Should an undergrad accept that some things don’t make sense, or study the foundation of mathematics to resolve this?

I’m a second year math student. And I’ve the following problem.

When I prepare myself for an exam, I can distinguish two phases. First I’m mainly interested in whatever is necessary to pass the exam. This means that I do not always read the theory too intensely, and I’m more focused on finding the “tricks” that I need to solve the problems.

Second phase: After some time, I feel like I’ve found most of the tricks, and at this point, I usually spend more time in understanding the theory. Just because, I know I will pass the exam, and I think it is fun to understand the theory better.

However, sometimes when I really try to understand the theory, I feel like understanding the foundation of something, doesn’t make me feel I understand it better, it makes me only feel like I never really understood it. And this kind of frustrates me, and I’m not sure if it is even smart to try to understand a concept in that way as an undergraduate student.

For example, I’m studying complex analysis. I skipped the formal definition of complex numbers a little bit, and I just began making calculation and showing if mappings are conformal/holomorphic etc. Now I’m back trying to understand the construction of the complex numbers as pairs of real numbers etc. My brain is getting more and more confused the more I think about it.

My brains goes like: “If you define $\mathbb{C}$ as pairs of real numbers with complex addition and complex multiplication in the usual way. Okay that makes sense, we clearly don’t have $\mathbb{R}$ as a subset of $\mathbb{C}$, but we can construct a bijection $f:\mathbb{R} \to \{(x,y):y=0\}:x\mapsto (x,0)$ that preserves every single property you want real numbers to have. So in that sense there is no difference between $\mathbb{R}$ and $\{(x,y):y=0\}$, and if we would define the real numbers as $\{(x,y):y=0\}$, then we have $\mathbb{R} \subset \mathbb{C}$. “Euphoric” Two seconds later, but wait… now I basicly define $\mathbb{R}$ as $\mathbb{R} \times 0$…. this doesn’t make sense…. “Frustration” (just to give an example, I wouldn’t post the whole list of things about theory of complex analysis that confuses me)

Sometimes I wonder if it there is any use of trying to understand mathematical concepts in this way as an undergraduate student. Should I just accept that it doesn’t totally make sense and go on with solving problems?

Mathematics is really about relations between things. Therefore, while constructions are good and useful, you should never take them very seriously.

Construction agnosticism

Consider the real and complex numbers. Given $$R\mathbb R$$, you can construct a ring which is isomorphic to $$C\mathbb C$$ by taking pairs of real numbers and defining addition and multiplication in the usual way. Note here that I say that you can define a ring isomorphic to $$C\mathbb C$$. We can ask the following question:

• Is the ring we have defined actually the ring of complex numbers $$C\mathbb C$$ itself?

But you shouldn’t ask yourself that question. (Just because you can ask a question, doesn’t mean you should.) It won’t do you any harm; it’s just not useful to.

What really matters? That there is a ring which has all the properties that we want from the complex numbers — such a ring exists. Then we can say: let $$C\mathbb C$$ be such a ring, and then study the maps $$f:C→Cf: \mathbb C \to \mathbb C$$.

When we say “the” complex number field, the word “the” is a red herring; we just want to talk about some ring which works that way. Similarly, we don’t really care about “the” real numbers. If $$C\mathbb C$$ is “the” complex number field, then it contains a principal ideal domain $$ZZ$$ coinciding with the abelian group generated by the multiplicative identity; a quotient field $$QQ$$ induced by $$ZZ$$; and a subfield $$FF$$ which is the analytic completion of $$QQ$$. We can then call these “the” real numbers. Just as we don’t care if “the” complex numbers consist of ordered pairs of objects from some ring $$SS$$, we don’t care if “the” real numbers are the ring $$SS$$ or some set of ordered pairs $$(s,0)(s,0)$$ for $$s∈Ss \in S$$ and $$00$$ the additive identity of $$SS$$. It just doesn’t matter — we just care about the proper subfield $$F⊆CF \subseteq \mathbb C$$ which has all of the same properties as the real numbers, so we may as well adopt the convention that this subfield is the field of real numbers.

This applies to the integers as well. The von Neumann construction of the natural numbers makes $$3={∅,{∅},{∅,{∅}}}3 = \{ \varnothing, \{\varnothing\}, \{\varnothing,\{\varnothing\}\}\}$$. Does this mean that $$33$$ “really is” a set which e.g. has the empty set as a member? Not really, because these ideas are totally irrelevant to what we care about the number $$33$$. We could consider any other “construction” of the natural numbers, in which case $$33$$ might not be a set at all (for instance, if we consider a set theory in which the natural numbers are atoms), in which case it is not only irrelevant to consider the maps $$f:3→3f:3\to3$$, but these would not even be defined. All we care about is that $$33$$ is part of a collection of objects $$N\mathbb N$$ which forms a monoid with some specific properties. The “true identity” of $$33$$ is beside the point.

Mathematical “interfaces” (in place of “foundations”)

If this ambiguity bothers you, you can think of it axiomatically as follows: treat each of the sets we care about — such as $$N\mathbb N$$, $$Z\mathbb Z$$, $$Q\mathbb Q$$, $$R\mathbb R$$, and $$C\mathbb C$$ — as underspecified objects, where we specify all of the properties about them that we could care about, and only those properties.

• von Neumann’s construction of ordinals describes a certain countable well-ordered monoid: we don’t define $$N\mathbb N$$ to be that monoid, but merely say that it is isomorphic to it, leaving further details to be filled in later.

• From equivalence classes of ordered pairs of elements of $$N\mathbb N$$, you can define an ordered ring $$ZZ$$, which contains a monoid $$M≅NM \cong \mathbb N$$ and which is the closure of $$MM$$ under differences. Now, $$N\mathbb N$$ can never be a subset of this set of equivalence classes; but it can be a subset of some other set. We never pretended to characterize precisely what object $$N\mathbb N$$ is, so who is to say that $$N\mathbb N$$ is not itself contained in
a ring which is isomorphic to $$ZZ$$?
Nobody, that’s who; without loss of generality we
may define $$Z\mathbb Z$$ to be a ring isomorphic to $$ZZ$$, and declare as
a refinement of the earlier specification that in fact $$N\mathbb N$$ is
contained in $$Z\mathbb Z$$.

• We may similarly declare that $$Z⊆Q\mathbb Z \subseteq \mathbb Q$$, where $$Q\mathbb Q$$ is isomorphic to the ring of equivalence classes of ordered pairs over $$Z\mathbb Z$$ in the usual way. We also declare that $$Q⊆R\mathbb Q \subseteq \mathbb R$$, where $$R\mathbb R$$ is isomorphic to the field of Dedekind cuts over $$Q\mathbb Q$$, or (equivalently) to the field of equivalence classes of Cauchy sequences, or any of the typical constructions of the real numbers. The set $$R\mathbb R$$ isn’t defined to be any particular one of these constructions, because (a) any of these constructions is as good as the others, and (b) we don’t really care about any of the details lying underneath any of the constructions, so long as the properties we care about hold for each.

You should think of these refinements as axioms which we add during the process of doing mathematics.
A definition is in the first place is only an axiom: one which defines a constant, such as defining ∅ by asserting ∀x:¬(x∈∅). These mathematical underspecifications — mathematical interfaces — are also axioms: having proven that a certain sort of monoid exists satisfying the Peano axioms, we assert that $$N\mathbb N$$ is such a monoid, saying nothing more until it suits us to; and similarly declaring that $$C\mathbb C$$ is a sort of number field of a kind which we’ve proven exists, and which happens to contain the field $$R\mathbb R$$ which we mentioned previously without quite defining it completely.

Fundamentally, this approach to mathematics is not really all that different from what we usually do: it merely substitutes complete descriptions (what Bertrand Russell would call simply “a description”) for objects, with partial descriptions. But pragmatically, in the real world as in mathematics, partial descriptions tend to be all that we care about (and in the real world, they are all that we ever have access to). Embracing this allows you to focus on what really matters.

If you are a mathematical “realist”, in which the real numbers has an identity separate from our descriptions of it and has some fixed location in the mathematical firmament, this sort of wishy-washiness as to the “exact identity” of these objects may bother you. After all, if you imagine the possible identities of the objects $$N\mathbb N$$, $$Z\mathbb Z$$, $$R\mathbb R$$, etc. as you subsume them into more and more complicated objects, it would seem that the set of objects with which a set such as $$N\mathbb N$$ could be identified recedes to infinity as our mathematical framework grows more elaborate. To this I can only say, “so much the worse for realism”. If you want the freedom to construct objects and only concern yourself with the relationships that matter, in the end it is better to abandon this preoccupation with the precise identity of a mathematical object, and engage in mathematics as the creative, descriptive, and above all incomplete and ongoing endeavor that it is.