We begin by exploring the theoretical significance of mathematics serving as a language by extending the notion of “language”, and then we use the concept of “foreign language learning” and analytic continuation of the Riemann zeta function to define “native speaker of mathematics”.
First of all, let’s provide a broader definition of “language”.
We might as well use the method of defining open sets in point set topology – let 𝓣 be a subset family of set X; if X and the empty set ∅ are in 𝓣, the intersection of any two (i.e., any finitely many) sets is in 𝓣, and the union of any subset family of 𝓣 is in 𝓣, then 𝓣 is called a topology of X. The set X with topology 𝓣 is called a topological space, and the elements (i.e., sets) in 𝓣 are called open sets. Why are they called open sets? Because the previously familiar open sets in metric spaces satisfy these three properties, while general topological spaces without metrics don’t have the notion of “distance” that was used to define original “open sets”. Therefore, ancient mathematicians used these three properties to define open sets in a larger “topology region”. Meanwhile, since open sets of metric spaces do satisfy these three properties, it is also true that open sets of topological spaces defined by these three properties can be validly applied to metric spaces. In this way, if we put all the open sets together in a metric space to form a set family, it certainly satisfies the above definition of “topological space”. Therefore, the concept of “topological space” is a larger space that contains “metric space” as its part.
Traditionally speaking, language serves the content, and we use language as a carrier to express certain meanings or emotions so that certain thoughts can be transmitted through this carrier. Following the above methodology, we call the carrier elements that meet the above properties language in a larger “topology” space, so that this language in the “topology” space also includes the traditional “language” as its part, meaning this definition is a sensible extension – therefore, when mathematical statements have the function of transmitting thinking, mathematics has also become a language, which belongs to the same category as English, Chinese, Spanish, etc.
Now that we’ve known the deeper meaning of “language”, let’s talk about “native speaker”. Suppose Noa’s mother tongue is Spanish, and she is trying to learn English as a second language. Her teachers are likely to convey (or imply) to her the following idea: “English is not your mother tongue, so you do not need to and cannot really understand the true flexible usage of this language; even if you can speak it fluently, you will never reach the true height of the authenticity of native speakers.” This is just like a vertical asymptote of a function, which is always and forever unattainable. Such a concept may even have become an unspoken presupposition for learning a second language. Under the influence of this explicit or implicit point of view, she may also become an unconditional acceptor of this view and pass it on to her students or subsequent generations. However, this idea has unconsciously become a barrier to her English Learning – the barrier is not the language barrier itself, but this kind of concept. With this concept, on one hand, she is more inclined not to integrate English into her life, because this view has regarded English as a simple learning goal rather than a language that really plays a role in daily communication. And at the same time, this goal is unattainable. On the other hand, she may not bother to research a language that is not integrated into her life the way she researches her mother tongue. In other words, when she was a kid, she imperceptibly learned Spanish in the most natural way through life and literature, she did not especially separate apart the listening, speaking, reading, writing, and grammar. However, English as a foreign language was divided into how much time she should spend in a day reciting words, practicing listening, practicing speaking in combination with pronunciation and intonation, learning inversion and verbs, etc., etc. In this way, she always learns English as a third party or layperson, rather than uses English as a confident user. Combining these two aspects, her English proficiency is more likely to be painstaking for many years with little progress. Ironically, this is really in response to the “maxim” that second language learners will never reach the level of their mother tongue, and she will continue to pass on her experience and “maxim”… In fact, as long as she modifies her concept here, she can break that helpless cycle – “I am a native English speaker.” In this way, it is natural for her to learn English as a native speaker of English, integrate English into her own life, and truly enter the “English region” from a layperson.
In order to better understand this conceptual evolution and the meaning of “English zone”, we hereby introduce another concept in mathematics — analytic continuation.
We shall use the Riemann ζ function as an example to better illustrate the application of this concept here. Historically speaking, the Riemann ζ function originated from the exploration of “the reciprocal sum of all natural numbers (from 1 to infinity) to the power of p”, also known as p-series. In order to make the p-series converge (that is, the infinite sum can tend to some number less than infinite), we obtain the value range of p in the complex number field ℂ is its real part being greater than 1; in other words, when p takes any complex number whose real part is greater than 1, it can be guaranteed that the infinite sum is a specific number rather than infinity. The so-called analytic continuation of the Riemann ζ function ζ(s) is to find a new analytic function whose domain contains the domain D (i.e., Re s>1) of the original function ζ(s) and these two functions should agree in D. It is not difficult to prove that the analytic continuation must be unique once it exists (See Property 2.1.7. in my master’s thesis). For simplicity, we also call this unique new function after analytic continuation ζ(s). And this new ζ(s) has its domain extended to the whole complex plane except point s=1.
With this definition illustrated by the example of ζ(s), the new “English region” can now be understood as the region {s: Re s≤1}\{1} after the analytic continuation of ζ(s). Once entering this area, Noa in the example above will really “own” English and become a “native speaker” who can improve English to the level of her mother tongue. Although this actually is based on the definition after the analytic continuation, then what? How can this affect her identity as a “native speaker of English” (the fact of convergence)? Some people are always told that they have blazed an unusual path. As a matter of fact, perhaps what they really did was just change a fundamental idea or concept at the beginning that’s taken for granted and unconditionally accepted by too many people; that’s how they made all the difference down the road.
In conclusion, taking English as an example, we’ve talked about the extension and great significance of “native speakers”. Making a slight transformation yourself by changing all above “English” to “mathematics” (the language in the extended “topological” space), and then reading the previous paragraph again, congratulations, you will independently deduce the definition of “native speakers of mathematics“. That is exactly what this article aims to convey at the end.
Ivan Zhanhu Feng
July 17, 2020
At Professor Garden
Last update: Oct 14, 2022