The Economics of Thought Expression: Why Mathematics and Programming Are Being Reevaluated in the AI Era

Rediscovering the Cost of "Expressing Thought"

Recently, through my dialogues with AI, I've been thinking deeply about the act of "expressing thought." When trying to convey complex instructions to AI, I unconsciously pursue efficiency in thought expression. The trial and error of selecting words, structuring sentences, and making revisions is essentially a conversion process of transferring what's in my mind to the external world.

This conversion inevitably incurs a cost. Ideas that seem clear in my head often don't convey as well when put into words. Could there be a more efficient method of expression? Driven by this question, I've been deepening my exploration of systems of thought expression that humans have developed throughout history.

To state my conclusion upfront, programming languages and mathematical expressions are likely the representation formats with the minimum data cost for conveying thought. This intuition has increasingly turned into conviction as I simultaneously experience both human-to-human and human-to-AI communication in this AI era.

The following diagram illustrates the relationship between methods of thought expression and their efficiency.

In this article, I reconsider the value of formal languages such as mathematics and programming from the perspective of the cost of expressing thought. Furthermore, I will explore why these skills become increasingly important in the AI era, delving into aspects of information theory and cognitive science.

Formal Languages and Natural Languages: The Efficiency of Thought Expression

The difference becomes more evident when considering an example that many people might use in daily life: email address spam filters.

For instance:

Natural language (88 characters in Japanese):
"Do not include suspicious words like 'free', 'earn money', or 'side job',
and only accept email addresses where the part after the @ sign consists of lowercase alphanumeric characters or hyphens,
ending with .com or .jp"

When trying to write these complex conditions in natural language without misunderstandings, concisely, and completely, it tends to become redundant and difficult to understand. However, in formal language (regular expression), it can be expressed in just 45 characters:

Formal language (45 characters):
/^(?!.*(無料|稼げる|副業)).*@[a-z0-9\-]+\.(com|jp)$/

As shown here, "formal languages" enable overwhelming information compression and precise condition specification for the same semantic content.

In programming and mathematics, specific patterns or processes can be expressed very concisely. This is a strength of structured languages. From an information theory perspective, formal languages have the characteristic of eliminating redundancy and maximizing information density.

Recently, I've noticed that I can express in a single sentence what would have previously required more than 10 lines of code when communicating with AI. For example, an instruction like "Read a CSV file, calculate the average of a specific column, graph the results, and save it" perfectly communicates to an AI that understands basic programming structures.

This phenomenon is fascinating from an information theory perspective. In computer science, there's a concept called Kolmogorov complexity, defined as the length of the shortest program (description) needed to generate certain data. Borrowing this concept, formal languages can be considered to have a higher compression ratio than natural languages.

Comparing the information efficiency of formal and natural languages reveals the following characteristics:

In information system development, specifications written in natural language tend to leave room for interpretation, reducing reliability. In contrast, formal description languages using mathematical notation eliminate ambiguity and enhance the reliability of descriptions.

From Ancient Philosophy to Information Theory: A Historical Perspective on Thought Expression

The pursuit of efficiency in thought expression could be said to mirror the history of human intellectual endeavors. It's widely known that Plato emphasized mathematics for philosophy.

There's a tradition that the entrance to Plato's Academy bore the inscription: "Let no one ignorant of geometry enter". Why was Plato so adamant about mathematics? I speculate that one reason might be because he considered mathematics a tool for clarifying thought.

For Plato, mathematical concepts, especially geometry, were thought to possess universal and perfect truth unlike the imperfections of reality. It was considered a spiritual activity pursuing truth through pure reason apart from the sensory world, and was viewed as something that leads the soul to a higher level.

This perspective resonates with today's information theory and cognitive science. The following shows the historical development of thought expression from ancient philosophy to modern information theory in chronological order.

What's interesting is that after this long history, we are once again facing the problem of optimizing thought expression. In the AI era, this issue is directly connected not only to efficiency but also to the possibility of human-AI collaboration.

Communication Efficiency with AI: The Value of Structured Expression

In communication with AI, efficient exchanges that maximize information while minimizing data volume significantly affect productivity. Structured expression is particularly important here.

Clarity and Conciseness are cited as basic principles of effective prompt engineering. Ambiguous instructions tend to lead to unpredictable results, while clear and concise instructions help AI focus on the intended task and generate accurate outputs.

What I've strongly felt while practicing these principles is the limitations of instructions using only natural language. Due to ambiguity, AI's interpretation can easily vary. In contrast, formal languages like programming languages were originally a means of giving clear instructions to computers, but they can also convey processing content precisely to AI.

In recent AI development projects, efficient communication is achieved using structured data formats like JSON, or specific notations like Markdown and Mermaid. These formats structure information and make it easier for AI to understand context and data relationships, thereby promoting more accurate and consistent output.

The following shows elements that enhance communication efficiency with AI.

A noteworthy aspect of communication with AI is process persistence and reusability. Saving instructions to AI as programs allows for reusing similar processes or directing only the changes, which can be more efficient than UI operations in some cases.

Recently, I've increasingly been inserting data into Notion not through UI operations but via API or MCP. The reason is simple: once I save the program for that process, I only need to communicate the modifications for similar processes. It takes effort initially, but once accustomed, it's more efficient than memorizing UI operations.

Technical Terminology and Context Compression: The Limits of Information Efficiency

In specialized fields, unique terminology systems (jargon) develop. This isn't merely for exclusivity but for information compression that embeds extensive context into short words.

For example, using the medical term "myocardial infarction" abbreviates the lengthy explanation: "a condition where blood flow to the heart muscle is blocked, causing tissue death in that area." Furthermore, among physicians, this term shares a broad context including severity and typical treatment approaches.

Technical terminology is used based on knowledge and background (context) shared within a specific community. This enables efficient communication of complex concepts or procedures in brief terms.

This mechanism can also be understood through the analogy of information compression. Just as data compression technology reduces capacity by eliminating data redundancy, technical terminology reduces data volume in communication by omitting repetitive detailed explanations and background information.

However, it's important to note that efficient communication through technical terminology is predicated on sharing that context. Using technical terms with someone who doesn't share this context risks miscommunication and misunderstanding.

The relationship between technical terminology and information compression can be illustrated as follows:

In the AI era, this information compression through technical terminology is becoming increasingly important. What I've keenly felt in my collaboration with AI is that whether you can use common technical terminology greatly affects efficiency. For instance, just saying "Validate using JSON schema" allows AI to understand a series of complex data validation tasks. However, for someone unfamiliar with this technical term, many more words would be needed to convey the same instruction.

Formal Languages and Thought Structures: Impact on Cognition

How do thought processes using formal languages (mathematics, logical symbols, code, etc.) differ from those using natural languages? From my own experience, I feel that using formal languages structures thinking more.

People with extensive programming experience tend to naturally think in structures like conditionals and loops even for everyday problems. This might be evidence that formal languages influence the very mode of thinking. Formal languages eliminate the ambiguity of natural languages and enable thinking based on strict rules. This is thought to contribute to the development of abstract and logical thinking abilities.

This perspective also connects to the broader context of hierarchical structure of thought. There's a model that captures our intellectual activities in a hierarchy of Data, Information, Knowledge, Intelligence, and Wisdom (DIKIW). In the AI era, this model is also helpful for understanding the relationship between humans and AI. While AI excels from data processing to providing knowledge, humans demonstrate their value in the domains of intelligence and wisdom. Learning formal languages is thought to have the effect of training thinking skills that traverse this hierarchy. I delve deeper into this topic in the article "Will AI Steal Human Intelligence? Redefining the Staircase from Data to Wisdom in the Generative AI Era".

Formal languages can also be viewed as "cognitive tools" that extend human cognitive abilities. By utilizing external representational systems such as characters and symbols, it becomes possible to handle more complex information and perform higher-level reasoning. Formal languages help externalize thought processes, express them in objectively verifiable forms, and can be expected to enhance the clarity and rigor of thinking.

From these characteristics, we can see the significance of learning mathematics and programming. It's not merely skill acquisition but perhaps the acquisition of cognitive tools that expand the very mode of thinking.

Skill Sets Required in the AI Era: Mathematics and Programming Reevaluated

According to the World Economic Forum's "Future of Jobs Report 2025," the core skill most valued by companies is "analytical thinking," followed by "resilience, flexibility, agility," and "leadership and social influence"¹.

As for technology-related skills, "technology literacy" and "AI and big data" are also listed as important skills. Technology-related occupations such as AI/machine learning specialists and big data specialists are among the most rapidly expanding occupations in terms of growth rate. According to the WEF report, demand for AI/machine learning specialists is predicted to increase by 40% by 2027².

However, there's also a risk that the emphasis on applied skills like AI and big data may undervalue the importance of foundational academic fields such as mathematics and programming that underpin them³. Depending on application tools without understanding these foundations may limit innovation capabilities and problem-solving abilities, potentially creating future vulnerabilities.

In the AI era, not only specific technical skills but also more universal abilities are valued. Complex thinking abilities, critical thinking skills, and strategic learning abilities – capabilities to adapt to change and continue learning independently – become essential.

The role of programming is also changing. With the development of AI code generation technology, there's a shift from writing code line by line to conveying intent to AI, verifying and modifying generated code, and focusing on higher-level system design and problem-solving.

However, the importance of programming fundamentals (logic, algorithms, data structures, language syntax, etc.) remains high. These fundamentals are essential for understanding, verifying, debugging, securing AI-generated code, and solving complex problems that AI cannot address.

Future engineers will need new skill sets in addition to fundamental knowledge:

The Future Brought by Optimizing Thought Expression

From the considerations so far, mathematics and programming can be said to be minimum cost media for thought expression, and their value is being reevaluated in the AI era. However, this is not simply acquiring skills in the conventional sense of memorizing programming languages, but rather the more essential meaning of acquiring structured thinking and expression.

What I'm particularly focusing on is that the economics of thought expression is beginning to work in collaboration with AI. Programmatic thinking and structured expression are extremely effective for efficiently conveying complex instructions to AI. This holds the potential to fundamentally change the productivity of thinking.

Conventional programming has been understood primarily in the context of instructions to computers. Today, however, its essence is expanding to the universal intellectual activity of structuring thought and optimizing expression. Programming concepts and structures (conditionals, loops, modularization, etc.) are powerful toolsets for organizing complex thoughts and expressing them efficiently.

Additionally, the conciseness and rigor of mathematical expression function as a common language for accurately conveying thought in communication with AI, which cannot tolerate ambiguity. The value of "clarification of thought" that ancient Greek philosophers found in mathematics may be reviving with new significance in the AI era.

Finally, the efficiency of thought expression relates not only to productivity but also to possibilities for creativity and innovation. As the cost of expression decreases, more thought experiments and creative attempts become possible. Through efficient communication with AI, humans can concentrate on higher-order thinking activities and engage in creating new value.

By recognizing the value of mathematics and programming as thought expression and repositioning them not merely as technical skills but as tools for thinking, I believe the possibilities for human intellectual activities in the AI era will greatly expand.

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