4.2 Abstraction of subject matter
4.2.1 A method of abstraction
The method described in (Floridi, 2013) involves a form of ‘levelism’ that is in- tended to be the main working method in the philosophy of information. It has broad applications to different fields of study. As Floridi’s method encompasses various other methods and it supports an epistemological approach, it generally suits our purposes here. However, the role of the method in the philosophy of information is rather different from its role in the philosophy of mathematics: in particular, we will see that the notions of ‘level of abstraction’ in the two fields do not always coincide. Thus, we make our own additional requirement to the definition later on. For now, note that levels of abstraction should be considered as “levels of observation or interpretation of a system”. This comes with the idea that such a level should not be considered independently, without a purpose or context. Surely, we can imagine that each foundational system for mathematics is created with a purpose in mind; the relation of levels of abstraction to this will come back in the last part of our argument. Consider, first, the following three-part definition adapted from (Floridi, 2013) that captures the notion of a level of abstraction.
Definition 7. (a) xis a variable of typeX (writtenx: X) ifxis a uniquely defined conceptual entity, andX is a set that comprises all values thatx
may take on.
(b) A typed variablex: X together with a statementαthat clarifies the fea- ture of the relevant system thatxrepresents (i.e. an ‘interpreted’ typed variable), is called anobservable.
(c) Lis alevel of abstraction (LoA)if it is a finite and non-empty set of observ- ables.
Thus, a LoA1 is essentially a set of ‘conceptual entities’, each capturing an
aspect of the system at hand. A distinction is made betweendiscreteobservables (observables whose type is a finite set) andanalogue observables (otherwise). Whereas most of the types that we will define will be analogue, we need to be slightly careful with their characterization as sets. Namely, we will often be ranging over set-theoretical or categorical objects in the metatheory, so that the type for us then comprises an externalcollection. Furthermore, thebehaviourof a system at a particular LoA is given by a predicate, that takes observables as values. Any instantiation of types for the observables that the predicate makes true is called a system behaviour. A LoA together with a behaviour is called a moderatedLoA.
1We write ‘a LoA’ with the pronunciation of LoA as the non-existent wordloain mind, instead of ‘an LoA’, which presumes the separate pronunciation of the letters ‘l’, ‘o’ and ‘a’.
Some examples. In order to provide a basic intuition for the concepts just for- malized, we provide some examples used by (Floridi, 2013). Suppose that our object of study is a human: then we could define a variable h (representing height) of typeRand interpreted by the unit of metres (making it an observ- able). The behaviour of the system is then given by the predicate0ăhă3, as
the length of humans is clearly bounded above and below. Alternatively, if we are evaluating wine, we could have observables forcolour,clarity,alcohol level, price, and so on, each with its own type. In this case, different LoAs will consist of the observables that suit our purpose for the wine: for example, the price of the wine is important if we would like to purchase a wine, but the colour of a wine is more important for the purpose of tasting it. Of course, observables between such LoAs could overlap. There are many more examples, but it should be clear that there is a certain level of freedom in the implementation of LoAs with respect to the analyzed system. This is part of why the method is so widely applicable, but it also requires us to justify the way we choose to apply it to mathematics later on.
The last relevant notion that we adopt from (Floridi, 2013) allows for the con- nection of different LoAs by means of a relation. Arelationbetween setsAand
Cis simply taken to be a subset ofAˆC. IfAhas a predicatepfor its observ- able,Rrelates it to the predicatePRppqonCthat holds just at thosec:Cthat are related byRto somea:Asatisfyingp. With this, a system can be discussed at various LoAs, as follows.
Definition 8. Agradient of abstractions (GoA)consists of the following. 1. A finite set of moderated LoAsLi(iďn).
2. A family of relationsRi,j Ď Li ˆLj (0 ď i ‰ j ă n). The family of relationsRi,jrelates the observables from each pairLi, Ljof distinct LoAs such that:
(a) Fori‰j,Ri,jis the reverse ofRj,i.
(b) The behaviour pj at Lj is at least as strong as the translated be- haviourPRi,jppiq, i.e.pjimpliesPRi,jppiq.
3. For each interpreted typex : X, y : Y inLi, Lj (respectively) such that
px:X, y:Yq PRi,j, a relationRxyĂXˆY.
Thus, a GoA establishes an explicit connection between the observables at multiple LoAs, allowing one to unambiguously reveal the way aspects of a sys- tem correspond to each other. Although this method acquires elegance from its simplicity, we cannot completely justify this way of relating LoAs for our pur- poses. The next section will therefore elaborate on the way in which we will apply the method to foundational systems. Note that (Van Leeuwen, 2014) ar- gues that, additionally,annotations(i.e. meta-data) should be added to a LoA in order to be able to express the “micro-structure” that exists between successive LoAs. This is meant to refine the gradient between such LoAs. The annotations should “describe how observables of an LoA are constrained or otherwise to be used, as a guide to a deeper insight or capability at this level”. We will not ex- plicitly use the method of annotations here, as we will provide our formalization with enough explanation that it will not give us any other benefits. This seems more relevant for looking at more concrete systems from different perspectives.