Hi, mathematics must be seen as a tool to model physics and many other areas. It is clear that some mathematics are far to be applie in pratice but for example is cosmology (black holes, relativity theory etc..,) the phenomena have been modeled mathematically first. Now, physical measurements allow to validate the mathematical model.

@Richard I totaly agree ! and also know it from my discipline, Rayleigh and Love waves were discovered after being derived mathematically, their characteristics were revealed before any instrument recorded them. Have a nice day,

@ Peter, What I meant is: people other than mathematicians, do speak more or less Mathematics, just as a foreign language! When mathematitions use their slang or jargon if you like,” forigners “from outside the discipline may very well feel lost. This is not in the best interest of science. After all jargon is deeper and more difficult than slang because I feel it is more limited to rather restricted sub groupes of Mathematicians. Some mathematicians will also feel lost !

The long history of mathematics generally lacks a distinction between pure and applied maths. Yet in the modern era of mathematics over, say, the last two centuries, there has been an almost exclusive focus on a philosophy of pure mathematics. In particular, emphasis has been given to the so-called foundations of mathematics — what is it that gives mathematical statements truth? Metamathematicians interested in foundations are commonly grouped into four camps.

Formalists, such as David Hilbert, view mathematics as being founded on a combination of set theory and logic (see Searching for the missing truth), and to some extent view the process of doing mathematics as an essentially meaningless shuffling of symbols according to certain prescribed rules.

Logicists see mathematics as being an extension of logic. The arch-logicists Bertrand Russell and Alfred North Whitehead famously took hundreds of pages to prove (logically) that one plus one equals two.

Intuitionists are exemplified by LEJ Brouwer, a man about whom it has been said that "he wouldn't believe that it was raining or not until he looked out of the window" (according to Donald Knuth ). This quote satirises one of the central intuitionist ideas, the rejection of the law of the excluded middle. This commonly accepted law says that a statement (such as "it is raining") is either true or false, even if we don't yet know which one it is. By contrast, intuitionists believe that unless you have either conclusively proved the statement or constructed a counter example, it has no objective truth value. (For an introduction to intuitionism read Constructive mathematics.)

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Plato and Aristotle as depicted in Raphael's fresco The school of Athens.

Moreover, intuitionists put a strict limit on the notions of infinity they accept. They believe that mathematics is entirely a product of the human mind, which they postulate to be only capable of grasping infinity as an extension of an algorithmic one-two-three kind of process. As a result, they only admit enumerable operations into their proofs, that is, operations that can be described using the natural numbers.

Finally, Platonists, members of the oldest of the four camps, believe in an external reality or existence of numbers and the other objects of mathematics. For a platonist such as Kurt Gödel, mathematics exists without the human mind, possibly without the physical universe, but there is a mysterious link between the mental world of humans and the platonic realm of mathematics.

It is disputed which of these four alternatives — if any — serves as the foundation of mathematics. It might seem like such rarefied discussions have nothing to do with the question of applicability, but it has been argued that this uncertainty over foundations has influenced the very practice of applying mathematics. In The loss of certainty, Morris Klinewrote in 1980 that "The crises and conflicts over what sound mathematics is have also discouraged the application of mathematical methodology to many areas of our culture such as philosophy, political science, ethics, and aesthetics [...] The Age of Reason is gone." Thankfully, mathematics is now beginning to be applied to these areas, but we have learned an important historical lesson: there is to the choice of applications of mathematics a sociological dimension sensitive to metamathematical problems.