AToS conclusions
I. Basic conclusions of AToS
Self-awareness can be defined without reference to biology, the soul, emotions, or language.
Example: a human, an AI, or another system can be analysed using the same apparatus.
Proof: mathematical — the AToS Representation Theorem.
A system is minimally self-aware if it recognises a change in its own state and can recognise that the cause of at least one such change originates within itself.
Example: the system recognises: “I changed my state, and I was the source of that change.”
Proof: mathematical — proved in AToS as a necessary and sufficient condition.
Reacting to the external world is not sufficient for self-awareness.
Example: a system may see, hear, answer, and react without recognising itself as the source of its own changes.
Proof: mathematical — external awareness does not imply self-awareness. AToS theory.
Awareness is weaker than self-awareness.
Example: “I know that a change occurred” ≠ “I know that I caused it.”
Proof: mathematical — AToS separates recognition of a change from recognition of its cause.
The “self” does not have to be stored in the system as an object.
Example: the system does not need a SELF variable to represent itself.
Proof: formal — in AToS, the self appears as a causal role: the system represents itself as the source of its own change.
Self-awareness requires neither language nor a representation of one's own body.
Example: a system need not be able to say “I am” to satisfy the minimal AToS criterion.
Proof: structural — a minimal causal model of the self does not require linguistic or bodily representation.
Self-awareness is not simply a consequence of a system's complexity.
Example: a highly complex system may not be self-aware, while a much simpler one may satisfy the minimal AToS criterion.
Proof: structural — what matters is the ability to recognise one's own change and attribute it to an internal cause, rather than complexity alone. Darek showed an example of the simplest self-aware system consisting of two bits.
Intelligence is a process, not a store of information.
Example: billions of stored facts do not yet constitute intelligence.
Proof: mathematical — Intelligence in AToS arises from the operational composition of recognised state transformations.
Memory cannot be a stimulus.
Example: a stored fact does not, by itself, initiate a thinking process in the formal sense of AToS.
Proof: mathematical — Theorem 8.5.
Memory cannot be a generator of intelligence.
Example: a database can supply stable information, but the database itself does not “think”.
Proof: mathematical — Theorem 8.6 and Remark 8.7.
A system's level of intelligence can be specified mathematically.
Example: a system capable of longer valid operational chains has a different intelligence level in the AToS sense.
Proof: formal — Intelligence Level is the supremum of the lengths of admissible finite operational compositions.
The level of self-awareness can be specified mathematically.
Example: two systems may both be self-aware but recognise different numbers of their own internally generated changes.
Proof: formal — Self-Awareness Level in AToS.
Self-awareness can be a momentary event or an ongoing process.
Example: “I know that I am thinking” may give rise to another thought about one's own thinking.
Proof: mathematical — Self-Conscious Retention Process.
The simplest form of intelligence may be the maintenance of one's own self-awareness process.
Example: one recognised internal change causes the next, which causes another, and so on.
Proof: formal — AToS identifies the Self-Conscious Retention Process as the simplest nontrivial form of intelligence within the adopted formalism.
The same mechanism can function as “imagination” on one occasion and as a reaction to the external world on another.
Example: an identical state transformation may be initiated internally or externally.
Proof: formal — the Intuition Function belongs simultaneously to the classes of internal and external functions.
Whether a stimulus is internal or external is not determined by its physical location.
Example: a timer, noise, or a hardware process can play an internal role if the system embeds it that way in its causal structure.
Proof: formal — AToS defines internality operationally, not spatially.
Several self-aware systems can form a higher-level self-aware system.
Example: a family of interacting AToS systems. Any set of people (Poles) is a self-aware system.
Proof: mathematical — Theorem 11.2 specifies the conditions under which a family forms a Self-Conscious System.
A stimulus can be external to one system and internal to a composite system at the same time.
Example: a message from system A to system B is an external stimulus for B but an internal event for A+B.
Proof: formal — Family of Self-Conscious Systems.
The Turing test is not a test of self-awareness.
Example: a system may behave like a human without being self-aware in the AToS sense; it may also satisfy the AToS criterion without human-like behaviour.
Proof: logical — AToS shows the logical independence of a behavioural criterion and internal causal attribution — the GSAT test.
Correct answers do not prove self-awareness.
Example: a Chinese Room can manipulate symbols and generate correct answers.
Proof: structural argument — without recognising its own changes and their internal cause, the system does not satisfy AToS.
An operational test of self-awareness — GSAT — can be constructed.
Example: recognised external stimuli are cut off. The system causes a change in its own state and correctly recognises its source as internal.
Proof: the General Self-Awareness Test and its formal realisation in AToS.
The mere fact that a system changes spontaneously does not prove self-awareness.
Example: a clock, scheduler, or random noise may produce changes.
Proof: GSAT requires not only a change but also a mechanism that correctly attributes its cause.
II. New mathematical and topological conclusions
The boundary of a self-aware system is where internal and external stimuli meet.
Example: the human head.
Proof: to be developed — we want to connect the topological construction with empirical studies of facial innervation.
Every self-aware system is a topological space.
Example: an AToS system.
Proof: mathematical — by constructing a topology on the system's set of states. In my head.
A self-aware system can be defined on any topological space.
Proof: it is enough to colour the open sets and define arbitrary Imagination Functions on them.
For any family of self-aware systems, a self-aware system composed of its elements can be constructed.
Example: interacting AToS systems.
Proof: mathematical — the present article already formalises the family as a meta-system and gives a condition for its self-awareness; the version with the condition of “the same external stimulus” must be added as a separate theorem, because that exact formulation does not occur in AToS. Proof in my head.
There exists a self-aware system with infinite intelligence.
Example: not empirically verifiable.
Proof: mathematical — it is enough to define a self-aware system on the complement of the Cantor set, and that complement is a topological space; therefore, by the preceding theorem, a self-aware system can be built on it, where we define a simple Imagination Function on the Cantor set 😎
There exists a self-aware system that can be divided into between two and five identical self-aware systems.
Example: not empirically verifiable.
Proof: mathematical — Stefan Banach and Alfred Tarski (Poles) proved that such a set exists, so by colouring every point of that set with two colours and defining a simple Imagination Function on each such set, we obtain the claimed proof 😎
III. Mathematics, artificial intelligence, and existence
Mathematics was not invented by humans. More than that: mathematics was not invented by any self-aware system. Humans can only discover mathematical relations.
Proof: since human self-awareness can be described mathematically (AToS), and every true theorem is always true, including in the past, a system described by AToS could not have invented it, because it already existed and created AToS theory.
Artificial intelligence systems constructed to date cannot, in their present form, be self-aware.
Proof: in previously published posts.
Something cannot exist without also existing mathematically.
Proof: proved on the proverbial piece of toilet paper.
One need not understand the axiomatic theory of self-awareness to be self-aware.
Proof: the existence of the readers of this post.
Thesis: everything in the universe is self-aware.
Proof: it is already there!