[Must-Read for University Freshmen] Where is that 'Mathematics' used in AI? The true meaning behind the first-year university curriculum
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As your new life begins, many students who have entered information science or science and engineering departments are likely expecting to encounter the latest AI technology. However, the first things listed on your schedule might just be 'Linear Algebra,' 'Calculus,' and 'Introduction to Computer Science'—all seemingly mundane foundational subjects.
Why, in an era where ChatGPT can freely manipulate language, must we still solve 'matrices' by hand? In this article, I would like to connect the university mathematics curriculum with the true nature of AI, based on academic facts.
1. 'Linear Algebra' is the language that quantifies the world
The first hurdle in university mathematics is Linear Algebra (matrices). In short, this is 'mathematics for calculating and processing large amounts of data (numbers) all at once'. This becomes the 'map of space' for AI.
I believe you learned about 'vectors' in high school mathematics. In high school, they were treated geometrically in 2D or 3D space, mainly as 'arrows with direction and magnitude.' However, in the world of AI, vectors are treated as sequences of data. For example, 1000-pixel image data is, to an AI, 'a 1000-dimensional vector with 1000 numbers lined up.'
In linear algebra, in addition to these multi-dimensional vectors, you will learn deeply about the concept of 'matrices,' where numbers are arranged in rows and columns. Matrices function as devices that apply transformations, such as 'rotating' or 'stretching,' to vectors.
Modern AI (neural networks) calculates and processes complex information by multiplying a vast number of matrices against input data (vectors). Understanding linear algebra and matrices is directly linked to understanding how AI 'represents' information.
2. 'Calculus' is the guideline for making AI smarter
AI cannot provide perfect answers from the start. It makes predictions on vast amounts of data, calculates the 'error' from the correct answer, and continues to adjust its own parameters so that the error is minimized.
This process of 'searching for the direction to minimize error' is, mathematically, an optimization problem called 'Gradient Descent.'
Partial differentiation is a method of calculus where, among many variables, 'only one variable is moved, while all other variables are treated as fixed constants.'
An AI with hundreds of millions of parameters must keep searching for 'which variable to move, in which direction, and by how much, to descend the error slope most efficiently.' The compass that makes this staggering calculation possible is optimization using partial differentiation (such as gradient descent).
3. 'Introduction to Computer Science/Programming' is the etiquette of implementation
AI does not work on theory alone. Understanding the hardware (CPU/GPU) that executes it and efficient data structures (algorithms) is essential.
'Introduction to Computer Science' teaches everything from 0 and 1 bit operations to memory management and OS mechanisms. While it may seem like old knowledge at first glance, when running large-scale AI models, low-level knowledge—such as how to efficiently exhaust computing resources—becomes the deciding factor for performance.
4. The significance of studying mathematics: Why is 'learning' important now?
The history of AI is also the history of the transition from 'humans writing rules (symbolism)' to 'learning from data (connectionism).'.
In the past, humans described features like 'has ears' or 'has whiskers' in an IF-THEN format, but this had its limits. The reason modern AI education emphasizes mathematics is that the approach of discovering complex laws that humans cannot define through mathematical 'weight adjustment (learning)' has won out.
Conclusion: Foundations are an 'investment,' not 'consumption'
The speed of AI evolution is extremely fast, and knowing how to use specific libraries or tools will become obsolete in a few years. However, the underlying mathematical and physical principles remain unchanged in that field.
EQUES Inc., which operates by leveraging the insights of the Matsuo Lab at the University of Tokyo, places extreme importance on this 'power of fundamentals'.
The exercises in basic subjects that you first-year students are working on now, which may seem boring at first glance, will surely become your strongest weapon for controlling the massive system that is AI with your own hands and building new intelligence in the future.
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EQUES Inc., a startup from the University of Tokyo's Matsuo Lab, is working to solve social issues by making full use of the latest AI technology.
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