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Reduction and surprisal, or, why water is wet

Last updated on 22 Jun 2018

In his classic work A System of Logic, which set up so many of the issues and problems of the modern field of the philosophy of science, John Stuart Mill wrote:

Not a trace of the properties of hydrogen or of oxygen is observable in those of their compound, water. [Bk III ch VI §1]

and drew from this the conclusion

This explains why mechanics is a deductive or demonstrative science, and chemistry not. In the one, we can compute the effects of combinations of causes, whether real or hypothetical, from the laws which we know to govern those causes when acting separately; because they continue to observe the same laws when in combination which they observed when separate: whatever would have happened in consequence of each cause taken by itself, happens when they are together, and we have only to “cast up” the results. Not so in the phenomena which are the peculiar subject of the science of chemistry. There, most of the uniformities to which the causes conformed when separate, cease altogether when they are conjoined; and we are not, at least in the present state of our knowledge, able to foresee what result will follow from any new combination, until we have tried the specific experiment.

If this be true of chemical combinations, it is still more true of those far more complex combinations of elements which constitute organized bodies; and in which those extraordinary new uniformities arise, which are called the laws of life. All organized bodies are composed of parts similar to those composing inorganic nature, and which have even themselves existed in an inorganic state; but the phenomena of life, which result from the juxtaposition of those parts in a certain manner, bear no analogy to any of the effects which would be produced by the action of the component substances considered as mere physical agents. To whatever degree we might imagine our knowledge of the properties of the several ingredients of a living body to be extended and perfected, it is certain that no mere summing up of the separate actions of those elements will ever amount to the action of the living body itself. [pp371f]

Here we have Mill’s statement of the basic problem, long before Lewes had coined the term “emergence” in 1875. “No mere summing up” of the to-be-reducing properties will give you the to-be-reduced properties. What does this mean?

Mill had a name for this, one not nearly so memorable: concurrence of causes [Bk III chapter X, §4], and he wrote

When the laws of the original agents cease entirely, and a phenomenon makes its appearance, which, with reference to those laws, is quite heterogeneous; when, for example, two gaseous substances, hydrogen and oxygen, on being brought together, throw off their peculiar properties, and produce the substance called water; in such cases the new fact may be subjected to experimental inquiry, like any other phenomenon; and the elements which are said to compose it may be considered as the mere agents of its production; the conditions on which it depends, the facts which make up its cause.

The effects of the new phenomenon, the properties of water, for instance, are as easily found by experiment as the effects of any other cause. But to discover the cause of it, that is, the particular conjunction of agents from which it results, is often difficult enough. In the first place, the origin and actual production of the phenomenon are most frequently inaccessible to our observation. If we could not have learned the composition of water until we found instances in which it was actually produced from oxygen and hydrogen, we should have been forced to wait until the casual thought struck some one of passing an electric spark through a mixture of the two gases, or inserting a lighted taper into it, merely to try what would happen. Besides, many substances, though they can be analysed, cannot by any known artificial means be recompounded: Further, even if we could have ascertained, by the Method of Agreement, that oxygen and hydrogen were both present when water is produced, no experimentation on oxygen and hydrogen separately, no knowledge of their laws, could have enabled us deductively to infer that they would produce water. We require a specific experiment on the two combined. [p440]

Mill’s argument here is simple. No knowledge of the properties of the parts (hydrogen and oxygen) would give us a knowledge by deduction of the properties of the whole (H2O). But something happened since Mill. Quantum mechanics and the periodic table happened. Now we can predict, to a high degree, what the properties of compounds are, depending upon our computational capacities (and I mean that literally – the capacities of our computers). For example, we often predict the microstructure of water itself just from a knowledge of the affinities and bonds of the H2) molecules (e.g., Paricaud et al. 2005). A philosophical argument can sometimes be overtaken by science and advances in computation.

So Mill’s point is that this is difficult, but not an in-principle impossibility. The properties of water are surprising to us based on what we know about hydrogen and oxygen in 1843, and due to limitations in what we can work out from them. It reminds me a little of this recent comic:

Abstrusegoose deduction

Of course Aragorn cannot do this (without some serious time on a supercomputer, anyway), but hey, it can be done. So if the wetness of water is a surprise, it is because we have some threshold for expectations of unsurprising and surprising results. What is this?

The surprisal value of a sequence (pieces of information in order) is roughly the inverse of expectation that it would occur by chance (see here). It is a measure of the amount of information expressed by a particular outcome, measured in bits (the negative logarithm of the probability of that event or sequence), given a distribution of outcomes and their probabilities. Now the surprisal of water having, say, a viscosity of some type, is relative to the framing assumptions we have for the ways elements interact when combined. It depends rather crucially on what we know and can work through.

If you only know the observable properties of these elements, then the idea that two gases would form a liquid at room temperature is surprising. The subjective surprisal value is high because the subjective probabilities are low. But once you understand the strong and weak bonds of these elements, and you have sufficient computational capacity to work out how these properties play out in large ensembles, you can predict the structure of water right down to the surface tension when impurities occur, and so even the biological properties of water. Once you have a fully formed quantum theory of the elements, then the liquidity of water is not longer a surprise, just a matter of working through the objective probabilities.

If one can derive a macro phenomenon from micro properties through theoretical deduction, then the emergence is simply subjective surprisal. So the remaining argument for objective emergence is, in my view, the downward causation argument, which I’ll address in the next post.


Mill, John Stuart. 1974. A System of Logic, Ratiocinative and Inductive: Being a Connected View of the Principles of Evidence and the Methods of Scientific Investigation, Books I–III. Edited by J. M. Robson. Vol. VII, Collected Works of John Stuart Mill. Toronto, Buffalo NY, London: University of Toronto Press/Routledge & Kegan Paul.

Paricaud, Patrice, Milan P?edota, Ariel A. Chialvo, and Peter T. Cummings. 2005. From dimer to condensed phases at extreme conditions: Accurate predictions of the properties of water by a Gaussian charge polarizable model. Journal of Chemical Physics 122 (24):14.



  1. It’s a small nit, but since you mentioned supercomputing I felt compelled to contribute something.

    You are (I think) conflating “sufficient computation” and “understanding” here:

    Once you have a fully formed quantum theory of the elements, then the liquidity of water is not longer a surprise, just a matter of working through the objective probabilities.

    More often than not, this isn’t the case.

    First, an example in your favor: Vasily V. Bulatov et al, Dislocation multi-junctions and strain hardening, Nature, v440, 27 April 2006. Crystal dislocation was understood well enough to code up a simulation, sufficient supercomputer time resulted in an unexpected result that was then validated on crystals in the real world. The emergent property her is “multi-junctions”, which were initially a emergent property of the simulation and only later found out to be an emergent property of the simulated material as well. Now that we have a better understanding of the process, mult-junctions are no longer seen as an emergent property but rather as an explicable, well-understood property.

    Good so far?

    The four-color theorem represents what I consider to be the common case. Based on a naive understanding of coloring maps, we have a high degree of surprise that no map needs more than four colors — this is an emergent property of topology. We can then simulate 1,936 different maps given sufficient computational power and show this result from first principles.

    What’s lacking here is insight. We have established we can get from a set of first principles A to a result B, but we’ve not appreciably reduced our surprise that this happens. In short, we haven’t increased our understanding.

    I see this in papers ranging from computer architecture simulations to ecosystem simulations. At one end of the process are wholly plausible first principles, at the other is an effect that the authors would like to convince me was non-obvious, and in the middle is a lot of supercomputer time. What’s lacking is the model: some formulation that would allow me to apply this result to a different set of inputs. Without that model (and without the understanding that a model demonstrates) I’d argue that the property is just as emergent as it was before the simulation started.

    • You have in fact supported my argument, which is that what counts here is our ability (or not) to grasp some properties based on knowledge of the parts. I never argued for a Chinese Room style syntactic account of understanding, merely of explanation. If I can explain one thing in terms of its parts, but at some point I cannot track how all these things work together in a large enough population or ensemble, has the ontology changed? Not at all; merely my ability to track inferences. When the deductive chain gets too long for us to hold in our heads, we call it “emergent”.

  2. The question is if explaining something as to its underlying causes explains the thing itself. Aristotle started his Metaphysics listing ancient theories of “the first causes” (nowadays it is quantum mechanics and genes, after next 500 hundred years – who knows). Yet the question he posited was not the causes but the “being” itself. This is the main point that Franz Brentano was discussing in his dissertation which Martin Heidegger considered as the crucial work and main question also of his own philosophy. The question of “being” was posited anew by Hegel and Husserl (inspired by Brentano as well) . Unless we know what the being is, what the life is then genetic or quantum mechanics won´t help us to “understand”. It would help us to techné. To understand is not the same as how to manipulate, how to process, how to reduce. Heidegger quoted Goethe: “Man suche nicht hinter den Phänomenen, sie selbst sind die Lehre.”

  3. jeff jeff

    Yes, you can theorectically predict many macro properties from micro with enough computer power, but all of that would be meaningless (and impossible) without first being able to acquire enough experience to correlate and calibrate the macro with the micro. You could never understand ‘wetness’ if you did not exist at the scale you do and have experience with things that are wet. Wetness is a subjective property that cannot be understood in terms of impersonal physical properties. You understand it much better by letting water run through your fingers.

    • Hi, Jeff.

      I’m going to disagree with you coming and going,

      Yes, you can theorectically predict many macro properties from micro with enough computer power

      I work with some of the largest supercomputers on the planet and I have not found this to be the case. Predicting macro properties requires abstraction, and that requires leaving out information, not adding more of it.

      As an example, let’s say you wanted to run a computer simulation of an environment that might be conducive to the formation of life. I can code up the physics and chemistry of such a systems, and given sufficient computing power I could even give you a molecular dynamics simulation of a mid-sized pond.

      Now, you tell me: what does life look like? I can tell you the location and chemical bonding of every atom in this word at any particular moment in time. Which bits of it are living, and how does throwing more supercomputing power at the problem allow you to make a better determination?

      Wetness is a subjective property that cannot be understood in terms of impersonal physical properties.

      This I don’t understand at all. If you hand me a beaker full of a particular compound, then I can test the viscosity, granularity, specific heat, etc. and come up with a very good prediction whether or not anyone else would consider it to be “wet” when it was poured over their fingers. If someone else didn’t consider it to be wet, I think the first explanation would be some sort of neurological condition rather than an idea that “wetness” is wholly subjective.

      • jeff jeff


        I agree with your first objection. You never know if you have enough information to do a an accurate enough simulation until you can compare it with what you observe at your own level of reality. But in any case, a simulation is always a simulation – not the real thing.

        Your second objection I’m not so sure about. Just because others may agree with my subjective feeling of wetness doesn’t necessarily make wetness an objective scientific fact.

  4. Bill Morse Bill Morse

    This reply was started in response to your first blog in this series It might still belong there, but it also addresses some concerns raised by other follows to this blog.

    Your claim that we can flatten reduction may be tempting, but you might want to consider why the ”layer cake” view is pervasive. I would argue that it is because that is how reduction actually achieves its successes. You are not going to get anywhere tyring to use Schrodinger’s equations to predict DNA structure, much less the behavior of elephant seals. Even if I granted you the ability to do that given enough computation power, it would be a waste of computation power. Given uncertainty and chaos, in fact details at lower levels become irrelevant at higher levels. We can’t yet predict protein folding, but even when we can it will be of little use in predicting the behavior of C. Elegans. The layer cake view started because of parsimony, and will continue because of computation efficiency.

    • I don’t reject the epistemic aspect of layercake reductionism – that is necessary, but also somewhat arbitrary as it depends on the current state of science and the relations between our theories (Nagelian reduction). But ontological reduction is the claim that all is merely physical, in one go. Any soecial fact, psychological fact, biological fact, or even chemical fact is just the way physics has worked out in that case.

  5. Me Me

    I’d like to read this article, but the text becomes covered by a white layer after I roll a little bit down. It’s on Windows, Chrome.
    You’d better fix that.

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