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In his 2007 book I Am a Strange Loop, Douglas Hofstadter uses an analogy based on a domino computer.

Indeed, it is possible to build logical gates made of dominoes (see e.g. here) and realize simple programs such as an adder (see here).

For definiteness, let’s imagine a “domino computer” able of determining whether an integer (entered in binary) is prime or not. The result is read on the last domino. If the latter falls, the number is prime. If it remains standing, the number is not prime.

Now, assume we enter the number 7 as input in the algorithm. The last domino falls.

How can we answer the following:

Why did the last domino fall?

Answer 1: Because the penultimate domino made it fall.

Answer 2: Because the number 7 is prime.

The first answer focuses on the physical character of the device. Each physical event is taken in a causal sequence of events and therefore has a complete causal explanation.

The second answer provides an explanation based on a mathematical property, ignoring the physical implementation of the algorithm in order to refer only to its function.

Hofstadter uses this analogy in order to address the notion of functionalism.

My puzzle is the following:

Although Answer 2 sounds very intuitive, it seems to imply a causal influence of an abstract mathematical property (being prime) of an abstract mathematical concept (the number 7) on a material object (the last domino).

How can we think of such a causal influence of an abstract property on the physical world?

Is anyone aware of references addressing this issue?

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  • 5
    "Seems" is the key word here. Answer 2 is correct, but it is a shorthand with misleading surface grammar. The full version is "because the machine is designed to drop the last domino on prime inputs and 7 is a prime input", and it does not ascribe any causal powers to numbers.
    – Conifold
    Feb 9, 2023 at 3:31
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    There is some literature on the role and nature of mathematical explanations in science, see e.g. Leng, Models, structures, and the explanatory role of mathematics in empirical science and Lange's book Because Without Cause: Non-Causal Explanations in Science and Mathematics.
    – Conifold
    Feb 9, 2023 at 3:56
  • But "explanation" is an human "process": what counts as an explanation is a "social construct" based on human intuition developed by the scientific community. Feb 9, 2023 at 8:03
  • If a seemingly simple question proves hard to answer, that usually means it was badly phrased.
    – Karl
    Feb 15, 2023 at 7:17
  • The very physical reason the last domino fell is that you pushed the first 3. It's 7 in binary, but only on the head of humans, our way of talking about it. To the physical world it's 3 dominos.
    – armand
    Feb 19, 2023 at 22:08

5 Answers 5

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How can we think of such a causal influence of an abstract property on the physical world? ... Is anyone aware of references addressing this issue?

Well, one way to address this issue is through computer science, where mathematical objects routinely are structured to control the real world, though the problem is much bigger. It's one of the central questions driving the philosophy of computer science. On the one hand, computers are physical artifacts, and on the other hand, there are mathematical abstractions that are at play, particularly in the use of programming languages. For instance, a more sophisticated style of programming languages relies on something called denotational semantics where programing language constructs are implemented as tools rooted in mathematical formalisms. Hence, a functional programmer of Haskell might be construed as writing software purely with mathematical object primitives:

Broadly speaking, denotational semantics is concerned with finding mathematical objects called domains that represent what programs do. For example, programs (or program phrases) might be represented by partial functions1 or by games3 between the environment and the system.

Now, it's plain as day that computers have a physical effect in the world, so the real question is 'what the heck?'. That's because depending on your metaphysical presuppositions, you may reject interactionism, which is the idea that abstractions and physical causation are connected. In effect, a computer evaluating objects of mathematical logic could be interpreted as participating in mental causation. Well, that's awkward if you believe the mind and body DON'T interact.

So, the real question is not how does mathematics have an impact on the physical world, but whether it does at all. It's a grand philosophical problem that extends far back to the Ancient Greeks, but may be most famous posed as Cartesian dualism. For an explanation of what you're after, why it's so tricky to suss out the relationship between mathematical objects and the physical extension and causality of the universe, read the response to Why is mind interacting with matter any more problematic than matter interacting with matter?

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Regarding "...realize simple programs", it's possible to design such a domino computer (but don't ask me exactly how) that computes any (and every) computable function whatsoever. That's called Turing Completeness, e.g., https://en.wikipedia.org/wiki/Turing_completeness, and it's surprisingly simple to design such machines, e.g., https://softwareengineering.stackexchange.com/questions/230538/. So you could arguably formulate your dilemma using much more complicated abstract examples.

Regarding "causal influence of an abstract mathematical property", consider an even simpler domino machine: Suppose you have two dominoes of length l and width w, and you stand them on end so that their height is also l. You place them a distance d apart, and tip the first one over in the second one's direction. Then if d<l the second domino will fall, but if d>l it won't. So is the mathematical property d<l a causal influence? Or is it just a description of the arrangement of the dominoes? It's the physical arrangement that causes the second domino to fall (or not). The d<l description is a consequence of that arrangement, not vice versa. You can't describe the situation until you observe the arrangement.

And your domino prime number computer is ultimately the same thing; just a slightly more complicated example. You say, "an integer (entered in binary)". Well, to "enter" that integer requires you to rearrange some of the leading dominoes in some way that represents the integer you have in mind. And then it's this particular domino arrangement that causes the last domino to fall (or not). Same situation for both our machines (and any other you conjure up): it's the physical chicken that precedes the mathematical egg.

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The point is that the dominoes have been arranged in a fashion such that they follow rules. And those rules allow a representation of rules of logic.

So the explanation is because the number 7 is prime, but that's not the WHOLE explanation. It has lots of steps and contextual information.

As an example: In the first vid you link, the Numberphile channel shows how to implement an AND gate. This implementation relies on the fact that a shorter chain of dominoes takes less time compared to a longer one.

So the explanation is not only that 7 is prime. It has many steps. Some of these steps look like so.

  • Convince people to get you many dominoes.
  • Convince people to help you set up many dominoes.
  • A domino can be stood on end and then tipped over.
  • A tipped over domino can knock over the next one.
  • You can chain these. The time the full chain takes is roughly proportional to the length.
  • You can stop the chain with a gap.
  • The chain can be set up to create a gap in a chain.

And so on and so on, many many steps.

  • Put dominoes in this pattern to make something that represents binary inputs up to length.
  • Put dominoes in this pattern to make something that represents addition of arbitrary inputs up to length.

And so on and so on, many many more steps.

  • Put dominoes in this pattern to make something that represents checking for prime of arbitrary inputs up to length.
  • Put inputs in this pattern to represent the binary form of 7.
  • Push this domino (or these dominoes) to start it.

And finally, after all of that hard work.

  • The last domino falls over because 7 is primary.

The key word here is represent. This should be thought of as "make present again." That is, the domino patterns make the things they represent become present again. The logic AND gate is made present again by the pattern of movement of the dominoes and the context we impose on it. The binary inputs make present again the idea of 7 based on their arrangement and the context we impose on them. And so on.

So the explanation is: The pattern of arrangement and movement of the dominoes, in the context we have imposed, represents a calculation of whether or not the input is prime, and the input 7 is prime, so the last domino falls.

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I guess it depends on which position in the spectrum of Philosophies of Mathematics do you stand.

For example, a Fictionalist would tell you that Answer 2 is a nice story, but no abstract objects or properties interferring in the real world whatsoever:

Fictionalism [..] is the view that (a) our mathematical sentences and theories do purport to be about abstract mathematical objects, as platonism suggests, but (b) there are no such things as abstract objects, and so (c) our mathematical theories are not true. Thus, the idea is that sentences like ‘3 is prime’ are false, or untrue, for the same reason that, say, ‘The tooth fairy is generous’ is false or untrue

So, in order to conclude there is some sort of causal interference, some sort of platonism must be assumed. If that is your case, then you may find Mathematical platonism and the causal relevance of abstracta interesting.

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Think of the dominoes as an extension of your phenotype (like a spider's web or the car you are driving). The pressing question is not what caused the last domino to fall but what caused the fall of the first domino representing "7".

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