The Water Paradox
The Water Paradox
It has been a while since we had a new paradox to cudgel our brains over. For your edification (and frustration) I will present what I call “the water paradox”. Like all paradoxes it aims to derive an absurdity from self-evident premises, thus demonstrating the auto-destructive powers of reason. We are not meant to accept the paradox as true (that’s why it’s a paradox), but to marvel at its existence. So consider the following principle: “Every object wholly composed of solid parts is solid”. That sounds right and examples confirm it: a rock is composed of solid parts and is solid, and similarly for a block of ice. Some things are not solid, such as molten metal, but they have non-solid parts: liquid things have liquid parts. If a substance has some solid parts, it is not wholly liquid; it is partly solid. If a sea is partly frozen, it is not liquid tout court; it is only partly liquid. It would be false to say of it, without qualification, that it is liquid. Someone could rightly reply that itisn’t liquid, though many of its partsare. To be liquid requires that allof it be liquid.
But is it true that what we routinely call liquid water is liquid with respect to its parts? What about its constituent molecules? The OEDdefines “solid” as “firm and stable in shape”, so that “liquid” means “not firm and stable in shape”. Drinking water is not firm and stable in shape, but its constituent molecules are—theyare not liquid. They slide over each other in so-called liquid water, but they are individually as solid as any solid object. So the molecular parts of water are themselves solid in both its solid and liquid state. But according to our principle, if all the parts of an object are solid, then so is the object: therefore there is no such thing as liquid water! That is paradoxical, since there is certainly a distinction between two states of water, which we mark with the terms “solid” and “liquid”.
Suppose that we were quite unperceptive about water and simply never notice that the water we drink and swim in has lots of little chunks of ice in it. If we were giants, these might be quite big chunks that are beneath our notice. Then we discover, to our surprise, the facts about this water: shouldn’t we conclude that we were wrong to suppose that our water is liquid? Shouldn’t we conclude instead that it is only partly liquid? It seemed liquid to us, but actually it isn’t. Well, science has discovered that room-temperature water is composed of unobservable solid parts, and so is not liquid after all. Imagine if you were a creature that could drink sand and swim in sand, so that sand seemed like a liquid to you: you would be within your rights to compare it to a liquid from a practical point of view, but it would be false to say of sand that it is a liquid. What if you could crunch up ice in your mouth and swallow it without melting? It would be solid, though drinkable. Isn’t that the way it is with water and us as things stand? Water seemsliquid to us, but on closer inspection it turns out not to be, since it is made of non-liquid parts. From a molecule’s-eye point of view, water is like so much sand—solid particles jostling around each other. Is a galaxy to be declared liquid because its parts move in relation to each other? Is the universe one big liquid? No, the universe is a solid object made of solid moveable parts. Isn’t that precisely what we have discovered water to be? Its liquidity is entirely superficial once you get down to the chemistry.
You might try to deny the premises of this argument. You might deny that molecules are solid, perhaps on the ground that they are parts of a liquid. But that seems hopeless given the empirical facts of chemistry, molecules being firm and stable objects; and anyway we can push the argument down to the atomic parts that compose molecules—they certainly aren’t liquid. Second, you might attack the main premise of the argument: you might claim that it is just not true that objects wholly composed of solid objects are solid—liquid water being a counterexample to this principle. You might say that liquidity merely requires the free motion of solid parts relative to each other, not liquidity all the way down. We have already seen that this is not the correct analysis of the concept of liquidity, since sand and galaxies are not liquids. But there is a further consideration: for consider substances that areliquid all the way down, unlike water–how should we describe such substances? Suppose Sis a substance that is very like water in its superficial appearance but whose physical nature is not atomic-molecular but continuous and infinitely malleable.Sis physically the way we assumed water to be before we discovered atoms and molecules: we thought everythingabout solid water (ice) melted when it was heated, not realizing that it has hidden components that resist melting. We can say that Sis super–liquid, meaning that it has no solid parts but is liquid through and through. Sis apparently moreliquid than water, as water with no bits of ice in it is more liquid than water with bits of ice in it. Sis wholly and completely liquid, pervasively liquid, right down to its fine structure, while water is liquid only superficially—when you look into it closely there is a lot of solidity there.
But do we really want to talk this way? What is this idea of one thing being more liquidthan another? Aren’t things either liquid or not? Isn’t it that Sis reallyliquid, but room-temperature water is not? On some planets the water is never liquid but always exists in a solid state (i.e. frozen): isn’t it the truth that water is never literally and objectively liquid, given its actual chemical nature? Eddington famously argued that matter is never really solid, given the amount of space present in atoms; his point was not merely that some things are more solid than others depending upon the amount of space they contain.We have discovered these things and they contradict our normal linguistic practices—they even challenge our concepts. We thought that matter is solid (dense, continuous), but it is not; we thought that water is liquid (in one of its forms), but it is not. Our ordinary concepts simply don’t apply. Those concepts were formed before we understood the nature of the physical world; they reflect our naïve pre-scientific understanding of nature. We had no idea that the parts of so-called liquids were solid, as we had no idea that so-called solids were mostly made up of space. Have we discovered that everything is really a gas—tiny particles widely separated in space? The principle I started with sounds correct on first hearing, indeed trivially true, but it leads quickly to the conclusion that nothing is liquid—nothing in our actual universe anyway. That is certainly disturbing and counter-intuitive, but maybe it is the sober truth. We can accordingly either abandon the word “liquid” as factually erroneous or retain it as a mere manner of speaking (like saying the sun rises). Our commonsense views of the physical world have been wrong before, and this is another example of that. Zeno argued paradoxically against the reality of motion, concluding that motion is not real; the present argument is designed to show, paradoxically, that liquidity is not real (both arguments are based on considerations about parts). It is rather as if “animal” meant “creature created by God” and then we discover that the things we call “animals” were not created in that way; the proper conclusion would be that no animals in that sense exist. We can craft a new word without the divine implication, and we could also replace “liquid” with some substitute that better reflects the facts, say “squishy”. What we can’t do is keep on talking in the old discredited way.
But why is this a paradox? Haven’t we simply discovered that nothing is liquid, as we have discovered that nothing is solid, or as Darwin discovered that there no divinely created animals? Our commonsense beliefs are just false. The same might be said of Zeno’s argument: it isn’t a paradox, just a demonstration that motion is unreal. We should simply stop saying that objects move: we live in a stationary world. Similarly, we should stop saying that substances are liquid: we live in a solid world (or a gaseous world if we follow Eddington). The trouble, however, is that the displaced beliefs are not so easily expendable: we can readily agree that there are no unicorns–but no moving objects! Some things stay still and some don’t: isn’t that just a fact? Likewise, is there no distinction between drinking water and ice? There is a distinction between moving and not moving, so we can’t just abandon the whole idea of movement—hence Zeno’s argument is a paradoxnot merely a non-existence proof. In the same way, the water paradox is not merely a proof that liquids don’t exist; it’s a genuine paradox because we can’t just abandon that idea. Some bodies of water are clearly different from other bodies of water—bathwater is different from frozen water. What word best captures this difference? The word “liquid” obviously, or some synonym; we can’t just dispense with the concept of liquidity. Hence we are reluctant to accept the argument against liquidity; we don’t just cheerfully accept a conceptual clarification. We want to protest that water is(often) liquid, no matter what the argument says. We are thus tugged in two directions. We might even be willing to contemplate accepting that some bodies of water are bothliquid and non-liquid, distinguishing two senses of “liquid”, or simply accepting the contradiction as true (as with diatheleism). We can’t just nonchalantly accept that drinking water isn’t liquid, as we can’t just nonchalantly accept that trains don’t move. These are genuine paradoxes not straightforward refutations of falsehoods.
It is a striking fact about the classic paradoxes (Zeno’s, the Liar, the Sorites, Russell’s) that they have been around for a long time and yet very little progress has been made with them. People periodically announce purported solutions, but there is little consensus and the core of the problem seems to remain, stubborn and defiant. Reason seems to undermine itself. Unreason we could understand leading to paradox—but reason! What is going on? Will we keep discovering new paradoxes while never solving the old ones? Might everythingturn out to be paradoxical on close analysis? Is paradox the rule rather than the exception? And what would this tell us about human thought? The fact that it isn’t too difficult to generate a paradox about liquid water is worrying—what’s next?
There is an ambiguity in the word “solid” in these discussions: it can either mean firm and stable in shape or dense in structure. In this essay I am using the first sense; Eddington was using the second sense (he didn’t deny that ordinary objects have a firm and stable shape).
Is one purpose of a good paradox to manifest the limitations of a static or absolutist psychological perspective or mode of thinking that is at odds with a dynamic or non-reductive aspect of reality?
Nothing is Liquid or Solid “all the way down”. Liquid and solid are relative concepts. Even though they are relative, we can still use them together with the word “is” and they can still reflect some feature of reality. (In fact, can you really imagine a world where some things were solid “all the way down”?)
I suspect the problem of free-will falls into this camp as well.
One reason so many of these paradoxes have not been grappled with properly I guess is because absolutism and reductionism are such strong psychological forces. Reductionism has become particularly powerful as a way of approaching the endeavour of understanding the world (i.e. to doing science), because it has proven so successful as a unifier in physical and biological sciences. (Though he doesn’t discuss classic paradoxes, the Nobel prize winning physicist Phil Anderson addresses this phenomenon in his book/collection of essays More and Different: Notes from a thoughtful Curmudgeon.)
Take Zeno’s paradox. The consensus (from people that should know better) appears to be that the arithmetisation of analysis (formalisation of limits etc) has resolved the paradox, namely that we, unlike Zeno, now know 1/2 +1/4 +… = 1. However, this “equality” is actually a statement that 1 is the mathematical limit of a particular sequence and amounts to saying: for any number 0.99….9 < 1, no matter how close to 1, I can find an integer n such that the finite sum 1/2 + 1/4 + … + 1/(2^n) is closer to (but obviously not equal to) 1. Hang on, isn’t this just Zeno’s statement?! This mathematical ‘observation’ is equivalent to Zeno’s – it just has been formalised in arithmetic. Therefore, it cannot not provide any insight into the paradox whatsoever.
Limits are taught in some high schools, and certainly in first year university maths degrees, so it is pretty remarkable (shocking, actually) that so many people have let themselves believe 1/2 +1/4 +… = 1 resolves Zeno’s paradox. The resolution of the paradox presumably comes from recognizing that motion is primitive, and not reducible to a sequence of positions as is tacitly assumed in the paradoxical mindset. What’s odd is that this insight is at the bedrock of physics: first there was Galilean relativity, then there was Newton’s formulation of the calculus and the formulation of gravity as a second order law (i.e. the ‘state’ of a particle comprises its position and motion, and the law of nature is a second order differential equation – critically, not a first order equation); and then we had Einstein’s theories of relativity. So, despite physics embracing motion as primitive, and developing and using calculus as a fundamental tool to describe nature, we still haven’t progressed much at an intellectual level with Zeno’s paradox. Maybe because philosophy is not taken seriously enough as a science?
I posted before responding to your question “Might everything turn out to be paradoxical on closer analysis?”
Probably yes, in the sense all non-trivial identities are necessarily not pure identities. Does 8 equal 5+3?
Another question: how much of this is a feature just of the mind, and how much a feature of nature independent of the mind (which may nevertheless be reflected in the workings of the mind through evolution)?
I tried to answer this question in “The Puzzle of Paradox” in Philosophical Provocations. Hard question.
Does numeral “8” refer to 4+4 and 8+3 and 7+1? That sounds like an odd thing to say, so “8=5+3” can’t be an identity statement.