Gödel’s First Incompleteness Theorem in Simple Symbols and Simple Terms

 

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The following piece explains a particular symbolic expression (or version) of Kurt Gödel’s first incompleteness theorem. It also includes a particular expression (or example) of a Gödel sentence (i.e., “This statement is false” — this link takes you to a humorous entry!). In terms of the actual symbols used, this representation of the theorem expresses a (slight) philosophical and logical bias. So it’s worth noting that almost every symbolisation of the theorem is unique — if sometimes only in tiny detail. (In logic and mathematical logic there’s the common phenomenon of various symbolic “dialects” competing with — or simply complementing — each other.) And this representation and explanation exclude all the other details which surround the bare theorem itself. Indeed this symbolic representation alone doesn’t prove or demonstrate anything. And even when the symbols are defined or interpreted, that’s still the case. In addition, it’s worth distinguishing the truth (and lack of proof ) of a Gödel sentence from any proof of the first incompleteness theorem itself — even if the two can’t be entirely disentangled!

Thus the following words don’t attempt to tackle the arguments and extra details which are required to establish the theorem. And neither do they extrapolate anything from it. However, even this basic approach is bound to leave out much detail. And that’s simply because this is a short introduction to a particular symbolisation of Gödel’s first incompleteness theorem.

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Three things need to be noted to begin with:

1. The first incompleteness theorem is essentially about systems and the truth-values of certain statements within those systems. (Alternatively, the first incompleteness theorem is about a particular system and a Gödel sentence within that particular system.)
2. Those systems and statements are arithmetical and therefore use natural numbers. (In other words, the first incompleteness theorem is not applied across the board — as it often is.)
3. Within those systems there are some true statements about natural numbers which cannot be proved within those systems. (Alternatively, within a given system there will be a true statement about natural numbers that cannot be proved within that system.)

To get to the core of Kurt Gödel’s first incompleteness theorem, let’s sum it up in its bare logical (or symbolic) form. This particular symbolism (just one among many) will hopefully capture what’s at the heart of the theorem.

Take the following symbolic representation from the logician and philosopher Professor Alasdair Urquhart (as found in his paper ‘Metatheory’):

G ↔ ¬Prov(⌜G⌝)

The following is a list of definitions of the symbols in the biconditional theorem above:

G = a Gödel sentence
↔ = if and only if (i.e., the biconditional symbol)
¬ = negation (or “not”)
Prov = provable
¬Prov = not provable
⌜G⌝ = The “code number” of the Gödel sentence G. (The superscripted Quine corners are — basically — quotation marks.)

Thus G ↔ ¬Prov (⌜G⌝ ) means:

The sentence “This sentence is false” is true if and only if it is not provable in system T (i.e., the system to which it belongs).

Or:

Gödel sentence G is true if and only if there is no proof of G in system T (i.e., the system to which it belongs).

So why is the symbol G put in brackets after the if and only if (i.e., ↔) sign and the sign (i.e., ¬) for negation? Why do we have the symbol ⌜ G⌝ rather than plain G? This is because the brackets (i.e., ⌜ and ⌝) symbolise self-reference or “quotation”. That is, firstly we have the symbol G, and then when we refer to G we get ⌜G ⌝.

Thus ⌜ G⌝ is a “code number”.

A code number is a number which is used to identify something. This means that ⌜ G⌝ is the code number of the Gödel sentence G (i.e., the symbol G without brackets). Furthermore, a Gödel number is a specific kind of code number. In mathematical logic, Gödel numbers are natural numbers which are assigned to statements (as well as to the individual symbols within those statements ) within a given system or formal language.

In terms of the biconditional symbol (i.e., ↔).

This symbolises that both sides of the equality sign (i.e., =) are logically equivalent in that both are either jointly true or jointly false. Note: this doesn’t also mean that they have the same meaning.

This is one expression of the aforesaid biconditional:

i) G 
is true if and only if 
ii) ¬Prov(⌜ G⌝) 
is true.

Alternatively, the inversion (i.e., since the theorem includes a biconditional):

i) ¬Prov(⌜ G⌝)
is true if and only if
ii) G
is true.

The Gödel sentence G (in this instance, “This statement is false”) is self-referential. That is, it refers to itself (or G refers to G). The archetype of this Gödel sentence is the Liar paradox; which is also self-referential. Indeed self-reference is at the heart of the whole show! Without self-reference we wouldn’t have a Gödel sentence or the problems and insights which arise from it. (See my ‘Why Empty Logic Leads to the Liar Paradox’.)

In addition, Gödel sentence G is true if and only if there is no proof of G. Ordinarily it’s taken that a mathematical statement P is taken to be true if and only if there is a proof of P. Gödel’s first incompleteness theorem is saying the literal opposite of that.

What’s also important here is to note the Gödel sentence’s position in a system (or theory). None of this makes any sense outside the context of the system (or theory) to which Gödel sentence G belongs. In other words, taking G entirely on its own makes no sense at all.

Is Roger Penrose a Platonist or a Pythagorean?

 

Roger Penrose is not only a mathematical physicist: he’s also a pure mathematician. So it’s not a surprise that Penrose expresses the deep relation between mathematics and the world (or nature) in the following way:

“[T]he more deeply we probe the fundamentals of physical behaviour, the more that it is very precisely controlled by mathematics.”

What’s more:

“[T]he mathematics that we find is not just of a direct calculational nature; it is of a profoundly sophisticated character, where there is subtlety and beauty of a kind that is not to be seen in the mathematics that is relevant to physics at a less fundamental level.”

Penrose is (rather obviously) profoundly aware of the importance of mathematics to (all) physics. Yet, more relevantly to this piece, he’s also aware that maths alone can sometimes (or often) lead the way in physics… and sometimes in a negative manner! So despite the eulogies to mathematics above, Penrose offers us these words of warning:

“In accordance with this, progress towards a deeper physical understanding, if it is not able to be guided in detail by experiment, must rely more and more heavily on an ability to appreciate the physical relevance and depth of the mathematics, and to ‘sniff out’ the appropriate ideas by use of a profoundly sensitive aesthetic mathematical appreciation.”

Platonism and Pythagoreanism in Contemporary Physics

Roger Penrose is a Platonist, not a Pythagorean. (Or at least he’s a Platonist in certain respects — see here, here and my ‘Platonist Roger Penrose Sees Mathematical Truths’ ) One reason why this can be argued is that Penrose admits that he

“might baulk at actually attempting to identify physical reality within the reality of Plato’s world”.

To the Pythagorean, the world literally is mathematical. Or, perhaps more accurately, the world literally is mathematics (i.e., the world is literally constituted by numbers, equations, etc.). That may sound odd. However, if we simply say that “the world is mathematical”, then that may (or does) only mean that the world can be accurately — even if very accurately — described by mathematics. The Pythagorean, however, states such phrases as “things are numbers”. He therefore establishes a literal identity between maths and the world (or parts thereof).

To the Platonist, on the other hand, the mathematical world is abstract and not at all the same as “physical reality”. (Plato often actively encouraged philosophers and mathematicians to turn their eyes — or souls — away from the physical world.) Yet it’s still undoubtedly the case that abstract mathematics — even Platonic mathematics — is a fantastic means to describe the world. Despite that, the Platonic world is still abstract and not identical to the physical world. In other words, there is no identity between the physical world and the Platonic world. However, there is an identity between the physical world and the Pythagorean world.

More generally, even a (at times) hard-headed positivist (see here) like Werner Heisenberg recognised the importance of the Pythagorean tradition in physics. He argued that

“this mode of observing nature, which led in part to a true dominion over natural forces and thus contributes decisively to the development of humanity, in an unforeseen manner vindicated the Pythagorean faith”.

All that may depend on what Heisenberg meant by the word “Pythagorean”. After all, it’s often the case that the word “Pythagorean” is simply used as a literal synonym for the word “Platonic”. Thus having said all the above, such distinctions between Platonism and Pythagoreanism (at least in these specific respects) may be a little vague or even artificial.

This may apply to Roger Penrose’s position too.

Take Penrose’s own (as it were) quasi-Pythagorean reading of the “complex-number system”. He writes:

“Yet we shall find that complex numbers, as much as reals, and perhaps even more so, find a unity with nature that is truly remarkable. It is as though Nature herself is as impressed by the scope and consistency of the complex-number system as we are ourselves, and has entrusted to these numbers the precise operations of her world at its minutest scales.”

Since the passage above is fairly poetic, it’s difficult to grasp exactly how Pythagorean it actually is. More clearly, surely the words

“[i]t is as though Nature herself is as impressed by the scope and consistency of the complex-number system”

are purely poetic — even if there’s a non-poetic “base” that’s expressed by the poeticisms. Penrose does, after all, prefix the statement above with the words “[i]t is as though”. So surely it can be said that Nature doesn’t need (or require) the complex-number system. It is us human beings (or physicists) who need that system in order to describe Nature.

In any case, Penrose’s strongest (possibly Pythagorean) claim is:

The complex numbers “find a unity with nature”.

Now is that “unity” also an identity? Not necessarily. After all, numbers may be united with Nature only in the sense that they can describe it perfectly. Saying that numbers are identical with nature, on the other hand, is something else entirely. As it is, the phrase “unity with nature” is hard to untangle. (Hence my use of the word “poetic” earlier.)

Having put a quasi-Pythagorean position on (at the least) complex numbers, Penrose then puts a (literally) down-to-earth position on the real numbers. Penrose writes:

“Presumably this suspicion arose because people could not ‘see’ the complex numbers as being presented to them in any obvious way by the physical world. In the case of the real numbers, it had seemed that distances, times, and other physical quantities were providing the reality that such numbers required; yet the complex numbers had appeared to be merely invented entities, called forth from the imaginations of mathematicians.”

Despite using the phrase “down-to-earth position” before the quote above, this passage is at least partly Pythagorean in that it states that

“distances, times, and other physical quantities [] provid[ed] the reality” which real numbers “required”.

This can be read as meaning that the real numbers are (as it were… or not) embodied in distances, times and other physical quantities. Yet — historically at least — it seems that complex numbers didn’t pass that Pythagorean test.

Examples: Paul Dirac, Etc.

It’s undoubtedly the case that various well-known (as well as largely unknown) physicists have often been led by mathematics when it comes to their theories. That is, they certainly haven’t always been led by experiments or by observation.

Take the case of Paul Dirac.

Dirac found the equation for the electron (see here). He also predicted the electron’s anti-particle (see here). Both the finding and the prediction came before any experimental evidence whatsoever.

Penrose calls Dirac’s finding of the equation for the electron an “aesthetic leap”. However, Penrose also says that it arose

“from the sound body of mathematical understanding that had arisen from the experimental findings of quantum mechanics”.

That basically means that although Dirac’s mathematics was (as it were again) pure, “the experimental findings of quantum mechanics” must still have been swirling around in Dirac’s head as he carried out his pure mathematics.

The Dirac case also shows us the to and thro between (pure) maths and experimental findings. That is, even if we have aesthetic and/or mathematical leaps, the mathematical physicists concerned were clearly still aware of the experimental findings which proceeded their abstract leaps. What’s more, Dirac’s own leaps were “made with great caution and subsequently confirmed in observation”. Indeed in both Dirac’s cases, confirmation came very quickly.

A purely philosophical slant can be put on the Dirac case. (Although I’m a little wary of shoehorning philosophical terms — or ways of thinking - onto what physicists have done.) As the philosopher James Ladyman (technically) puts it:

“Sophisticated inductivism is not refuted by those episodes in the history of science where a theory was proposed before the data were on hand to test it let alone suggest it... Theories may be produced by any means necessary but then their degree of confirmation is a relationship between them and the evidence and is independent of how they were produced.”

We can now say that in Dirac’s case there was no “data [] on hand to test it let alone suggest it”. Actually, the last clause (“let alone suggest it”) may be a little strong in that previous experiments in (quantum) physics must surely have suggested various things to Dirac. The thing is, Dirac still had no (hard) data to back up his prediction or equation. Despite that, Dirac’s theories were “produced by any means necessary” (or by any mathematical means necessary) and only then were they confirmed.

To get back to Penrose.

Penrose goes into more detail elsewhere when he says that in the cases of Dirac’s equation for the electron, Einstein’s general relativity and “the general framework” of quantum mechanics,

“physical considerations — ultimately observational ones — have provided the overriding criteria for acceptance”.

Opposed to that, Penrose goes on to say that

“[i]n many of the modern ideas for fundamentality advancing our understanding of the laws of the universe, adequate physical criteria — i.e. experimental data, or even the possibility of experimental investigation — are not available”.

Penrose then concludes by saying that

“we may question whether the accessible mathematical desiderata are sufficient to enable us to estimate the chances of success of these ideas”.

All above shows us that Penrose is still acknowledging that (in a basic sense at least) the mathematics comes first. That is, Penrose believes that any “acceptance” of the “ideas” for “our understanding of the laws of nature” often comes after the (pure) mathematics. (That’s if the maths is ever truly pure in that previous experiments, observations, physical theories, etc. will — or may — be swilling around in the mathematical physicist’s head.) To repeat: the mathematical speculation (or theorising) comes first, and only then do physicists expect the “physical considerations” to provide the “overriding criteria for acceptance”.

When it comes to many (or some) “modern ideas” (Penrose mainly has string/M theory in mind — see here), on the other hand, “physical criteria” are “not available”. Yet that was also true — as we’ve seen — of the examples which Penrose himself cites (i.e., quantum mechanics, general relativity and Dirac’s equation for the electron). In these example, physical criteria were not available at the times these ideas were first formulated. This means that the observations, confirmations, experiments, etc. came after — even if very soon after.

So what if the experiments haven’t been done? Which precise experiments must guide the physicist? And what if there are no currently relevant or possible experiments which can guide the theoretical physicist? Of course it can now be argued that if there are no relevant, actual or possible experiments (or observations), then in what sense is any given physicist — even if mathematical physicist — doing physics at all?

String Theory and Penrose’s Twistor Theory

Despite Penrose’s emphasis on the fundamentally important role of maths in physics (which is hardly an original emphasis), Penrose is still highly suspicious of the nature of string theory.

Although Penrose doesn’t always name names, he still stresses “the mathematics that is relevant to physics”. He warns that

“if it is not able to be guided in detail by experiment, [it] must rely more and more heavily on an ability to appreciate the physical relevance and depth of the mathematics, and to ‘sniff out’ the appropriate ideas by use of a profoundly sensitive aesthetic mathematical appreciation”.

This squares with what British science writer and astrophysicist John Gribbin has to say.

Gribbin too talks in terms of what he calls a “physical model” of “mathematical concepts”. He writes (in his Schrodinger’s Kittens and the Search for Reality) that “a strong operational axiom” tells us that

“literally every version of mathematical concepts has a physical model somewhere, and the clever physicist should be advised to deliberately and routinely seek out, as part of his activity, physical models of already discovered mathematical structures”.

Yet even in Gribbin’s case, it’s still clear that a “mathematical concept” comes first and only then is a “physical model” found to square with it.

Penrose’s words on his own twistor theory are also very relevant here in that after criticising string theorists for seemingly divorcing their mathematics from experiment, prediction, observation, etc., he then freely confesses that he’s — at least partly - guilty of exactly the same sin.

Firstly, Penrose tells us about the pure mathematics of twistor theory. He writes:

“Yet twistor theory, like string theory, has had a significant influence on pure mathematics, and this has been regarded as one of its greatest strengths.”

Penrose then cites a couple of very-specific examples:

“Twistor theory has had an important impact on the theory of integrable systems [] on representation theory, and on differential geometry.”

And then we have the mathematical aesthetics of twistor theory:

“Twistor theory has been greatly guided by considerations of mathematical elegance and interest, and its gains much of its strength from its rigorous and fruitful mathematical structure.”

Finally, the confession:

“That is all very well, the candid reader might be inclined to remark with some justification, but did I not complain [] that a weakness of string theory was that it was largely mathematically driven, with too little guidance coming from the nature of the physical world? In some respects this is a valid criticism of twistor theory also. There is certainly no hard reason, coming from modern observational data, to force us into a belief that twistor theory provides the route that modern physics should follow… The main criticism that can be levelled at twistor theory, as of now, is that it is not really a physical theory. It certainly makes no unambiguous physical predictions.”

So how does Penrose extract himself from this problem? Well, to be honest, he doesn’t go into great detail — at least not after these specific passages.

The obvious question to ask now is this:

What is twistor theory doing right that string theory is doing wrong?

Is the answer to that question entirely determined by how close each theory is to “the nature of the physical world”? But don’t we (as it were) get to the physical world only through theory? As Stephen Hawking once put it:

“If what we regards as real depends on our theory, how can we make reality the basis of our philosophy? But we cannot distinguish what is real about the universe without a theory… Beyond that it makes no sense to ask if it corresponds to reality, because we do not know what reality is independent of theory.”

In any case, perhaps it’s the case that (as Penrose may believe) the mathematics of twistor theory is superior to the mathematics of string theory.

String theory particularly has been criticised for not making “unambiguous physical predictions”. Yet here’s Penrose saying exactly the same thing about his own twistor theory.

Finally, there probably never is (to use Penrose’s own words) “a hard reason” to “force” us to believe any physical theory — at least not in the early days of such theories. This obliquely brings on board the largely philosophical idea of the underdetermination of theory by data in that the “modern observational data” which Penrose mentions will never be enough to force the issue of which theory to accept. In other words, whatever observational data there is can be interpreted (or theorised about) in many ways. Alternatively, the same observational data can produce — or be explained by — numerous (often rival) physical theories.

What is Water? A Philosophical Inquiry into Natural Kinds

 

The classic case of a natural kind is water. This natural kind throws up many problems. These problems have been debated many times in — mainly analytic — philosophy. One main focus in this debate has been on the differences between water’s “microscopic” (or “microstructural”) properties and its “macroscopic” properties. More specifically, there’s been some kind of philosophical opposition which has been made between the microstructure of water and its “classical” (or macroscopic) properties. Added to that (though related to macroscopic properties) is the emphasis which has been made on our phenomenological (or phenomenal) experiences of water.

Even those who accept that there are natural kinds still acknowledge that some of the things which are taken to be natural kinds today weren’t taken to be so a hundred years ago (or even more recently). Now facts like that alone don’t give one a reason to be sceptical about the reality of natural kinds. Take mathematics as a similar case. Mathematicians (not just Platonists or realists) believe that there are determinate answers (or “solutions”) to mathematical “problems” even if in the past mathematicians have sometimes got the answers wrong or we don’t have the correct answers today. Similarly, even if we make mistakes about natural kinds; surely they must still be real.

But let’s firstly start off with one definition of “natural kinds” from the Internet Encyclopaedia of Philosophy:

“[I]t is commonly assumed that, among the countless possible types of classifications, one group is privileged. Philosophy refers to such categories as natural kinds. Standard examples of such kinds include fundamental physical particles, chemical elements, and biological species… Candidates for natural kinds can include man-made substances, such as synthetic elements, that can be created in a laboratory…. Groupings that are artificial or arbitrary are not natural; they are invented or imposed on nature. Natural kinds, on the other hand, are not invented, and many assume that scientific investigations should discover them.”

I won’t go into great detail about the definition above because much of it isn’t relevant to what will be discussed later. However, it can be said that it’s certainly the case that as a chemical substance, water (or a sample of H₂O molecules) is “privileged” by… well, (at the very least) philosophers. Yet water is privileged precisely because it’s seen — by philosophers — as being the classic natural kind. So that’s a kind of circular situation.

The passage above also says that natural kinds “are not invented” (i.e., unlike “groupings that are artificial or arbitrary”). There’s a problem here too. What a natural-kind term refers to (or is supposed to refer to) may not be invented. However, the natural kind term itself, and all the definitions which “belong” to it, most certainly are invented. (In a strong sense, this is a case of applying an anti-realist position to natural kinds.) And that may account for the problems we encounter in many discussions of natural kinds — that confusion (or conflation) of what natural kind terms refer to and natural-kind terms (along with their definitions) themselves. What natural-kind terms refer to may certainly be real (at least in most cases). Nonetheless, it’s taking them as natural kinds that’s the (philosophical) problem. And that problem lies at the heart of this piece.

Now considering what was said in the Internet Encyclopaedia of Philosophy definition of the term “natural kinds” above, it may seem strange to consider the possibility that what natural kinds are taken to be may be a somewhat contingent or even arbitrary matter. But, of course, many philosophers will now immediately state that if natural kinds are the result of contingent and/or arbitrary decisions (or even if they have a contingent and/or arbitrary nature), then they can’t be natural kinds at all!

All Samples of Water Contain Micro-Organisms

The philosopher George Bealer picks up on the microscopic-macroscopic-properties distinction (mentioned at the beginning) when it comes to water. He cites the classic twin-earth example of the opposition between XYX-as-water and H₂O-as-water. 

Readers must note here that Bealer’s following “thought experiment” is extremely artificial. However, that often doesn’t matter in philosophy because such cases are chosen to illustrate broader philosophical issues. And, in this instance, that broader philosophical issue is the — possibly? — contingent and/or arbitrary nature of natural kinds.

In his paper ‘Propositions’, Bealer firstly cites the possibility (or actuality) that

“all and only water here on earth is composed entirely of certain micro-organisms”.

Even if the factual claim that all samples of water contain “certain micro-organisms” is false, we can still argue that every sample of water will contain at least some constituents other than H₂O molecules — whether that includes “foreign” ions, dust, tiny bits of plastic, or Bealer’s micro-organisms. Now does that matter when it comes to to the nature of the natural kind water?

(It turns out that “nearly every body of water” here on Earth does contain microorganisms — see here. Despite that, one would intuitively believe that chemists must surely be able to create “pure water”. Yet, as it turns out, even chemists can’t do that — see here.)

Bealer states his broader position thus:

“If, like live coral or caviar, all and only water here on earth is composed entirely of certain micro-organisms, then on a twin earth a stuff which contains no micro-organisms whatsoever but which nevertheless contains the same chemicals as those found in samples of water on earth would not qualify as water.”

The general point here is that what we take water to be may be a contingent and/or even an arbitrary matter. Of course Bealer isn’t arguing that water here on earth isn’t H₂O. Or, more correctly, he isn’t arguing that all samples of water don’t contain mostly H₂O molecules. Bealer is saying that the fact that water contains mostly H₂O molecules is only a part of this (possible) story. In his example, literally all samples of water here on earth also contain micro-organisms. Now if every sample of water does contain micro-organisms, then why aren’t these micro-organisms part of the essence of water? Or, alternatively, why aren’t these micro-organisms constitutive of the natural kind water?

Now if we provisionally accept that all samples of water contains micro-organisms, then if what passes for water on Twin Earth doesn’t contain any micro-organisms, then surely it can’t be water. (At least it can’t be water, according to people on earth.) Yet here there seems to be an obvious distinction which can be made here between water (or H₂O molecules) and what else we may find in water. And that’s still the case even if we always find the same given x (other than H₂O molecules) in water.

For example, simply because all trees have fungi, moulds or lichen on them, that doesn’t mean that we have a joint tree-fungi/etc. natural kind. Similarly, if every sample of gold has microscopic particles of dirt on it, that wouldn’t mean that we have a joint gold-dirt natural kind.

The fact that a distinction can be made between water and what else is found in every sample of water may not matter when it comes to Bealer’s central point. That is, why do we (or why do philosophers) exclude these micro-organisms from the natural kind water if they occur (or exist) in literally every sample of water? Having said that, just as all samples of water contain micro-organisms (at least in this case), so every example of water also contains hydrogen atoms. Yet although each H₂O molecule includes two hydrogen atoms, they’re not actually the same thing. Similarly, why conflate micro-organisms and water simply because the former can be found in all samples of the latter?

Let’s look at this another way.

Even if water (or a collection of H₂O molecules) is diluted with whiskey (to invert things for this example), then it’s still water that’s being diluted. Similarly (as earlier), if every tree has fungi/mould/lichen/etc. growing on it, then that doesn’t mean that trees literally are fungi/mould/lichen/etc; or that taken together trees and fungi/mould/lichen/etc. constitute a joint natural kind.

So a summary of Bealer’s argument can be posed in the form of this question:

What if literally every sample of water contains something that isn’t H₂O?

Well, that depends. In one case, a sample of water may contain x and another sample may contain y. So, yes, it may be the case that every sample of water contains something that’s over and above H₂O molecules. But what if water always contains the same x that’s over and above H₂O molecules? That seems to be Bealer’s argument.

Bealer further stresses the contingent and/or arbitrary nature of natural kinds in his Twin Earth example. He continues:

“[T]hen on a twin earth a stuff which contains no micro-organisms whatsoever but which nevertheless contains the same chemicals as those found in samples of water on earth would NOT qualify as water.”

Now we’re in the absurd (or simply possible) situation in which all the samples of a substance that’s entirely made up of H₂O molecules may not be deemed — by people on earth at least — to be water! Why is that? It’s because these samples don’t contain any micro-organisms. Yet intuitively it would seem that the “water” (note the scare quotes) on Twin Earth has more right to be deemed a natural kind (or simply as water) than water on actual Earth. (Still bear in mind the supposition that all samples of water here on earth contain micro-organisms.) Indeed why can’t Twin-Earthers reverse the earthling position by claiming that water on earth is not water precisely because each sample of it contains micro-organisms!

The Microscopic and Macroscopic Properties of Water

Bealer offers us another possibility.

He makes a distinction (mentioned in the introduction) between water’s microscopic (or microstructural) and its macroscopic properties (i.e., rather than between water’s possible different microstructural properties). Bealer writes:

“If every disjoint pair of samples of water here on earth have different microstructural compositions but nevertheless uniform macroscopic properties, then on a twin earth a stuff which has those same macroscopic properties would qualify as water.”

We can of course ask about the chances that “every disjoint pair of samples of water” could have “different microstructural compositions”. Wouldn’t the chances of this be virtually zero? And why choose pairs — rather than triples or n-tuples — of water samples in the first place? The passage above also seems to assume that water can be water even if each member of “every disjoint pair of samples” has a different microstructural composition. In any case, here it’s being argued that microstructure isn’t the only factor to consider when it comes to water’s being water. Indeed if each member of every selected pair of samples contains a different microstructural composition, then microstructure simply can’t be a factor at all!

Bealer stresses water’s macroscopic properties in this example.

The water on Twin Earth has the same macroscopic properties as the water here on Earth. However, is it likely that a chemical substance on Twin Earth with a completely different microstructure would have the same macroscopic properties as water here on Earth? Well, it is of course possible. That is, I’m assuming here that Bealer is at least partly referring to experiential (or phenomenal) properties — such as water’s transparency, wetness, liquidity, variant temperature (as registered by the sensory-systems of human beings), thirst-quenching qualities, etc. (The properties of water which chemists cite are very different to these. They include polarity, surface tension, cohesion, adhesion, evaporative cooling, etc.) Now could something that isn’t made up of H₂O molecules be wet, transparent, quench thirst, etc. in exactly the same way that water here on Earth is and does? Well, as before, I presume that all this is possible. (What biological or physiological effects would Twin-Earth water have if a earthling drank it?)

So now we can sum up Bealer’s position with another simple question:

When it comes to natural kinds, why shouldn’t macroscopic properties (i.e., rather than exclusively microstructural properties) be what is important?

Again, isn’t it somewhat arbitrary and/or contingent that philosophers see only microstructural properties as being constitutive of natural kinds (i.e., at least when it comes to natural kinds like water), rather than seeing macroscopic properties in the same way?

Are There Different Kinds of Water?

One needn’t be a chemist or a layperson to find the position that “there are other kinds of water” odd (as some philosophers have done). More concretely, if these other kinds of water share nothing with H₂O molecules, then why are they water at all? Alternatively put, is it possible that “not all water has the same microstructure”? All that may depend on what’s meant by the words “share nothing”. For example, in one scenario it is the case that all rival samples of water do share macroscopic properties; though not microstructural properties. (We can of course debate how the sharing of macroscopic properties actually cashes out.)

In actual fact, not all water here on Earth is made up exclusively of H₂O molecules. As Alex Barber puts it in his book Language and Thought:

“[I]ndeed, we knew this already, since it would surely be stipulative to deny that heavy water (D₂O) is really water.”

Here we’re back to our contingent and/or arbitrary (i.e., “stipulative”) decisions concerning natural kinds. After all, it’s quite possible that some chemists don’t see D₂O (or “heavy water”) as water. In other words, what’s to stop them from deciding that D₂O isn’t water? More clearly and obviously, if D₂O isn’t H₂O, then surely D₂O and H₂O can’t both be water. However, it is the case that the molecules H₂O and D₂O do share some things — they both include an oxygen atom, protons, electrons and other chemical/atomic elements/forces. But does all that matter? Is all that enough?

(A D₂O molecule includes a “heavy” hydrogen atom. It’s heavy because it contains an extra neutron in its nucleus, along with the standard proton. The light hydrogen atom, on the other hand, only contains a single proton.)

Of course one way to “solve” this particular problem is simply to see water’s macroscopic properties as being constitutive of it being a natural kind. Thus, in this case at least (as stated), both H₂O and D₂O do have exactly the same macroscopic properties! (Or do they? Yes; H₂O and D₂O have the same macroscopic properties when it comes to the sensory — or phenomenal — experiences of human beings. But they don’t do so when it comes to chemical analysis — see here.)

So if D₂O is water, then why can’t Twin Earth’s XYZ (along with H₂O and D₂O) also be water? After all, XYX does, at least hypothetically, have the same macroscopic properties. (At least it’s taken to do so in the philosophical literature.)

Conclusion

It’s precisely because of these problems that some philosophers have argued that the word “water” is not a natural kind term at all. However, that may well still mean that the symbol “H₂O” itself does actually symbolise a natural kind. So what natural kind does symbol “H₂O” symbolise? Water? Perhaps, then, in order to avoid this circularity all we really have left is this:

The symbol “H₂O” = (or refers to) H₂O

(Of course the mathematical identity/equality sign above can’t be taken literally. A symbol can’t literally be identical to what it symbolises.)

Or, at the the very least, all we have is this:

The symbol “H₂O” = a molecule made up of two hydrogen atoms and one oxygen atom (plus lots of other molecular, quantum, bonding, etc. stuff)

So is heavy water (or D₂O) a genuine natural kind? And if we take H₂O as a natural kind, then are both D₂O and H₂O genuine natural kinds? Indeed are they the same natural kind (if with slight microstructural differences)? But, again, which one is truly water? Both? Alternatively, perhaps neither is a natural kind.