"Subjective Chronology" — An Intellectual Adventure to Unravel "Perceived Time" with Mathematical Formulas [Column] Science of Thought (No. 86)

🧠 3 things you will learn in this column

1. The discovery that "how we perceive time" can be structured with mathematical formulas
   Emotion, focus, freshness, memory, boredom—you will find that your perceived time actually has a structure that can be "factorized".

2. You can visualize the invisible "mental time density" with graphs
   By viewing subjective time not as a "flow" "flow" but as a "density" "density", you can see how daily experiences connect to memory and satisfaction.

3. You can rethink your time as something that can be "designed"
   While learning tips to increase the density of subjective time, let's gain a new perspective to make your day more "dense and memorable."

⏱️ [Introduction] "Why does the same hour feel so different?"

⏱️ Introduction | The true nature of "Wait, it's already this time..."

In the morning, before you know it, it's time to leave.
In the afternoon meeting, no matter how many times you check the clock, it doesn't move.
The conversation with a friend at night passed by in an instant.

Even though it should be the "same hour," the way it feels is completely different.
That is the mysterious phenomenon called "subjective time."

⌛ What on earth is time?

We understand time as "numbers on a clock."
But in reality, isn't the "mental perception" behind it what determines
the quality of your time?

• Even in the same 60 minutes, if you are excited, it feels "short"

• If you are bored, it feels "long"

• But what we can remember later is usually the time that was "dense"

What if time is not just something that flows, but a "perceivable phenomenon"? If we capture its true nature with mathematical formulas, what kind of scenery will we see?

📐 Questions to explore in this column

In this column, with the theme of "subjective time," we will think about
how it is felt, how it changes, and how it can be designed.

And, while factorizing elements such as emotion, focus, freshness, memorization, and boredom, I will guide you on a journey to mathematically visualize your own "landscape of time."

🧠 Why look at time with mathematical formulas?

Mathematical formulas are not just a list of cold symbols.
They are a lens that gives "contours" to invisible phenomena.

This column is not just a theoretical explanation.
The meaning of feeling, the joy of thinking, and how to savor the time we live.
It is an attempt to approach such questions carefully, using a little bit of mathematics.

🎯 There are three goals for this column.

1. To intellectually re-examine familiar sensations

2. To visualize invisible time using mathematical formulas

3. To take home your own "blueprint of time"

Now, let's set off on a journey together.
The "time you feel" is richer, more structured, and freer than you think.

📐 [Chapter 1] Let's Factorize Subjective Time

What does it mean to say "time was dense"?

There are classes where you feel, "What, only 30 minutes have passed!?"
And there are experiences where you even feel a sense of time disappearing, like, "I realized it was already night...!"

Even with the same "60 minutes",human "perceived time" varies.
What exactly influences this "subjective time"?

🧩 Let's factorize: The structure of perceived time

Here, a mathematical formula makes an appearance.
This is an attempt tomake your daily "sense of time" clearer than words.

Subjective time density $${D(t)}$$ can be expressed as follows:

$${D(t) = C_0 + k \cdot \frac{E(t) \cdot A(t) \cdot N(t) \cdot M(t)}{1 + B(t)}}$$

Each factor has the following meaning:

• $${E(t)}$$: Emotion
  → Joy, surprise, anxiety, etc.; the more your emotions are stirred, the denser time feels.

• $${A(t)}$$: Attention
  → The degree to which attention is directed. **When you concentrate, time feels "packed."**

• $${N(t)}$$: Novelty
  → First-time experiences and unexpected events leave an impression and stretch the sense of time.

• $${M(t)}$$: Memory
  → The more you can "recall" it later, the more that moment **"remains" as subjective time**.

• $${B(t)}$$: Boredom
  → Boredom is a factor that dilutes subjective time. It is in the denominator because it has the power to diminish the experience.

• $${C_0}$$: Constant (Base Sensation)
  → No matter how little you feel, time never reaches zero. It is the minimum sense of time.

• $${k}$$: Scaling factor (Coefficient)
  → A constant that adjusts the "sensitivity to time" based on age, culture, and individual differences.

⏳ Time is "Density"

What is interesting about this formula is that it does not measure the "quantity of time," but rather depicts the "thickness of time = density."

For example, try to recall an experience like this:

• The first day of your first trip abroad → Time felt long and dense

• An afternoon spent aimlessly scrolling on your smartphone → Time felt fast and thin

This is because the concentration of Emotion × Focus × Novelty × Memory changed the "volumetric feel" of time itself.

📊 Time that remains in memory is proof that time was flowing

Strangely enough, days when you feel "that day was long" are days with many memories. In other words,
subjective time is accumulated as "density of experience".

If so, how can you design a day to make it "dense"? Conversely, in what state does
"time melt away and disappear"?

This formula might just provide the answer to such questions.

🔍 The "Landscape of Time" revealed by the formula

When you factorize an abstract concept like "time" and view it as a formula, the sense of intellectual engagement increases significantly.
It is like giving contours to the fog.

In the next chapter, we will plot this density $${D(t)}$$ on a graph. Once the formula can depict even the "flow" of time, your "perspective on perceived time" will surely change.

📈 [Chapter 2] What I saw when I turned subjective time into a function

Trying to depict "flowing time" on a graph

In the previous chapter, I "factorized" the
"density" of subjective time into elements such as emotion, focus, novelty, and memorization.

So, what happens if we can depict how this density changes over time?

—Yes, by treating it as a function, the "landscape of time" that was previously invisible begins to emerge.

⏳ Subjective time is the "density of accumulated experiences"

Usually, time is treated as an "objective thing" measured by a clock.
However, our actual physical experience is the result of the "intensity" of each passing moment accumulating.

Therefore, subjective time $${T_s}$$ can be expressed as follows:

$${T_s = \int_{t_0}^{t_1} D(t) \, dt}$$

It is like the image of summing up the "density of experience at each moment," just as one might accumulate the strength of coffee cup by cup.

Here, $${D(t)}$$ is the subjective time density at that point in time.

For example, think of it as something like the "temperature of the heart" at that moment, where the higher the temperature, the more "intensely time is felt."

📐 Mapping a "Day of the Heart" with Mathematical Formulas

Now, let's actually simulate a single day.
The following graph calculates and visualizes a hypothetical "daily time density":

🧭 An example of a virtual daily life (weekday):

• 8:00 | Morning coffee and small moments of wonder → Emotion and focus increase

• 10:00–12:00 | Work focus zone → Time density peaks

• 12:30–13:30 | Lunch break daydreaming time → Density drops

• 15:00 | Afternoon re-focus → Memory and concentration intersect again

• 20:00 | Emotions run high during conversation with a friend → Subjective time becomes intense again

• Night | Relaxed time → Density is slightly lower as the day ends

🧠 Formula for subjective time density $${D(t)}$$ (Recap)

$${D(t) = C_0 + k \cdot \frac{E(t) \cdot A(t) \cdot N(t) \cdot M(t)}{1 + B(t)}}$$

Each factor fluctuates over time and possesses its own characteristic "time dependency."

This is much like how plants adjust the intensity of their photosynthesis depending on the time of day; our state of mind also changes according to time.

Emotion $${E(t)}$$: Intensity of emotion
→ It tends to rise sharply due to external stimuli and then decay over time.
→ Mathematical model: exponential decay

🌟Metaphor: Like a sparkler, it scatters sparks all at once at first, but then gradually fades away—that is the rhythm of emotion.

Attention $${A(t)}$$: Degree of concentration
→ It gradually decreases over time, but also shows waves of concentration (periodicity) like the Pomodoro technique.
→ Mathematical model: Gaussian or sine curve (periodic function)

🌟Metaphor: Concentration is the rhythm of the tide, ebbing and flowing. It visits repeatedly, sometimes strongly, sometimes gently.

Novelty $${N(t)}$$: Novelty
→ As the same tasks or environments continue, one gradually gets used to them, and the sense of freshness diminishes.
→ Mathematical model: Reciprocal function

$${\frac 1 {1 + βt}}$$

🌟Metaphor: You are captivated by a view you see for the first time, but if you take the same road every day, the "scenery" becomes just a background—that is the decay of novelty.

Memory $${M(t)}$$: Ease of remaining in memory
→ It reflects the "depth" of memory generated by the combination of emotion, concentration, and novelty, averaged over time.
→ Mathematical model: moving average

🌟Metaphor: Memory is not a single photograph, but like film exposed little by little, it is burned into the mind repeatedly and becomes stronger.

Boredom $${B(t)}$$: Degree of boredom
→ As emotion and concentration fade, boredom increases and rises monotonically. Conversely, it decreases when E, A, and N are high.
→ Mathematical model: Monotonically increasing function or inverse function

e.g. $${\frac1 {E + A + N + 1}}$$

🌟Metaphor: Boredom is like a bathtub where the water level rises quietly; before you know it, it gradually submerges our senses.

By combining these, we can visualize the invisible "way we perceive time" using mathematical formulas.

🎯 Why is this important?

We usually talk about time intuitively, calling it "long/short" or "busy/free." But that is the integral result of "what we were feeling and how we were feeling it."

In other words, time is not a "flow," but something you "create".

• With emotion

• With focused attention

• By experiencing new things

• By spending time in a way that leaves memories

—time designed in such a way becomes high-density, meaningful time.

🧭 Hint for the next chapter: "Let's design your day"

In the next Chapter 3, I will introduce a method to actually
map out your own day using a "time density graph" based on this model.

Moving from time that is "just spent" to time that is "self-designed." As a first step, why not "visualize" your subjective time?

🧠 [Chapter 3] Let's map out your day using a "time density graph"

The concept of designing the "quality of time"

In this chapter, using the thoughts and mathematical models covered so far, we will delve into the perspective of
visualizing and designing "how you spend your time".

Every day, we feel like "I have no time," "it went by in a flash," or "today was long." That depends not simply on
"whether the clock has moved", but on how much your heart was engaged with that time.

If so, it should be even more interesting if you could "visualize" the density of your own time.

🗺️ Step 1: Self-diagnosis of subjective time density

First, try to be conscious of the following five elements throughout your day.

• Emotion: Were your feelings stirred in that moment?

• Attention: Was your consciousness firmly focused?

• Novelty: Were there any new discoveries or stimuli during that time?

• Memory: Will you be able to remember it later?

• Boredom: Were you just "getting through" the time?

Let's try to rate these on a 5-point scale (0-4) for each time period.

Try keeping a log casually, as if it were a thermometer for scoring your own mental condition.

📝 Step 2: Try keeping a daily log (Example)

For example, try self-evaluating your "Emotion, Attention, Novelty, Memory, and Boredom" for each time period on a rough 5-point scale (0-4).

☀️ 8:00–9:00 | Morning routine and commute

• Emotion (E): 3 (My mood was lifted by my favorite music)

• Attention (A): 4 (I was focused on getting ready for work)

• Novelty (N): 2 (I listened to a new podcast)

• Memory (M): 3 (The topic was memorable)

• Boredom (B): 1 (It wasn't that boring)

▶ During this time, emotion and attention were high, and there was some novelty.
It felt like time was "packed tight."

💼 10:00–11:00 | Desk work during business hours

• Emotion (E): 2 (I was proceeding somewhat matter-of-factly)

• Attention (A): 3 (I was focused)

• Novelty (N): 1 (Routine work as usual)

• Memory (M): 2 (Might leave a faint memory)

• Boredom (B): 2 (Started to feel a bit sluggish)

▶ Time that was lacking in emotion and freshness despite being focused.
It felt more like "work progressed" rather than "time passed."

🍽️ 13:00–14:00 | Post-lunch hazy time

• Emotion (E): 1 (No particular emotional movement)

• Attention (A): 1 (Too sleepy to focus)

• Novelty (N): 0 (Completely routine)

• Memory (M): 1 (Cannot recall what I was doing)

• Boredom (B): 3 (Quite boring)

▶ A low-density blank zone. This time slot is a classic example of "leaving no memory behind."

🔁 15:00–16:00 | Afternoon focus time

• Emotion (E): 3 (Felt a sense of fulfillment)

• Attention (A): 4 (Immersed in the task)

• Novelty (N): 2 (New ideas emerged)

• Memory (M): 4 (Likely to be able to look back on this later)

• Boredom (B): 0 (Not bored at all)

▶ The peak of ideal time density.
A moment where I felt, "I didn't realize it was already evening, but that was a rich period of time."

🌙 20:00–21:00 | Phone call with a friend or reading time

• Emotion (E): 4 (My heart was moved)

• Concentration (A): 3 (I was able to get absorbed in the story or book)

• Novelty (N): 3 (I had unexpected discoveries)

• Memory (M): 4 (It will definitely remain in my memory)

• Boredom (B): 1 (I was hardly bored at all)

▶ Emotion × Freshness × Memorability A time that leaves a "lingering emotional aftertaste" when these are combined.
In such moments, even the next day, you can feel that "last night was good".

In this way, when you log the "emotional movement and density of time" of a day as a score,
where your "quality of time" increases and where it is lost emerges.

It is like a weather map recording the "atmospheric pressure changes of the heart" throughout the day, allowing you to see the changes within yourself.

📊 Step 3: Seeing the "Shades of Time" in a Graph

When you turn it into a bar graph or line graph, you will make discoveries such as the following:

• "Dense Time": Time where emotion, concentration, and novelty are all present (= an unforgettable moment)

• "Time that completely slipped away": A zone without emotion or concentration, which did not remain in memory

• "Time where boredom took center stage": A time period where the experience was diluted and time just "flowed by"

💡 Step 4: The Concept of "Time Design"

What becomes visible through this analysis is the perspective that
time is not something you "spend," but something you "can design".

For example:

• You tend to get sleepy right after lunch → Add novelty to increase stimulation

• Emotions are easily moved at night → Use it for important conversations or reading

• If you feel bored, consciously increase either emotion or concentration

Subjectively "designing" time is also an attempt to change the very density of your life's experience.

✨ Recommended Exercise: Try keeping a "Today's Time Density Log"

1. Whether on paper or your smartphone, try taking notes every 3 to 4 hours.

2. Score your emotion, focus, novelty, memory, and boredom on a scale of 0 to 4.

3. Try calculating your own $${D}$$.

4. Look back at your day's "density map" and think about "what influenced it."

Even just doing that will help you see the "habits of your time perception."

📌 Next Preview: Summary and Questions for the Reader

In the final chapter, while looking back on this journey of subjective chronology, we will head toward the question of "how to
"live" time."

What kind of time did you "spend" today, and what kind of time did you
"feel"?

🏁 [Summary] Time is not a "number," but a "density"

Beyond the idea that "a day is 24 hours"

"I was busy today," "it went by in a flash," "it was a dense time" — we describe our daily events through our
"sense of time."

But that is not just the progress of clock hands. It is
proof that your heart was moved and a trace of a dense experience.

⏱️ Time is not something that flows, but something that is felt

In this column, I have factorized what we call "subjective time."

• Was your emotion moved?

• Were you able to focus (Attention)?

• Did you feel freshness (Novelty)?

• Will it remain in your memory (Memory)?

• Was it not boring (Boredom)?

The formula derived by combining these elements:

$${D(t) = C_0 + k \cdot \frac{E(t) \cdot A(t) \cdot N(t) \cdot M(t)}{1 + B(t)}}$$

And by accumulating this density, our "day as experienced" is formed.

📊 The "Other Time" Revealed by the Formula

Through this formula and graph, we were able to find
"which moments the heart engaged with, and where time slipped away." we were able to find.

It could be said that this is an act of "reclaiming the 'contours' of your own life from the flow of time that simply passes by."

🧭 Where is your time heading right now?

The goal of this column is not merely to explain a mathematical formula.
Rather, it lies in "gaining a perspective on how you perceive your own time" through that formula.

• What kind of time do you feel is "dense"?

• What emotions or focus were present during that time?

• And today, what kind of time was a "memorable minute"?

✨ Designing "Dense Time"

Finally, I would like to offer these words.

May "time with depth" be etched onto the pages of your life.
Time that cannot be measured by numbers, but that you can certainly feel you have "lived." And that time,
will continue to be freely designable depending on how you choose and perceive it.

If this column has given even a slight change to your "view of time," I would be very happy if you could let me know with a Like 💙, Follow ✅, or Comment 💬.

Thank you very much for reading until the end.

(This column was structured and written while utilizing ChatGPT. It is part of an experiment to explore a new form of knowledge where generative AI and human thought collaborate.)

Profile | Kazuomi Matsunaga

🧭Thinking Navigator

Biography:Graduated from Kyushu University, Faculty of Engineering (Materials Engineering) -> Engineer at a metal materials manufacturer in the Chukyo region (Production Engineering) -> Advertising sales for a local free paper in Nagasaki Prefecture -> University staff member (<- currently here). Living in Nagasaki Prefecture, in my 50s. Both of my children have left the nest to attend university, and I am currently living with my wife.
I am a writer who lets my thoughts play daily at the intersection of "questions" and "curiosity." Recently, I have been writing about my own unique perspectives on themes such as education, storytelling, psychology, AI, and design, centered around my serialized column, "The Science of Thought." I am exploring "free knowledge design" that transcends existing frameworks. My motto is "Imagination opens up the world."
In my professional life, while working in university and education-related fields, I value carefully observing "the moment someone starts to 'like' something" and "the scenes where questions begin to sprout."
🧭 I would be happy if I could deliver articles that leave you with a "new perspective" after reading them.
📮 Impressions and messages of support are always welcome!