The Geometry of Meaning
What is the meaning of space? Why does the sky still remind you of "her"?
A very real mathematical proof on space
Previously, I introduced how abstract space can be. Specifically, how your mind can be represented as a vector space. Today, we expand that space.
To all my mathematical readers:
Claim: Space allows for meaning to form
Definitions require structure
Structure requires relationships
Relationships require a space where they can exist
Thus: meaning emerges through relationships held in space
QED*
*Please do not attempt to rigorously (dis)prove this I’m trying to do something okay.
Finding yourself through “set therapy”
Really edging on set theory right now. But we won’t get into the fact that everything is basically a set today.
Today is about relationships and how meaning forms. Yes, I tricked you. We’re going to work on your relationship skills and why you attach meaning to everything.
And to do that we must start at the foundation. The structure. The home that contains everything. We’re going back to your childhood home (therapists love me).
That “childhood home” is your mind. It is the space that contains every thought, belief, and memory about you and your interactions with the outside environment. It’s the safe space that allows you to retreat to when the world becomes too loud.
It is a space only you can access.
A space that others will never be able to know as deeply as you do.
In that mind space, memories form through historical, time-dependent relationships, which then becomes your memory space. And within that memory space is something that starts to resemble identity. And then, oh look, you.
To represent this clearly convoluted, nested relationship, we impose structure. And what is all of this if not a set within a set within a set within a set? A collection of sets.
Computationally, we can represent this hierarchy as a collection of arrays.
import numpy as np
rng = np.random.default_rng()
# all numbers are arbitrary bc everything in life is arbitrary
n_thoughts = 100
thought_dim = 4
# thoughts living rent-free in your mind space
thoughts = rng.normal(size=(n_thoughts, thought_dim))
# now only some thoughts may become memories
memory_indices = rng.choice(n_thoughts, size=40, replace=False)
memory_space = thoughts[memory_indices]
# and only some memories become identitifers
identity_indices = rng.choice(len(memory_space), size=15, replace=False)
identity_space = memory_space[identity_indices]
# you as a compression of your identity
you = identity_space.mean(axis=0)
# your mind as the container of it all
mind_space = [
thoughts,
memory_space,
identity_space,
you
]But all this is just hierarchy! Where is the meaning?
As much as I’d love to be the person who solves the philosophical human crisis of meaning, I, too, am part of the system, and therefore, cannot be decoupled from it.
However, let us try by looking at the relationships between your thoughts.
The hierarchy above showed us how things were contained. But a matrix on the other hand shows us how its contents are related.
And what does a collection of arrays allow us to build? A matrix!
Consider you have 5 thoughts (how insightful of you).
import numpy as np
import re
# your riveting thoughts
thoughts = [
"the sky looks nice today", # T1
"the sky is the same color as when I met her", # T2
"her", # T3
"i need to go buy dog food", # T4
"the sky is blue" # T5
]
# extracting words (tokens) from your thoughts
def tokenize(text):
return set(re.findall(r"\b\w+\b", text.lower()))
tokens = [tokenize(t) for t in thoughts]
# building up your thought-tokens matrix
n = len(thoughts)
T = np.zeros((n, n))
# finding the jaccard similarity between your thought-tokens but in matrix form
for i in range(n):
for j in range(n):
intersection = tokens[i] & tokens[j]
union = tokens[i] | tokens[j]
T[i, j] = len(intersection) / len(union) if union else 0
print(np.round(T, 2))
[[1. 0.15 0. 0. 0.29]
[0.15 1. 0.1 0.06 0.27]
[0. 0.1 1. 0. 0. ]
[0. 0.06 0. 1. 0. ]
[0.29 0.27 0. 0. 1. ]]
# your thoughts-relationship matrix output With 0 being no lexical overlap and 1 being the thought itself, your matrix output shows us how T1, “the sky looks nice today” has the highest Jaccard similarity to T5, “the sky is blue” (0.29). And it was actually T5, “the sky is blue” that had some similarity to T2, “the sky is the same color as when I met her” (0.27).
Technically speaking, Jaccard similarity is more about word similarity rather than semantic analysis which is probably what you were hoping for. Because if you look at T1, “the sky looks nice today” or T5, “the sky is blue” to T3, “her” the similarities are 0 (and yet here you are still emotionally connecting the sky to her).
The subjective intimacy of cognition
Lexical analysis tells us T1 (“the sky looks nice today” ) and T5 (“the sky is blue” ) hold no relationship to T3 (“her”), yet another analysis may suggest otherwise. Because “meaning” is not inherently stored at the surface level of just word overlaps. It’s encoded in the historical connotation underneath them (amongst other things).
While we can continue to play around with your 5 thoughts semantically, that doesn’t really correlate to your own personal memory analysis does it?
Lexical analysis gives us a surface level glance. Semantic analysis may give us some more connotative understanding. But at the same time it’s still not you.
It’s the model’s attempt at understanding something that’s not inherently itself.
The model’s representation of your thoughts is not your own experience of your thoughts.
When we study cognition, many like to take an objective stance on it. Removing the intimacy of what is inherently a subjective phenomenon. In the name of rigor of course. The thousands of cited literature can theorize and explain what cognition is, but it will never be able to tell you what cognition feels like.
Consider this:
You don’t just have thoughts.
You have a space of thoughts.
And when the space changes, everything inside it starts to feel different too.Not because the thoughts themselves necessarily have to change (though yes you can change your mind).
But rather, the relationships between the thoughts did.
When something in life changes—your perspective, your job, your goals, or you meeting “her”—does just that one thing change alone or does everything else slightly shift with it?
That’s your mind space changing.
The main point without the deep dive is that meaning emerges from relationships between representations within a structured space.
However, the sort of meaning you wish to extract is dependent on what analysis you choose to conduct. Ie you have to first intentionally define what meaning is in order to understand it accurately. Otherwise those around you might just lexically assume that you are over her, meanwhile you’re still up at night reminiscing “The Good Times”.
But even then, the analysis will still contain some limits. Because despite how delta-epsilon close your friends try to understand, the inherently subjective nature of cognitive meaning cannot be completely measured objectively. Ie I will never fully understand how much “she” meant to you and why the sky still reminds you of her.
Space is constantly expanding
I’ve been suggesting that a certain “her-vector” may reorganize your mind space. Sounds romantic on the surface, but is actually just linear algebra underneath. Basis change anthropomorphized as a person.
When we analyze a well-defined, bounded space, it gives us the ability to analyze the relationships contained in that space. By further defining the “basis” of analysis, we delve deeper into the relationships contained in that space.
In a sense, the space remains bounded but is “expanded” through the depth of the analyses. Space itself stays fixed. But through understanding the relationships that make up the space, our understanding of that space deepens. So metaphorically speaking, the space isn’t expanding outward but rather inward.
Changing a basis does not change the underlying space.
It changes the representation and understanding of the space.
Nothing changed. And yet everything changed.