics (this sort of thing happens frequently). Pretty weird how intimately connected our innermost "mind" and the outermost "universe" really are. This is a profound mystery!

Bruce Bennett, my advisor in grad school, defines mathematics as "unified consciousness theory". As you come to master a branch of mathematics, it's as though you've grown a new abstract organ of perception through which you may then view the world. You've grown a new "mind's eye" that can perceive realities literally inconceivable without this new organ of perception."

---Rafael Espericueta

Professor of Mathematics

Bakersfield College

@Arno Gorgels

Thanks Sir

Regards,

Nitish

Mathematics reveals hidden patterns that help us understand the universe.

Here is a post from the Math Forum that might interest those following this thread:

Stripped to its barest essence, mathematics is the derivation of

theorems from axioms.

So what does that mean?

It means that mathematics is a collection of extended, collaborative

games of 'what if', played by mathematicians who make up sets of rules

(axioms) and then explore the consequences (theorems) of following

those rules.

For example, you can start out with a few rules like:

A point has only location.

A line has direction and length.

Two lines interesect at a point.

and so on, and then you see where that takes you. That's what Euclid

did, and ended up more or less inventing geometry. And that's what

other mathematicians have done over the centuries, inventing

arithmetic, and number theory, and calculus, and group theory, and so

on.

It's a little like what you do when you invent a board game like

chess. You specify that there are such-and-such pieces, and they can

move in such-and-such ways, and then you let people explore which

board positions are possible or impossible to achieve.

The main difference is that in chess, you're trying to win, while in

math, you're just trying to figure out what kinds of things can - and

can't - happen. So a 'chessamatician', instead of playing complete

games, might just sit and think about questions like this:

If I place a knight (the piece that looks like a horse, and moves

in an L-shaped jump) on any position, can it reach all other

positions?

What is the minimum number of moves that would be required to get

from any position to any other position?

http://mathforum.org/library/drmath/view/52350.html

Mathematics uses symbols and numerical values to explain the potency of scientific innovations.

@Arno Gorgels:

Yes, from what Stewart writes in the attached article, math has a profound role in nature

(see the 5 revolutions pointed out by Stewart).

@Arno Gorgels:

Another good place to look for the role of maths in nature is

http://ww2.odu.edu/~jadam/publications.html

Thanks Professor James Peters,

Math is the language of nearly everything.

@Arno Gorgels: ...Are the axioms decisive or the derivations?

If you mean by derivations, the theorems that spring from the starting point provided by axioms, then you may find the following quote interesting:

Euclid’s presentation is

axiomatic in that it puts a number of statements (the axioms, although he

did not call them that) at the beginning and develops the rest of the theory

from them via deductions and definitions.

D. Schlimm, Axioms in mathematical practice, Philos. Math. (III) 21, 2013, p. 37ff.

In answer to your question, it seems that without the starting point provided by axioms,

writing down and proving theorems is bit difficult.

Arno,

Please state Cantor's axioms.

Arno,

Among the Z-F axioms, we can start by taking a look at the axiom of choice :

Given any collection of nonempty bins, regardless of the number of bins, we can choose an object from each bin.

Okay, now we can test the idea that the AC is taken from observations of nature. On the surface, the AC does reflect your idea. However, if the number of bins is very large, it may not be possible to choose an object from each bin, especially if some of the bins are not reachable.

Here is an example: Let the bins be stars in the visible universe. There is only one star that is reachable (at the present time and for the forseable future), so the axiom of choice fails.

What is mathematics? A human art! Like novel writing. Just as there are criteria what a good novel is, there are criteria what good math is. Math grows since contributing to math is extremely satisfying for the contributor. Having solved a mathematical problem which respected mathematicians did not succeed to solve so far causes feelings of bliss, comparable to the feelings of a mountain climber who reached an exceptionally demanding summit. This keeps mathematics growing. Just as the mountain climber, the mathematician is free to choose his goals, but when he finally moved into his selected wall of his selected mountain, then he has to be extremely carefull to avoid all the dangers and insurmountable obstacles in his way. If these difficulties would not exist, there would be no bliss associated with success.

What is Mathematics? Very simple: Mathematics is the science of structure, order, and relation that has evolved from elemental practices of counting, measuring, and describing the shapes of objects. It deals with logical reasoning and quantitative calculation, and its development has involved an increasing degree of idealization and abstraction of its subject matter. Since the 17th century, mathematics has been an indispensable adjunct to the physical sciences and technology, and in more recent times it has assumed a similar role in the quantitative aspects of the life sciences.

Because the space, the distance, and the regular movement of universe all is maths. Distance make lines and points, and the structures out of random make a "cadence", and the language of cadences is Maths. The letters of words are maths too.

All H have the same size, magnitude. All C, all O, all N... We are maths.

Arno,

There is an implicit, incorrect assumption in my example, namely, to choose a star, the star must be reachable.

If we have set of bins (each containing a star), no matter how many, we can a star in one of the bins, even though the star we choose is not reachable. Hence, the AC holds.

Assuming that the universe is intrinsically mathematical, I'd tentatively define mathematics as the study of the fundamental relations of our universe and beyond ("beyond" because mathematics can model things that don't directly pertain to the physical universe, yet have a logical, exact behavior to them).

The Wikipedia definition of Mathematics is "the study of quantity, structure, space and change", following from the definitions given by the Oxford English Dictionary, G. T. Kneebone and others. I'm not quite sure how that stacks up to modern mathematics, that has grown to be ever more abstract and complex.

The computer scientist Edsger Dijkstra said that the world would be better served by mathematics if its sole definition was "the art and science of effective reasoning", and some thoughts by Gottfried Leibniz somehow leaned towards that direction long before Dijkstra. While that is a good way to view the mathematical method itself (in my opinion), I believe this definition ignores the subject matter of mathematics to focus on the method.

I have to say I really liked what Ulrich said, for it is true that mathematics is intelecutally demanding, but at the same time utterly satisfying once one's problem is solved, and I believe this sentiment was shared by Polya in the 20th century (if you still didn't read How to Solve It, it's an awesome read). I just can't see how his answer would work as a definition of mathematics.

@James

If we work within the well-defined mathematical world in which all sets are finite (i.e. in the 'Grothendieck universe of hereditarily finite sets'), choice works always, not by axiom but by construction. Since there are certainly only finitely many stars, no problem with AC can arrise.

@Fabricio,

as you certainly suspected my previous answer was not intended to define Mathematics, but only to illuminate one of its aspects.

@Ulrich,

Yes, I agree.

Mathematics is like art in the sense that it gives pleasure and unlocks new ways of discovering new beautiful things

“A mathematician, like a painter or poet, is a maker

of patterns. If his patterns are more permanent than

theirs, it is because they are made with ideas.” ---- G.H. Hardy

Mathematical is a well-defined abstract representation with objects or symbols consisting of a collection of variables and rules governing their values or has some meaningful logic.

Mathematics is the expression of logic.

@Joydip Dhar:

An excellent quote from Hardy! And what Hardy wrote many decades ago is reminiscent of what David Mumford writes about patterns in his ICM 2002 paper,

i.e., we derive patterns from perceived patterns.

What is the source of your Hardy quote?

@Ana María Sánchez Peralta:

Please explain a bit further in what sense mathematics is the expression of logic.

Vitaly,

Perhaps your conception of mathematics can be expanded to include not just natural numbers but also sets, relations and axioms that identify relations between sets.

Vitaliy,

Actually, no. We can identify set operations such intersection and union that do not rely on the natural numbers to have meaning.

Vitaly,

take into account that orthodox set theory does not rely on presenting the elements of a set in a way that one could 'count them during this presentation'. Instead it allows to define elements simply as those objects that have a certain property (which has, of course, to be defined by set theoretical expressions). So, the number of these elements comes not primarily into the play. It may need a difficult investigation to decide whether there is an element at all, or finitely many elements, or infinitely many. So in orthodox set theory natural numbers play no visible role.

This is completely different when one restricts sets to the 'heriditarily finite sets'. Such a restriction defines a framework for that part of mathematics that deals with real-world problems.

This is the direction of 'strict finitism' in the ongoing discussion on the foundations of mathematics.

Mathematical intuition and the cognitive roots of 1mathematical concepts

vy Giuseppe Longo, Arnaud Viarouge

http://en.youscribe.com/catalogue/educational-resources/education/internship-reports/mathematical-intuition-and-the-cognitive-roots-of-mathematical-1722009

''as soon as we impose, in our practice of communication and understanding of the world, maximal conceptual stability and invariance (we hate absolutes), we are doing Mathematics. Otherwise, it is prose, whether we know it or not.''

It would be interesting to test experimentally this hypothesis with young children years before they even learn to count. We would ask then to do certain tasks cooperatively but without seeing each other so they would have to invent effective ways of communication. It would be interesting to see if they would gradually invent naturally a form of primitive mathematical language.

Louis,

Nice observation.

Here is an article by Tom Brady (The New York Times) that corroborates what you have observed:

Manil Suri, a professor in Baltimore, Maryland, thinks math can be fun and that on some level humans crave the order it brings. Like many mathematicians, he believes humans are wired for it.

To prove his point, Professor Suri suggests that we envision a sequence of regular polygons: a hexagon, an octagon, a decagon and so on, and imagine the number of sides increasing indefinitely. "Eventually, the sides shrink so much that the kinks start flattening out and the perimeter begins to appear curved," he wrote in The Times. "And then you see it: what will emerge is a circle, while at the same time the polygon can never actually become one.

"The realization is exhilarating - it lights up pleasure centers in your brain," Professor Suri added. "This underlying concept of a limit is one upon which all of calculus is built."...

The interplay between the primitive pleasure center of our brains and the auditory cortex - the part of our brain where we hear and imagine music - helps us to decipher the abstract relationships between sounds, according to Zattore and Salimpoor.

Louis,

The NY Times story about music and mathematics was pointed out to me by Prof. Som Naimpally.

Arno,

e.g. ZFC

Vitaly,

recall what you do in order to show that sets A and B are equal: You assume x \in A and show that this implies x \in B and vice versa. You don't compare elements in any way. What you write is correct for the kind of combinatoric set theory children do at school.

Arno,

I try to write what I mean and vice versa: A representative example for what I called 'orthodox set theory' is the version of set theory given by the axioms of Zermelo and Fraenkel together with the axiom of choice, commonly abbreviated as ZFC. This version is orthodox since the opus magnum of mathematical orthodoxy 'the Bourbaki' is based on it. My favorite version is that of Bernay and v. Neumann on which Catagories and Functors are usually based. As Wikipedia shows there are many more modern approaches to set theory. Some of them I would not call orthodox.

Vitaly,

as it looks to me, your notion of counting is so wide that it becomes meaningless.

Consider data structures such as vector, map (in C++,say). What would be the the advantage of those if there would exist no iterators for them. You could not count their entries! Considering this, you will appreciate the full meaning of the fact that set theoretical axioms don't provide an iterator for the elements of sets. No counting, only random sampling! Or, making use of specific structures of specific sets that were carefully built as to provide an iterator.

Quantitatively, mathematics describes/communicates the structural dynamics of the universe (the form of function); while qualitatively, physics explains the phenomenological processes or functions of those structures (the function of form).

From a mainstream standpoint, symbols and numbers (and the manipulation thereof, i.e. mathematics) are two information sets that emerge during my evolution as Homo sapiens and development as a child that help me describe, control, and predict objects and events.

From a more theoretically appropriate perspective, I am the symbols and numbers themselves, and I feed back upon and forward to describe myself to myself. That is, I am mathematics:

https://www.researchgate.net/publication/233865314_Complete_and_Consistent_Theory_of_the_Universe?ev=prf_pub

Conference Paper Complete and Consistent Theory of the Universe

Vitaly,

Perhaps you will agree that symbols can be non-numeric. For example, assume we observe a set pattern that we denote by P. A set pattern is a collection of sets in some sense near a given set (call it M for pattern motif). Then we can consider sets A, B in pattern P. Notice so far I have not considered the cardinality of P, A, and B.

It will help if you explain your thinking concerning notation as the beginning of counting.

Vitali,

it was in my mind too, during my first year at university. Then I learned that sets as collections of objects are illusions for infinite sets. They are merely properties (predicates) and set theory is nothing but a very convenient and universal language to define and manipulate predicates.

About irrationals: . Is it so difficult to accept that for 'uncompletable' objects 'exact' simply means that the completion can algorithically carried further step by step, where each step gives one decimal digit accuracy more? This works for sqrt(2) as one learns in school. If your mind is longing for the exactness of your childhood, imagine a representation of sqrt(2) with more digits than there are baryons in the universe.

James Peters, "Please explain a bit further in what sense mathematics is the expression of logic".

Logic is based in inference. The inference is the process by which conclusions are drawn from correct premises. Sure correct premises are maths for example.

http://es.wikipedia.org/wiki/L%C3%B3gica (In Spanish is better).

1+1: 2

Arno, the thesis here is on the role of infinite sets as collections of objects.

If you have an argument thereto, fine. But please retain verbiage.

Dear Ana María Sánchez Peralta,

Many thanks for your explanation. I understand what you mean, now.

Arno,

Perhaps you will accept that ZF set theory is about sets of abstract objects, either abstract points or sets of abstract points. Abstract points are identified by their location. I set theory is often extended to sets of non-abstract objects such as picture points that have both location and meaurable content. Perhaps that is what you mean when you write that set theory is about real and abstract objects.

Arno,

Very interesting! Please suggest a paper that discusses well-ordered vacuum quanta.

Dear Erik, we are all. I'm looking from me all, and I put it in me to interpret for me. (I know I am what I not understand yet too). What can we do from this perspective? (To rest a little?) But we must continue walking with our limited body looking smallest details of all.

Vitali,

>

Be sure, if your point of view would provide a clearer view of the matter we struggle with, it would have the better right. Equal right for all views independent of their sharpness and explanatory power would not be a good principle for science. So we can't but let arguments fight for dominance and not protect ourselfes behind rights. To such rights we may resort as persons but our arguments have to stand for themseves.

To be completely explicit, in the matter of 'uncompleted objects' you had to point out a weakness of my proposed solution in order to let your pessimistic view survive as a reasonable one.

Ana Maria,

In doing mathematics, one looks outwardly rather than inwardly. The secrets of the universe appear to lie in the mathematics that we invent, since we are, after all, part of the universe. I sometimes wonder about a mathematics that will illuminate the pleasures of our experience.

Well, I have to say I'm a bit baffled that the idea of infinite sets still seems to be more or less unacceptable to some mathematicians.

It feels strange because it makes perfect sense in a theory of computation context, in which it is possible to prove there are countably infinite turing machines, but uncountably infinite functions (which is basically stating there is an uncountably infinite set of non-computable functions).

Maybe infinite sets, and the very hierarchy of infinites introduced by Cantor, are in their essence of abstractional utility, even if they don't "manifest" in nature.

But does it manifest only in differential infinities or in actual infinities as well?

Mathematics is wonderful! If nothing else, it keeps entertained with all sorts of very interesting problems.

Prof. Peters,

It is the way students enjoy mathematics. I used to, my son continued this family tradition and enjoy solving mathematical problem using in his technology course. However, when he will be old enough to look back at the origin, invention and formalisation of mathematics, perhaps he will not as happy as he is now.

Vitaly,

to me, your responds appear as random remarks with virtually no connection to what I said. It's not fun to continue an discussion under these circumstances.

Dear Fabricio.

One can find what appear to be elements of the microcopic world, tending toward infinitessimally small magnitudes (close to but never exactly equal to zero). I mention this because so far the discussion has been about huge as opposed to small unboundedness in nature. We have only to look at seeming unboundedness in either the small direction or increasing size direction in nature and realise that there one can imagine an inspiration for Cantor to conceptualise the cardinality of the natural numbers as one of the infinities.

Issam Sinjab wrote: ...mathematics is the science of patterns.

That conception of mathematics takes us in a good direction, since every theorem in mathematics can be viewed a representation of a pattern that yields a tangible result, demonstrated in the proof of the theorem.

Arno and Fabricio,

Perhaps you both know that Cantor corresponded with the Youngs (husband and wife) during the writing of their book The Theory of Sets of Points. The first appendix

(note 1) concerns Cantor and with very detailed references to Cantor's work.

Arno,

Three parts of the Young's book may interest you:

Preface to 2nd Edition contains two letters from Cantor and the 9 March 1907 letter congratulates the Youngs concerning their book, signaling the theory of sets as a separate mathematical discipline, which was chiefly Cantor's creation.

ch. VII. Cantor's Numbers, which sets forth Cantor's notion of a well-defined set, and the modified definition by the Youngs.

Appendix, starting on page 284, which defends the theory of derived and deduced sets without the use of Cantor's s numbers.

Arno,

Many thanks for more details about Cantor's theory. This is tremendously interesting and important in getting to know the fabric and weave patterns of mathematics.

The book by the Youngs is very thorough, presenting a view of set theory that complements Cantor's theory.

Ulrich Mutze,

Sir,

With all due regards I plead you not to leave the thread. We need your expertise to make us wise. It most time happens in a discussion to put one's argument, the person often do not listen voices of being cautious and patient, meaning thereby let other put their point of view in the end we reach a concrete and agreed results are overlooked or interpreted as argument against the one who has no regard for age, experience and expertise to prove oneself.

We need your words of wisdom and guidance irrespective of how harshly you try to make us wise. We all on this thread reading and appreciating you.

Ulrich,

Good afternoon from a sunny corner of the University of Manitoba!

I also hope you will stay with us and contribute your comments for various topics that have arisen and continue to arise in this thread. I does seem that the principal topic for this thread is of great interest for many of us. I am very grateful to you and others for your gracious comments.

James

Nice platform for simple discussion.

As Mathematics is the abstract study of topics such as object, numbers, structure, space, and change. I feel it has inherent beauty, sense and intelligence. Every research in our scientific world are mathematical footprint.

"The question, "What is mathematics?", can have many interpretations. It can mean (and has most often been taken to mean), "What is the nature of mathematical objects?" It can mean, "What is special about how we reason in mathematics, or about the methods of mathematics?" What it has rarely meant in recent years, but what I would argue it should mean is, "What is the common nature of those subjects that are classified as mathematics which causes us to lump them together under the same name?" For example, biology can be described as the study of living organisms. This is the common property of all the subjects to which the name "biology" is applied. Political science is the study of the ways communities govern themselves. The question is, what is an equivalent description of mathematics? "

---------------------------------------------------------------------------------

What is Mathematics I: The Question, by Bonnie Gold, Monmouth University .

http://sigmaa.maa.org/pom/PomSigmaa/WhatMathI.htm

@Isaam:

Under that interpretation of the question, I'd define Mathematics as "the study of formal abstractions whose properties can be deduced with the usage of definitions, axioms and inference rules", but that's one hell of a logic-ridden definition, maybe there are better ones.

Issam,

Your suggestion is excellent! I would say the common property among areas of mathematics is the proof of theorems based axioms that set forth conditions such as nearness of a set to itself in proximity space theory and definitions of key terms.