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Comment by rob74 | Hacker News Reader

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.

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It is abstract in the strict sense, of course. Every science is, as "abstract" simply means "not concrete". All reasoning is by definition abstract in the sense it all reasoning by definition involved concepts, and concepts are by definition abstract.

Numbers, for example, are abstract in the sense that you cannot find concrete numbers walking around or falling off trees or whatever. They're quantities abstracted from concrete particulars.

What the author is concerned with is how mathematics became so abstract.

You have abstractions that bear no apparent relation to concrete reality, at least not according to any direct correspondence. You have degrees of abstraction that generalize various fields of mathematics in a way that are increasingly far removed from concrete reality.

Right? Math is abstraction at its very core. Ridiculous premise acting as if this is anything but beyond ancient.

Mathematics arose from ancient humans need to count and measure. Even the invention\discovery of Calculus was in service to physics. It has probably only been 300 years or so since Mathematics has been symbolic, before that it was more geometric and more attached to the physical world.

Leibniz (late 1600s) helped to popularize negative numbers. At the time most mathematicians thought they were "absurd" and "fictitious".

No, not highly abstract from the beginning.

Sorry what? Ancient humans invented symbols to count. How is that not symbolic?

Geometry is “attached” to the physical world… but in an abstract way… but you can point to the thing your measuring maybe so it doesn’t count…

Abstraction was perfected if not invented by mathematics.

Symbolic here refers of doing math with place holders, be it letters or something. Ancient world had notations for recording numbers. But much less so to do math with them. Say like long division.

Almost from the first time people started writing about mathematics, they were writing about it in an abstract way. The Egyptians and the Babylonians kept things relatively concrete and mostly stuck to word problems (although lists of pythagorean triples is evidence for very early "number theory"), but Greece, China and India were all working in abstractions relatively early.

In particular, ancient Greek geometry at least after 300 BC proceeded from axioms, which is a central component of the abstract approach.

> Leibniz (late 1600s) helped to popularize negative numbers.

Wasn't that imaginary numbers?

Archimedes did Calculus before Newton.

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

He didn't connect the dots, so no he didn't do calculus even if he did some things related to it.

> My understanding was that mathematics was abstract from the very beginning.

It wasn't; but that's a common misunderstanding from hundreds of centuries of common practice.

So, how has maths gotten so abstract? Easy, it has been taken over by abstraction astronauts(1), which have existed throghout all eras (and not just for software engineering).

Mathematics was created by unofficial engineers as a way to better accomplish useful activities (guessing the best time of year to start migrating, and later harvesting; counting what portion of harvest should be collected to fill the granaries for the whole winter; building temples for the Pharaoh that wouldn't collapse...)

But then, it was adopted by thinkers that enjoyed the activity by itself and started exploring it by sheer joy; math stopped representing "something that needed doing in an efficient way", and was considered "something to think about to the last consecuences".

Then it was merged into philosophy, with considerations about perfect regular solids, or things like the (misunderstood) metaphor of shadows in Plato's cave (which people interpreted as being about duality of the essences, when it was merely an allegory on clarity of thinking and explanation). Going from an intuitive physical reality such as natural numbers ("we have two cows", or "two fingers") to the current understanding of numbers as an abstract entity ("the universe has the essence of number 'two' floating beyond the orbit of Uranus"(2)) was a consequence of that historical process, when layers upon layers of abstraction took thinkers further and further away from the practical origins of math.

[1] https://www.joelonsoftware.com/2001/04/21/dont-let-architect...

[2] https://en.wikipedia.org/wiki/Hyperuranion

I think it is fair to say that it was always an abstraction. But, crucially, it was built on language as much as it was empiricism.

That is, numbers were specifically used to abstract over how other things behave using simple and strict rules. No?

> That is, numbers were specifically used to abstract over how other things behave using simple and strict rules. No?

Agree that math is built on language. But math is not any specific set of abstractions; time and again mathematicians have found out that if you change the definitions and axioms, you achieve a quite different set of abstractions (different numbers, geometries, infinity sets...). Does it mean that the previous math ceases to exist when you find a contradiction on it? No, it's just that you start talking about new objects, because you have gained new knowledge.

The math is not in the specific objects you find, it's in the process to find them. Rationalism consider on thinking one step at a time with rigor. Math is the language by which you explain rational thought in a very precise, unambiguous way. You can express many different thoughts, even inconsistent ones, with the same precise language of mathematics.

Agreed that we grew math to be that way. But there is an easy to trace history on the names of the numbers. Reals, Rationals, Imaginary, etc. They were largely named based on their relation to the language on how to relate them to physical things.

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