An Incredibly Short Tale About Human Imagination

By Dr Mridu Prabal Goswami (Indian Statistical Institute, North-East Centre, Tezpur)

Author's Note: The views expressed in this piece are personal. The author
is extremely thankful to Mousumi Kalita Sachdeva and Abinash Panda.

Author's Preface: This piece is only a snapshot of the evolution and the articulation of human imagination that uses the language of Mathematics as the medium of expression. Being an outsider, i.e., not having any formal degree in Mathematics, is a serious limitation of the author. The emotional conflict that ensues from this limitation about whether to write this piece remains unresolved. Naturally, this piece does not intend to teach Mathematics. This piece only carries a personal wish that developers of this language feel what they do is so incredibly beautiful that even an outsider can appreciate it.

Yuval Noah Harari, the author of 'Sapiens A Brief History of Humankind', says language has made homo sapiens what it is now (see Araki. N. (2019): Sapiens and Language, Bull. Hiroshima Inst. Tech. Research, 53, 1-10). The tool of language has enabled Homines sapientes to express the belief that Homines sapientes are a space-faring species. A tiger however strong physically it is, may not have the language to express the wish "tigers were a space-faring species'' (The author does not intend to disrespect tigers and or any other animal species. They are mentioned only to bring out the juxtaposition of two species). Homines sapientes have formed, articulated, and long started their efforts to actualize such wishes. The language of Mathematics is the language that will be used to design ships that may enable the homines sapientes to visit far-off places. Two of the most important rules of this language are the rules of calculus and numbers. The following caveat is noteworthy: tigers may have the ability to hide technology from humans. However, an offended tiger may roar in displeasure and assert that tigers also use calculus and numbers to optimize the numbers and distances of possible prey from their locations during their hunt. While this maybe true, homines sapientes have written down the rules of calculus and numbers, which tigers are welcome to verify if they can and have the time to do so. If the tigers have also written down their calculus, then maybe it is advisable to make their research open access so that other species can also access the benefits. Since the author is not familiar with any journal, book, or lecture video on Mathematics by the tigers or any other species, the author shall limit their attention to Mathematics by homines sapientes. However, the author shall make occasional references to the tigers so that the conversation with another species is not lost.

It maybe feasible to start a discussion of the history of human imagination when the wheel was invented. The circular geometry of this object might have helped a generation of human ancestors to transport goods in their carts quite easily compared to the generations before who did not have access to the benefits of this circular geometry. Due to the incapability of the author to go that far back in time and talk about human imagination, this tale starts with the invention of what is currently known as modern calculus (The author does not have any opinion about whether Mathematics is discovered or invented. The choice of the word `invented' is the outcome of a random experiment.)

Imagine a function \(f:{1,2}\rightarrow \mathbb{Q}\) defined by \(f(1)=5,f(2)=10\), where \(\mathbb{Q}\) denotes the set of rational numbers. Further, let \({1,2}\) be not some arbitrary set of symbols, and instead
\({1,2}\subseteq \mathbb{Q}\). Let us ask an undefined question, "Is \(f\) continuous?''. We can change the status of the question from undefined to defined by defining a distance metric on \(\mathbb{Q}\). Let the distance between \(x,y \in \mathbb{Q}\), be \(d(x,y)=|x-y|\). Now we can apply the standard definition of continuity of a function for metric spaces that uses convergent sequences.
Thus, we need to check if \(x_{n}\rightarrow x\), where \(x_n\in {1,2}\) for all \(n\), and \(x\in {1,2}\), then \(f(x_{n})\rightarrow f(x)\), where \(f(x_{n})\in \mathbb{Q}\) for all \(n\), and \(f(x)\in \mathbb{Q}\). Since the only sequences \(\{x_{n}\}_{n=1}^{\infty}\) that converge in \({1,2}\) are the sequences for which either \(x_n=1\) after some \(n\) or \(x_n=2\) after some \(n\), the only sequences that are relevant in the range of the function are those with \(f(x_n)=5\) after some \(n\) or \(f(x_n)=10\) after some \(n\). Since sequences that are eventually constant converge, the function \(f\) is continuous.
However, the function does not "look'' like a function that has no breakpoints.

Let us do a thought experiment. Let us imagine that one of the homines sapientes a long time ago is sitting on the bank of a river, this homo sapiens who has nothing much to do picks up a pebble and draws a curve on the sand, has an epiphany about curves with no breakpoints, returns home with the exuberance of writing down Mathematics of no breakpoints, also passes a smirk at the resting tiger while returning home, sits down to write down the Mathematics, but then realizes that a notion of the continuum is not at its disposal yet. The snooty sapiens who had passed a smirk at the resting tiger realizes the importance of the continuum. If the imaginary snooty sapiens could invent distance metrics and continuous functions without inventing the continuum, it could be considered remarkable. Possibly this did not happen, and the homo sapiens might have received a smirk back from the resting tiger. However, the news that maybe conveyed to the resting tigers is that homines sapientes did invent the continuum called the set of real numbers denoted by \(\mathbb{R}\). Under the distance function \(d\) defined above, \(\{1,2\}\) and \(\mathbb{Q}\) are subspaces, i.e., special cases of \(\mathbb{R}\) with distance function \(d\). But hang on, another curious sapiens asks, if continuity of a function can be defined without using the continuum, then can integration and differentiability also be defined without using the continuum.

Consider the function \(f\) again. Let us assign weights to \(1\) and \(2\). Let the weights be \(\omega(1)=\frac{1}{3}\), \(\omega(2)=\frac{2}{3}\) respectively. Then define \(\int_{{1,2}}f(x)d\omega(x)=f(1)\omega(1)+f(2)\omega(2)=1\frac{1}{3}+2\frac{2}{3}=\frac{5}{3}\). However, making sense of a differentiable function in the standard sense is difficult without the continuum. Naively, let for \(a\in \{1,2\}\) the derivative of \(f\) at \(a\) be \(f'(a)=\lim_{x\rightarrow a}\frac{f(x)-f(a)}{x-a}\). This seems outrageous. Since the only sequence that converges in \(\{1,2\}\) are those sequences that are constant after some \(n\), the ratio \(\frac{f(x)-f(a)}{x-a}=\frac{0}{0}\) after some \(n\) and thus undefined. To make sure that \(f'(a)\) is defined, sequences that converge to \(a\) that are eventually not constants are required. The continuum does the job. Homines sapientes have defined continuum, proved its properties, (The axiomatic construction of the continuum by itself provides its properties, thus "defined the continuum, proved its properties'' may not be considered a correct way of articulating the imagination behind the continuum. However, given the nature of this piece that it is a fable the author shall not be pedantic.) used it to understand the effect of small changes i.e., derivative, and have used it in real-life applications.
Imagining the continuum that cannot be seen, for example, no human
has possibly seen \(x\) where \(\pi=x\), and using the continuum for the advancement of homines sapientes possibly vindicates human ancestor's smirk.

A homo sapiens who could imagine something as incredible as the continuum would have murmured its properties in utter despair if the sapiens were unable to demonstrate the incredible imagination to other humans. The incredible news is that the sapiens did not suffer from such despair. In addition to the real numbers, the sapiens also has the access to another four incredible objects that are also created by the humans: paper, ink, pen, and letters which together produce the most incredible linguistics tool, in ways that are completely mysterious to the author, known as Proof'. By applying this tool every sapiens can demonstrate its imagination to any other fellow homo sapiens. Further, by definition of a proof, the fellow sapiens can verify that the proof provided by a sapiens by the means of which the sapiens claims to have demonstrated its imagination is indeed what the claim says it to be. Proof makes sure that a claim is not substantiated by any subjective interpretations. (A proof is defined within an axiomatic system, as before the author shall not be pedantic.) It maybe true that each lonely philosopher tiger can imagine axioms and deductions. However, if they cannot express their imaginations by writing proofs, then maybe it is natural to see tigers murmuring in despair. Thus, it maybe considered acceptable to say, "The invention of Proof, in particular, has made homo sapiens what it is now''.

It seems that home sapiens has transcended its ability to write proofs:
the homines sapientes have now come to possess the ability to create fairies. These fairies are called AI or artificial intelligence. These fairies can write and verify proofs written by humans. The concern is that these fairies, though they are manifestations of human imaginations, may start writing proofs more efficiently than humans. If the latter happens, then what is the point of Mathematicians? Number theorist Andrew Granville brings up this in a Quanta Magazine documentary, "When Computers Write Proofs, What's the Point of Mathematicians?''. Whether the fairies will surpass homines sapientes intelligence is a matter for another discussion. Whether, as articulated by Yuval Noah Harari, the presence of the fairies has ushered in the end of human history is a matter for a different discussion. (See Mustafa Suleyman & Yuval Noah Harari's debate: What does the AI revolution mean for our future?) But so far, it seems that homines sapientes have done incredibly well by inventing Proof, and thus may have atoned for the snooty ancestor's smirk.