Some colleagues say you cannot separate applied from pure maths. Other claim the two fields have nothing in common, quoting Hilbert to support that view...

Both views are correct, in part at least, and the reason is that the question is not being well posed. Do you mean "why separate pure vs applied maths" from an administrative/sociological viewpoint or from a philosophical/epistemological view point?

In the first case the separation does exist (in case of doubt try to convince your dean - who may be a chemist - to hire 5 more mathematicians doing algebraic topology, try a second time later but suggesting he hires people in numerics-simulation-modelisation... and compare the reaction in both case). On the other hand, it is probably almost impossible to separate pure from applied math on an epistemological level (number theory is a perfect example of a "pure" field with impressive applications (say in cryptology).

To summarize: the question should definitively be better focused.

As maths is a very broad or wide subject so there became a need to be separated, bold fields are totally different, by separating these, one can easily say that i have interest in pure maths or in applied maths. without separating what will he say that i have interest in maths or not interest in maths.

See if you have any gravity to fields of the Applied or Pure Maths and see your Objectives and dreams

actually look to your background in Analytical mathematics

Can Mathematics be separated? Can we have the application of mathematics (so called applied mathematics) without the help of the basic concepts of mathematics (so called pure mathematics)? Not at all. Mathematics as a whole cannot be separated.

I guess that I would ask the question about what kind of filter would I use to distill mathematics into "pure math." As I read texts from either discipline, there often appears to be at best a very blurry boundary. Perhaps, the interesting point is how the truly remarkable problems get posed. Why did geometry develop as it did? Was there a drive from the practical aspects of computation in the construction of large edifices that begged for the generation of more abstract principles until there was an initial distillation into Euclid's elements? Often, the driving force to create a branch of mathematics is some problem as stated in the world. Einstein's approach to Relativity was not formulated in terms that Minkowski used to reformulate in terms of non-Euclidean geometry. The use of the Fourier series was driven by theories of heat energy and that gave rise to the need to validate the approach in a more general analytic approach. That analytic approach was then abstracted to develop a more rigorous sense of geometry (back to the problems driving the original Euclidean formulation.

What I'm getting to is that the problems of the world seem to have been the driving force to quantify and then abstract the underlying predictive capabilities of the models.

Pure mathematics is mathematics that studies entirely abstract concepts. It has been described as "that part of mathematical activity that is done without explicit or immediate consideration of direct application," although what is "pure" in one era often becomes applied later.

One central concept in pure mathematics is the idea of generality; pure mathematics often exhibits a trend towards increased generality. These means generalizing theorems or mathematical structures can lead to deeper understanding of the original theorems or structures; generality can simplify the presentation of material, resulting in shorter proofs or arguments that are easier to follow. One can use generality to avoid duplication of effort, proving a general result instead of having to prove separate cases independently, or using results from other areas of mathematics; generality can facilitate connections between different branches of mathematics. Category theory is one area of mathematics dedicated to exploring this commonality of structure as it plays out in some areas of math.

Pure mathematicians are often driven by abstract problems. To make the abstract concrete, here are a couple of examples: “are there infinitely many twin primes” or “does every true mathematical statement have a proof?” To be more precise, mathematics built out of axioms, and the nature of mathematical truth are governed by predicate logic.

A mathematical theorem is a true statement that is accompanied by a proof that illustrates its truth beyond all doubt by deduction using logic. Unlike an empirical theory, it is not enough to simply construct an explanation that may change as exceptions arise. Something a mathematician suspect of being true due to evidence, but not proof, is simply conjecture.

Applied mathematics is a branch of mathematics that concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Thus, "applied mathematics" is a mathematical science with specialized knowledge. The term "applied mathematics" also describes the professional specialty in which mathematicians work on practical problems; as a profession focused on practical problems, applied mathematics focuses on the formulation and study of mathematical models. In the past, practical applications have motivated the development of mathematical theories, which then became the subject of study in pure mathematics, where mathematics is developed primarily for its own sake. Thus, the activity of applied mathematics is vitally connected with research in pure mathematics.

Sometimes, the term "applicable mathematics" is used to distinguish between the traditional applied mathematics that developed alongside physics and the many areas of mathematics that are applicable to real-world problems today.

There is no consensus as to what the various branches of applied mathematics are. Such categorizations are made difficult by the way mathematics and science change over time, and also by the way universities organize departments, courses, and degrees. Many mathematicians distinguish between "applied mathematics", which is concerned with mathematical methods, and the "applications of mathematics" within science and engineering.

Applied mathematics is a branch of mathematics that concerns itself with the application of mathematical knowledge to other domains. Such applications include numerical analysis, mathematics of engineering, linear programming, optimization and operations research, continuous modelling, mathematical biology and bioinformatics, information theory, game theory, probability and statistics, financial mathematics, actuarial science, cryptography and hence combinatorics and even finite geometry to some extent, graph theory as applied to network analysis, and a great deal of what is called computer science.

Applied mathematicians are typically motivated by problems arising from the physical world. They use mathematics to model and solve these problems. These models are really theories and, as with any science, they are subject to testifiability and falsifiability. As the amount of information regarding the problem increases, these models will possibly change.

Pure and applied are not necessarily mutually exclusive. There are many great mathematicians who tread both grounds. .

Pure and applied mathematics are totally different disciplines, whose methodologies, goals, and deliverables are utterly separate.

No one has put it better than David Hilbert : "We are often told that pure and applied mathematics are hostile to each other. This is not true. Pure and applied mathematics are not hostile to each other. Pure and applied mathematics have never been hostile to each other. Pure and applied mathematics cannot be hostile to each other because, in fact, there is absolutely nothing in common between them."

Applied mathematics should properly be called 'mathematical modelling' and is a branch of engineering.

Mathematics includes many pure and applied courses. Every mathematician must know the fundamentals of both types of courses, especially in pure mathematics foundation courses. Later on as per his/her interest they can select few courses in further study and limited to a particular area/topic in case of research work.

So, in my opinion the pure and applied mathematics are inseparable sister disciplines.

It is not necessary because the difference between Pure and Applied Mathematics is often the difference in motivation and emphasis. They are not disjoint because their activities are overlapping with each other. Applied mathematicians often use significant ideas from Pure Mathematics to build the mathematical models. On the other hand, Pure Mathematicians often use ideas from Applied Mathematics as the basis for their abstract research. Pure Mathematicians are concentrating on the question such as "What is going on here?" and Applied Mathematicians are concentrating on the question such as "What can I do with this method?". However, the question can often lead to the same direction. Therefore, there is no advantage of separating Pure and Applied Mathematics because in fact, the distinction between them is often blurred in practice. In other words, there is no clear distinction between Pure and Applied Mathematics.

This separation is temporary since every pure math eventually become applicable.

I think it is compulsory because of pure and applied mathematics are different by nature and application.applied mathematics is useful in engineering and other courses and raised a question why.but pure mathematics develope models for applied mathematics.

@Ivan Alim: Pure Mathematicians are concentrating on the question such as "What is going on here?" and Applied Mathematicians are concentrating on the question such as "What can I do with this method?".

Instead of "What is going on there?", you may want to consider "What results (theorems) and proofs emerge from the theory". For example, dense subsets have no counterpart in the physical world (as far as I know). We can then follow Taimanov's example and consider continuity in the context of dense subsets.

Pure and Applied Mathematics can not be separated. Applied mathematics offers applications to pure mathematics. Pure mathematics offers answers to applied mathematics. Together the two mathematical expands the horizons of knowledge.

Some colleagues say you cannot separate applied from pure maths. Other claim the two fields have nothing in common, quoting Hilbert to support that view...

Both views are correct, in part at least, and the reason is that the question is not being well posed. Do you mean "why separate pure vs applied maths" from an administrative/sociological viewpoint or from a philosophical/epistemological view point?

In the first case the separation does exist (in case of doubt try to convince your dean - who may be a chemist - to hire 5 more mathematicians doing algebraic topology, try a second time later but suggesting he hires people in numerics-simulation-modelisation... and compare the reaction in both case). On the other hand, it is probably almost impossible to separate pure from applied math on an epistemological level (number theory is a perfect example of a "pure" field with impressive applications (say in cryptology).

To summarize: the question should definitively be better focused.

This is a very interesting  discussion topic. I saw the post only recently. 

First of all, let us see the basis of the labelling .  The existing distinction is, based on the motive of pursuing   mathematics. The division occur whether the subject is pursued for the sake of clarity/beauty  or for utility.

Those portion of mathematics which is useful to solve the needs of other disciplines (science, engineering and others ) began to be categorised as 'applied mathematics'.

 Those parts ( or branches ) of mathematics that  are pursued for the sake of  itself (despite no immediate utility to other disciplines of knowledge), do come under the banner of 'pure' mathematics. The pursuit of 'pure' mathematics from the sake of achieving greater clarity. many a times such studies are fuelled by the aesthetics of its underlying structures also. All such mathematical topics pursued with these considerations, come under the banner of 'pure' mathematics.

Thus, the distinction arise solely from the point of view of pursuing mathematics from two different stand-points.

As pointed out by others in the forum, what is considered 'pure' today could find an application in the future. .i.e. when an abstract principle finds a concrete expression for a real world system, it becomes part of the applied mathematics.

Eminent mathematicians like Courant suggest the importance of pursuing mathematics from both the standpoints. Intellectual health of scientific society is determined by the proper balance in pursuing these two approaches.