Is Computation Abstract or Concrete?

John Schwenkler kindly asked me to blog about my new book, Physical Computation: A Mechanistic Account. I am grateful for the invitation.

The original motivation for the research that led to the book was to make progress on the vexed question of whether cognition involves computation. That seems to require clarity on what computation amounts to. That’s what this book aims to deliver.

A preliminary source of confusion is that computation is a mathematical notion, and most mainstream philosophers notoriously think of mathematical notions as referring to abstract objects. Abstract objects, in turn, are objects that exist “outside” spacetime, where they have neither causes nor effects. This view is known as platonism in the philosophy of mathematics.

But computation is also supposed to somehow characterize physical systems, such as the one on which, chances are, you are reading this post. What gives? Is computation abstract or concrete?

It’s both, of course. On one hand, computation is a feature of at least some concrete, physical systems. Some physical systems perform (physical) computations. (That’s what my book is about.)

This rather pedestrian point has been surprisingly difficult to grasp for some philosophers, perhaps because they are so used to think of mathematical notions as referring to abstract objects, and they follow mainstream philosophy in thinking of abstract objects as categorically different from concrete ones.

On the other hand, computation is also abstract at least in the  sense that in order to capture what a physical system is computing, we must abstract away from many of its more specific physical characteristics, such as its physical composition and even the specific physical variables that implement the computational vehicles being manipulated.

What about abstract objects? Is computation also an abstract object? As I pointed out, many philosophers think of mathematical objects as abstract objects, and computation  is surely a mathematical notion. So many philosophers of computation think of computation as an abstract object. But what philosophers of computation often seem to forget is that the very notion of abstract object is rather obscure and contentious. There are other options in philosophy of mathematics besides the positing of abstract objects.

I find abstract objects unintelligible. Partly because I find them unintelligible, I don’t believe there are any. In writing the book, at some point I drafted an argument that there are no abstract objects, with a sketch of an alternative (nonplatonistic) ontology of mathematics. But a wise friend (Carl Craver) and an anonymous referee for OUP advised me to drop that material, as it went too far from the core of the book and was potentially distracting, not to mention insufficiently worked out.

I followed their good advice. My book is officially neutral between platonism about computation and antiplatonism. For all the book says, there may well be computations qua abstract objects. Regardless of your ontology of mathematics, my book proposes a mechanistic account of what it takes for a physical system to perform a computation. Hopefully this account helps clarify the foundations of the computational sciences, including computer science and (computational) cognitive neuroscience.


  1. This is genuinely exciting, thanks for writing it: your book is an immediate entry in my reading list.
    I am excited because I’ve recently tackled the challenge of “whether cognition involves computation” and ended up so close to what you seem to hint here. I didn’t dare to go as far as saying that computations only exist when physically performed, but I can testify that I was tempted (spoiler: I do say that abstract/mathematical computations are epistemological constructs, but then, what concept isn’t?).

    Thus, I have two things to propose:
    1) my ruminations are published at these addresses, naturally I’d love to hear your thoughts: Sergio’s Computational Functionalism and Sergio Resurgent.
    2) Given my own temptations, I would be absolutely thrilled if you could share the part you have decided to leave out of your book. This comment here comes with a mandatory email address, please feel free to use it in case you would like to send me a file privately. I understand I’m a complete stranger asking for something very sensitive, so I expect an “I’d rather not” reply. Please don’t hesitate to confirm my expectations, I know I’m asking a lot.
    In all cases, thanks again.

  2. Ken

    Hi, Gualtiero,
    I’ve not noticed the emergence of this sense of “abstract” in the philosophy of science.

    “On the other hand, computation is also abstract at least in the sense that in order to capture what a physical system is computing, we must abstract away from many of its more specific physical characteristics, such as its physical composition and even the specific physical variables that implement the computational vehicles being manipulated.”

    I don’t doubt it’s out there in the literature, but I’m not really sure what it is. It sounds as though being abstract in the foregoing sense is a matter of how one describes things. That it is a linguistic affair. But, I would have thought that whether or not a process is a computation is not a linguistic matter. Processes are, or are not, computations quite apart from languages. So, this is a matter of clarity about the metaphysics. (And, I guess if you are worried about the existence of abstract objects, you are already worrying about metaphysics.)

  3. Gualtiero Piccinini

    Thanks, Ken. I think it’s definitely a linguistic affair, a matter of how we describe things. But if such descriptions capture something objective about the world, as I think they do, there is also something in the world that corresponds to such descriptions. So it’s also a matter of metaphysics. I talk a little bit about this in the book. I have a more detailed treatment of this matter in a couple of papers, one co-authored with Corey Maley (in New Waves in the Philosophy of Mind) and one co-authored with Worth “Trey” Boone (which is unpublished but I’m happy to share it if anyone is interested). The bottom line is that IMO systems possess causal powers, and different subsets of the total set of causal powers that something has correspond to different descriptions. Roughly, the more abstract a description, the smaller the subset of causal powers it captures.

  4. Like Sergio, I’m also very impatient to receive this book, primarily because I have my own peculiar way to ‘explain away’ intentional characterizations of computation (and because I’m a big fan of yours and Craver’s work).

    The fact is, the ‘curse of dimensionality’ is as much a problem for the human brain as it is for researchers studying the human brain. Given metacognitive parochialism, it simply stands to reason that philosophers, reflecting upon their own cognitive activity, would be liable to confuse differences in signal for differences in being. The simple fact (and it is a fact) is that we’re natural in such a way that we cannot conceive ourselves as natural. The way to naturalize computation, I think, consists understanding this incapacity, our need to work with low-dimensional posits, as opposed to positing some miraculous capacity to intuit some ghostly beyond.

    But I just have to say: from phenomenology to eliminativism in the space of a week! Long live the Brains Blog! 😉

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