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Subjectivity in Mathematics and Problems of Defining Objectivity as Opposite to Subjectivity
by Sven Nilsen, 2020
In an episode of Star Trek Voyager, Tom Paris breaks the warp speed limit and told his experience as “existing everywhere and any time in the universe at once”. The idea of objectivity as “uniform subjectivity” is kind of the same, not absence of subjectivity, but an unbiased representation of it. This is him, right before he de-evolves into a prehistoric creature, altering his DNA as a side effect of traveling this fast.
In this blog post I will discuss the philosophy of absolute objectivity and problems it introduces in mathematics. I also introduce “uniform subjectivity” to talk about objectivity in a different sense than opposition to subjectivity.
This idea is based on my experience with working on Avatar Extensions, which has a gradually changed how I think about objectivity vs subjectivity in mathematics.
Disclaimer: I do not go into philosophical literature in this post, instead I emphasize logical argumentation.
Objectivity as opposite to individual subjectivity
From the Wikipedia article on objectivity:
In philosophy, objectivity is the concept of truth independent from individual subjectivity (bias caused by one’s perception, emotions, or imagination).”
From the article on objectivity in the Internet Encyclopedia of Philosophy:
The terms “objectivity” and “subjectivity,” in their modern usage, generally relate to a perceiving subject (normally a person) and a perceived or unperceived object. The object is something that presumably exists independent of the subject’s perception of it. In other words, the object would be there, as it is, even if no subject perceived it. Hence, objectivity is typically associated with ideas such as reality, truth and reliability.
With other words, objectivity is often defined as an opposite, or independence, to individual subjectivity.
There are many uses of the words “subjective” and “objective”, but I will not go into all of them here. The use of “subjective” and “objective” I refer to is more to question the contextual framework we use to think about these terms.
In particular, I oppose the position of defining “objective” in terms of the opposite, or independence, of “subjective”. I find this approach very problematic, because it uses specific human capabilities, or functions like a complementary set. It is not very informative of what we actually mean by something being objective.
To explain what I find lacking in this approach, one can think about the definition of prime numbers: A prime number can be defined as not being a composition number, but when the definition of a composition number is missing, I would be unsatisfied with a such definition. There are three ways of resolving this:
- Provide a definition of prime numbers in terms of lacking divisibility
- Provide a definition of composition numbers in terms of divisibility
- Try to think of numbers that are like or almost primes
Either 1) or 2) would suffice.
The key insight is that a definition should guide the intuition toward concrete examples of the existence of certain mathematical properties, or the lack of thereof.
For example, I see no problems using a working definition of subjectivity as human experience. Within this definition, one can talk about objectivity in some contexts. The problem is when one is trying to grasp objectivity in a much broader sense.
For example, is there a problem that objectivity as opposite to subjectivity produces wrong results? No, not directly. However, it might lead to exclusion of forms of subjectivity that are sound and meaningful:
"The sky is blue" is a statement that is both objective in some sense, but also subjective,
showing that it is not always easy to separate what is objective from what is subjective.
This raises the question: How can we understand the interplay between the objective and subjective in a deeper way?
For elaboration about the scientific subjectivity of “the sky is blue”, see footnote [1].
However, I am unable to provide a clear definition of subjectivity in a much broader sense, and hence a definition of objectivity as an opposite to subjectivity is not very useful. The options 1) and 2) are off the table. With objectivity as “uniform subjectivity”, which I will elaborate on later in this article, I make an attempt at option 3).
I believe an ideal definition of objectivity should be without reference to things that are specific to being a human. Fortunately, one does not need to look very far: The word “bias” is already mentioned in the Wikipedia article.
One might use bias to generalize this approach of defining objectivity.
Objectivity as opposite to bias
Bias can be studied outside the framework of human physiology.
Algorithms can have bias, e.g. a biased 3D renderer, which means it does not converge to the rendering equation. This is not because the algorithm simulates brain activity, or it fails to be “objective” in itself some ways. On the other hand, the bias comes from the mathematical description of the algorithm, which produces results that a computer program can compare to another mathematical description we associate with “objective”, although in a subjective way.
Again, the ideas of objective and subjective are mixed together. The difference is that this time, there is no human in the loop. There are only mathematical descriptions and computer programs.
Thererfore, a computer scientist might find a definition of objectivity, as opposite to bias in general instead of subjectivity, more satisfying.
However, I think this definition too is problematic.
It is very hard to come up with a general definition of bias that is not involving some kind of human-centric point of view.
From the Wikipedia article on bias:
“Bias is a disproportionate weight in favor of or against an idea or thing, usually in a way that is closed-minded, prejudicial, or unfair.”
I would definitively label this definition of bias as human-centric.
This definition of bias seems to sound more biased than what I actually mean when I use the word “bias”.
OK, maybe I can try to use bias in the sense of algorithms. However, instead of working around problems that are specific to the definition of bias, I want to go back to the definition of objectivity. I believe this a problem that will occur in many different approaches and I would rather have a method that works for all of them.
Is it possible to elaborate on a definition of objectivity that is free of any human-centric point of view?
Absolute objectivity
The intuition that people have about something being objectively true, is what they mean when they refer to it as a “fact”.
A fact is something that just is, independent of whether one believes it or not.
Notice that, again, I have failed to eliminate the human-centric point of view, since I included the word “believe”.
However, imagine that I had a magical machine with these properties:
- I can program the machine with any problem and any goal
- The program will produce an answer immediately with a solution, if it exists
I program the machine with the goal “find a non-human-centric definition of objectivity, starting with objectivity being opposite of subjectivity”.
Now, consider the possibility that this machine outputted a solution.
By reading this new definition created by the machine, I might get an understanding of what absolute objectivity is.
The way I imagined how it would feel like to experience absolute objectivity, was as if I were floating in an empty universe, existing everywhere, yet not being aware of anything in that universe, because being aware means I am processing information from a subjective point of view.
This might seem like a contradiction: How can one exist everywhere, yet not experience anything?
As an example, one could point to a chair, say the chair exists, yet it does not experience anything. Hence, the chair is referred to as an object, and this idea generalized is what we think of as “objective”.
Notice that all of a sudden, there is a change from a descriptive notion of objectivity, to an intrinsic notion of objectivity, which only can be described as the absence of experience. With other words, we do not have any concepts to talk about how it is to be like a chair, without explaining it as an absence of human experiences we are familiar with. This shows that we are failing to grasp what absolute objectivity is in itself, because we can only relate to it as an opposite of subjectivity.
Now, consider the possibility that the machine answers “There exists no solution”.
The possibility of absolute objectivity as an impossibility
Since I do not have access to a definition of objectivity that excludes all forms of human-centric point of views, I might consider the consequences of the possibility that a such definition is impossible.
What kind of consequences would this bring e.g. to mathematics?
For example, when I write a computer program that operates on natural numbers, I have a feeling that the program is “objective” in some sense, although I can not explain precisely what I mean by that.
There are two philosophical positions that I can imagine would be relevant in a such situation:
- I assume that absolute objectivity exists, although I am unable to find a definition for it
- I assume that absolute objectivity is impossible
The second position is quite interesting.
When some degree of subjectivity is always necessary
The idea that mathematics might not be free of subjectivity, is thought provoking.
It is also interesting to see whether one can refine the language we use in philosophy, to more accurately address the ideas around subjectivity and what it means.
For example, to develop a theory of consciousness, I could start with a notion of human-level subjectivity. As I reduce the consciousness into processes happening in the brain, I simultaneously reduce the notion of subjectivity to other forms. When I get down to the particle level of physics, I would start to understand something about how subjectivity is related to particles.
It is already common in physics to describe the world seen from a subjective perspective: Coordinate systems.
Coordinate systems and subjectivity
A coordinate system is a way to representing the states of a geometric system relative to a frame of reference.
Mathematically, it means that certain physical states have some nice mathematical properties.
For example, if I stand at the North Pole with a map that uses the North Pole as origo,
my own position would be (0, 0, 0), the same as the North Pole.
The coordinate (0, 0, 0) is related to the subjective experience I have by standing at the North Pole,
but it is less subjective when you think about it as just a point in a coordinate system.
Still, the idea that one can think about certain coordinates as more subjective than others, means that when I choose to represent a physical system with a coordinate system, I am also choosing some degree of subjectivity that follows with the choice of the coordinate system.
From an algorithmic perspective, some descriptions of the world state require less information than other world states.
At meta-level, I could invent a rule “The position of the observer is at origo”.
In principle, I could choose any point as (0, 0, 0), hence creating an observer at any point.
This means, that every point has the same amount of potential subjectivity.
I can use this idea to come up with a weaker notion of objectivity.
Objectivity as uniform subjectivity
In a system where I can not get rid of subjectivity, I might define objectivity as following:
Objectivity as uniform subjectivity, means that the potential of subjectivity is the same everywhere
With other words, I no longer define objectivity as the opposite of subjectivity, but as a balanced subjectivity.
Another way to think about it, is as objectivity being an arena, where subjectivity can take place as an activity. With other words, objectivity is kind of a starting point and subjectivity as choices being made from that starting point. When subjectivity takes place uniformly, it preserves some weaker notion of objectivity.
Absolute objectivity might be thought of as a special case where the potential for subjectivity is zero everywhere. Although this might seem like uniform subjectivity providing a definition of absolute objectivity, there is no new information about absolute objectivity, since zero potential for subjectivity is identical to objectivity as opposite of subjectivity. Likewise, if absolute objectivity is impossible, then as a consequence, uniform subjectivity requires a non-zero potential for subjectivity.
For example, in the presence of human subjectivity, I can talk about objectivity in the sense of having potential, being experienced by any human.
- “The sky is blue”, something that feels like a “fact”, but is subjective to human experience, our sensory perceptions and language faculties of the brain
- “For every day, there is a tomorrow”, something that feels like a “fact”, but is subjective to human semantics of time, missing some edge cases
Notice that I do not exclude any form of subjectivity to claim I am talking about something objectively. Objectivity no longer means the opposite of subjectivity or the absence of it.
When we are talking about objects, as existing in themselves, independent of subjective experience, I feel it is more precise to approach it as a counter-factual to something we start to think about from a subjective point of view. First, we assume that objects have properties that holds for all observers, which is true even if the set of observers is zero. Second, we assume that the properties that objects have in themselves, are independent of any individual observer. Notice that this does not violate the idea of objectivity as uniform subjectivity. Even when talking about a set of zero observers, counter-factually the set could be non-zero. Hence, the potential for subjectivity is still the same. However, when one claims that the properties objects have in themselves are best understood with a zero set of observers, one runs into problems, because removing the observers does not make it easier to understand what objects are in themselves.
For example, when learning about quantum mechanics, some people might believe that the objective description of a particle, is when the particle is unobserved. Other people might believe the objective description of a particle is when the particle is observed. However, to preserve the potential of subjectivity uniformly, one must include the description of the particle when unobserved, observed or partially observed. This is because observation is a potential subjectivity, which can not be increased in some place without reducing it in another place. If the potential subjectivity is the same everywhere, then a description of an objective system must include all possible states of observation/unobservation.
I do not claim that objective existence is impossible, only that the definition of it as an opposition to subjectivity is problematic. When broadening the definition of what objectivity can mean, one can also find that it supports circumstances that includes subjectivity.
Uniform subjectivity and Avatar Extensions
When working on Avatar Extensions, I get a feeling of the mathematics, that I am thinking about some kind of uniform subjectivity.
It means, I no longer feel comfortable with thinking about objectivity as an opposition to subjectivity.
What this means for generalizing mathematics, I imagine as the following example: One might introduce arbitrary amounts of subjectivity, yet keep the mathematics from being unsound, by making the subjectivity uniform.
With other words, it might be possible to make mathematics arbitrary subjective, without losing the weaker notion of objectivity.
This idea of looking at objectivity through a new lens of uniform subjectivity, feels like a conceptual breakthrough for me in the philosophy of mathematics.
Footnotes
[1] “The sky is blue” can be thought as subjective both from a human experience point of view and a scientific view. From a scientific view, the color of the sky depends on it chemical composition, the color of the source of light, and the angle of the light. Measuring the light with an optical spectrometer, the light is “blue” scientifically when it falls within the EM-spectrum that is defined as blue. However, it is not sufficient to determine that there is some light within the blue EM-spectrum. One must also measure for the entire visible EM-spectrum, in case there are other parts of the visible spectrum that have a higher density than the blue EM-spectrum. Indeed, this happens sometimes, for example in a red sunset, a gray cloudy day, or when a duststorm brings yellow particles into the athmosphere, or when aurora borealis shines green in the night sky, or for rainbows within a narrow angle range. Yet, there are some contexts which “the sky is blue” can be treated as objective, even from a scientific view point. The subjectivity in the sentence “the sky is blue” is such of a kind that similar conditions that makes the sky look blue occurs regularly, such as in many places during the day when the sky is clear. This satisfies the notion of objectivity as “uniform subjectivity”.