This should be a comprehensive assessment. Based on applications and based on possible impact on further researches.

This should be a comprehensive assessment. Based on applications and based on possible impact on further researches.

Mathematical results in general are indeed looked at both their places in expanding mathematics itself ( current and opening possibilities for further research) and their applications in other disciplines. But from pure mathematical point of view, mathematical results should be appreciated just for the sake of expanding mathematics and mathematical research.

Does appreciation lead to high ranking? I mean, is it one of the factors that lead mathematical results to higher ranks?

Like the other contributors, I think mathematics should serve both, application and "basics". You are "payed" for application, maths gives satisfaction for research results in fundamentals.

Why shouldn't mathematical results be ranked in the same way results are ranked in other disciplines?

Mathematics can be just theoretical, while other disciplines are not necessarily so. That is why the ranking procedure should possibly be different for mathematics. 

Dear Hemanta,

Mathematics has several faces. It is a body of knowledge, but it is also a human activity and creativity, a language and a tool to deal with many kinds of problems. Mathematical investigations involve searching for patterns, formulating, testing, and justifying conjectures, reflecting, and generalising.

As you are well aware, mathematics is generally categorised into pure and applied mathematics. However, it does not mean that pure mathematics cannot be applied mathematics. In fact, most of the applied mathematics consists of pure mathematics, the result of research in pure mathematics not the consequence to model a complex system or structure to apply to solve a particular problem in engineering, business or in any other field of science.

As far as question of ranking mathematical findings is concerned, it depends not only on the nature of research, whether applied or pure or applied as well as potential impact of findings within the field and outside the field. On the basis of its application in engineering and other sciences it may conveniently ranked as:

1. Highest relevance

2. Very high relevance

3. High relevance

4. Medium relevance

5. Low relevance

Mathematical research in terms of international standard, internal impact, international interest in the problem investigate and probability of its publication in international journals of repute. As such, it may be:

1. Outstanding

2. Excellent

3. Very Good

4. Good

5. Insufficient

Mathematics worth its salt conveys not only results (theorems and proofs) that have been found but also the connections of the results with the work of others.   In other words, the basis for new results should be made clear.    A good example of this approach to providing a basis for new results in given in

D. Valtorta, On the p-Laplace operator on Riemannian Manifolds, Ph.D. thesis, Universita Degli Studi Di Milano, 2014:

http://arxiv.org/pdf/1212.3422.pdf

See, for example, the comments about Theorems 1.2.1 and 1.2.2, page 6.

Mathematics worth its salt conveys two things, according to Professor James Peters. How to decide whether some result is worth its salt? 

Worth of mathematical results may be measured in two directions (I)its originality and  how it initiates further investigations (ii) its immediate or future relevance in applied disciplines

"how to decide whether some result is worth its salt ?"

This is a difficult question considering that the score of any given ranking method may very well vary in time. For instance, a mathematical finding with seemingly no obvious direct application could turn out to be extremely important years later. Such a case happened before and certainly will happen again.

The importance of a mathematical finding is analogous to the amount of old theories that are mede unnecessary! 

If we commit to rank discoveries as publications, we will surely do it in one way or another, as there are plenty of methods associated with scientometry. We may even find an optimal method for measuring discoveries as pure knowledge, which of course will take into account (perhaps as a weighted mean)  application, simplicity, adequacy, filling gaps, philosophical commitments and several other criteria. However, my concern is not the method of measuring but the motivation for such measuring in the particular field of mathematics. From an epistemological view, it seems that ranking discoveries in mathematics is senseless. From both a nominalist and platonist perspective, mathematics is infinitely expansive and the creation of mathematical objects (either considered as having a realistic expression, purely abstract or fictional), axioms and theories is boundless. Mathematician's art and skills create new objects and theories, however they lack the epistemological content necessary in any objective ranking. A mathematical object does not create new knowledge, either interpreted in physical terms or not. It is just a construction (logical, symbolic, syntactic, etc.) whose utility is to be found or not in the future as application (within mathematics or within sciences) and that could represent or not a part of the reality. Even if the construction finally finds physical systems to represent or to be applied to, we can say it was already there before being discovered (constructed). So, what would be the motivation for ranking such constructions? In my opinion, there is no scientific motivation and no criteria either, except exclusively within mathematics. But if the ranking criteria are forced to stay within mathematics, we can no longer talk of objectivity - somehow, mathematics is useful only applied. On the contrary, if you asked the same question for a physical science, not only that we have epistemological criteria available, but other social criteria would also add to these.  

Dear Lehtihet,

Yes, indeed some results due to Ramanujan were found to be useful many years later.

Dear Professor Drossos,

Yes, it has happened quite a few times. But then to accept the corrections that can make an old theory redundant, scientific temperament is required, and scientific temperament, I think, has become a thing of the past! 

Dear @Hemanta, our rienf @Mohammad gave  very fine explanation! You have got very good answers! I am not going to discuss on criteria, but to add some more details on Mathematics Discoveries in different fields of science and its applications, in order to enhance the discussion!

http://www.nsf.gov/discoveries/index.jsp?prio_area=9

It depends who is the person who rank those findings. If a Mathematician does this task, then he/she cares about the Mathematics as a whole and historical continuous structure. If a Physicist does that, then he/she cares about what new can we get from that finding, where exactly could we use it. If a Philosopher does it, then he/she cares about what is the new entry in human thought, brought by the new finding.

Dear Professor Jacic,

In another thread, I remember to have observed that some people do not like to accept that mathematical discoveries are discoveries after all! Indeed, I have used the term mathematical results instead of mathematical discoveries just to avoid an unnecessary debate. Thank you for enhancing this discussion. 

Hemanta, it is a semantic choice, but I do not consider mathematical "discoveries" as discoveries. The reason is that only physical sciences can claim a discovery, as having some empirical nature. Mathematical findings are constructions. One can object that proving a conjecture for instance and proof in general are discoveries, but they lack empirical components (for naming them as such). A proof is a (discovered within mathematics) logical path, but not a discovery in sense that creates new knowledge.

Well!

Professor Jacic, as I have said, people from the physical sciences for example, do not accept that mathematical discoveries are discoveries after all! Your comments? 

Can we have an example of Mathematical Result which can be categorized as a discovery in the sense Barboianu have narrated.

I think the question is *why* should we rank such things. Like all rankings, they exist so that those that are incapable of judging quality can quantitatively justify their actions. 

This is the main unsolved mathematical problem nowadays. If you are interested, try to read one of my preprints "on cordinatization of mathematics" on www.arxiv.org

Dear Michael Small,

Those who go for quantity, leaving aside quality, can hardly think about doing anything original. Such people remain satisfied either modifying or generalizing mathematical results established by others. They always follow the leader; they can not lead. Such results are in general not of high quality.

Those who go for quality, leaving aside quantity, are interested in doing something original, which would be later either modified or generalized by the first group of people mentioned above. Results established by the those who do not go for quantity are usually of high quality. Such people lead others; they can not be led easily.

The first group consists of mathematical workers, while the second group consists of mathematicians. 

In all branches of knowledge, this clustering is valid. For example, Alexander Graham Bell was a scientist; all those who modified his instrument known as telephone are scientific workers.

Dear Subrata,

For example, calculus of integration was discovered by Newton. This is not just an usage of the word; calculus was discovered; no other word can explain it. 

In terms of another remarkable mathematical finding, I suggest considering the geometry flowing from the Postulates in Euclid's elements.    If we consider what has happened in response to Euclid's fifth postulate by Bolyai and introduction of non-Euclidean geometry (lines on a curved surface):

http://www.ams.org/notices/200509/rev-osserman.pdf

or by Riemann, who introduced an alternate to the parallel postulate (no parallel lines) and ushered in the study of manifolds:

http://www.radford.edu/~wacase/math%20116%20section%207.4%20new%20v1.pdf

the end result is a remarkable mathematical achievement.

Dear Professor Peters,

Beautiful examples! How about such other examples outside geometry?

Dear Hemanta, the best examples that I can think of now and mention in that preprint are from graph theory (if that is what you mean)

Dear Peteris,

The FCT for example? But it was originally proved using computers, and Paul Halmos did not accept that kind of proofs as proofs.

Dear Hermanta,

Another astonishing finding can be found in the work by Felix Hausdorff in his 1914 monograph Grundzuge der Mengenlehre (Foundations of Set Theory), later translated into English by translators in the American Mathematical Society.    Hausdorff dedicated his book to Georg Cantor.   See the 1920 review:

http://www.ams.org/journals/bull/1920-27-03/S0002-9904-1920-03378-1/S0002-9904-1920-03378-1.pdf

Perhaps the greatest achievement in this little book is the introduction of topological spaces (ch. VIII).   In the first edition of this book, a theory of ordered sets and Lebesgue integral are introduced.

It is important to notice that the second edition of Hausdorff's Mengenlehre (appearing in 1927) was completely rewritten. ordered sets and the Lebesgue integral were abandoned and his theory of topological spaces was completely revised.   The American Mathematkcal Societry translation is based on the second edition.

Dear Professor James,

Perhaps would be wrong word here to be used; introducing the notions of topological spaces and of measure theory were indeed great mathematical events. Hausdorff was Hausdorff; he was unique, a great mathematician..

Dear Hemanta,

it seems that we are talking about different things. In that preprint I am trying to promote the idea that proofs, mathematical concepts, theories etc. should themselves be mapped to simpler mathematical objects. The history of mathematics is a mathematical object itself.

thanks for so many examples of discoveries in mathematics infact the list will long. Can we add method of Least squares estimation,  simplex for LPP..beacuse of their application value

Dear Peteris,

The examples supplied by those who are taking part in this discussion are actually the mathematical discoveries that are worth the salt. I think, we are on the right track. 

Dear Subrata,

Least squares estimation was indeed a great mathematical discovery. However, regarding the simplex algorithm, I have reservations. 

The simplex algorithm is actually an extension of the Gauss-Jordan algorithm of solving linear system of equations. We should perhaps salute the Gauss-Jordan algorithm first before saluting the simplex algorithm!

Least squares estimation and simplex algorithm are therefore not of the same level. The first one is an original thing, while the second one is application of another discovery. 

@Hemanta K. Baruah:  How about such other examples outside geometry?

So far, I have limited the consideration of outstanding mathematical achievements to Euclid (with consequent results from Bolyai, Lobachevsky, and Riemann) and Felix Hausdorff

http://www.genealogy.ams.org/id.php?id=46991

The first and second editions of his little book Grundzüge der Mengenlehre, 1914, later translated by the American Mathematical Society as Set Theory, 1857.  Hausdorff is considered the founder of set-theoretic topology.    His work on set theory fits within the purview of remarkable mathematical achievement that includes

> theory of ordered sets and introduction of cofinality (introduced by Hausdorff):

A subset B of a set A is cofinal, provided

For every $a\in A$, $a \leq b$ for some $b\in B$.

http://en.wikipedia.org/wiki/Cofinal_(mathematics)

>distance between a point and a set (this metric introduced by Hausdorff is a precursor of the metric introduced by E. Cech during his famous seminar, 1936-1939):   Let $A, B$ be nonempty sets in a space $X$, then the distance between $A$ and $B$ (denoted by $D(A,B)$) is defined by

\[

D(A,B) = inf\left\{ d(a,b): a \in A, b \in B \right\},

\]

where $d(a,b)$ is a measure of the distance between $a$  and $b$.

>topological spaces (Hausdorff coined this term):

http://en.wikipedia.org/wiki/Topology

So, in response to your questions about examples outside geometry, I suggest that proximity space theory introduced by E. Cech (late 1930s) and Efremovich (early 1930s)  is a remarkable mathematical achievement.   This theory was initially viewed as a form of infinitesimal geometry by Efremovich.   But in hands of Efremovich's student, Smirnov, closeness of sets becomes abstract and, in some sense, geometry independent.   It all depends on how we interpret closeness and the axioms that a proximity relation must satisfy.   

Personally, I think giving a proximity relation an infinitesimal closeness setting is better than what Smirnov proposed.   

Could we have a short break for following dish?

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 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”.  

Sincerely yours,

Geng

Well!

@ Geng Ouyang : Mathematically and Philosophically , the ZERO exists it is not true to say "nothing and not being". If zero were not there, mathematics and science would not have developed.Philosophically it is something like whether God exists? If I assume , God exists and people are happy so it is there.

Thank you, dear Mr. R. C. Mittal ,

ZERO as a member of “mathematic language”, it does exist in our science when we have a positional numeric system and it served as a place order.”------we created and defined something in universe for our science.

Now the exactly same thing happens to “a new number symbol 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 meaning of “zero”.

Sincerely yours,

Geng

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.

The discussion has started to take a very different turn! Let us continue it please!

Dear Mr. Hemanta K. Baruah, you are right.

There are roughly two kinds of mathematical findings; one is in the theoretical area while another is in the applying area. The findings in the applying area bring immediate “value and rank” while some of the findings in the theoretical area may bring future “value and rank”. If, however, the mathematical findings can be in between, it would be wonderful.

But that is OK, it is science life.

Geng

Dear Geng,

You mean, findings in applied mathematics get immediate attention, while those in pure mathematics take longer durations to be recognized! I think, you are right. I never thought about it earlier.

Dear Hemanta，

Our science history tells us some of those findings in pure mathematics are born as they must according to the mathematical law, but when and how to be accepted and put into applying area are decided by different factors in our human life-----they even never know how and whether are accepted, ranked or not.