How "concrete" is mathematics, even when it's formal, rather than natural science?

How "concrete" is mathematics, even when it's formal, rather than natural science?

So because it relates to natural science, then it's unreasonable to view it as "entirely abstract" or "entirely formal". Therefore mathematics is also not 100% formal science, but a part of it constitutes formality, whereas e.g. application of mathematics has many more aspects than just formality.

Perhaps multi-disciplinarity is the proper way for a modern man to think about sciences? That they work in tandem, rather than as individuals.

• I’m not sure ‘actual’ is the right word here: like all necessary statements, mathematical statements are true in the actual world. Perhaps you mean concrete? As in: Is mathematics only about abstract objects, like the number 5; or is it also about concrete objects, like the five coins in my pocket?’ Would that be a fair way to rephrase your question? Nov 24 '17 at 13:00
• @MarkOxford If one removes maths from the coins, then one cannot understand coins as having a circle shape, a diameter (and a radius), a height, a thickness, physical strength etc. etc. So if one removes maths from the coin, then one cannot understand coins as physical objects? Nov 24 '17 at 13:30
• No science is 100% formal, and all other sciences it have both formal and material aspects. In mathematics this is crudely reflected by separating it into pure and applied parts. Empirical sciences have formal aspects too, for instance geometric optics or Hamiltonian dynamics in physics are mostly formal (except for guidelines for applying calculations to practice), but they are most prevalent in mathematics, and their prevalence decreases as we move from hard to soft sciences. That things have many aspects and connect to other things is a platitude, it is not very useful without specifics. Nov 25 '17 at 5:29
• For a recent book: Bob Clark, Wittgenstein, Mathematics and World, Palgrave (2017). Nov 25 '17 at 19:45

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.

Mathematics is the formal study of axiomatic systems and that which can be derived from them through logical consequence. The results are purely defined by the logic and the axioms. Historically, mathematics evolved from attempts to study the nouminal world. Therefore a lot of our mathematics are well suited to describing this type of reality. However, there is no reason that we have to use those axiomatic frameworks, and it also is difficult to tell whether or not an axiomatic framework truly represents reality.

For instance, I do not need to use a mathematical system that assumes logical consistency. This would allow a statement to be both true and false at the same time. This flexibility may fit reality, in which case reality is a lot harder to understand than we normally think, or it might just be a fun abstract thought process.

So, mathematics is not concrete at all. It is made concrete by the way in which it was constructed and how desire to take mathematical constructs and figure out if and how they can be applied to reality.