Scientific Processes and Social Processes

Abstract

We clarify the notions scientific process and social process with structuralist means. Three questions are formulated, and answered in the structuralistic, set-theoretic framework. What is a scientific process, and a process in science? What can be meant by a non-social process? In which sense a non-social process can be a part of a scientific process in social science? We are specifically interested in social processes. Our answers use the notion of the generalized subset relation applied to set-theoretical structures, and the set of structuralistically reconstructed empirical theories.

Appendix

x is a potential model of collision mechanics ($x \in M_{p}({\bf CCM})$) iff x has the form $\langle P,T,\hbox{I}{\bf R},v,m \rangle$, and (1) P is a set (of particles), (2) T is a set (of points of time), (3) IR is the set of real numbers, (4) $v: P \times T \rightarrow \hbox{I}{\bf R}^{3}$ is a function (the velocity function) and (5) $m: P \rightarrow \hbox{I}{\bf R}$ is a function (the mass function).

Omitting formal details, we define three ECSs τ₁, τ₂, τ₃ for the base sets and auxiliary sets for CCM as follows. τ₁(P, T, IR) = P, τ₂(P, T, IR) = T, and τ₃(P, T, IR) = IR. We define two ECSs for the base relations for CCM as follows. $v \in \tau_{4}(P,T,\, \hbox{I}{\bf R}) = \wp (P \times T \times \hbox{I}{\bf R}^{3})$ and $m \in \tau_{5}(P,T,\hbox{I}{\bf R}) = \wp (P \times T \times \hbox{I}{\bf R})$.

The formal expression $\langle p,t,\alpha_{1},\alpha_{2},\alpha_{3} \rangle \in v$ says that $\langle \alpha_{1},\alpha_{2},\alpha_{3} \rangle$ is the function value of $v: v(p,t) = \langle \alpha_{1},\alpha_{2},\alpha_{3} \rangle$, and $\langle p, \alpha \rangle \in m$ says: m(p) = α.

• D1 Let $x = \langle D_{1},\ldots,D_{m},R_{1},\ldots,R_{n} \rangle \in M_{p}$ be a potential model of T typified by the list $\langle \tau_{1},\ldots,\tau_{m},\tau_{m+1},\ldots,\tau_{m+n} \rangle$ of basic ECSs.

• • a. y is a structure of states in x typified by a state signature $\langle \tau_{1}^{\prime},\ldots,\tau_{t}^{\prime},\tau_{t+1}^{\prime},\ldots,\tau_{t+u}^{\prime} \rangle$ iff there exist $D_{1}^{\prime},\ldots,D_{t}^{\prime},R_{1}^{\prime},\ldots,R_{u}^{\prime}$, there exist an injective function $\xi : \{1,\ldots,t\} \rightarrow \{1,\ldots,m\}$ and there exist an injective function $\zeta : \{1,\ldots,u\} \rightarrow \{1,\ldots,n\}$ such that the following holds:

    • 1. $\forall i \leq t \exists a_{i} (a_{i} \in D_{\xi(i)} \wedge D_{i}^{\prime} = \{ a_i \})$

    • 2. $\forall j \leq u \exists r_{j} (r_{j} \in R_{\zeta(j)} \wedge R_{j}^{\prime} = \{ r_j \})$

    • 3. $\forall j \leq u ( r_{j} \in \tau_{j}^{\prime}(D_{1}^{\prime},\ldots,D_{t}^{\prime}))$

    • 4. $y = \langle D_{1}^{\prime},\ldots,D_{t}^{\prime},R_{1}^{\prime},\ldots,R_{u}^{\prime} \rangle$.

  • b. s is a state in x typified by state signature $\sigma = \langle \tau_{1}^{\prime},\ldots,\tau_{t}^{\prime},\tau_{t+1}^{\prime},\ldots,\tau_{t+u}^{\prime} \rangle$ iff there exists a structure y of states in x typified by σ such that y has the form $\langle U_{1},\ldots,U_{t+u} \rangle$, and for all j ≤ u exists r _j such that

    • 1. U _t+j  = {r _j }, and

    • 2. $s = \langle r_1,\ldots,r_u \rangle$.

  • c. S(x, σ) is the class of states (or the state space) in x typified by σ.

  • d. S(T) is the class of all possible states in T.

• D2

  • a. KP is a kind of process iff there exist S, S _b , S _e and caus such that the following conditions hold:

    • 1. $KP = \langle S,S_{b},S_{e},caus \rangle$

    • 2. S is a non-empty set, (a set of states)

    • 3. $\emptyset \neq S_{b} \subseteq S$ and $\emptyset \neq S_{e} \subseteq S$, (sets of ‘beginning-’ and ‘end-states’ of processes)

    • 4. $caus \subseteq S \times S, (caus(s,s^{\prime})$ means: s is a direct cause of $s^{\prime}$)

    • 5. $\forall s,s^{\prime}(caus(s,s^{\prime}) \rightarrow s \in S_{b} \wedge s^{\prime} \in S_{e})$

    • 6. for all $s,s^{\prime},s_{o}$, if $s,s^{\prime},s_{o}$ are pairwise different, then $caus(s,s^{\prime}) \rightarrow \neg (caus(s,s_{o}) \wedge caus(s_o,s^{\prime}))$

    • 7. $\forall s,s^{\prime}( (s \neq s^{\prime} \wedge caus(s,s^{\prime})) \rightarrow \neg caus(s^{\prime},s))$.

  • b. p is a process of kind $\langle S,S_{b},S_{e},caus \rangle$ iff

    • 1. $\langle S,S_{b},S_{e},caus \rangle$ is a kind of process

    • 2. there exist s _b , s _e such that

      • 2.1. $s_{b} \in S_{b}$ and $s_{e} \in S_{e}$

      • 2.2. $\langle s_{b},s_{e} \rangle \in caus$

      • 2.3. $p = \langle s_{b},s_{e} \rangle$.

• D3 Let $T = \langle \langle M_{p},M,M_{pp},L,\ldots\rangle, A, I \rangle$ be a theory, x a model of M and $\sigma = \langle \tau^{1},\ldots,\tau^{r} \rangle$ a state signature in T.

  • a. p is a theoretical process in x typified by σ iff there are sets S _b , S _e , caus and S(x, σ) and the following holds

    • 1. S(x, σ) is a state space for x typified by σ

    • 2. $\langle S(x,\sigma),S_{b},S_{e},caus \rangle$ is a kind of process

    • 3. p is a process of kind $\langle S(x,\sigma),S_{b},S_{e},caus \rangle$.

  • b. PC(T) is the class of theoretical processes in T iff $PC(T) = \{ p / \exists x \in M \exists \sigma \in E(T)^{*}( p$ is a theoretical process in x typified by σ)}.

• D4 Let T be a theory, $x= \langle D_{1},\ldots,D_{m},R_{1},\ldots,R_{n} \rangle \in M_{p}$ and let $\langle \tau_{1},\ldots,\tau_{m+n} \rangle$ be the list of basic ECSs for T.

  • a. $x^{\prime}$ is a generalized substructure of $x, x^{\prime} \sqsubseteq x$, iff there are $D_{1}^{\prime},\ldots,D_{m}^{\prime}, R_{1}^{\prime},\ldots,R_{n}^{\prime}$ such that

    • 1. $x^{\prime} = \langle D_{1}^{\prime},\ldots,D_{m}^{\prime},R_{1}^{\prime},\ldots,R_{n}^{\prime} \rangle$

    • 2. $\forall i \leq m ( D_{i}^{\prime} \subseteq D_i)$

    • 3. $\forall j \leq n ( R_{j}^{\prime} \subseteq R_{j} \wedge R_{j}^{\prime} \in \tau_{j}(D_{1}^{\prime},\ldots,D_{m}^{\prime}) )$.

  • b. z is a generalized partial model iff there exist $y \in M_{pp}$ such that $z \sqsubseteq y$. The class of generalized partial models is denoted by M ^gen _pp .