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.
Complaining about jargon is lazy. Most communications about complicated things are not aimed at the layman, because to do anything useful with the complicated things, you tend to have to understand a fair amount of the context of the field. Once you're committed to actually learning about the field, the jargon is the easiest part: they're just words or phrases that mean something very specific.
To put it another way: Jargon is the source code of the sciences. To an outsider, looking in on software development, they see the somewhat impenetrable wall of parentheses and semicolons and go "Ah, that's why programming is hard: you have to understand code". And I hope everyone here can understand that that's an uninformed thing to say. Syntax is the easy part of programming, it was made specifically to make expressing the rigorous problem solving easier. Jargon is the same way: it exists to make expressing very specific things that only people in this subfield actually think about easier, instead of having to vaguely gesture at the concept, or completely redefine it every time anybody wants to communicate within the field.
Abstraction isn't to gatekeep; it's to increase the utility. It's the same as "dependency inversion" in programming: do your logic in terms of interfaces/properties, not in terms of a particular instance. This makes reasoning reusable. It also often makes things clearer by cutting out distracting details that aren't related to the core idea.
People are aware that you need context to motivate abstractions. That's why we start with numbers and fractions and not ideals and localizations.
Jargon in any field is to communicate quickly with precision. Again the point is not to gatekeep. It's that e.g. doctors spend a lot of time talking to other doctors about complex medical topics, and need a high bandwidth way to discuss things that may require a lot of nuance. The gatekeeping is not about knowing the words; it's knowing all of the information that the words are condensing.
Formal reasoning is the point, which is not by itself abstraction.
Someone else in this discussion is saying Euclid's Elements is abstract, which is near complete nonsense. If that is abstract our perception of everything except for the fundamental [whatever] we are formed of is an abstraction.
I love how you lot just redefine words to suit your purpose:
https://www.etymonline.com/word/formal
"late 14c., "pertaining to form or arrangement;" also, in philosophy and theology, "pertaining to the form or essence of a thing," from Old French formal, formel "formal, constituent" (13c.) and directly from Latin formalis, from forma "a form, figure, shape" (see form (n.)). From early 15c. as "in due or proper form, according to recognized form," As a noun, c. 1600 (plural) "things that are formal;" as a short way to say formal dance, recorded by 1906 among U.S. college students."
There's not a much better description of what Euclid was doing.
What you mean is someone has redefined the word to suit their purpose, which is precisely what I pointed out at the top.
Edit to add: this comment had a sibling, that was suggesting that given a specific proof assistant requires all input to be formal logic perhaps the word formal could be redefined to mean that which is accepted by the proof assistant. Sadly this fine example of my point has been deleted.
Every mathematician understands what a formal proof is. Ditto a formal statement of a mathematical or logical proposition. The mathematicians of 100 years ago also all understood, and the meaning hasn't changed over the 100 years.
> The mathematicians of 100 years ago also all understood, and the meaning hasn't changed over the 100 years.
Isn't that the subject of the whole argument? That mathematicians have taken the road off in a very specific direction, and everyone disagreeing is ejected from the field, rather like occurred more recently in theoretical physics with string theory.
Prior to that time quite clearly you had formal proofs which do not meet the symbolic abstraction requirements that pure mathematicians apparently believe are axiomatic to their field today, even if they attempt to pretend otherwise, as argued over the case of Euclid elsewhere. If the Pythagoreans were reincarnated, as they probably expected, they would no doubt be dismissed as crackpots by these same people.
Not all proofs are formal, and most published papers are not formal in the strictest sense. That is why they talk about "formalizing" a proof if there is some question about it. It is that formalization process which often finds flaws.
No, abstraction is the point and formal reasoning is a tool. And yes, what Euclid did is obviously abstraction, I don’t know why so you consider this stance nonsense.
Can you say how mathematics is inherently abstract in a way consistent with your day-to-day life as a concrete person? Or is your personhood also an abstraction?
I could construct a formal reasoning scheme involving rules and jugs on my table, where we can pour liquids from one to another. It would be in no way symbolic, since it could use the liquids directly to simply be what they are. Is constructing and studing such a mechanism not mathematics? Similarly with something like musical intervals.
Of course I can. I frequently use numbers which are great abstraction. I can use same number five to describe apples, bananas and everything countable.
> to describe apples, bananas and everything countable
An apple is an abstraction over the particles/waves that comprise it, as is a banana.
Euclid is no more abstract than the day to day existence of a normal person, hence to claim that it is unusually abstract is to ignore, as you did, the abstraction inherent in day to day life.
As I pointed out it's very possible to create formal reasoning systems which are not symbolic or abstract, but due to that are we to assume constructing or studying them would not be a mathematical exercise? In fact the Pythagoreans did all sorts of stuff like that.
> An apple is an abstraction over the particles/waves that comprise it, as is a banana.
No, you don’t understand what abstraction is. Apple is exactly arrangement of particles, it’s not abstraction over them.
> hence to claim that it is unusually abstract
Who talks about him being unusually abstract (and not just abstract)?
> is to ignore, as you did, the abstraction inherent in day to day life.
How am I ignoring this abstraction when I’ve provided you exactly that (numbers are abstraction inherent in day to day life).
I’m sorry but you seem to be discussing in bad faith.
> Apple is exactly arrangement of particles, it’s not abstraction over them.
No. You can do things to that apple, such as bite it, and it is still an apple, despite it now having a different set of particles. It is the abstract concept of appleness (which we define . . . somehow) applied to that arrangement of particles.
> I’m sorry but you seem to be discussing in bad faith.
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.