[LAW Document 72 Hours] Are you a "Differentialist"? Or an "Integralist"?

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In his article "On Happiness - Part 4: 'Differentialists' and 'Integralists'," Ken Kusunoki explains the differences in how people perceive happiness by dividing them into

"Differentialists"

and

"Integralists."

He explains:

"A 'Differentialist' is a type of person who feels happiness by looking at the change between two relatively close points—for example, having achieved something, gotten through a challenge, or seen an improvement in their own evaluation—and the magnitude of that change.

"An 'Integralist' is a type of person who feels happiness not from the rate of change, but from the total area accumulated from the past.

Of course, this is not about which is better or worse, but simply a difference in human types; I believe people can be divided into those who seek happiness through the magnitude of the integrated value and those who seek it through the magnitude of the differentiated value."

In my distant memories, weren't differentiation and integration mutually complementary relationships, and so on?

It is a great learning experience to use mathematics to speak elegantly and take a bird's-eye view of things (^^)

In the sense of learning the logical reasoning essential for living without being deceived in the modern age, as a precursor to Toshiya Sato's

"Space Alien Chipmunk Learns Medical Statistics" (Iwanami Science Library) by Toshiya Sato,

perhaps

"Expanded New Edition: An Introduction to Mathematics for Survival" (Yorimichi Pan! Se) by Noriko Arai

feels like a must-read for everyone (^^)

Conversely, if you have awakened to the fun of mathematics itself, after this book, I recommend Hiroshi Yuki's

"Math Girls" by Hiroshi Yuki.

is recommended.

The methods for solving various problems in Newton's "Principia," written in the 17th century, strictly follow Newton's own approach.

And this is, in fact, extremely difficult.

The reason it is difficult, ironically, is that they were done using elementary concepts—that is, concepts and vocabulary that had already been established before Newton.

Perhaps it is easier to understand if I say it is like how the crane-and-tortoise problem is difficult for those who do not know equations.

Even for problems that a modern person would solve using algebra and calculus, Newton insists on solving them geometrically.

Even though Newton himself was one of the discoverers of calculus, the Principia does not contain Leibniz-style symbols like "dx/dt," nor even Newton-style symbols like "x'" (read as "x prime").

It is strictly

“when the difference is made infinitely small”

a phrase that, in modern times, would use

“lim”

which appears only in words, not in mathematical formulas.

Even so, it is a relief that the arithmetic and square root symbols used in the occasional mathematical formula are the same as those used today, but the extremely famous "SECTIO III, PROPOSICIO XI. PROBLEMA VI.," the derivation of the inverse-square law from an elliptical orbit, is no exception to this.

The proof for this diagram, which appears on "Original P. 118, English Translation P. 207," is truly elegant, but when it comes to clarity, it falls far short of the method appearing in

“Introduction to Mathematics, Vol. 2” (Iwanami Shinsho) by Hiraku Toyama

where the ellipse is expressed in polar coordinates, a differential equation is set up, and it is solved calmly.

Of course, the differentiation is in the Leibniz-style expression, not the Newton-style.

If you compare the original text with the English translation, you will see that the number of mathematical formulas increases as you go toward the latter.

And the clarity follows this order as well.

"Mathematical formulas are difficult."

If you feel that way, I highly recommend comparing this book with an "Introduction to Mathematics."

You will inevitably realize just how much mathematical formulas simplify things.

At the same time, you will notice how conservative Newton was as an author.

His modest use of formulas is one example (prioritizing words (in Latin, of course!) over formulas).

The fact that it took him twenty years to publish the discoveries he made in his twenties is another.

And above all,

"Philosophiae naturalis principia mathematica"

is the title itself.

It is "Philosophiae naturalis"

but it is not

"scientia"

.

The word "science" did not yet exist, and Newton did not approve of coining new words.

Even though he later became the Master of the Mint (where coins are made).

Newton himself

would likely respond if called a

"Natural Philosopher," but

if called a

"Scientist," he would not.

"Huh?"

That might have been the reaction.

Partly for that reason, Newton is,

"the first scientist"

rather than,

"the last alchemist"

is how he is often called in recent years.

However, there is a sense of discomfort there, just as there is in calling Julius Caesar,

"the first emperor"

instead of

"the last leader of the Republic."

It feels the same.

"It"

Even if there is no word to express it,

"it"

is undoubtedly what it is.

Calling him "the zeroth modern scientist" fits best.

With that said, I read a simpler book on calculus and woke up my long-dormant

"mathematical brain."

With the desire to awaken it, I decided to (briefly) pull myself together!

Hiroyuki Nagano, 'Calculus Once More'

Point 1:
Differentiation demonstrates its power when finding the maximum or minimum values of a function, or when calculating approximate values.

Point 2:
Differentiation and integration are inverse operations of each other.

Point 3:
Differentiation means 'dividing into minute parts,' while integration means 'stacking up the divided parts.'

Point 4:
In physics, as well as in economics, sociology, and ecology, anyone who takes the stance of trying to unravel phenomena that change due to certain factors will inevitably have to face differential equations.

Yusuke Tomishima, 'Calculus You Can Understand Just by Looking'

Point 1:
Calculus is mathematics for predicting the future.

We capture small changes with differentiation and stack those changes up with integration to predict the future.

Point 2:
Differentiation and integration of road curves.

When a driver steers at a constant pace, we calculate the direction of travel at each and every moment (differentiation).

By calculating the trajectory the car will draw as a result (integration) and determining the shape of the road, accidents are reduced.

Point 3:
Differentiation and integration in weather forecasting.

We divide the atmosphere into a fine grid (differentiation) and calculate the air pressure, temperature, and humidity for each part.

We add those results together (integration) to predict the weather.

Point 4:
Differentiation and integration in planetary exploration.

Based on Tsiolkovsky's formula, we control the rocket's propulsion direction and speed (differentiation) and calculate how to change them to reach the destination (integration).

Toshiyuki Kobayashi, 'Differentiation and Integration to Build Fundamental Strength'

Fumiharu Kato, 'Suken Lecture Series: University Liberal Arts Calculus'

Fumiharu Kato, 'Chart Institute Series: University Liberal Arts Calculus'

・
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When you encounter a difficult problem that you cannot immediately understand, the following approach is also considered effective.

1. Break it down into smaller, easier-to-think-about units and analyze them (a differential way of thinking)

2. Aggregate those analyses to consider the whole (an integral way of thinking)

Points to keep in mind when you encounter something.

That is,

・Things you have encountered in the past but have forgotten, or

・Seeing someone else handle something with ease that you gave up on because you couldn't understand it

Be careful not to be impressed by that and just accept it as it is (^^;

Don't take anything at face value,

'Is that really true?'

2024 was a year where I once again realized the importance of doubting and thinking for myself (^^)