>But as the abstraction ladder has shown, all we know are abstractions.

You read this line as something like, “everything that exists is an abstraction.” I haven’t read the book you’re excerpting, but that’s not how I read the quote. I read it as emphasizing that everything that we know is abstract.

I imagined this working something like this: imagine life without abstractions. For now, I’m going to cache that out as, “Imagine life without concepts.” What would that life be like? It would be a life full of particulars, of sense experiences that are completely disconnected from another. There would be no basis to collect certain sense experiences together under one concept.

That would also mean that we wouldn’t know anything useful. Maybe, depending on what “know” means, we would still know some things — I’m imagining the memories of what a particular sense experience was like at a particular time — but we’d be unable to describe it, and we’d be unable to transfer that knowledge to any other context.

Everything that we know involves abstraction. But I don’t think that everything that exists is an abstraction.

Imagine we have some ladder of abstraction (though, more properly, it’s some sort of lattice–there are multiple rungs above and below every rung, and that’ll be important later); there are three things we can do on our ladder.

We can do work on a rung, we can move up a rung, or we can move down a rung. When we manipulate statements on a particular rung, the work we do obeys the (grammatical) rules of that rung, and it’s concrete. When we do work to move up a run on the ladder, we’re making our statements more general and the work we do is synthesis. When we move down a rung on the ladder, we’re making our statements more particular and the work we do is analysis.

In math classes we spend the enormous bulk of our time teaching concrete operations at a level of abstraction that isn’t motivated to our students. If we spent less time teaching what turn out to be fairly abstruse concrete relations among disconnectedly general ideas, and spent more time teaching our students how to do the tricky and demanding work of analysis and synthesis on a smaller vocabulary of general statements, I suspect we’d serve them well.

Abstraction = generalization?
I am blessed with a daughter who was “bad at math”, who observed that those “good at math” generalize in a specific way that she did not “get”. So, using triangles to calculate the height of a flagpole – yes, seen that, can do all flagpoles; the height of a building -yes, seen that, all buildings no problem; the height of a tree -? Because she did not group all of them into a “tall thing” class. Once the “tall thing” connection was explicitly made, she could do it, also street lights, statues, (some difficulty with flying airplanes, modification of tall things to include things where the top is high off the ground) but she was unable to make that leap herself, because a tree does not look like either a building or a flagpole. The specificity got in the way. Without that generalization, math becomes ridiculous.
In math we talk about looking for patterns, are we really talking about looking for “abstractions”? And how do we teach looking for generalities? Do we really do that at all, or do we somehow expect kids to pick this up along the way? I suspect the latter.

So then, off on my own wild tangent of thoughts, is it much more difficult for kids with disrupted lives to predict with patterns? If your life is unpredictable, you stop looking for a pattern to predict what will happen next. Life is in fact a series of disconnected events playing at random.
I also wonder whether this is tied to reading difficulties. The word “table” looks nothing like a table, but it has to generalize to all tables – 4 legs, 3 legs, one center post, high, low, cupboard underneath, a cardboard box… and then, for crying out loud, we use the exact same word to mean something totally different in math!
Someone else’s area of expertise…

Any philosopher’s out there? I think Aristotle talked about this same process with knowing in general. Rather than Platonic ideals that were held in our head, Aristotle said we observed, and abstracted generalizations.

Interesting that humans have witnessed this for thousands of years, and we are still trying to figure it out.

Dan: My favorite location is the intersection of language and mathematics. There’s a nice donut shop right on the corner that I like!

Similar to the use of the ladder of abstraction, I use the lattice of specificity (my phrase) with students, which is akin to the power set, P(S), of set S, which I do not share with my students.

When discussing mathematics with my students, and they name something using a less specific descriptor, like ‘expression,’ instead of ‘quadratic binomial equation,’ or ‘number,’ instead of ‘leading coefficient,’ I frequently explain the need for more specific language to disambiguate their contribution. I illustrate this frequently using a taxonomicesque representation like “mammal – human – male – student – name.”

While this may not exactly be what you envision when you discuss abstraction, the parallel practice of specificity may help students better understand the abstraction rungs on any specific ladder.

FYI, if you look at the Brian Harvey video he starts talking about abstraction at ~8:45.

I really like his comment “an abstraction is about controlling complexity”

Have a look the work done by David Tall On cognitive development (http://homepages.warwick.ac.uk/staff/David.Tall/themes/cognitive-development.html)
His body of work is vast in trying to understand how we acquire mathematical thinking. I find it speaks a lot to the process of moving up and down the ladder and acquiring the ability to connect abstraction at different levels. His whole work is inspiring and helps me to understand why some students progress and others get stuck on a specific rung.

Since you mention the abstraction of a chair, you might enjoy the musings of an interesting philosopher named J. Krishnamurti, who used to ask his students to point out a chair in the room, and then having done so, would complain that they didn’t point to a chair they pointed to some strangely constructed wooden object. A chair is an abstraction. Then he would ask again, and they would refrain from pointing, and he would say that any child could see that was a chair over there, and any fool could sit on it.. I think he was trying to get at a similar point that you are — that abstractions sit at different levels in some ladder of consciousness.

You can read some of his conversations here: http://www.jkrishnamurti.org/krishnamurti-teachings/view-text.php?tid=41&chid=1

I can’t help but be reminded of the “Van Hiele” levels of abstraction which deal specifically with geometry. I was exposed to this interesting research because it is cited as a basis of Michael Serra’s Discovering Geometry curriculum. In that context, he tried to address the assumption made by the standard high school geometry curriculum that all students were ready for the abstraction known as proofs. I’m not aware of a direct analogue to algebra, but I’d be interested to hear about it.