Big Picture for Everyone
The relationship between the equation model and
base vectors for mapping human semantic space.
September 20, 2020
Most humans seem to learn by contextualizing what they perceive to answer questions, like “Why?”,
“How?”, “What?”, “Where?”, “When?”, “Who?”, to help them orientate themselves in their lives. Most
formal problem solving in the modern science is done through the use of equation model, widely recognized
as F(X) = Y, where equality sign “=” is used to separate the terms in search of parameters to satisfy
the equality. Is there a semantic relationship between these basic human questions and the equation model,
and if so, then what kind? We explore this question, and arrive at a qualitative categorical relationship,
where broadly Fis answered by questions “What?”, “Where?”, “When?”, the Xis answered by “Who?”
and “How?”, and the Yis answered by the question “Why?”. The result, while simple, may help us to
deﬁne the base vectors for mapping human semantic space, and to create intuitions on how to convert
human perceptions (“qualia”) and conceptions (“semasia”) into formal mathematical problems.
The equation model F(X) = Y, which describes triplet (F, X, Y )is widely known as functional form of
equation. It is easy to see that with the triplet deﬁned in arbitrary spaces, it is suﬃciently abstract to talk
about any goals Y, with respect to any world F, pursued by any process that varies X.
However, in general, it seem to be hard for most humans and their groups to convert everyday problems
into mathematical ones. The majority of humans seem to be unable to reduce their everyday problems into
mathematical ones, and apply the formulas they’ve learned at their schools to arrive at the solutions of their
problems. That motivated us to seek for a more concrete semantic decomposition of the components of the
equation model into the more concrete concepts widely used by modern humans independent of their personal
backgrounds, to create more of the general consensus in their inter-coordination.
To maximize the applicability of this pan-disciplinary categorical relationship, we aimed to relate the equation
model with ≤8human concepts that describe its components, because that is the upper limit of short-term
memory for most humans, and that way may be helpful for humans to use it for practical purposes.
Note: For a practising mathematician it may occur natural to think of Fas the rule deﬁning relationship,
and to realize, that it is the universe that deﬁnes it, and to think of humanity’s approximation of
the universe’s rule with the totality of our mental models and theories as its statistical estimate
Fthat is their model of the world, and think of Yas goals (set of desired properties deﬁning the
desired states), and think of themselves as probable processes X(t)over time tthat parametrize the
world with actions in ways that the value of F(ˆ
X(t)) approaches the their imagined ˆ
many may still ﬁnd themselves confused as to how do the basic human questions relate to this
interpretation of the world through the concept of equation. We further explore and elucidate this
If we take a look at how do human kids understand the world, we may notice that they ask a lot of questions,
and there seem to be just a few basic question categories that they use to do it. These categories of inquiry are
identiﬁed by some of the basic questions like “Why?”, “How?”, “Who?”, “When?”, etc. When humans
grow up and get entrepreneurial, they try to answer the same questions using methodologies like the so-called
“Lean Canvas”: “Problem”, “Solution”, “Customers”,..., and usually continue asking these questions in
context of other organization frameworks about all sorts of things to help theirselves move in their state spaces
until the end states of their lives.
Initially, we tried to identify the technical concepts that, in our opinion, correlate best with what the equation
model implies, namely, realizing that the equation F(X) = Yhas two parts separated at the equality sign:
one part that speciﬁes the desired condition to satisfy =Y, and another part that describes the state F(X)
dependence on our actions X. From that, it was quite clear to us, that the conditions =Ypart has to
correspond to concepts that humans associate with things that they seek. We named that by the word
“Goals” (because that is what is broadly recognized as formal deﬁnition of the concept of "goal" – a set
of conditions that when satisﬁed, what is to be referred to as “goal” is said to be “achieved”), but later we
realized that humans tend to use a variety of other related (synonymous, inverse, hypernymous, or strongly
otherwise related) concepts to describe or imply goals, such as the concepts referred to by words “Problems”,
“Challenges” , etc., and many others. We realized that the most prominent among the question categories
that people ask when they speak about goals, is “Why?”:
F(X) = Y
Continuing, it was clear that what remains is a decomposition of “Goals” into actions. For example:
#A monkey wanted a banana. (Y - Goal)
→So it had an idea to use a stick to reach it. (Arrow 1)
99K So it had actually taken the actual stick and performed an action. (Arrow 2)
#A human wanted to the Moon. (Y - Goal)
→So it had an idea to use a rocket to reach it. (Arrow 1)
99K So it had actually made the rocket and performed an action. (Arrow 2)
So, we asked ourselves: What parts of the equation do these arrows (→Arrow 1, 99K Arrow 2)
represent, and what human questions answer them? We thought that these have to be the concepts
of an “Idea” and “Plan” (project). There we had our model:
This model is just a set of simple words known to most people, corresponding to more technical words like
intent, principle and action, that we saw in examples of the behavior of living beings, and how they operate
(i.e., how humans and monkeys decompose their intents into actions). This way, it is easy to see that these
concepts (goal, idea, plan) correspond to questions “Why?”, “How?” and “What?” (grammatical accusative
case, or a so-called “object” in subject–object–verb language model).
99K Plan ("What?")
We later identiﬁed other words (synonyms), that people use to answer those questions, such as:
#Goal: “Problem”, “Challenge”, “Question”, “Category”, “Domain”, “Concept”, “Intent”,...
→Idea: “Solution”, “Opportunity”, “Answer”, “Transformation”, “Algorithm”, “Principle”, “Method”,...
99K Plan: “Organization”, “Company”, “Group”, “Delivery”, “Activity”, “Actions”, “Project”,...
On a side note, you can see the reﬂection of that in our current website, as the top titles like “Question”,
“Idea”,“Project” that, communicate these categories in such a way that they feel naturally actionable to
most English speakers – it’s easy to ask a question, suggest an idea, and start a project.
We call the questions or categories like these – “semantic base vectors”, and we just explained 3of them (we
came up with 6in total, described below).
So, when someone asks a question like “Why?” – what they are asking, is for the other party (whom they
are asking) to project whatever they are doing or planning onto the axis deﬁned by the “Why?” dimension
to elucide the answer.
These three vectors were suﬃcient for basic understanding. However, let’s take a look how do they correspond
to parts of the equation model. We had:
F(X) = Y
When mathematicians speak, what they mean by writing equations, is that we write the de-facto circumstances
that we have in the world on one side of equality sign (“=”), and the desired circumstances on the other side:
F(X) = Y
“What we have” “What we want”
That is the semantics of equation as a concept. Pragmatically, to solve an equation, one or the other side is
transformed, until we have the equality not as a condition, but as a fact already.
From here, it is easy to notice the concepts of “Idea” and “Plan” (introduced through examples of a monkey
and a human, moon and banana) in the equation model.
In context of the equation semantics above, the obvious answers to questions “What we have?” and “What we
want” would be:
F(X): “What we have?” is:
→We have the world. (world state 1)
Y: “What we want” is:
→We want the goals. (world state 2)
We already connected the “Y” with concept of “Goal”, so we leave it out and focus on the “F(X)”,
— asking: What are F and X, if the F(X) is the world?
The most natural answer to this question that we’ve got, is that:
→Fis “remaining world without agent (doer, or process) itself”
→Xis “the agent (doer, or process), that is part of the world”
By looking at what human questions identify these categories, we match up:
X : “Who?” (agent)
F : “What?” (world)
However, up above in the introduction of the model, we observed that the concept of “Plan” answers the
question “What?” (we said — grammatical accusative case, or a so-called “object” in “subject–object–verb”
language model). So, how do these mappings relate?
It is easy to see that the so-called “subject” (of “subject–object–verb” language model) corresponds here to
agent (X), and the so-called “object” of the model corresponds to world (F), so that we correctly interpret
that agent acts on world: we say – “agent parametrizes the world, seeking to make it equate to goals”. That is,
so that agents like humans or monkeys parametrize the world (F) with what they are (themselves as world
parameters) in terms of their actions (selves as processes), to achieve their goals (the following future
states of the world).
The fact that both semantic categories of “Plan” and “World” are answers the same question of “What?”
do not contradict: by answering the same question (“What?”) they are both just being part of the world (as
is “agent”), but “agent” has this extra category of “Who?” that is useful to people’s minds to help ﬁgure
out causal social relationships, and that is probably why people have this semantic dimension or question.
In context of mathematics and optimization theory, it is widely known that what we mean by “X” in many
contexts is not just a parameter, but an optimizer, or a process, that searches for parameters (just like agents
do), and we identify analogies:
X : “Who?”, (agent) – “Process”, “subject” – the Human, the Monkey,...
F : “What?” (world), “Plan”, “object” – the Banana, the Moon,...
For clarity: in the context of language model (“subject-verb-object”) – “X” part of the equation corresponds
to “subject+verb” part, while “F” part of the equation corresponds to “object” part. That is true because
we consider agent (X) here as both a part of the world (“subject”), and an actor doing (“verb”).
We go back to ﬁrst insights, and plug in what we have:
#Goal ("Why?") – Y –having "imaginary (desired) Banana in monkey’s mouth" is cool.
→Idea ("How?") – ???
99K Plan ("What?") – part of F –"the Banana in the real world, and steps of getting Banana".
Then, and ask ourselves:
– Where do the steps of getting Banana come from?
– What does the X correspond to as “sub ject+verb”?
– Does X really correspond to Ideas?
With a little bit of abstraction, namely, the concept of “class and instance” (as in object-oriented-programming)
(or the concept of a “random process and its realization” , as in process theory), we realize that an “Idea”
has to be an abstract principle (that emerges as a representation that is a result of mental processes) to be
able to be an “actions generator”, or “plan steps generator” for it all to make sense. Thus we deﬁne
an “Idea” as an abstract prescriptive principle “X”, that can generate actions (and then the actual actions
(like plans and projects), must be the concrete realizations of its process “ ˆ
X(t)”(samples of a random
variable deﬁned by an “Idea”)).
With these abstractions in place, we can indeed say that “X” really correspond to “Idea”, and write:
#Goal ("Why?") – Y
→Idea ("How?") – X
99K Plan ("What?") – part of F – X(t)
Now that we have elucidated the ontological anatomy of the equation model –
F(X) = Y
We ﬁnally have a mapping between mathematical equation model and the basic human questions asked by
However, we said that when human kids understand the world, they also ask the questions like “Who” and
“When?”, one of which is obviously not covered in the among our three semantic base vectors. Do we need
more semantic vectors?
We thought – yes. Namely, because the current “What?” in the “F” part is too broad – it can be pretty much
anything, and from experience, we know that people move in the 3+1-dimensional spacetime, where there
are a couple of very useful coordinate systems and reference systems, that people use to talk about things
as parts of the world, like identities, space and time, that as we know are answers to questions “Who?”,
“Where?”,“When?”, and therefore we include them as the remaining 3dimensions, totalling 6.
You might notice that we arrived to the fact that both “How?” and “Who?” correspond to “X” somehow.
What does it mean? Well, think about it: the “X” as a process is as much an agent as it is an algorithm. So,
“algorithmness” of it is addressed by “How?” and “identityness” of it is addressed by “Who?”.
There, we arrive at our core model:
“Who?”, “Where?”, “When?” DOING “What?”, “How?”, “Why?”
F(X) = Y
where, Fis world, Xis processes, Yis goals, and where F(X)is:
It’s 6questions that answer basic questions about world’s processes, and closely corresponds to and mean-
ingfully decompose common philosophical decompositions of the world, like:
subject – verb – object
substance – entity – essence
world – operations – dreams
(entities – processes – conditions)
Note: In a broader sense, any world is really about “What does what?” rather than “Who does what?”,
because processes are more general than agents – it is suﬃcient to focus on processes, and their input-
output quality to understand the world and its systems, in a similar way how it is suﬃcient to understand
sentences as pairs like (object, subject+verb) and not triples like (subject, verb, object).
Obtaining and retaining simple human-understandability of world’s systems and processes. Transforming
semantic spaces to forms more conducive to human understanding. Embedding of databases into the space
of the semantic vectors by labeling tables, collections and entities with the question categories. Creating
database indexes optimal for human interpretation and application of the equation model for information
retrieval, machine learning, machine reasoning and search for actions to achieve goals. Simple interpretability
to most people:
Latest revision at: https://mindey.com/equation-semantics.pdf.