Tech stack/hierarchy in mathematics

What are layers?

Consider the OSI model. In technology, in building a complex system, one often divides the system into layers with simple exterior appearance, but complex interiors. The layers are basically modules that have a direction: whereas "modules" are egalitarian, and can be assembled in many directions, layers have a fixed pecking order, and can only be assembled in one fixed direction.

Layers = Modules affected by gravity

What is gravity? Gravity is the reality-mentality strain, when reality is far away from mentality. Take for example tho OSI model.

Layer 1 is on the physical material, where electrodynamics, friction, heat dissipation, signal distortion, waveguides... live. This is the world of physicists and base material.

The higher up, the closer to the human mind. Until we arrive at the application layer, where humans spend most of their time interacting with the Internet.

This motif of "high vs low" is a universal metaphor, and can be seen everywhere. The basic structure is the same:

• High, expensive, beautiful, rare, imaginary, cultural, abstract, general, universal, human-understandable, stratosphere.
• Low, cheap, ugly, plentiful, real, physical, concrete, specific, provincial, inhuman, rock bottom.

This metaphor can be found outside of technology. Examples include

• In social hierarchy, the low class interacts physically with non-human physical matter, while the high class interacts verbally with humans.
• In Marxist sociology, the society has a substructure and a superstructure. The substructure works with physical matter, while the superstructure works with people.
•  Low entertainment interacts directly with emotions, while high entertainment interacts with potentialities and abstractions of emotions, or ideas that relate to emotions.

Mathematical modules

Knowing about modules and layers in technology, we turn to modules and layers in math. This video shows it pretty well. You could also read The Map of Mathematics by Quanta. Or Mathematics Subject Classification.

So those are modules. How about hierarchy? As before, hierarchies emerge when there is "gravity". There are two kinds of mathematical hierarchies, which I give below.

Example 1: the towering abstraction of modern algebraic geometry

I cannot emphasize enough how abstract modern algebraic geometry has become. Most of it must be blamed on Grothendieck, the revolutionary. The way of Grothendieck is explained as follows:

Grothendieck describes two styles in mathematics. If you think of a theorem to be proved as a nut to be opened, so as to reach “the nourishing flesh protected by the shell”, then the hammer and chisel principle is: “put the cutting edge of the chisel against the shell and strike hard. If needed, begin again at many different points until the shell cracks—and you are satisfied”.

The first analogy that came to my mind is of immersing the nut in some softening liquid, and why not simply water? From time to time you rub so the liquid penetrates better, and otherwise you let time pass. The shell becomes more flexible through weeks and months — when the time is ripe, hand pressure is enough, the shell opens like a perfectly ripened avocado! A different image came to me a few weeks ago. The unknown thing to be known appeared to me as some stretch of earth or hard marl, resisting penetration ... the sea advances insensibly in silence, nothing seems to happen, nothing moves, the water is so far off you hardly hear it ... yet finally it surrounds the resistant substance.

Deligne describes a characteristic Grothendieck proof as a long series of trivial steps where “nothing seems to happen, and yet at the end a highly non-trivial theorem is there”. 

According to Grothendieck, to solve a problem, one must find the right viewpoint, where the problem become trivial. In order to get to the right viewpoint, he built a massive landscape out of abstraction upon abstraction.

In 1958, when Grothendieck (aged 30) announced a massive program to rewrite the foundations of geometry, he assembled a coterie of brilliant followers and conducted a seminar that met 10 hours a day, 5 days a week, for over a decade. Grothendieck talked; others took notes, went home, filled in details, expanded on his ideas, wrote final drafts, and returned the next day for more. Jean Dieudonne, a mathematician of quite considerable prominence in his own right, subjugated himself entirely to the project and was at his desk every morning at 5AM so that he could do three hours of editing before Grothendieck arrived and started talking again at 8:00. (Here and elsewhere I am reporting history as I’ve heard it from the participants and others who followed developments closely as they were happening. If I’ve got some details wrong, I’m happy to be corrected.) The resulting volumes filled almost 10,000 pages and rocked the mathematical world.

The scale of abstraction is palpable in the sheer amount of prerequisites one needs in order to get to even understand statements of theorems in modern algebraic geometry, which uses jargons like "scheme" "nilpotent" "stack" "germ" "stalk" and so on. One ambitious project to lay down the fundamentals of modern algebraic geometry, the Stacks Project, currently has 6933 pages and is still a work in progress.

Kevin Buzzard wrote:

They are working on results about about objects which in some cases take hundreds of axioms to define, or are even more complicated: sometimes even the definitions of the objects we study can only be formalised once one has proved hard theorems. For example the definition of the canonical model of a Shimura variety over a number field can only be made once one has proved most of the theorems in Deligne’s paper on canonical models, which in turn rely on the theory of CM abelian varieties, which in turn rely on the theorems of global class field theory... I once went to an entire 24 lecture course by John Coates which assumed local class field theory and deduced the theorems of global class field theory. I have read enough of the book by Shimura and Taniyama on CM abelian varieties to know what’s going on there. I have been to a study group on Deligne’s paper on canonical models. So after perhaps 100 hours of study absorbing the prerequisites, I was ready for the definition of a Shimura variety over a number field. And then there is still the small matter of the definition of etale cohomology.

The hierarchical structure of algebraic geometry does not come from any real kind of gravity. Rather, algebraic geometry is more like an archipelago in the land of mathematics. Students start at the shore of conic sections, proceed to cubic curves, then projective geometry, then complex projective geometry, and so on. Each island is reachable only from the previous island. Each island can stand alone, but for humans who are born with intuitions about curves, but not schemes and abelian varieties, the only path to reaching the depths of the island chain is by going one by one through the chain.

When climbing a mountain, one starts at the bottom, because of gravity. When traversing an island chain, one starts at the shore, not because of gravity, but because one is born on the shores. An algebraic geometer who is trying to reach the end of the island chain feels the pull of gravity from their human birth. Gravity comes from the on-shore location of human intuition in the land of mathematics.

I will call this kind of hierarchy "archipelagic hierarchy".

Example 2: the many guises of mathematical quantum mechanics

Now we know what are modules and hierarchies, we can look at an example hierarchy: mathematics of quantum mechanics. I am not talking about how Newtonian physics is an approximation of relativistic quantum mechanics. I'm only talking about exact results.

There are many equivalent ways to define quantum mechanics as a mathematical structure, and yet they have different flavors and are generally taught differently. From what I see, there are these layers:

1. Monoidal categories
2. C*-algebra
3. Operator theory
4. Hilbert spaces
5. Square-integrable-function spaces and matrices

The kind of quantum mechanics that students learn in university are almost invariably the last kind, and for engineers, this is the only kind they really use. More ambitious students learn Hilbert spaces, and some quantum mechanics textbooks actually start with Hilbert spaces. Operator theory and C*-algebras are reserved for graduate students with at least a full-year course on abstract analysis. Monoidal category quantum mechanics is a research topic that has only appeared in a very recent textbook.

The reason it takes such a terribly long and winding path for a student of QM to reach monoidal category QM is not because it is hard, but because historically, this is how mathematical QM developed, and it seems ontogeny recapitulates phylogeny. In fact, the monoidal category formulation of QM is so visually striking and simple that Coecke had the audacity to title a paper Kindergarten Quantum Mechanics.

This hierarchy, however, can be justified in one way: each level can abstract the level below, and represent the level above. Just like groups abstract symmetries, and symmetries represent groups, Abstract operator theory is represented by abstract Hilbert spaces, which are represented by square-integrable-function spaces and matrices.

Here, the gravity is in the abstract-representation direction. I call this kind of hierarchy "representational hierarchy".

There is however no necessity to climb the representational hierarchy in order. It's entirely possible to start directly with category theory, and learn the representations afterwards, or simultaneously. Computers have been programmed to directly reason in category theory formulation of QM, in the Quantomatic project. Considering this, one might see that the gravity of this hierarchy as historical, since the hierarchy follows the historical development of QM, rather than the logical development.

Speculation on future math

To keep modern society from collapsing, the economy must grow. To keep the economy growing, the researchers must keep delivering. To get keep new researchers coming, knowledge must be taught to new generations at an increasing pace. In the long past, university degrees in math taught nothing more than calculus, geometry, and classical mechanics. Now it is functional analysis, abstract algebra, category theory, probability and statistics, combinatorics, among others.

As time goes on, the amount of learning necessary keeps growing, unless knowledge can be refactored so that it takes shorter time to learn enough to rush to the cutting edge. There is hope that this can be done, by several methods.

Cleaning inside a module: a lot of mathematicians don't do breakthroughs, but polishing. They carefully review what has been done, and straighten out the proofs, arrange them into a logical sequence, assess their relative importance and intuitiveness, write textbooks. These kind of solid polishing is how we got from Newton's Fluxions to AP Calculus textbooks.

Refactoring between modules: this is similar to polishing inside a module, in that it is mainly work to make the logical connections between structures more legible, rather than creating new structures.

Paradigm shift: a revolution in how the mathematical modules are structured can make things easier to learn. The heliocentric system is much more elegant than the epicycle-on-epicycle system. Category theory and type theory provides a much more powerful language for organizing all the equivalences of mathematical structures. 

I foresee several paradigm shifts in mathematics: 

• Type theory and category theory together would usurp set theory and take over the foundation of mathematics. Set theory is so 20th century!
• Computer formalization of mathematics would fundamentally change how math is published and trusted. Instead of basing trust on word of mouth and personal verification, trust in proofs would be based on computer certificates. In 10 years it would be common for mathematicians to formalize the abstracts of their papers, and in 20 years, the entirety.
• Nonclassical logics, such as intuitionistic logic, linear logic, internal logics of toposes, etc, become widely recognized as useful. They would not replace classical logic, but mathematicians would recognize that alternative logics are not philosophically suspicious, and are very useful in certain contexts.
• Similarly, rigorous use of infinitesimals, like synthetic differential geometry and nonstandard analysis, would come out of obscurity and become part of the standard toolkit of the working mathematician.
• Geometric algebra and analysis would feature prominently in linear algebra, replace vector analysis, and become the language of choice in parts of differential geometry, Lie theory, and complex analysis. It would also become the main tool for a new edition of Geometrical Methods of Mathematical Physics.

Change the curriculum: remove outdated knowledge to make way for the new. Rearrange course material. Add new viewpoints.

Some ideas include:

• Remove exact arithmetics (such as 12 * 13 = 156) and replace it by inexact arithmetics and reasoning with orders of magnitude in the style of Fermi estimates.
• More programming (I particularly like LOGO Turtle).
• Teach (monoidal) category theory by string diagrams.
• Teach intuitive calculus before "precalculus", the kind of intuitive calculus that can be done by just drawing diagrams, the kind that Newton, Cavalieri, and Archimedes did.
• Teach geometrical algebra and analysis, instead of this current mess of vector analysis, analytical geometry, complex analysis, etc.
• Teach basic type theory and dimensional reasoning (which relies on type theory) in combination with high school physics.

I'll write more about how to reform mathematics.

Cramming: just make students devote longer time to studying. It can and has happened. In early 20th century, most people did not go to high school, but as time goes on, more and more people go to universities. This chart from the world bank shows it rising from 10% to 38% in the past 50 years:

South Korea is an extreme example, with 68% of all people eventually attending tertiary education. This heavy education fueled South Korea's rise in economic productivity, but even South Korea cannot squeeze much more learning power out of its people.

Presumably, cramming is going to stay. People would have to keep studying throughout their lives. This does not necessarily mean that they would take longer to start working. Rather, education of the future must provide a very flexible base upon which to quickly learn and un-learn new things. Scientists and other professionals of the future are going to be particularly stressed, as they are people closest to the firehose of new information. More and more of their time would be spent learning new tools to manage the firehose, as well as ways to de-stress, work without burnout.