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.
How has mathematics gotten so abstract?
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Re: How has mathematics gotten so abstract?
#62Earlier quoted context omitted.
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.
It is absolutely a science, a formal science. What it isn't is an empirical science. The "symbol pushing" is a methodological tool, and a very useful one that opened up the possibility of new expansive fields of mathematics. (Of course, it is important to always distinguish between properties of the abstraction or the tool from the object of study.)
Well, we are talking about pure mathematics and there is not much Popperian scientific method in it.
Re: How has mathematics gotten so abstract?
#63>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…
> 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.
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).
Re: How has mathematics gotten so abstract?
#64There was a time, not that long ago in human history, that zero was "so abstract".
Sure even 500 years ago negative numbers were "absurd" in western mathematics and even in eastern mathematics where they were used they were more thought of as credits and debts than just abstract numbers.
Re: How has mathematics gotten so abstract?
#65How 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, simp…
> 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 acco…
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?
Re: How has mathematics gotten so abstract?
#66I 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.
Re: How has mathematics gotten so abstract?
#67How 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, simp…
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.
Re: How has mathematics gotten so abstract?
#68>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…
> 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 mu…
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.
Re: How has mathematics gotten so abstract?
#69The 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.
Re: How has mathematics gotten so abstract?
#70How 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, simp…
Right? Math is abstraction at its very core. Ridiculous premise acting as if this is anything but beyond ancient.