Perhaps a concrete example will help. Suppose we have two mind-independent forms, A and B. Form A is tracked by the abstractions of statistical mechanics (equilibrium, temperature, phase, entropy, etc.). Form B is tracked by the abstractions of particle physics (field, particle, charge, gauge symmetry, etc.).
As far as I am aware, the abstractions of statistical mechanics cannot be “reduced” to the abstractions of particle physics. Typical reduction strategies like identity, translation, elimination, deduction, etc. all fail here. For instance, there is no way to define or derive phase changes from the equations of particle physics.
So what warrants the conclusion that form A is reducible to form B given that our knowledge of forms A and B is exhausted by the abstractions of statistical mechanics and particle physics respectively?
I think we are coming at this from different angles. When I agreed that abstractions were not reducible to other abstractions, I meant that abstractions qua abstraction, not qua content, were not reducible. It is not clear to me exactly what reducing one abstraction to another even entails, in this sense.
For instance, we agreed that high level software programs were reducible to assembly language or machine code. But is the concept of “high level program” reducible to the concept of “assembly language”? Clearly not. They describe different constructs, with different semantics and different relationships to hardware. How would you reduce these concepts, or any concepts?
I believe there is a very close mapping between abstractions and words, such that most words are named abstractions. So lets look at an analogous version of your argument, substituting words for abstractions.
Many philosophers argue that all understanding is conditioned by language. Lets assume this proposition. Further, I claim that words are not generally reducible in any meaningful sense to other words. Given all these, does it follow that we cannot know that anything is reducible? I don’t think so, no more than it would follow we cannot know water is wet, because words aren’t wet. Words, even if they truly condition all of our understanding, don’t form an impenetrable barrier such that we cannot know anything nonverbal.
Or maybe you would argue they would, given these presuppositions?
What exactly did you mean when you said that abstractions weren’t reducible to other abstractions?
It’s not clear to me what exactly you think the “concept qua concept” is over and above the “concept qua content”. Perhaps you could elaborate on that?
Also, high-level languages are reducible to machine code precisely because their constructs (“for-loop”) can be precisely translated to machine code instructions (“jump/branch”) while preserving the semantics. So, yes, some higher level concepts are reducible to lower-level concepts, but many aren’t. As mentioned above, statistical mechanics is not reducible to the particle physics in the way Java is reducible to x86.
Actually, I’ve realized I need to amend my claim. Upon further reflection, I don’t think that the computer language case is a genuine case of reduction at all. What’s really happening is that algorithmic concepts like “for-loop” are realizable in both high-level languages and low-level languages. Compilation isn’t reduction, it’s a translation from one implementation to another that guarantees operational equivalence.
Perhaps you could clarify what you mean when you say that the higher level “reduces” to the lower in this case?
Yes, this is a really good thing to discuss. The distinction I have in mind is “template” vs “instantiation”. Abstractions in my conception pick out slices of the world. The picking is “concept qua concept”, that which is picked out is “concept qua content”. If an abstraction is considered as a boundary, the content is the area inside. If an abstraction is considered as a filter, the content is that which passes. If an abstraction is considered a specification, the content is the fulfillment.
The concept “one million dollars” picks out sums of money equal to (or greater than) one million dollars. These sums can be spent, while the concept “one million dollars” is worth nothing. These sums are trivially reduced to dollars; one million dollars is nothing more than a sufficient quantity of dollars. While the concept “one million dollars” is non-reducible; “one million dollars” specifies more than “dollars”, and it is not one million “dollar” concepts, it is one concept.
What is confusing is that everything I have said remains an abstraction. “That which ‘one million dollars’ picks out” is no more spendable than “one million dollars”. Both are, in fact, specifications. This is a necessary consequence of language and of thought. Neither can grasp reality; their medium is stand-ins.
But, there is a difference between “one million dollars” and “that which ‘one-million dollars’ picks out”. The first carves out something specific. The second is parasitic on the first; if it is specification, that specification is just whatever it is the first carved out. The first defines, the second is what it is that has been defined.
Lets call the first “normative abstraction” and the second a “token abstraction”. The first makes rules, while the second is just there because the reality the rule picks out cannot be spoken or thought. The second is a token stand-in for that reality. Normative abstractions are “concept qua concept”, token abstractions, along with the reality they stand in for, are, “concept qua content”.
Oh and I agree with you, computer language is not a straightforward case of reduction. I would maybe characterize it a little differently: High level concepts like for-loops are specifiable in high level and low level languages. The reason to specify them this way is so the high level concept can be realized in machine code.
Thanks for laying that out. You’ve made a distinction between a concept’s “template” and its “instantiation”. The template is something like a specification that defines the “boundary” of the concept, whereas the instantiation is a concrete “something” that satisfies the specification. This makes sense to me.
I think where I get tripped up is when you go on to say that specifications can’t “grasp reality; their medium is stand-ins”. Specifications ultimately pick out “token abstractions”, which are stand-ins for the underlying “reality the rule picks out”.
So we seem to have a three-tiered structure: (1) specifications, which pick out (2) token abstractions, which stand-in for (3) things in the world. Both (1) and (2) are mental, whereas (3) is physical, and our knowledge of (3) is completely exhausted by (1) and (2). Is that right?
If yes, my question back to you is: where does reducibility live in this three-tiered structure?
Back. Sorry for the delay, I was travelling and then dealing with some serious jet lag.
Not specifications, it is thought and language can’t grasp reality. Abstractions are specifications that pick out reality. But I can only verbally gesture at what an abstraction picks out, I can’t deliver the physical thing to you in words, nor can I think it. Thought and language just lack that casual power. To communicate or think it, it must be replaced with a token stand-in for that reality. This is what I mean by token abstraction.
On the one hand, there is the reality that “trees” picks out, the woody leafy organic entities. On the other hand, there is what I just wrote, which yet another abstraction: “the reality that ‘trees’ picks out, the woody leafy organic entities”. The distinction is between reality, and language and thought gesturing at that reality.
I’m not sure if (2) is what (1) picks out, or if (1) picks out (3), and (2) is the expression of (3). But it may not make a difference.
I am uneasy with “completely exhausted”. Knowledge is of (3), not (1) and (2). (1) and (2) are how we know (3). I think you were making the same mistake in our indirect realism discussion. Because we know something through an intermediary is not to say that the intermediary is a barrier. Even if the intermediary is “exhaustive” in the sense that everything we know must pass through it.
The whole point of this distinction is to be precise about what can and cannot be reduced. (3) is what is reducible, not (1), not even (2). Is it clear now why your argument doesn’t work? Just because (1) and (2) are not reducible, doesn’t prevent us from knowing that (3) is. The reducibility of (3) is just another fact about it, knowledge of that fact is no more inhibited by the non-reducibility of (1) and (2) than knowledge of anything at all about (3).
I was thinking about what exactly it even means to say that something is “reducible”. Here is a minimal definition.
To say that “Y is reducible to X” just means “Y is a form X can take”. Computers are a form transistors can take. Transistors are a form the right elements can take. A million dollars is a form dollars can take. A house is a form plywood can take. Life(2) is a form Life(1) can take.
But specifications are thought and language. So how is it that they can grasp reality but thought and language can’t?
Perhaps not. But I think it probably depends on what you think (2) is, and what it means for (2) to “express” (3). You’ve said that (2) is a “stand-in”, but what does this mean exactly? Perhaps you could elaborate?
It depends on what you mean by “barrier”. In our previous discussion my point was simply that indirect realism gets the phenomenology wrong. I wasn’t denying the possibility of indirect knowledge.
But how can we know this fact unless it is reflected in (1) and (2)? Could you clarify how your account handles this?
In my opinion, this is much too weak. Even Plato himself would have accepted that “Y is a form X can take”. And if your definition can’t rule out Platonic realism, I don’t think it can work as an adequate definition of reductionism.
“Grasp” is a poor choice. “Be” reality. Thought and language aren’t reality, they can only point at it. (2) is just the verbal expression of (3).
Lets not worry too much about (2), it might not be important anyway.
It is reflected, just not in the reducibility of (1) and (2) themselves, as abstractions. Knowledge and understanding is not exhausted in (1) and (2). They are the building blocks that allow knowledge at all, not just of individual things, but of classes of things. Knowledge is expressed as connections between abstractions. These connections represent various relationships between the connected abstractions.
“Computers are made of transistors” expresses a linkage between the abstractions “computer” and “transistor”. The linkage represents a compositional relationship between the two. Obviously, what the linkage is about is (3), not (1) and (2). Computers are made of transistors, not the concept “computers”.
“Computers are reducible to transistors”, whatever it is that “reducible” precisely means, is another such linkage.
True, “form” is much too ambiguous.
Here, for “Y to be a form X can take” means that Y is X in the right quantity, proportions, and arrangement.
You gave a revised definition of reduction that I think is better than the first. That said, I want to set that aside for a moment and just focus on the topic of knowledge and concepts.
You said that knowledge is not exhausted in (1) and (2) because there are also “connections” that express knowledge. I would say that you have called “connections” are what is typically referred to as “relations” in the philosophical literature, so I will use that term going forward. My question to you is: can’t relations also be captured by concepts? Consider that “made of” is a concept that expresses the composition relation. Likewise, “reducible to” is a concept that expresses the reduction relation. If that’s right, then I don’t think that the appeal to relations shows that our knowledge of the world exceeds what is captured by (1) and (2). Would you agree?
I agree that relations, such as “is made of”, are also abstractions. And I agree that much if not all of our knowledge is formulated with abstractions. Whether this means that knowledge cannot exceed what is “captured” by abstractions is less clear.