Gestalt Therapy Meets Set Theory

If you’ve been following my blog, you’ve probably noticed my shift towards psychology in the past period. And before that, I spent a lot of time tackling mathematics and proofs, especially set theory.

Gestalt therapy focuses on how awareness organizes figure and ground in contact with the world. Set theory, with its emphasis on partitions and transitions, gives us a playful but precise language to represent these processes.

This blog post will be a conjuction of both ideas.

Formal Definition

For a person $p$ at time $t$ :

$I(p,t)$ = inner environment (thoughts, feelings, sensations, experiences)
$O(p,t)$ = outer environment (people, objects, events)
$E(p,t) = I(p,t) \cup O(p,t)$ , the complete environment

At time $t$ , awareness organizes into figure and ground:

$FG(p,t) = (f(p,t), g(p,t))$

where

$f(p,t) =$ figure (focus of experience) at time $t$ , $f(p,t) \subseteq E(p,t)$
$g(p,t) = E(p,t) \setminus f(p,t)$ , the ground (surrounding context supporting the figure)

with the invariants:

$f(p,t) \cap g(p,t) = \varnothing$ – an element is either in the figure or the ground
$f(p,t) \cup g(p,t) = E(p,t)$ – figure and the ground make up the complete environment

Due to the very dynamic nature, there’s no point in coming up with a fixed set of inference rules; as Perls would say:

I don’t know what the theory will be next because I’m always developing and simplifying what I’m doing more and more.

Cycle of Experience

Gestalt describes a natural cycle: need -> awareness -> contact -> assimilation -> withdrawal. In set theory:

Need emerges, that is, a new element $i_\text{need} \in I(p,t)$ arises: $FG(p,t) = (\{ i_\text{need} \}, E(p,t) \setminus \{ i_\text{need} \})$
Awareness expands. The environment offers an object $o \in O(p,t+\delta t)$ (could also be $o \in I(p,t+\delta t)$ ) to meet the need: $FG(p,t+\delta t) = (\{ i_\text{need}, o \}, E(p,t+\delta t) \setminus \{ i_\text{need}, o \})$
Contact. Attention narrows on the object. $FG(p,t+2\delta t) = (\{ o \}, E(p,t+2\delta t) \setminus \{o\})$
Assimilation. The need dissolves. $i_\text{need} \notin I(p,t+3\delta t)$
Withdrawal / rest. Awareness relaxes back into ground. $FG(p,t+4\delta t) = (\{ \sigma \}, E(p,t+4\delta t))$ , where $\sigma$ is rest/diffuse awareness (non-demanding figure)

And, that is Gestalt in a nutshell.

When I rush, I interrupt this cycle. Presence allows it to complete. Where do you get stuck – ignoring, blocking, rushing?

Contact Disturbances

Disturbances are misalignments between the inner and the outer environments:

Introjection ( $o \in O(p,t) \;\;\Rightarrow\;\; o \in I(p,t)$ ) – taking in outer as inner. For example, adopting others’ rules unconsciously.
Projection ( $i \in I(p,t) \;\;\Rightarrow\;\; i \in O(p,t)$ ) – placing inner into outer. For example, attributing one’s anger to another.
Retroflection ( $i \in I(p,t) \;\;\Rightarrow\;\; i' \in I(p,t)$ ) – turning impulses back onto self. For example, anger turned into self-criticism.
Deflection ( $o \in O(p,t) \;\;\Rightarrow\;\; o' \in O(p,t),\; o' \neq o$ ) – keeping outer at a distance. For example, joking to avoid contact.
Confluence ( $I(p,t) = O(p,t)$ ) – boundaries blur. For example, “we feel exactly the same”.

Examples

Example 1: Natural Figure Shift (Healthy Contact)

Initial configuration:

$I(p,t) = \{i_\text{hunger}, i_\text{curiosity}\}$
$O(p,t) = \{o_\text{food}, o_\text{book}\}$

Situation:

$FG(p,t) = (\{i_\text{curiosity}\}, \{i_\text{hunger},o_\text{food},o_\text{book}\})$
$FG(p,t+\delta t) = (\{i_\text{curiosity}, o_\text{book}\}, \{i_\text{hunger},o_\text{food}\})$
$FG(p,t+2\delta t) = (\{i_\text{hunger}\}, \{i_\text{curiosity},o_\text{food},o_\text{book}\})$
$FG(p,t+3\delta t) = (\{i_\text{hunger}, o_\text{food}\}, \{i_\text{curiosity},o_\text{book}\})$
$FG(p,t+4\delta t) = (\{\sigma\}, \{i_\text{curiosity},o_\text{book},o_\text{food}\})$ with $i_\text{hunger}$ dissolved, i.e. $i_\text{hunger} \notin I(p,t+4\delta t)$

Interpretation: curiosity led to book led to hunger led to food.

Example 2: Projection

Initial configuration:

$I(p,t) = \{i_\text{anger}\}$
$O(p,t) = \{p, p'\}$ (two people)

Situation:

$FG(p,t) = (\{i_\text{anger}\}, \{p,p'\})$ – p notices their anger
$FG(p,t+\delta t) = (\{p'\}, \{i_\text{anger},p\})$ – anger remains in the ground, but awareness attributes it to $p'$ .

Interpretation: anger shifted outward.

Example 3: Internal Dialogue – Talking vs Listening

Initial configuration:

$I(p,t) = \{ i_\text{talk}, i_\text{listen} \}$
$O(p,t) = \{ p, p' \}$ (two people)

Situation:

$FG(p,t) = (\{ i_\text{talk}, p \}, \{ i_\text{listen}, p' \})$ – self-talk dominates; listening and the other person are ground.
$FG(p,t+\delta t) = (\{ i_\text{talk}, i_\text{listen}, p \}, \{ p' \})$ – contact emerges as one starts to listen.
$FG(p,t+2\delta t) = (\{ p' \}, \{ i_\text{talk}, i_\text{listen}, p \})$ – the other person becomes figure; talk/listen step back to supportive ground.

Interpretation: internal dialogue balances talk and listen, letting figure shift for real contact.

Conclusion

Gestalt embraces paradox and ambiguity.

Set theory allows to precisely describe a moment. But, while analysis can help:

In the objective account of human behavior, the individual is portrayed as “an instrument operated by a kind of remote control,” shaped by environment and social pressures, which renders modern man “almost a bystander in his own life” – Perls

Set Theory misses felt sense. It will explain the what and the how, but true understanding comes from experiencing, not mathematical symbols or written words.

Awareness can be drawn as sets, but must be lived as presence.

Growth starts with differentiation — noticing what is mine and what is not, what belongs to now and what belongs to then. It’s becoming aware of the boundaries between experiences, of what stands in figure and what stays in ground. When I sense the tension in my shoulders, the sound of the wind, or the thought in my mind, I’m not analyzing — I’m simply separating one thing from another to see it clearly.

Without this, everything blurs together. Once I differentiate, awareness shapes a clear figure from its ground. Differentiation is how awareness breathes.

Gestalt Therapy Meets Set Theory