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The modern world of mathematics is divided into different categories and if you are so lucky as to meet real‐life mathematicians and engage them in a conversation, they will typically tell you that they are either mathematicians or applied mathematicians. You have probably heard of mathematics, but what is applied mathematics? A quick look on the Internet will give you conflicting definitions. It will also reveal that applied mathematics has found its place in modern academia. As such it is recognized by international scientific societies, journals, and the usual conferences. What is so special about applied mathematics? How is it different from mathematics, or any other scientific discipline?
Mathematics
Let us start with mathematics itself. Whereas philosophers still ponder the best definition, most scientists and mathematicians agree that modern mathematics is an intellectual discipline whose aim is to study idealized objects and their relationships, based on formal logic. Mathematics stands apart from scientific disciplines because it is not restricted by reality. It proceeds solely through logic and is only restricted by our imagination. Indeed, once structures and operations have been defined in a formal setting, the possibilities are endless. You can think of it as a game with very precise rules. Once the rules are laid out, the game of proving or disproving a statement proceeds.
For example, mathematicians have enjoyed numbers for millennia. Take, for instance, the natural numbers (0,1,2, …) and the familiar multiplication operation (×). If we take two numbers p and q together, we obtain a third one as n = p × q. A simple question is then to do the reverse operation: given a number n can we find two numbers p and q such that n = p × q? The simple answer is: of course! Take p = 1 and q = n. If this is the only possible way that a natural number n larger than 1 can be written as a product of two numbers, then n is called a prime number. Mathematicians love prime numbers and their wonderful, and oftentimes, surprising properties. We can now try to prove or disprove statements about these numbers. Let us start with simple ones. We can prove that there exist prime numbers by showing that the natural numbers 2, 3, and 5 have all the required properties to be prime numbers. We can disprove the naive statement that all odd numbers are prime by showing that 9 = 3 × 3. A more interesting statement is that there are infinitely many prime numbers. This was first investigated c.300 BC by Euclid who showed that new larger prime numbers can always be constructed from the list of all known prime numbers up to a certain value. As we construct new prime numbers the list of prime numbers increases indefinitely. Prime numbers have beautiful properties and play a central role in number theory and pure mathematics. Mathematicians are still trying to establish simple relationships between them. For instance, most mathematicians believe there are infinitely many pairs of prime numbers that differ by 2, the so‐called twin‐prime conjecture (a conjecture is a statement believed to be true but still unconfirmed). For example, (5,7), (11,13), and (18369287,18369289) are all pairs of primes separated by 2, and many more such pairs are known. The burning question is: are there infinitely many such pairs? Mathematicians do believe that it is the case but demonstrating this seemingly simple property is so difficult that it has not yet been proved or disproved. However, at the time of writing, a recent breakthrough has taken place. It was established that there exist infinitely many pairs of prime numbers that differ by 246. This result shook the mathematical community and the subject is now a hot topic of modern mathematics.
Through centuries of formalization and generalization, mathematics has evolved into a unified field with clear rules.
