[Tazerenix]

>Today, mathematics is regarded as an abstract science.

Pure mathematics is regarded as an abstract science, which it is by definition. Arnol'd argued vehemently and much more convincingly for the viewpoint that all mathematics is (and must be) linked to the natural sciences.

>On forums such as Stack Exchange, trained mathematicians may sneer at newcomers who ask for intuitive explanations of mathematical constructs.

Mathematicians use intuition routinely at all levels of investigation. This is captured for example by Tao's famous stages of rigour (https://terrytao.wordpress.com/career-advice/theres-more-to-...). Mathematicians require that their intuition is useful for mathematics: if intuition disagrees with rigour, the intuition must be discarded or modified so that it becomes a sharper, more useful razor. If intuition leads one to believe and pursue false mathematical statements, then it isn't (mathematical) intuition after all. Most beginners in mathematics do not have the knowledge to discern the difference (because mathematics is very subtle) and many experts lack the patience required to help navigate beginners through building (and appreciating the importance of) that intuition.

The next paragraph about how mathematics was closely coupled to reality for most of history and only recently with our understanding of infinite sets became too abstract is not really at all accurate of the history of mathematics. Euclid's Elements is 2300 years old and is presented in a completely abstract way.

The mainstream view in mathematics is that infinite sets, especially ones as pedestrian as the naturals or the reals, are not particularly weird after all. Once one develops the aforementioned mathematical intuition (that is, once one discards the naive, human-centric notion that our intuition about finite things should be the "correct" lens through which to understand infinite things, and instead allows our rigorous understanding of infinite sets to inform our intuition for what to expect) the confusion fades away like a mirage. That process occurs for all abstract parts of mathematics as one comes to appreciate them (expect, possibly, for things like spectral sequences).

[pdpi]

> Pure mathematics is regarded as an abstract science, which it is by definition.

I'd argue that, by definition, mathemtatics is not, and cannot be, a science. Mathematics deals with provable truths, science cannot prove truth and must deal falsifiability instead.

[myrmidon]

You could turn the argument around and say that math must be a science because it builds on falsifiable hypotheses and makes testable predictions.

In the end arguing about whether mathematics is a science or not makes no more sense than bickering about tomates being fruit; can be answered both yes and no using reasonable definitions.

[AlexandrB]

Mathematical "truth" all depends on what axioms you start with. So, in a sense, it doesn't prove "truth" either - just systemic consistency[1] given those starting axioms. Science at least grapples with observable phenomena in the universe.

[1] And even this has limits: https://en.wikipedia.org/wiki/Gödel%27s_incompleteness_theor...

[abdullahkhalids]

Mathematical proofs are checked by noisy finite computational machines (humans). Even computer proofs' inputs-outputs are interpreted by humans. Your uncertainty in a theorem is lower bounded by the inherent error rate of human brains.

[tiahura]

Mathematics is a science of formal systems. Proofs are its experiments, axioms its assumptions. Both math and science test consistency—one internally, the other against nature. Different methods, same spirit of systematic inquiry.

[lo_zamoyski]

It's not an empirical science, but it is a science, where "science" means any systematic body of knowledge of an aspect of a thing and its causes under a certain method. (In that sense, most of what are considered scientific fields are families of sciences.) Mathematics is what you'd call a formal science with formal structure and quantity as its object of study and deductive inference and analysis as its primary methods (the cause of greatest interest is the formal cause).

[nitwit005]

A proof is just an argument that something is true. Ideally, you've made an extremely strong argument, but it's still a human making a claim something is true. Plenty of published proofs have been shown to be false.

Math is scientific in the sense that you've proposed a hypothesis, and others can test it.

[The_suffocated]

Somewhat tangential to the discussion: I have once read that Richard Feynman was opposed to the idea (originally due to Karl Popper) that falsifiability is central to physics, but I haven't read any explanation.

[Aardwolf]

I'm not sure if it deals only with provable truths? It even deals with the concept of unprovability itself, if the incompleteness theorem is considered part of mathematics

[mcmoor]

It isn't, that's why it's in own section in STEM, and rightfully so. It's a higher tool that without it, science would come to a screeching halt.

[ubj]

Science involves both deductive and inductive reasoning. I would in turn argue that mathematics is a science that focuses heavily (but not entirely) on deductive reasoning.

[_9ptr]

He probably means science in a wider sense as opposed to the anglo-american narrower sense where science is just physics, chemistry, biology and similar topics.

[weinzierl]

Pure mathematics is just symbol pushing and can never be science. It is lot of fun though and as it turned out occasionally pretty useful for science.

[griffzhowl]

I agree in general but

> Euclid's Elements is 2300 years old and is presented in a completely abstract way.

depends on what you mean by completely abstract. Euclid relies in a logically essential way on the diagrams. Even the first theorem doesn't follow from the postulates as explicitly stated, but relies on the diagram for us to conclude that two circles sharing a radius intersect.

This is a thought-provoking paper on the issue by Viktor Blasjo, Operationalism: An Interpretation of the Philosophy of Ancient Greek Geometry https://link.springer.com/article/10.1007/s10699-021-09791-4

which was recently the subject of a guest video on 3blue1brown https://www.youtube.com/watch?v=M-MgQC6z3VU

[ndriscoll]

Not only is intuition important (or the entire point; anyone with some basic training or even a computer can follow rules to do formal symbol manipulation. It's the intuition for what symbol manipulation to do when that's interesting), but it is literally discussed in a helpful, nonjudgmental way on Math Stack Exchange. e.g.

https://math.stackexchange.com/questions/31859/what-concept-...

Other great sources for quick intuition checks are Wikipedia and now LLMs, but mainly through putting in the work to discover the nuances that exist or learning related topics to develop that wider context for yourself.

[nkrisc]

> The next paragraph about how mathematics was closely coupled to reality for most of history and only recently with our understanding of infinite sets became too abstract is not really at all accurate of the history of mathematics. Euclid's Elements is 2300 years old and is presented in a completely abstract way.

I may be off-base as an outsider to mathematics, but Euclid’s Elements, per my understanding, is very much grounded in the physical reality of the shapes and relationships he describes, if you were to physically construct them.

[empath75]

Quite the opposite, Plato, several hundred years before Euclid was already talking about geometry as abstract, and indeed the world of ideas and mathematics as being _more real_ than the physical world, and Euclid is very much in that tradition.

I am going to quote from the _very beginning_ of the elements:

Definition 1. A point is that which has no part. Definition 2. A line is breadthless length.

Both of these two definitions are impossible to construct physically right off the bat.

All of the physically realized constructions of shapes were considered to basically be shadows of an idealized form of them.

[gaze]

The only things that are weird in math are things that would not be expected after understanding the definitions. A lot of the early hurdles in mathematics are just learning and gaining comfort with the fact that the object under scrutiny is nothing more than what it's defined to be.