📂JOF

2026 Summer Omakase: A Paradigm Shift

Introduction

We often think that science advances by accumulating knowledge steadily, like stacking bricks one by one. But Thomas Kuhn argued that when we look closely at the history of science, there are moments where this is not the case. At a certain moment an entire old worldview collapses, and we come to see the same phenomena with completely different eyes—a leap. He called this a paradigm shift. For this summer’s course, we have prepared moments where the very perspective of a field is turned entirely upside down, ranging from physics and biology through analysis and probability theory all the way to machine learning.

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Classical Mechanics → Relativity

In the world of classical mechanics, time and space are absolute. No matter which observer looks, clocks tick at the same rate and the length of a ruler does not change. The coordinates of two observers moving at different speeds are connected simply, as below, through the Galilean transformation.

$$ \begin{pmatrix} x^{\prime} \\ t^{\prime} \end{pmatrix} = \begin{pmatrix} 1 & -v \\ 0 & 1 \end{pmatrix} \begin{pmatrix} x \\ t \end{pmatrix} \implies x^{\prime} = x - vt, \quad t^{\prime} = t $$

Notice here that the transformation does not touch time $t$ at all. As can be seen from the bottom row of the matrix being $(0, 1)$, no matter how much we transform the coordinates, time always comes back as $t^{\prime} = t$. Time was nothing more than an absolute cosmic backdrop flowing fairly for everyone. But once it was revealed that the speed of light is always constant regardless of the observer’s state of motion, this seemingly solid worldview began to shake to its very roots. In the Lorentz transformation that took its place, not only space but even time becomes intermingled depending on the observer’s state of motion.

$$ \begin{pmatrix} x^{\prime} \\ t^{\prime} \end{pmatrix} = \gamma \begin{pmatrix} 1 & -v \\ -v/c^{2} & 1 \end{pmatrix} \begin{pmatrix} x \\ t \end{pmatrix} \implies x^{\prime} = \gamma (x - vt), \quad t^{\prime} = \gamma \left( t - \frac{v}{c^{2}} x \right) $$

Here $\gamma = 1 / \sqrt{1 - v^{2}/c^{2}}$ is the Lorentz factor, which grows larger as the speed approaches the speed of light. As the lower-left component of the matrix, which was $0$ in the Galilean transformation, now becomes $-v/c^{2}$, the spatial coordinate $x$ begins to intervene in determining the time $t^{\prime}$. As a result, two events that occurred simultaneously for one person are no longer simultaneous for another (loss of simultaneity), moving clocks run slow (time dilation), and moving rulers shrink (length contraction). The stage of absolute time and space disappeared, and a spacetime that looks different to each observer took its place.

"Time and space are absolute for everyone"
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"Time and space are relative to the observer"

• Galilean transformation
• Lorentz transformation
• Time dilation
• Length contraction

Classical Mechanics → Quantum Mechanics

Around the same time, classical mechanics also collapsed in an entirely different direction. The world governed by Newton’s laws of motion is thoroughly deterministic. If we know exactly a particle’s current position and momentum, we can in principle work out the entire past and future trajectory of that particle by solving the equations. The particle has a fixed position at every moment and moves along a fixed path.

But when we descend into the microscopic world smaller than the atom, this certainty wavers. In quantum mechanics, which governs this world, the electron we believed to be a particle behaves like a wave, and the state of a particle is described not by a fixed position but by a wave function. Moreover, this wave function tells us not ‘where the particle is’ but only ‘what the probability is of finding it somewhere.’ Furthermore, according to the uncertainty principle, it is fundamentally impossible to know position and momentum exactly at the same time.

$$ \Delta x \Delta p \geq \frac{\hbar}{2} $$

A particle that had moved along a determined trajectory turned into something that exists only within a cloud of probability. And the gaze of the physicists who thus delved into the microscopic world would soon fling open the door of an entirely different field.

"A particle has a fixed position and trajectory"
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"The state of a particle is described by a probabilistic wave function"

• Matter wave
• Uncertainty principle
• Schrödinger equation
• Wave function
  • Probabilistic interpretation of the wave function

Classical Biology → Molecular Biology

Around the time quantum mechanics was shaking up the landscape of physics, Erwin Schrödinger, one of its protagonists, thought that combining biology and quantum mechanics would become the cutting edge of science, and published a book titled What Is Life? James Watson, inspired by this book, became interested in the structure of DNA, and later, together with Francis Crick, revealed that the structure of DNA is a double helix.¹ This was published in the prestigious journal Nature as a short paper of about one page titled Molecular Structure of Nucleic Acids: A Structure for Deoxyribose Nucleic Acid.

For a long time, biology was a discipline that observed and classified living organisms and described life phenomena as they were. Even though it was known what was inherited and how, exactly which substance realized it and how remained veiled. With James Watson and Francis Crick’s discovery, how genetic information is stored and replicated began to be explained at the molecular level, and classical biology, which had remained at observation and classification, leaped into molecular biology, which can explain the phenomena of life science at the molecular level.

"Observe and classify life at the level of the individual organism"
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"Explain life phenomena through mechanisms at the molecular level"

• DNA, RNA, chromosomes
• The central dogma of molecular biology

Function: Correspondence of Numbers → Relation Between Sets

Originally, to mathematicians a function was a correspondence sending a number to a number, nothing more and nothing less. In other words, a function was thought of as being like a single formula. But as mathematics and engineering developed, objects like ‘something that takes a function as input and returns a single number’—for instance, something that assigns to a function $f$ its integral value $\int f dx$—began to be needed.

$$ \begin{aligned} \text{classic function:} \quad & \text{number} \longmapsto \text{number} \\ \text{new object:} \quad & \text{function} \longmapsto \text{number} \end{aligned} $$

The mathematicians of the time did not dare call this a function, and cautiously called it a ‘function-like thing’ (fonctionnelles). This is the starting point of the concept we today call a functional. This awkwardness arose because the definition of a function was not broad enough. Later, as rigorous set theory took hold, a function was redefined, escaping the narrow frame of ‘a correspondence of number to number,’ as a binary relation between the elements of two sets that satisfies certain conditions. The operator that acts on the wave function in quantum mechanics is also a representative example of such a mapping that takes a function as input.

"A function outputs a number when you put in a number"
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"A function is a binary relation between sets"

• Linear functional
  • Why a functional came to be named a functional
  • Generalized function
• Binary relation
• Sets and propositional functions

Riemann Integral → Lebesgue Integral

The Riemann integral that we first learn as undergraduates begins with finely dividing the domain. It divides the $x$-axis densely, erects a thin rectangle over each interval, and adds up their areas. For most well-behaved functions there is no problem at all, but in the face of a pathological function like the Dirichlet function, which takes the value $1$ on rationals and $0$ on irrationals, this method becomes powerless. No matter how finely we divide the domain, every interval contains both rationals and irrationals together, so the sum of the areas taken from above and the sum of the areas taken from below hardly narrow down toward $1$ and $0$ respectively.

The Lebesgue integral flips the idea around. It divides not the domain but the range. It asks, “If we gather up all the points whose function value is around this much, what is the size of that set?” But to do this, we must be able to measure ’the size of the set of gathered points.’ It is exactly at this point that the concept of a measure becomes necessary, and measure theory, which rigorously generalizes length and area, is born. The Lebesgue integral covers all the functions that the Riemann integral handled while also covering a far broader world, so it can be called a true generalization of the Riemann integral.

"Divide the domain and add up the areas of rectangles"
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"Divide the range and add up the measures of the preimages"

• Riemann integral
  • Dirichlet function
• Lebesgue integral
  • The Lebesgue integral as a generalization of the Riemann integral
• Definition of a measure

Probability: Frequency → Integral

When we first learn probability, we calculate like this. The value with the number of all possible cases in the denominator and the number of cases of the event of interest in the numerator is the probability that the event occurs. It’s the idea that the probability of rolling an even number on a die is $3/6$. This is an intuitive definition rooted in the frequency with which an event occurs.

$$ P(A) = \frac{|A|}{|S|} $$

But this definition soon runs into a limit. This is because when the number of cases is infinite, or in a continuous situation such as picking any real number at random from the interval $[0, 1]$, the ‘denominator’ does not hold. Having just learned in measure theory how to measure the size of a set, we are now ready to define probability anew. We simply need to define probability as a measure whose total measure is $1$. Then probability becomes a measure, and the expectation becomes none other than an integral.

"Probability is the ratio of cases in which an event occurs among all cases"
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"Probability is a measure whose total measure is 1, and expectation is an integral"

• Probability and the addition rule of probability
• Probability defined measure-theoretically
• Expectation defined measure-theoretically

Frequentism → Bayesian

If the foundation of probability has changed, then there are also different branches in the attitude with which one reasons about the world using that probability. In frequentism, the parameter is regarded as a fixed unknown constant. The sample in our hands is merely one observation obtained by chance in order to find out that constant, and it is believed that as the sample grows larger it draws closer to the truth. Their way is to weigh “if this experiment were repeated infinitely,” imagining even data not yet obtained.

Bayesianism starts from the exact opposite position. It regards the parameter itself as having a distribution, and looks only at the data currently in hand. It merely updates the belief held before beginning the analysis—that is, the prior distribution—with the data to obtain the posterior distribution.

$$ p(\theta | y) = \frac{p(y | \theta) , \pi(\theta)}{p(y)} $$

This perspective, in which each time new data comes in yesterday’s posterior distribution becomes today’s prior distribution, sequentially updating the belief, differs in its very attitude toward the world from frequentism, which chased after a fixed truth.

"The parameter is a fixed constant"
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"The parameter follows a distribution, and its value is updated with data"

Interestingly, this contrast looks similar to the relationship between classical mechanics and quantum mechanics seen earlier. Classical mechanics and frequentism both regard the object we wish to know (the position and momentum of a particle, or the parameter) as being fixed at a single determined value. Quantum mechanics and Bayesianism, on the other hand, regard that object not as a fixed value but as something following a certain distribution. Two entirely different fields have passed through the same paradigm shift of ‘seeing the value one seeks as a distribution.’

Existing theory		Paradigm shift
Classical mechanics: position and momentum are fixed values	→	Quantum mechanics: position and momentum are distributions (wave function)
Frequentism: the parameter is a fixed constant	→	Bayesianism: the parameter follows a distribution

• Proof of Bayes’ theorem, and the prior and posterior distributions
• The Bayesian paradigm

Classical Machine Learning → Deep Learning

In traditional machine learning, human domain knowledge and intuition were absolute. How did you get a computer to distinguish pedestrians on the road? Engineers of the time had to design and extract useful information, that is, features, directly from the raw data. They extracted features such as where in an image the pixel values change abruptly (edge detection) and which direction of brightness change is strong (HOG), using filters that humans devised mathematically by hand. Handed the numbers that humans had thus carefully carved out, the model would then perform only a statistical classification. How well the models of the past worked ultimately depended on how well humans designed the features.

Deep learning flips this division of roles. Instead of humans deciding in advance which features are important, an artificial neural network learns the necessary representations on its own from the raw data. For example, a convolutional neural network learns directly from the data what corresponds to the filters that humans used to design by hand. Behind the ability of neural networks to be fitted so flexibly to data lies the Cybenko theorem, which states that a sufficiently large neural network can approximate any continuous function as accurately as desired. The place where humans carved out features came to be replaced by data and optimization.

"A person directly designs useful features"
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"The features themselves are learned from the data"

• Deep learning
• Artificial neural network
  • Multilayer perceptron
  • Convolutional neural network
• Cybenko theorem (universal approximation theorem)