This reminds of of that one time when I was on a date with a girl from the history department who somehow bemusedly sat through my entire mini-lecture on comparing infinite sets. Twenty years and three kids later, she'll still occasionally look me straight in the eye and declare "my infinity is bigger than your infinity."

>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).

How has mathematics gotten so abstract? My understanding was that mathematics was abstract from the very beginning. Sure, you can say that two cows plus two more cows makes four cows, but that already is an abstraction - someone who has no knowledge of math might object that one cow is rarely exactly the same as another cow, so just assigning the value "1" to any cow you see is an oversimplification. Of course, simple examples such as this can be translated into intuitive concepts more easily, but they are still abstract.

Proposed rule: People writing about the history of mathematics, should learn something about the history of mathematics.

Mathematicians didn't just randomly decide to go to abstraction and the foundations of mathematics. They were forced there by a series of crises where the mathematics that they knew fell apart. For example Joseph Fourier came up with a way to add up a bunch of well-behaved functions - sin and cos - and came up to something that wasn't considered a function - a square wave.

The focus on abstraction and axiomatization came after decades of trying to repair mathematics over and over again. Trying to retell the story in terms of the resulting mathematical flow of the ideas, completely mangles the actual flow of events.

This article explores a particular kind of abstractness in mathematics, especially the construction of numbers and the cardinalities of infinite sets. It is all very interesting indeed.

However, the kind of abstractness I most enjoy in mathematics is found in algebraic structures such as groups and rings, or even simpler structures like magmas and monoids. These structures avoid relying on specific types of numbers or elements, and instead focus on the relationships and operations themselves. For me, this reveals an even deeper beauty, i.e., different domains of mathematics, or even problems in computer science, can be unified under the same algebraic framework.

Consider, for example, the fact that the set of real numbers forms a vector space over the set of rationals. Can it get more abstract than that? We know such a vector space must have a basis, but what would that basis even look like? The existence of such a basis (Hamel basis) is guaranteed by the axioms and proofs, yet it defies explicit description. That, to me, is the most intriguing kind of abstractness!

Despite being so abstract, the same algebraic structures find concrete applications in computing, for example, in the form of coding theory. Concepts such as polynomial rings and cosets of subspaces over finite fields play an important role in error-correcting codes, without which modern data transmission and storage would not exist in their current form.

I think the title is a little tongue in cheek. The rest of the blog post develops the Foundations of arithmetic in a clear, well-grounded manner. This is probably a really good introduction for someone about to take a Foundations course. I say this having just Potter's "Set Theory and it's Philosophy" which covers the same material (and a lot more obviously) in 300 some pages. Another good introduction is Frederic Schuller's YouTube lectures, though already there you can start to see the over abstraction.

Infinity is a convenience that pays off in terseness. There's constructive mathematics, but it's wordy and has lots of cases. You can escape undecidablity if you give up infinity. Most mathematicians consider that a bad trade.

Just drop the axiom of infinity and quit whining.

https://en.wikipedia.org/wiki/Ultrafinitism

The French Bourbaki school certainly had a large influence on increasing abstraction in math, with their rallying cry "Down With Triangles". The more fundamental reason is that generalizing a problem works; it distills the essence and allows machinery from other branches of math to help solve it.

"A mathematician is a person who can find analogies between theorems; a better mathematician is one who can see analogies between proofs and the best mathematician can notice analogies between theories. One can imagine that the ultimate mathematician is one who can see analogies between analogies."

-- Stefan Banach

The definition of bijection is much more interesting than comparing cardinals. Many everyday use cases where (structure-preserving) bijections make it clear that two apriori different objects can be treated similarly.

More generally, mathematics is experimental not just in the sense that it can be used to make physical predictions, but also (probably more importantly) in that definitions are "experiments" whose outcome is judged by their usefulness.

My mental representation of this phenomenon is like inverted Russian dolls: you start by learning the inner layers, the basics, and as you mature, you work your way into more abstractions, more unified theories, more structures, adding layers as you learn more and more. Adding difficulty but this extreme refinement is also very beautiful. When studying mathematics I like to think of all these steps, all the people, and centuries of trial and errors, refinements it took to arrive where we are now.

Discussions of this sort can easily get chaotic, because people tend to conflate intuitiveness and concreteness. Sometimes the whole point of abstraction is to make a concept clearer and more intuitive. The distinction between polynomial function and polynomial is an example.

I used to be a physicist and I love math for the toolbox it provides (mostly Analysis). It allows to solve a physical model and make predictions.

When I was studying, I always got top marks in Analysis.

Then came Algebra, Topology and similar nightmares. Oh crap, that was difficult. Not really because of the complexity, but rather because of abstraction, an abstraction I could not take to physics (I was not a very good physicist either). This is the moment I realized that I will never be "good in maths" and that will remain a toolbox to me.

Fast forward 30 years, my son has differentials in high school (France, math was one of his "majors").

He comes to me to ask what the fuck it is (we have a unhealthy fascination for maths in France, and teach them the same was as in 1950). It is only when we went from physical models to differentials that it became clear. We did again the trip Newton did - physics rocks :)

This article can also be written as "The unreasonable effectiveness of abstraction in mathematics."

There was a time, not that long ago in human history, that zero was "so abstract".

>Next, consider the time needed for Achilles to reach the yellow dot; once again, by the time he gets there, the turtle will have moved forward a tiny bit. This process can be continued indefinitely; the gap keeps getting smaller but never goes to zero, so we must conclude that Achilles can’t possibly win the race.

Am i daft, eventually (Very soon) Achilles would over take the turtles position regardless of how far it moved... I am missing something?

I found it a bit ironic that the author introduced C code there as an aid, but didn't incorporate it into their argument. As I see it, code is exactly the bridge between abstract math and the empirical world - the process of writing code to implement your mathematical structure and then seeing if it gives you the output you expect (or better yet, with Lean, if it proves your proposition) essentially makes math a natural science again.

None of that was even the abstract stuff. It is all models of sizes, order, and inclusion (integers, cardinals, ordinals, sets). Not the nastier abstractions of partial orders, associativity, composition and so on (lattices, categories, ...).

How has blog posts authors gotten so uneducated or/and clickbaiting?

Math in its core has always been abstract. It’s the whole point.

Isn't this true for many other fields of study?

Given the collective time put into it, easier stuff was already solved thousands of years ago, and people are not really left with something trivial to work on. Hence focusing on more and more abstract things as those are the only things left to do something novel.

It's always been abstract. They'll say to me, "Give me a concrete example with numbers!"

I get what they're saying in practice. But numbers are abstract. They only seem concrete because you'd internalized the abstract concept.

One could also say the opposite. It's not abstract at all, just a set of rules and their implications. Plausibly the least abstract thing there is.

On the other hand, two cookies plus three cookies, what even is a cookie? What if they're different sizes? Do sandwich cookies count as one or two? If you cut one in half, does you count it as two cookies now? All very abstract. Just give me some concrete definitions and rules and I'll give you a concrete answer.

I believe that abstraction is recursive in nature which creates multiple layers of abstract ideas leading to new areas or insights. For instance our understanding of continuity and limit led to calculus, which when tied to the (abstract) idea of linearity led to the idea of linear operator which explains various phenomena in the real world surprisingly well.

I feel like a great deal more credit should be given to Cauchy and his school, but I understand the tale is long enough.

The Peano axioms are pretty nifty though. To get a better appreciation of the difficulty of formally constructing the integers as we know them, I recommend trying the Numbers Game in Lean found here: https://adam.math.hhu.de/

I like Peano, but he was using Grassmann's definition of natural numbers

The number 1 is what a cow, a fox, a stone ... have in common, oneness. Mathematics is abstraction, written down.

I wish the scroll bar was a little less invisible.

Unlike Zeno's famous example the paradox which does better at explaining the problem is https://en.wikipedia.org/wiki/Coastline_paradox which Mandelbrot seemed particularly keen on.

The tendency towards excessive abstraction is the same as the use of jargon in other fields: it just serves to gatekeep everything. The history of mathematics (and science) is actually full of amateurs, priests and bored aristocrats that happened to help make progress, often in their spare time.

"Indeed, persistently trying to relate the foundations of math to reality has become the calling card of online cranks." <-- Hm??? I'm getting self-conscious. Details?

I believe mathematics was much tamer before Georg Cantor's work. If I had to pick a specific point in history when maths got "so abstract", it would be the introduction of axiomatic set theory by Zermelo.

I personally cannot wrap my head around Cantor's infinitary ideas, but I'm sure it makes perfect sense to people with better mathematical intuition than me.

What else is it supposed to do?