Mathematics is a Philosophy and not a Science
I will try to show you that mathematics is under philosophy and not under science.
Why does it matter to the community?
There is so much confusion going on where to place mathematics, some people think mathematics is a science, most not even bother. However, it is my belief that to work seriously in something academic we have to know where we are in the tree of human knowledge, in my point of view we are under Philosophy.
In this blog we will define what is mathematics, we will explain why mathematics is not science but a philosophy and in the end, we will argue that whether it should move from the science school to the humanities school.
What is mathematics?
Mathematics is the science of structure, order, and relation that has evolved from elemental practices of counting, measuring, and describing the shapes of objects. It deals with logical reasoning and quantitative calculation, and its development has involved an increasing degree of idealization and abstraction of its subject.
Why it isn't a science?
A Science is the systematic study of the structure and behaviour of the physical and natural world through observation, experimentation, and the testing of theories against the evidence obtained. It involves a pursuit of knowledge covering general truths or the operations of fundamental laws.
Being a mathematician is rarely doing:
- Observation of the natural world
- Experimentation in physical world
- Testing theories against evidence from nature
Let me argue why this is the case.
Observation
We as mathematicians rarely do natural observation, in pure areas it is not even considered (unless you consider exploitation of computational methods as an observation, which to be honest does not really count with the same weight of a scientific observation as these are the ultimate proof whereas in maths these are either for counterexamples or creating intuition) and in applied areas we let that to engineers and scientists; We just model logically what they have gathered either computational or formally.
Experimentation
There is no such thing as an empirical experimentation in mathematics, after all it is not an empirical subject but a deductive one, we use axioms and deduction as justification not experiments.
Testing theories against evidence in nature
We do not test our theories against nature, we test our theories against our axioms, logic and counterexamples, this is simply not standard scientific practice.
The myth of formal science
Formal science is a branch of science that studies abstract structures described by formal systems. (Some even call it mathematical sciences 🍌, this for me is completely off the trolley, and see that institutions such as Oxford University have masters dedicated to this kind of sciences, whilst in practice I think they got it right in allocating an education to these subjects, please change that name to something that I propose in the end, good job nonetheless! Love you Oxford! #penrosebestmathematician).
So what is an abstract structure ?
In formal science, an abstract structure refers to a conceptual framework that is defined by a set of rules or axioms within a formal system. These structures are not tied to any physical objects, but are instead purely theoretical constructs. Some areas that are in this definition are as following:
- Mathematics
- Logic
- Theoretical Computer Science
- Theoretical Physics
- Information Theory
OK, so according to this definition, the study of normative theories of morality is a formal science and therefore let's put the moral department in the next office to the physics department. (Of course I am kidding) This definition is in my opinion nonsense, not because of the “formal” but because of the “science” term. A science is attached to the notion of reality, a formal science needs not by definition. On the other hand, there is no scientific method in mathematics, this is such a big deal.
Being distached from reality in your work is a sign that you are not working in a science.
In this sense, the sociology department or even the economics one have more reason to be nearest to the physics department than the mathematics one.
Pragmatic reasons to be associated with science departments
The necessity of rigour
The greatest reason that maths is so connected to the other science departments has to do with the fact that science uses mathematics heavily to express rigour in their statistical/mathematical modelling and conclusions in their papers and courses. This has created a demand for mathematics departments to be nearby for lectures and scientific collaboration (Note the term scientific collaboration, not mathematical collaboration), therefore in practice maths will be normally nearby to help out others.
Funding
Let’s not compare the funding that science and engineering school have in comparison with the humanities and arts. This has obvious benefits when it comes to do better research and overall well-being of the field, so in a sense mathematicians take the hit for the better.
Why it is a philosophy?
Philosophy, derived from the Ancient Greek term meaning “love of wisdom,” is a systematic study of fundamental questions concerning topics such as existence, reason, knowledge, value, mind, and language. It involves rational and critical inquiry, reflecting on its own methods and assumptions. Philosophy emphasizes clarity, precise arguments, and rigorous reasoning to explore a wide range of questions about the nature of ourselves and the world we live in.
It is clear to me that mathematics is the systematic study of the human reason, where it is involved in rational and critical inquiry, reflecting on its own methods and assumptions. This is a way more natural definition for mathematics (and formal “sciences” more generally) in a sense this is the real home, in fact in old Greece this was the case.
Mathematics in a humanities school?
While theoretically this should be the solution, in practice we need to serve our science/engineering friends and being with a humanities school might prejudice the development of students and research collaboration for paperwork reasons. I would argue nonetheless that we should not be in a science building as well, given that it might confuse students and professionals what their meta role in the development of the field is; For these reasons I propose the following concept.
School of Formal Philosophies
This school would be a school of the so-called (Formal Sciences), however from the arguments that we have presented I would consider it better named (Formal Philosophies). In this way we could be collaborating with our science/engineering friends and in the same way have our own space where feel metaphysically connected.
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