Mathematical Emotions: Ambivalence

Notebook LM talks about this blog. 20250718

"Partly happy because it is the same as it ever was, and partly sad because it is the same as it ever was."Notebook LM talks about this blog.

An intriguing phrase. A mathematical model?
How would you? WHY would you? 

Curiosity: 

Binary State Model Define two emotional states: 

H : Happiness, triggered by “it is the same as it ever was.” 

S : Sadness, triggered by the same phrase. 

Let X be the state of the world (“it is the same as it ever was”). 

Then: P ( H | X ) > 0 P ( S | X ) > 0 

The probability of being happy or sad, given the state X, are both nonzero. 

In short: Modeled mathematically, this phrase is a beautiful example of how the same input can yield multiple, even contradictory, outputs—a concept that appears in probability, logic, psychology, and philosophy. 

"Ambivalence" is most commonly used in psychology and everyday language to describe mixed feelings, the mathematical modeling of ambivalence appears in several fields, often under different names or formalizations. 

Here are some ways this concept is represented mathematically: 

Probability & Decision Theory 

Mixed States/Probabilistic Preferences: 

In decision theory, agents may have probabilistic (not deterministic) preferences, reflecting uncertainty or ambivalence between choices. 

For example, a probability vector P = ( p 1 , p 2 ) where neither p 1 nor p 2 is 1 or 0 models ambivalence between options. 

Fuzzy Logic: 

Rather than binary true/false, elements can partially belong to sets (e.g., "happy" at 0.7, "sad" at 0.4), directly modeling ambivalent or overlapping states. 

Quantum Mechanics Superposition: 

A quantum state can be in a superposition of basis states (e.g., both "on" and "off"), analogous to being in two emotional states at once. 

Psychometrics & Social Sciences Emotional Response Function: 

Define an emotional response function 

E ( X ) : E ( X ) = α ⋅ H ( X ) + β ⋅ S ( X ) 

where 

H ( X ) and S ( X ) 

are the “happiness” and “sadness” functions triggered by X , and α , β ∈ [ 0 , 1 ] are the weights for how strongly you feel each. 

For your phrase: X = “the same as it ever was” 

Both H ( X ) > 0 and S ( X ) > 0 

So your emotional state is a mixture, not pure happiness or sadness. 

Ambivalence as Overlap 

Let’s define the set of triggers:

A is for happiness, and B is for sadness: 

X ∈ A ∩ B

 

That is, “the same as it ever was” is in both sets, so it causes ambivalence. 

Temporal Model:

 

Suppose happiness and sadness are functions of time and change: 

Let f ( t ) be the state at time t . If f ( t ) = f ( t − 1 ) for all t , then the world hasn’t changed. 

Define: 

H ( f ( t ) − f ( t − 1 ) ) = h 1 if f ( t ) = f ( t − 1 ) (happy for stability) 

S ( f ( t ) − f ( t − 1 ) ) = s 1 if f ( t ) = f ( t − 1 ) (sad for lack of change) 

Both can be nonzero for the same condition. 

You feel both emotions simultaneously for the same reason. 

This is going into the Science Fair software in the next drop.