Formal and concrete are adjectives qualifying a concept which is otherwise less permanent over time and space. In pure mathematics (distinct from applied mathematics) consistency will suffice as an adjective. In nature science matter is studied. The word matter, as the prime content of nature, indicates that nature is currently studied only as material nature. What is studied excludes concept of consciousness or awareness, which alone exists by itself and for ever. Only such existence can be called real.
Names and forms appear in consciousness which is mind of person identifying with a body in mind. Mathematical ideas appear in mind as what are logically connected by intellect in mind. Many systems of mathematical logic are created and they appear separately in mind. Their creation is to design of predispositions and latent tendencies of the person related to other persons and objects in mind.
Axioms are unchanging logic created by intellect as connection of mental ideas. Formal study is logical analysis. The ideas can be (and are) correlated by intellect with concepts of material objects. In the West, matter is mistaken by many persons to be the reality and worse still to be the sole reality. Therefore ideas and concepts in mind are mistaken to be representations of material reality, only failing which they are purely imagined ideas or concepts. Pure mathematical ideas are purely imagined.
Whitehead and Russel defined Peano arithmetic to be mentally logical. Godel introduced ambiguity into it by mixing logical concept of quantitative cardinal numbers with the different logical concept of ordinal numbers which represent spatial order or sequential order. Such order to which free numerals are pegged falls outside the logic of cardinal numbers. Thus Godel claimed that the logic of Paeno arithmetic was both true and false, consistent and inconsistent. Consistency is lacking in multivarious logic of perceived concepts created in physics. The resulting inconsistency is named uncertainty in material reality.
Correlating concepts, which are called perceived material conceptual reality, with mental concepts of mathematics makes the correlated mixture inconsistent or uncertain; not so the purely logical systems of mathematics. Mathematics by itself is consistent and never concrete in the sense that perceived concepts mistakenly to be material outside the mind are said to be concrete. Whether created concept is concrete or consistent the creative construction is involuntary and creation in the mind is by predispositions and latent tendencies of mathematicians and physicists.