Patrick Reany
2 December 2023
I was wanting the answer to a question I had had at the back of my mind for sometime: Is category
theory itself a part of mathematics? Having been informed by BingChat that there is no generally
accepted definition of mathematics, I didn't lose hope for a quick resolution to the question by
engaging ChatGPT with my quest. So, undaunted, I asked ChatGPT what it thought about it, trying
hard not to prejudice its answer. In fact, I tried so hard to not prejudice the answer, that I gave
the LLM the choice of a category of 'premathematics', which I had envisioned to be something akin
to symbolic logic or maybe even set theory -- afterall, without a definition of mathematics, how do
we know that set theory is a division of mathematics. But, it turns out that apparently no one has
ever heard of the discipline of 'premathematics'.
Anyway, if it makes you feel any better, before this essay is over, I will have established not only
that symbolic logic, set theory, and category theory are all parts of mathematics, but so is chess!
I had this conversation with ChatGPT 3.5 on 27 November 2023.
I'll deal with the discussion in two parts. The first is how ChatGPT reacts to my inquiry (is it realistic
or insightful?). Then I'll deal with the definition of mathematics (in a short presentation) so that I
can answer the question myself.
Patrick:
Is category theory mathematics, pre-mathematics, or metamathematics? Justify your answer, please.
ChatGPT:
Category theory is a branch of mathematics, not pre-mathematics or metamathematics. It is a formal
and abstract mathematical theory that deals with the relationships between mathematical structures
and the ways in which these relationships can be preserved. Category theory provides a unifying
framework for various branches of mathematics by abstracting common features and patterns.
Here's a brief justification for why category theory is considered a branch of mathematics:
Formalism and Axioms: Category theory is built on a formal set of axioms and rules, just like other
branches of mathematics. It establishes definitions, theorems, and proofs within its framework.
Mathematical Structures: Category theory studies mathematical structures and their relationships. It
doesn't exist in isolation but applies to various mathematical objects and theories. Categories are
used to describe and relate objects and morphisms in different mathematical domains.
Applications in Mathematics: Category theory has found applications in various branches of mathematics,
such as algebra, topology, and logic. It provides a common language and perspective that allows
mathematicians to express and understand mathematical concepts more broadly.
While category theory has connections to logic and foundations of mathematics, it is not considered
metamathematics. Metamathematics typically deals with the study of mathematical theories, their
structure, and their properties. Category theory, on the other hand, is more concerned with the
relationships between different mathematical structures and the abstraction of these relationships.
In summary, category theory is firmly situated within the realm of mathematics and serves as a
powerful tool for understanding and connecting diverse mathematical concepts and structures.
Patrick editorial: It seemed to me that ChatGPT has given me an enthymeme, that is, an argument
with a hidden or unstated assumption, that being the definition of mathematics. Now, I had honestly
thought that by now the mathmematics community had a respectible definition of mathematics, but it
seems that I am wrong on that. Ultimately, how does one decide if category theory is mathematics
if you don't have a definition of mathematics?
Patrick:
I'm trying to proffer a definition of mathematics that will settle this question I posed. Classical
mathematics seemed to care a lot about number and geometry, which category theory seems not much
interested in, per se. On the other hand, if I define mathematics too broadly, it will include formal logic
as well. (Would that be a good thing or a bad thing?) So, my definition that would justify calling category
theory as mathematics is given as: Mathematics is the study of objects and methods on them that lend
themselves to conjectures and formal proofs.
Patrick editorial: When I used the term 'methods' I was thinking by analogy to object-orientedChatGPT:
programming. If you prefer, you can think of it as transformations, relationships, or operations, etc.
Classical Mathematics:
Classical mathematics, with its focus on number theory and geometry, indeed involves the study of
objects (numbers, geometric shapes) and methods (algebraic operations, geometric constructions).
Conjectures and formal proofs are integral to classical mathematical reasoning. Mathematicians formulate
conjectures based on observations and then provide rigorous proofs to establish the truth of these conjectures.
Category Theory:
Category theory fits well into your proposed definition. While it may not directly focus on traditional
objects like numbers or geometric shapes, it deals with abstract structures and their relationships,
providing a framework for expressing and proving mathematical ideas.
Category theory allows for the formulation of conjectures and the development of formal proofs,
making it a legitimate part of the study of mathematics under your definition.
Formal Logic:
Formal logic can also be considered part of mathematics under your definition. It involves the study of
logical systems, propositions, and the rules of inference.
Conjectures in logic may involve the consistency of logical systems or the provability of certain statements,
and formal proofs are essential for establishing logical validity.
In summary, your definition appears to encompass a broad range of mathematical activity, including
classical mathematics, category theory, and formal logic. It emphasizes the common thread of studying
objects and the methods applied to them, with an emphasis on the formulation of conjectures and the
development of formal proofs. This definition is sufficiently inclusive without being overly restrictive
or overly broad, and it allows for the recognition of diverse areas within the mathematical landscape.
So, not only did ChatGPT accept my working definition of mathematics as plausible, it gave me back
an essay to show that if one accepts the definition, then one can fit the three topics of classical
mathematics, category theory, and Formal (symbolic) logic within it.
Let's look one more time at my vague definition: Mathematics is the study of objects and methods on
them that lend themselves to conjectures and formal proofs. It took me less than a minute to craft this
definition. It's vague for two reasons: 1) it had to be very general to be sure to include category
theory, and 2) I knew the drawback of being too concrete, which is to leave out subjects I would want
included.
So, I agree with ChatGPT with the reasons it gave that classical mathematics, category theory, and
symbolic logic would be included in mathematics by my definition. And for similar reasons, it must
include set theory. But ChatGPT did not spend much time to analyze whether or not it included
controversial claims, one being that chess is mathematics. Think about it. Chess pieces are its objects.
They are governed by chess rules, which are its methods. One can make chess conjectures that can be
tested by formal proofs, such as the declaration of a mate in 6 moves. If it's true, it can be
proved true by demonstration.
I'm not the only one to wonder if chess is mathematical. See,
https://chess.stackexchange.com/questions/27259/has-anyone-attempted-to-characterize-chess-mathematically
Under the title "Has anyone attempted to characterize chess mathematically?", postoronnim claims that
chess has "completeness and elegance that are found in mathematical proofs and theories." I would agree
that chess has the qualities of completeness and elegance, but what do those qualities have to do with
mathematics, per se? The implication is that mathematics is also about "completeness and elegance",
but so what? I don't think that those descriptions bear on the essence of mathematics. They help to
characterize both mathematics and chess, but they don't define either one of them. We must be careful
not to substitute a partial definition (or characterization) for an actual definition.
Personally, I don't tend to think of chess as mathematics, but my definition seems to include it.
However, my tendency on how to regard chess is subjective. I see no objective reason not to think
of chess as a form of mathematics, and those who play it as engaging in recreational mathematics,
perhaps. My own feeling on this is that chess is more like symbolic logic in that they both are more
about conjecture and proof, rather than about number or geometry. Sure, one can use mathematics to
analyze chess, such as by geometry, statistics, and probability, but none of those analyses really bear
on the question, Is chess a form of mathematics.
Also, despite some controversy about whether statistics is mathematics, my definition seems to
include it.
ChatGPT concluded that I had hit some sweet point, where the definition is neither too broad nor
too restrictive. For my part, I think that the definition is a little too broad, though I don't
know how to fix it.
Okay, so what makes for a good definition? Let X be some thing or concept that we're trying to define.
Let x be any thing or concept. A good definition is suppose to create a set X by which we can test whether
for arbitrary thing x, that x is an element of X or not. One way to do this is to just list or enumerate
the elements of X. For classical mathematics, we had only two elements: number theory and geometry.
But it would be unthinkable to define mathematics by enumeration today. Instead, we need a rule-based
definition for that. The definition I gave fits the bill. But in general, a good definition should tell us
enough true things about X (hopefully concisely) that the set is well-defined, meaning that we can decide
all the things that are in the set and all the things that are not in the set.
A lot of clever mathematicians have made a lot of clever and beautiful claims about Mathematics. Some
of these are merely poetry, such as mathematics being the queen of the sciences. Most of the others are
characterizations of mathematics. A characterization will say something true of X, but will
leave out enough truths about X so that the end result falls short of a proper definition. For example,
I can claim that logical deduction is a tool of rational thought, which is true (so far as I'm concerned).
But that claim doesn't go far enough to define rational thought.
Now we return to the question of whether statistics is mathematics. My definition seems to include it.
For one reference, see:
https://stats.stackexchange.com/questions/136862/is-statistics-mathematics
What are the objects of statistics? Statistical measures, such as averages, means, modes, standard
deviations, etc. What are its methods? Algebra, extraction of roots, integral calculus, etc. Can one form
conjectures with these and prove them formally? Yes.
Here's one.
Joe wants to get an A in his biology class. To do so, he needs at least a 90% average over ten exams on
the semester. (All exams contribute equally to the average.) He currently has 89% average over 9 exams.
The minimum score he needs to get on his tenth exam to get at least a 90% average on ten exams is 99%.
To prove this, just do the algebra.
I suppose that some people might prefer to consider that the data are the objects of statistics, but
I prefer to go one step higher. If one looks at the listings of the major theorems of statistics/probability
online, the data is suppressed.
On 1 December 2023, I asked ChatGPT to help me out, and it did.
Patrick:
I need an example of a purely statistical theorem to show that statistical theorems stress statistical
quantities and their algebraic relationships, rather than mere statistical data.
ChatGPT:
Certainly! One classic example of a statistical theorem that emphasizes statistical quantities and their
algebraic relationships is the Central Limit Theorem (CLT).
The Central Limit Theorem states that the sum (or average) of a large number of independent and
identically distributed random variables, no matter the shape of their original distribution, will be
approximately normally distributed. This theorem is crucial in statistics because it allows practitioners
to make inferences about population parameters even when the underlying distribution of the population
may be unknown or non-normal.
Mathematically, if X1, X2,...,Xn
are independent and identically distributed random variables with
mean μ and standard deviation σ, then the distribution of the sample mean X‾
approaches a normal
distribution with mean μ and standard deviation σ / √ n
as n (the sample size) becomes large.
This theorem highlights the importance of statistical quantities (mean and standard deviation) and their
algebraic relationships (standard deviation of the sample mean) in understanding the behavior of sample
statistics. It demonstrates that the shape of the distribution of the sample mean is determined by these
parameters, rather than the specific details of the original distribution of the individual random variables.
Conclusion:
One advantage of defining a thing by enumeration is that you don't have to think very hard. I suppose that if
Euclid had been asked why mathematics is number theory and geometry, he could have replied, "Just because."
On the other hand, defining a set by the common characteristics shared by the elements of the set takes away
some of the apparent arbitrariness of the definition that can appear in a thing defined by enumeration. Also,
a drawback of definition by enumeration is that to include another element to the set, the set (the definition)
must be modified accordingly. However, new elements can be added implicitly to a set defined by a rule or rules
automatically (without changing the definition), and that's how we got chess added in among the subjects of
mathematics.
But is my definition any good? Well, I think it's way better than no working definition. It ignores the
philosophical/ontological problems that seem to have frustrated some other definitions being generally
accepted. Since I'm not a philosopher of mathematics, I won't let those matters stop me from positing
some practical definition.