Generic framework

(Metaphorical definition) Mathematics is the intellectual discovery of nature's eternal, immutable infrastructure.

By "infrastructure", I mean that the discovery ultimately targets fundamental, ubiquitous ideas that we sense when observing appropriate configurations of things. (I am deliberately imprecise here, in the spirit of the quotation under the title of this essay.) By "intellectual", I mean that the process of discovering happens entirely within one's mind.

All ideas in mathematics are tied to observations. For example, when we look at {apple, apple, apple}, or {orange, orange, orange}, we sense "threeness", denoted by the symbol 3. When we observe the edge of a ruler, we sense a "straight line segment", even in spite of knowing that it would not look smooth through a magnifying glass. Looking at a computer network plan, we sense the idea of a "graph". Imagining a row of natural numbers, starting from 0 and fading away in the distance, can lead us to sense what we'd call the "set of natural numbers", denoted by ℕ. When we throw the dice, we sense "randomness". When we think of all humans ever born, we sense "potential infinity" (the actual set is finite at any given point in time, it's never completed, and may grow indefinitely). When we write down √-1, we sense something weird as if a number whose square is -1 existed (it sounded really weird before complex numbers became widely accepted). When we ask if there may be a positive quantity smaller than any positive real number, we sense the "infinitesimally small".

The ideas in the previous paragraph are sensed with varying vagueness. While "threeness" is pretty clear, √-1 and "infinitesimally small" feel like guessing. Since all the ideas are unclear to some (varying) extent, we need to invent theories in order to describe them and their relationships.(1) (2)

As for the methodology, the starting point of mathematical theories are fundamental, ubiquitous ideas (incl. logic) about which we have accumulated so much and so consistent day-to-day experience that makes it possible to confidently rely on our intuition. We just close our eyes, and in an iterative process, come up with new ideas based on the already available ones, think out or take note of assumptions about the ideas we have, and discover the logical consequences of the assumptions.(3) The only constraint is that the resulting theories must constitute conceivable stories about our world. As a minimum, statements including "clear" ideas (e.g. natural numbers, finite sets) should coincide with our experience, and statements including "unclear" ideas (e.g. √-1, infinitesimals, infinite sets) should not lead to known contradictions.(4) (5)

The reason for mathematics transcending cultures and millennia is that humans have always had very similar experiences about the fundamental, ubiquitous ideas from which mathematics emerged. An alien civilization, if any, might develop a mathematics very different from ours, provided they exist in a very different environment and/or have very different sense organs.

The merits of a mathematical theory are assessed based on its beauty, success, and consistency. With regard to the development of mathematics, the first aspect is the most important guiding principle. The reason is, as once somebody put it, that beauty is felt as a result of sensing a deep law of nature. (I am deliberately imprecise here too.) As opposed to other sciences, the aim in mathematics is thus not to pinpoint and test, but rather to reflect laws of nature. That is, mathematics is both a science and an art.

In the following, standpoints of various philosophical schools are presented and commented. They can be used to customize the framework outlined above. It's a matter of personal preference.

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(1) Isn't "threeness" crystal clear?
Good question. I don't think it is.

(2) Is √-1 a ubiquitous idea?
No, it's neither fundamental nor ubiquitous, but it bears a close relationship to such ideas, which at the end of the day makes it interesting for mathematical discovery. On a related note, the various kinds of mathematical spaces (vector spaces, topological spaces, etc.) are all about discovering what emerges from the interplay between the fundamental, ubiquitous ideas of set, relation, and operation.

(3) What about discoveries where computers are used?
There is a philosophical concept of the "extended mind", which for the mathematical practice means that the mind can be aided by things like pencil and paper, calculator, or computer. Such extensions boost already existing capabilities of the mind, especially the memory and the ability to derive logical consequences.

(4) What if it turns out that a story is not true?
No theory is true, there are only circumstances under which it cannot be falsified. The story will continue to have its own life in our mind and can be further developed via logical deductions.

(5) Wouldn't it be more productive to allow experiments in mathematics?
Thought experiments are allowed, of course. Others would be of very limited use, since they would either be rendered superfluous by logical proofs (due to the "unreasonable effectiveness" of logic in mathematics), or just couldn't be performed at all (e.g. due to infinity involved).