by Sven Nilsen, 2023
This question came up during a reading group on Hegel, which I unfortunately could not get to. Kent Palmer mentioned to me that my name came up during this reading group. I was going to listen to the recording anyway, because I find these Hegel reading groups like gold mines of ideas. Sometimes, I listen to the same recording multiple times.
“Is mathematics subordinate to philosophy?”
This question can be interpreted in many ways.
First, I would like to approach this question using abstract corruption, because this is a subject of great interest to me.
Abstract corruption is the study of what happens logically when we lie or get corrupted, or similar activities. There are many applications of abstract corruption, for example in Game Theory with partial knowledge, where some players have more power than others.
When I interpret the question in the sense of abstract corruption, I think about philosophers that want to establish dominance in their social hierarchy. By claiming that mathematics is subordinate to philosphy, one is essentially “gaslighting” people who view their work primarily as mathematical in nature and who despise philosophical debate.
By the way, I am not one of them. I enjoy both mathematics and philosophy.
This form of establishing dominance is also found among chimpanzees. It is a very interesting phenomena, where a chimp might e.g. physically hit another chimp who does not dare fight back, to show the other chimps that the one that hits is stronger and better fit to lead the tribe.
It is not always the physically strongest chimp that leads the tribe. Sometimes, the chimp who hits the other chimp, is weaker, but has enough social status that the other chimps will protect the weaker chimp from potential attacks.
Abstract corruption here, is due to the weaker chimp having more power in its social game, despite being vulnerable one-on-one.
So, one can imagine some philosopher, confident in their surrounding tribe of philosophers, making a snarking off-hand remark about the people over in the mathematics department:
“Mathematics is subordinate to philosophy”
However, besides this form of abstract corruption, is there anything more to this question?
The basic problem of using a word like “subordinate” is that it signals a role of mathematics in relation to philosophy.
I believe that 99.9% of mathematics is done simply because people enjoy it. There are very few incentives to do mathematics socially, so enjoyment is my primary hypothesis:
There are also natural obstacles, such as: Not enough time.
However, the same arguments could be made about philosophy!
For example, in many countries today, several people who are intellectual are put under surveillance by the government, to make sure that they do not cause problems. Does it work? No.
Still, with all these obstacles in place, some people still choose to do mathematics and some choose to do philosophy.
Why? I think the answer is: Enjoyment.
There are other things that are enjoyable, such as skydiving. Does it makes sense to say:
“Is skydiving subordinate to philosophy?”
I think not. Therefore, in my opinion, not all enjoyable things are subordinate to philosophy. What makes it reasonable to say “mathematics” instead of “skydiving”?
Most of mathematics is done without the accompanied thought, that it ought or is, subordinate to philosophy. However, philosophy might claim to study enjoyment itself and the unthinking philosophical activity of e.g. mathematics, or even skydiving. This sounds like a proof, right?
Now, this is where I get to the core of the problem, where I approach the question by logical reasoning:
Once we are able to define formally the proposition, how mathematics is subordinate to philosophy, we can use mathematical tools to reason about it. So, the formal proof of subordination must come from mathematics.
In fact, anything we can define formally gets consumed by mathematics. A lot of previous philosophical ideas have turned into mathematics and are applied every day without accompanied philosophical thoughts.
It is like, philosophers constantly move on to new domains where mathematics over time increases influence in the subject of study.
Most of applied mathematics is simply hardware, algorithms and data structures, running unsupervised as computers. There are more computers in the world than people, including all forms, but counting them is a hard problem.
If you take all the books printed about philosophy on top of each other and compare it to a stack of all computers, I would guess that the stack with computers is much higher. However, in digital form, all the books printed about philosophy ever could fit in your palm.
Thinking is slow and hard, which is why we use mathematics to assist human thought. There are very few people who contribute to philosophy and very many machines that do information processing. The economic incentives are aligned to make mathematics more influential over time, because who can afford to hire a philosopher by demand?
I think that mathematics is an important tool in philosophy, but as an activity there is too much mathematics going on (due to enjoyment) or applied mathematics (due to economic incentives) that the word “subordinate” seems ill fitting.
It might be more accurate to phrase it like this:
“Out of the vast mathematical knowledge we have today, only a tiny fraction of it is used in philosophy.”
However, one can object to this statement too, because philosophy draws on knowledge from almost every field of mathematics.
From the perspective of philosophy, it might seem that mathematics is just a useful tool. Philosophy seems, to philosophers, as the “main thing” they do, which makes it easy to believe everybody ought to think that way.
This does not imply that mathematics is “subordinate” to philosophy in particular.
How about this question:
“Is mathematics subordinate to some subset of activities performed by mankind?”
The above question should be true if “Mathematics is subordinate to philosophy” is true.
I believe that mathematics can, from a certain perspective, not be concerned about mankind at all, neither about philosophy, nor any subset of activities we do.
Thinking is very rare in the world and its existence is fragile. Mathematics in the world is neither rare nor fragile, only in the hands of humans.
Mathematics does not care whether it is subordinate or not, because in its form, it is not like the thoughts of chimpanzees and their relatives. It is possible to use mathematics, to grasp some of it with our minds, but it also seems to have a nature on its own. We only get a glimpse of it, not being able to possess it fully and not able to dominate it with our will, at least not entirely.
Mathematics is kind of like a black hole when it comes to ownership. You can always claim to own a black hole, move it around using gravity, extract energy from it etc. The black hole can also destroy you, if you fall into it. So, claiming that a black hole is “subordinate” in some way, implies that you have control over what you are doing.
However, mathematics is also like the idea of black holes in general. Can you claim to own the idea of black holes? I do not believe that makes sense. A person might claim to be the first who thought about black holes in history, but black holes exist out of reach, out of influence, such that the only control we have over it is in the abstract domain.
To me, the idea that mathematics, ought to be or is, subordinate to philosophy, seems to appeal to philosophy as an open-ended process that claims ownership and dominance of abstract thought. It is related to the controversial idea that as soon as we get to have control over something, we can think about it as subordinate and inferior to ourselves by just being a tool.
Is mathematics like that? I am feeling skeptical about this perspective.
Is it possible for philosophy to be subordinate to mathematics? In what sense?