Communications of the ACM
On the Evolution of Languages: Natural, Mathematical, and Computational
The evolution of natural languages (such as English, Chinese or Hebrew) has occupied researchers over the last century (e.g., Chomsky, 1965; Pinker, 2007; Deacon, 1997). In particular, the nature of biological vs. cultural evolution has been of interest (e.g., Mesoudi, 2007). In contrast to biological evolution, which is measured in millions of years, cultural evolution occurs over the scale of thousands of years, and often much less (think e.g., of the evolution of English from Shakespeare around 1600 to the present; or the evolution of Hebrew from Ben Yehuda at the turn of the 19th century to the present). In this blog, we explore the (cultural) evolution of languages. Specifically, we compare the evolution of natural languages, which are used by people all over the world, with two artificial languages: the language of mathematics and programing languages.
It is somewhat surprising to observe that while computer languages, which appeared a mere 80 years ago, keep evolving under the selection pressure of the capabilities of the human brain, the mathematical language, which exists for more than 2300 years, has not. And we are asking Why.
We are inspired by Deacon's (1997) book "The Symbolic Species: The Co-Evolution of Language and the Brain," which addresses two great puzzles concerning language. One, if we believe that our language facility has evolved by Darwinian natural selection, how is it that we don't find 'simple languages' in other species? This puzzle will not be addressed here. (His answer, in a nutshell, is that it is not language but rather symbolic thinking which has evolved, and language has been a by-product.)
The second puzzle is a famous one, that has been addressed by many researchers all the way back to Chomsky (1965): How can we explain the miraculous ease by which children naturally and successfully learn and master the enormously complex language system, without any direct instruction?
Here is Deacon's (1997) elegant solution to the puzzle of the unreasonable effectiveness of children's language learning. According to Deacon, it is not the brain that evolved to make language learning possible (as claimed e.g., by Chomsky's famous student, Pinker (2007)), but rather it is language itself that evolved to fit the learning capabilities of children's brains, because otherwise it would not have been replicated. Specifically, languages evolve, and since the mechanism for their replication between generations is children's learning, there is a strong selection pressure on languages to conform to the learning capabilities of children's brains.
We note that language is a tool, and like all tools, its evolution is subject to two (often conflicting) evolutionary pressures. On the one hand there is the task-oriented pressure: the tool must be suitable for the relevant task. On the other hand, there is the user-oriented pressure: the tool must fit the user's natural capabilities. The final evolutionary 'solution' is some compromise between these two pressures. See for example the three tools portrayed in Figure 1, where the user-oriented parts are the handles, which fit the capabilities of the human hand. The task-oriented parts — the tip of the screw-driver, the container of the mug, and the cutter of the can-opener — are seen to fit the tool's main purpose.
Figure 1. Selection pressures in the evolution of tools: task-oriented vs. user-oriented pressures
In computer science too, we see a trend in the evolution of programming languages, from machine-friendly assembly languages (in the 1940's), through human-friendly 'high-level' languages (in the 1960-70s) and object-oriented languages (in the 1980s), to a scripting approach (in the 1990s) and tools for creating programs without as much programming (in the onset of the millennium).
The development of this plethora of programming languages in such a short time can be explained by the need to bridge the gap between the task-oriented pressure and the user-oriented pressure. The closer a programming language is to the current problem domain, the easier it is for the human programmer to get the machine to accomplish the task (06langs, n.d.). In other words, selecting a language that fits our needs facilitates our thinking, in accordance with the following paraphrase of Benjamin Whorf: Language crucially influences the way we think and what we can think about. Yet, in some cases, we need to adjust our preference to fit the environment we work in, similarly to when we travel to new places (Sherman, 2015).
Mathematics, too, is a tool that goes through (cultural) evolution, and the same analysis could be expected to apply. Somewhat paradoxically, we don't see such evolution in the language of mathematics. On the contrary, while the mathematical language has become more and more precise and formal under the task-oriented pressure, it has been getting less and less user-friendly, as the following brief description of the evolution of calculus illustrates.
The original calculus invented (independently) by Newton and Leibnitz at the 17th century was intuitive and imprecise. In fact, it contained logical contradictions and inconsistencies, such as the definitions of infinitesimals and continuity. It took the mathematical community more than 150 years to set the record straight and put calculus on firm and rigorous foundations. The new calculus was definitely more powerful, but it carried a high price tag, paid in the currency of user-friendliness: the historical push to suppress the intuition of time and process, and replace it by a timeless algebraic notation, has resulted in the notorious Epsilon-Delta formalism — the dread of undergraduates all over the world.
Thus, the task-oriented pressure has worked admirably well, pushing mathematical language towards ever-increasing power via decontextualization, generalization, abstraction, formalism, and rigor. However, all these developments also made mathematics substantially less user-friendly, at least for external users (such as scientists and engineers), and for the general population of learners. The case of research mathematicians themselves is more subtle and controversial, but a case could be made (elsewhere) that the culture of modern mathematics rejects user-friendly 'mutations' even when they exist.
We might be tempted to say all this is because it's a formal language, but so are programming languages. So whence the difference? In what follows, we suggest three possible reasons for this difference, miraculously all beginning with M.
1. The Mental explanation. This argument highlights the clash between human natural thinking (fundamentally contextualized, see below) and mathematical thinking (stressing decontextualization). While our ability to use natural language is deeply contextualized and thus consonant with 'human nature', modern mathematics is supposed to be totally de-contextualized, which squarely clashes with 'human nature'. "The fundamental computational bias of human cognition" from cognitive psychology is relevant here.
[Our natural intuitions] are highly contextualized, personalized, and socialized. They are driven by considerations of relevance and are aimed at inferring intentionality by the use of conversational implicature even in situations that are devoid of conversational features […] The primacy of these mechanisms leads to what has been termed the fundamental computational bias in human cognition […] – the tendency toward automatic contextualization of problems. (Stanovich & West, 2000; emphasis added)
In computer science, by comparison, this problem is alleviated by the plethora of programming languages, which enables us to think contextually by selecting the specific language suitable for solving a given problem.
2. The Market explanation. In computer science, poor communication simply costs a lot of money! And it can be measured in the price of software development (programmers' hours, debugging, updating, customer dissatisfaction, etc.). In mathematics, poor communication may make the life of the mathematician harder, and the life of millions of math learners (students) and users (engineers) miserable, but it doesn't cost money, at least not in a directly accountable way. Nor does it show up in any tangible and measurable way. Professional mathematicians are accustomed to putting in the necessary effort and students have no power to change the situation.
3. The Macho explanation. In the spirit of the satirical publications "Real Men Don't Eat Quiche" (Feirstein, 1982) and "Real Programmers Don't Use PASCAL" (Post, 1982), we might sometimes encounter "Real Mathematicians Don't Need Sugar-Coated Formalism." It needs to be emphasized, however, that the macho attitude is at the level of community, not individuals. Many mathematicians we know are personally gentle, shy and modest. The macho attitude does show up only in the general culture of the research mathematical community.
A final thought: Who is selecting whom?
Instead of people applying selection pressure on tools, as in natural and programming languages, in mathematics things seem to have gone the other way around: mathematics is applying selection pressure on people! The fact that mathematical thinking goes in certain ways 'against human nature' (Leron & Hazzan, 2009) has been exploited by various institutions to select people — quite literally! — by their performance in mathematics. For example, students in prestigious college departments, such as computer science, are selected and filtered through their grades in school math. One relevant question that should be asked in the context of ACM is how this selection criterion shapes the evolution of programming languages, as well as their accessibly to a wider public.
References
Chomsky, N. (1965). Aspects of the Theory of Syntax. MIT Press
Deacon, T. W. (1997). The Symbolic Species: The Co-evolution of Language and the Brain, W. W. Norton & Company.
Feirstein, B. (1982). Real Men Don't Eat Quiche, Pocket Books.
Leron & Hazzan (2009). Intuitive vs analytical thinking: four perspectives. Educational Studies in Mathematics 71, pp. 263–278.
Mesoudi, A. (2007). Biological and Cultural Evolution: Similar but Different. Biological Theory 2(2), pp. 119–123.
Pinker, S. (2007). The Language Instinct: How the Mind Creates Language. Harper and Collins.
Post, Ed (1982). Real Programmers Don't Use Pascal. https://www.ecb.torontomu.ca/~elf/hack/realmen.html
Sherman, M. (2015). Why Are There So Many Programming Languages?, the OVERFLOW, https://stackoverflow.blog/2015/07/29/why-are-there-so-many-programming-languages/
Stanovich, K. E. and West, R.F. (2000). Individual differences in reasoning: Implications for the rationality debate? Behavioral and Brain Sciences 23 (5), pp. 645-65; discussion: pp. 665-726.
06langs (n.d). Lecture 8: Programming Languages, https://www.cs.princeton.edu/courses/archive/fall09/cos109/06langs.pdf
Uri Leron is professor emeritus at the Technion's Department of Education in Science and Technology. In his quest for the origins of mathematical intuition and for ways to bridge the gap with its formal face, he has applied insights from cognitive science, computer science, and evolution. Orit Hazzan is a professor at the Technion's Department of Education in Science and Technology. Her research focuses on computer science, software engineering, and data science education. For additional details, see https://orithazzan.net.technion.ac.il/.
Comments
James Higgins-Thomas
I feel like there's an aspect that I'm not sure how fits into your three suggested explanations: continuity. Mathematics, unlike computer (or even natural) languages captures a continuous evolution of ideas over time and there is a need - a driving force - that the early learnings are not lost or forced to be recreated with time. An area that comes to mind here (and which could use some modernization!) is the classic "PEMDAS", which formalizes ambiguity in expressions. Making these rules more intuitive isn't necessarily difficult, but it would be disastrous to anyone consulting old texts. Consequently, math does evolve, but generally through addition and not evolution.
Computer languages (and natural ones) do not have the same historic pressures. Yes, it would be nice if we, today, could read old English or Latin or whatever, but that pressure to be able to do so isn't quite the same. And it certainly isn't strong amongst the majority of us (we do see, however, that languages have been retained past their cultural "death" by scholars and others who do have a need/desire to retain this historic knowledge).
Retention of historical tools isn't unique to Mathematics, either. Its hard to know how much is fact vs allegory, but the story of why railroad tracks are set at roughly 4'8" comes to mind (because of the width of two standard draft horses). Truth or not, there are many engineering constants that are driven more by continuity than by other factors. Even the US's stubborn clinging to the Imperial measurement system is such. We're unlikely to change our power standard in quite a while, motor vehicles are constrained in width and height, and so forth. I believe that the needs of Mathematics to retain this continuity is even greater than those posed by other disciplines and contributes significantly to the ossification of what has gone before and is now considered "settled" syntax and semantics.
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