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数学的目的

I want to overview and understand all of math. I'm taking two approaches.

• Math discovery I'm documenting and organizing the ways of figuring things out in mathematics.
• Map of math I'm creating a map that show how the branches of math unfold to yield its key concepts and results.

Each discipline has a House of Knowledge that systematizes its ways of figuring things out. That House of Knowledge reflects the observer that the discipline supposes. Mathematics is that discipline which considers an observer in the most unrestricted sense. Thus mathematics pervades all disciplines.

The nature of math

• The basic driver of math is ambiguity, as in the use of variables.
• Mathematics is based on generalization, abstraction.
• Logic and category theory are examples of abstraction, separating general arguments about argumentation or about mathematical content from particular methods in particular branches.
• Category theory introduces a black box approach that emphasizes the external transformations rather than the internal structure, the system of internal subsystems.
• Fundamental lemma in category theory: Any mathematical object can be characterized by its universal property - loosely by a representation of the morphisms to or from other objects of a similar form.
• Grammar (natural language) always has a meaning. Mathematics is that which obeys logic, but need not necessarily have a meaning.
• Mathematics teaches us how unconditionality expresses itself in conditions, which is possible conditionally.
• Jeigu yra matematinis apibrėžimas (pavyzdžiui, baigtinė tiksli seka), kuris galioja vienam žmogui, tai tai galioja visiems. Šitą mintį mąsčiau bekliedėdamas, besirgdamas.
• Mathematics is the study of structure. It is the study of systems, what it means to live in them, and where and how and why they fail or not.

Emily Riehl, Category Theory in Context. Preface: "Atiyah described mathematics as the “science of analogy.” In this vein, the purview of category theory is mathematical analogy." For me, adjunctions are the analogies.

读物

Courant, Robbins. What Is Mathematics?

Philosophy of math

• Who knows two? essay
• David Corfield: Towards a Philosophy of Real Mathematics. Introduction.
• https://www.researchgate.net/publication/280387376_Real_numbers_A_critique_and_way_forward

Older Notes from Math Future

Joe, I very much like your focus on "why" and your many beautiful examples.

They bring to mind a few more reasons for "why" we have math:

• As the basis for a caste system based on how much math you have passed. A way of controlling who is in what profession, for example.
• As a way of treating people differently (through rates, credit scores, incomes, etc.) without them having full knowledge or even understanding what its all about.
• As a way of making our systems just incomprehensible enough to most people so that they can't argue with them. For example, most people think that banks loan out money based on the deposits they have. But actually, the central banking system and participating banks are chartered by the government to create loans in an amount ten times or more than whatever assets a bank has; but nobody creates the money needed to pay the interest on those loans, which grows exponentially; which might be all right if the economy itself grew exponentially; but we have thereby legislated the need to grow exponentially, naturally or (when that fails) otherwise; thus the pressure to (artificially) monetize everything in sight; and to prey on the most vulnerable (a major reason why ghettos persist, I think). So that bubbles (based on money for money's sake) are inevitable. Similarly, the recent housing crisis was an application of math.
• Math also lets us model realities in ways that let us suspend thinking about the underlying meaning. Which is essential for modern warfare.

Computers (and all systems) likewise allow us to ignore the underlying meaning. Social software is in a large part a way to avoid human contact by controlling it in very rigid channels.

Joe, my examples are negative, but I think the positive side would be math for citizenship.

I suppose that a distinction can be made between what must be taught-learned and what should be optional. I once thought that what really need to be taught is ethics, what is right and wrong. For example, language should be taught as a way of empathizing with others and ourselves, of caring about them. Math should be taught as the study of systems, especially the systems that we find ourselves in. It's morally essential for citizens in our modern world to distinguish between linear, exponential and periodic behavior and appreciate the implications. Overall, I imagine having a required school of just maybe two hours a day but that focused only on what is agreed to be absolutely essential. Which I think would include drill of "math facts" (multiplication tables, etc.) And most adults as well would be required to regularly show competence. Then the rest of education would all be optional.