David Chalmers' Thermostat and its Experiences

[The words "experience" and "consciousness" are used interchangeably in this piece; even though they aren't synonyms.]

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David Chalmers says that “information is everywhere”. Is that really the case?

As some linguists (or pedants?) have said: “If everyone is brave, then no one is brave.” The point being made here is that a term only makes sense if it can be distinguished from non-examples. However, my example is an adjective (“brave”) applied to human persons. The word “information” is a noun. So saying “information is everywhere” is roughly equivalent to saying “dust is everywhere”. 

Information is surely a characteristic of things, events, conditions, etc: rather than a thing in itself. However, none of this may matter. A prima facie problem with the omnipresence of information may fade away on seeing what David Chalmers - and other “information theorists” - have to say about information.

In addition: if, as Chalmers argues,

“experience itself as a fundamental feature of the world, alongside mass, charge, and space-time”

then, by definition, experience can't be exclusive to humans or animals generally. Something that's a “fundamental feature of the world” must literally be everywhere; just as Chalmers says about “mass, charge, and space-time”.

This means that Chalmers' linkage of experience to information is thoroughly non-biological.

Chalmers also links experience - therefore information - to thermostats. A thermostat isn't alive; yet it can still be seen as a (to use Chalmers' words) “maximally-simple” information system.

Scott Aaronson (referring to Integrated Information Theory), for one, states one problem with the experience-is-everywhere idea in the following passage:

“[IIT] unavoidably predicts vast amounts of consciousness in physical systems that no sane person would regard as particularly ‘conscious’ at all: indeed, systems that do nothing but apply a low-density parity-check code, or other simple transformations of their input data. Moreover, IIT predicts not merely that these systems are ‘slightly’ conscious (which would be fine), but that they can be unboundedly more conscious than humans are.”

Here again it probably needs to be stated that if experience/consciousness = information (or that information – sometimes? - equals experience/consciousness), then experience/consciousness must indeed be everywhere.

However, there's the remaining question: 

Is it the case that information actually is experience or is it that information brings about experience? 

If it's the latter, then we'll simply repeat all the problems we have with both the emergence of one thing from another thing and the reduction of one thing to another thing.

There's also hint of this problem when Chalmers asks us “[w]hy should this sort of processing be responsible for experience?” Here Chalmers uses the word “responsible” (as in “responsible for experience”). In other words, firstly we have processing: then we have experience. So it seems - in this context at least - that processing isn't the same thing as experience. (It is responsible for experience.) And if processing is responsible for experience, so is information. Thus information and experience can't be the same thing.

This may simply be, however, a grammatical fact in that even if information is experience, it can still be grammatically correct to say that “information is responsible for experience”.

What is Information?

The word 'information' has massively different uses; some of which tend to differ strongly from the ones we use in everyday life. Indeed we can use the words of Claude E. Shannon to back this up. He wrote:

"It is hardly to be expected that a single concept of information would satisfactorily account for the numerous possible applications of this general field."

The most important point to realise is that minds (or observers) are usually thought to be required to make information information. However, information is also said to exist without minds/observers. Some philosophers and physicists argue that information existed before human minds; and it will also exist after human minds disappear from the universe. This, of course, raises lots of philosophical and semantic questions.

It may help to compare information with knowledge. The later requires a person, mind or observer. The former (as just stated), may not.

If we move away from David Chalmers, we can cite Guilio Tononi as another example of someone who believes that consciousness/experience simply is information. Thus, if that's an identity statement, then we can invert it and say that 

information is (=) consciousness.

Consciousness doesn't equal just any kind of information; though any kind of information (embodied in a system) may be conscious (at least to some extent).

Indeed, according to Tononi, the mathematical measure of that information (in an informational system) is φ (phi). Not only are systems more than their parts: those systems have various degrees of "informational integration". The higher the informational integration, the more likely that informational system will be conscious. Or, alternatively, the higher the degree of integration, the higher the degree of consciousness.

Integrated Information Theory (IIT) isn't only close to Chalmers' view when it comes to information-equaling-experience, Tononi is also committed to a form (there are many forms) of panpsychism.

The problem (if it is a problem) with arguing that consciousness/experience is information, and that information is everywhere, is that (as has just been said) even basic objects (or systems) have a degree of information. Therefore such basic things (or systems) must also have a degree of consciousness. Or, in IIT speak, all such things (systems) have a “φ value”; which is the measure of the degree of information (therefore consciousness) in the system. Thus Chalmers' thermostat may also have a degree of experience. (Or, for Chalmers, “proto-experience”.)

Clearly we've entered the territory of panpsychism here. Not surprisingly, Tononi is happy with panpsychism; even if his position isn't identical to Chalmers' panprotopsychism.

Interestingly enough, David Chalmers - in one paper at least - doesn't really tell us what information is or what he means by the word “information”. He does tell us, however, that “information is everywhere”. He also tells us about “complex information processing” and “simpler information-processing”. I suppose that in the case of a thermostat, we can guess what information is. Basically, heat and cold are information. Though is heat and cold information for the thermostat? Indeed does that matter? Or is it the case that the actions which are carried out on the heat or cold (by the thermostat) constitute information? Or, perhaps more likely, is it the physical nature (its mechanical and physical innards) of a thermostat that constitutes its information?

To slightly change the subject for a second.

John Searle has a problem with the overuse of the word “computation”. He cites the example of a window as a (to use Chalmers' words again) “maximally-simple” computer. Searle writes:

“... the window in front of me is a very simple computer. Window open = 1, window closed = 0. That is, if we accept Turing’s definition according to which anything to which you can assign a 0 and a 1 is a computer, then the window is a simple and trivial computer.”

Searle's basic point is that just about anything can be seen as a computer.

Indeed computers are everywhere – just like Chalmers' experience. Does this tie in with Chalmers' position on information and maximally-simple information-processing?

In other words, does a window contain information? By that I don't mean the information that may exist in a window's material and mechanical structure. (According to many, a window - being a physical thing - must contain information.) I mean to ask whether or not a window - like a thermostat - has information qua a technological device which is designed to be both opened and shut?

Searle will of course conclude that this is an example of information-for-us.

Searle also has something to say about information (not just computers). He writes:

“[Koch] is not saying that information causes consciousness; he is saying that certain information just is consciousness, and because information is everywhere, consciousness is everywhere.”

This appears to be the same as Chalmers' position. Needless to say, Searle has a problem. He concludes:

"I think that if you analyze this carefully, you will see that the view is incoherent. Consciousness is independent of an observer. I am conscious no matter what anybody thinks. But information is typically relative to observers...

“...These sentences, for example, make sense only relative to our capacity to interpret them. So you can’t explain consciousness by saying it consists of information, because information exists only relative to consciousness.”

As for thermostats, Searle has something to say on them too. He writes:

"I say about my thermostat that it perceives changes in the temperature; I say of my carburettor that it knows when to enrich the mixture; and I say of my computer that its memory is bigger than the memory of the computer I had last year."

This means that this is a Searlian way (as with Dennett) of taking an “intentional stance” towards thermostats. We can treat them - or take them - as intentional (though inanimate) objects. Or we can take them as as-if intentional objects.

The as-if-ness of windows and thermostats is derived from the fact that these inanimate objects have been designed to perceive, know and act. Though this is only as-if perception, as-if knowledge, and as-if action. (Indeed it's only as-if information.) Such things are dependent on human perception, human knowledge, and human action. Perception, knowledge and action require real - or intrinsic - intentionality: not as-if intentionality. Thermostats and windows have a degree of as-if intentionality, derived from (our) intrinsic intentionality. However, according to Searle, despite all these qualifications of as-if intentionality, as-if intentionality is still ‘real’ intentionality; though it's derived from actual/real intentionality.

To get back to Searle's position on information.

For one, it's certainly the case that some – or even many – physicists and mathematicians don't see information in Searle's strictly philosophical or semantic way. In addition, Integrated Information Theory's use of the word 'information' also receives much support in contemporary physics. This support includes how such things as particles and fields are seen in informational terms. As for thermodynamics: if there's an event which affects a dynamic system, then that too can read as being informational input into the system.

Indeed in the field called pancomputationalism, (just about) anything can be deemed to be information. In these cases, that information could be represented or modeled as also being a computational system.

Information may well become information-for-us to such physicists. However, it's still information before it becomes information-for-us.

Perhaps all this boils down to the definition of the word 'information'. The way that some physicists define the word will make it the case that, in Searle's terms, information need not be "observer-relative". On Searle's definition, on the other hand, the word 'information' is defined to make it the case that information must be – or always is – relative to persons (or minds).

Is there anything more to this dispute that rival definitions? Perhaps not. However, in one sense there must be one vital distinction to be made. If information also equals experience, then information not being dependent on human beings makes a big difference. It means that such information is information - and therefore experience - regardless of what we observe or think. However, this is the panpsychist's view; and the physicists just mentioned (those who accept that information need not be observer-relative) don't necessarily also accept that information is the same as experience. Indeed I suspect that most physicists don't believe that.

Thus we now have three positions:

i) Information is relative to observers. (Searle's position.)

ii) Information exists regardless of observers; though it isn't equal to experience. (The position of some physicists and philosophers.)

iii) Information exists regardless of observers and it is also equal to experience. (Chalmers' position.)

A Thermostat and its Experiences

Firstly, let me offer Wikipedia's definition of a thermostat:

“A thermostat is a component which senses the temperature of a system so that the system's temperature is maintained near a desired setpoint...

“A thermostat exerts control by switching heating or cooling devices on or off, or by regulating the flow of a heat transfer fluid as needed, to maintain the correct temperature...”

What does Chalmers himself mean by the word 'information' when it comes - specifically - to a thermostat? He writes:

“Both [thermostats and connectionist models] take an input, perform a quick and easy nonlinear transformation on it, and produce an output.”

As previously stated, in terms of the thermostat at least, information is information-for-us; not information for the thermostat itself. After all, thermostats respond to temperature because we've designed them to do so. Nonetheless, whatever it's doing (even if designed), it's still doing. That is, the thermostat is acting on information. When it's hot, it does one thing. And when it's cold, it does another thing.

Thus does a thermostat have as-if information (to use Searle's term, which is usually applied to intentionality)? Or does it have real (first-order) information? In other words, does the fact that a thermostat is designed by human beings automatically stop it from having experiences which are themselves determined by its informational innards?

After all, humans are also - in a strong sense - designed by their DNA and we certainly have experiences. Thermostats are designed by humans: do they have experiences?

Finally, in one piece Chalmers tackles the case of NETtalk and asks us whether or not it does (or could) instantiate experience. He writes:

“NETTALK, then, is not an instantiation of conscious experience; it is only a model of it.”

Of course we can now rewrite that passage in the following way:

A thermostat, then, is not an instantiation of conscious experience; it is a model of it.

The question is, then, whether or not Chalmers has mixed up models with realities (as it were). NETtalk is certainly more complex than a thermostat. However, Chalmers has often argued that complexity in itself (in this case at least) may not matter.

The Appeal of Simplicity & Complexity

Chalmers plays up simplicity. He also plays down complexity. For example, Chalmers writes that “one wonders how relevant this whiff of complexity will ultimately be to the arguments about consciousness”. He goes further when he says that

“[o]nce a model with five units, say, is to be regarded as a model of consciousness, surely a model with one unit will also yield some insight”.

I presume that a thermostat has more than “one unit”; though we'd need to know what exactly a unit is.

Chalmers also makes what seems to be an obvious point – at least it seems obvious if one already accepts the information/experience link. He writes:

“Surely, somewhere on the continuum between systems with rich and complex conscious experience and systems with no experience at all, there are systems with simple conscious experience. A model with superposition of information seems to be more than we need - why, after all, should not the simplest cases involve information experienced discretely?”

Can we go simpler than a thermostat? Perhaps we can if this is all about information; though that would depend on our position on information. What about a dot on a piece of paper which is then made completely blank (i.e., at a later stage when the dot has been erased with a rubber)?

Chalmers also gives a biological (or “real life”) example of this phenomenon. He writes:

“We might imagine a traumatized creature that is blind to every other distinction to which humans are normally sensitive, but which can still experience hot and cold. Despite the lack of superposition, this experience would still qualify as a phenomenology.”

At a prima facie level, it does indeed seem obvious that complexity matters. After all, many theorists have made a strong link between the complexity of the brain and consciousness. Chalmers himself acknowledges the (intuitive) appeal of complexity. He writes:

“After all, does it not seem that this rich superposition of information is an inessential element of consciousness?”

Of course Chalmers then rejects this requirement for complexity.

Having said all that, we can also quickly consider Phillip Goff's argument here. He argues that there may be “little minds” (or seats of experience) in the brain, and all of them, on their own, are very simple. Now, of course, we have the problem of the “composition” (or "combination") of all these little minds in order to make a big mind. 

What is Simple Experience?

When Chalmers says that

“[w]here there is simple information processing, there is simple experience, and where there is complex information processing, there is complex experience”

what does he mean by “simple experience”? What is a simple experience? How simple can an experience be? Can we even imagine (or conceive) of such a thing?

I suppose I can imagine a very simple pain. (Pain can certainly be experienced.) Or would that only be a mild pain; rather than a simple pain? (Some philosophers have argued that there needs to be more than phenomenology for a pain to be pain.)

What about a simple visual experience? Well, a thermostat can't have such a thing. So what simple experiences can a thermostat have? A thermostat is designed to physically react to the temperature. However, does it feel the temperature? (We can think of feels which are either strongly dependent on sense organs or feels which are purely mental/experiential in nature.) Does a thermostat experience its innards working? That is, does it experience itself taking in information and then responding to that information? But what could that possibly mean? In order to experience itself taking in information, perhaps the thermostat would need to be both an “it” and also an it capable of experiencing itself as an it. That, surely, goes way beyond simple experience.

What has just been said may also apply to a single-celled organism. Does it experience taking in information and then responding to it? Can it feel that information? It can't see or touch it. So what is the experience of information (or the taking in and responding to it) when it comes to a single-celled organism? Sure, causal things happen within a cell. However, things happening within a cell don't - in and of themselves - tell us that it has an experience of things happening within it.

Now what about a mouse? A mouse has a brain and sensory organs. So, obviously, it's vastly different to a single-celled organism and a thermostat. Nonetheless, the idea of a very-simple experience is still problematic.

Deflating Gödelised Physics, With Stephen Hawking (1)

This piece isn't about deflating Kurt Gödel's metamathematics or even deflating his own comments on physics. It's about deflating other people's applications of Gödel's theorems to physics.

Indeed Gödel himself wasn't too keen on applying his findings to physics – especially to quantum physics. According to John D. Barrow:

“Godel was not minded to draw any strong conclusions for physics from his incompleteness theorems. He made no connections with the Uncertainty Principle of quantum mechanics....”

More broadly, Gödel's theorems may not have the massive and important applications to physics which some philosophers and scientists believe they do have.

For and Against Gödelised Physics

Some scientists are unhappy with the claim that Kurt Gödel's theorems can be applied to physics. Others are very happy with it. More explicitly, many people in the field claim that Gödel incompleteness means – or they sometimes simply suggest - that any Theory of Everything must fail.

For example, way back in 1966, the Hungarian Catholic priest and physicist, Stanley Jaki, argued that any Theory of Everything is bound to be a consistent mathematical theory. Therefore it must also be incomplete.

On the other side of the argument, in 1997, the German computer scientist, Jürgen Schmidhuber, argued against this defeatist - or simply modest/humble – position. Strongly put, Schmidhuber says that Gödel incompleteness is irrelevant for computable physics.

Thus, despite such pros and cons, it's still the case that many physicists argue that Gödel incompleteness doesn't mean that a Theory of Everything can't be constructed. This is because they also believe that all that's needed for such a theory is a statement of the rules which underpin all physical theories. Critics of this position, on the other hand, say that this simply bypasses the problem of our understanding of all these physical systems. Clearly, that lack of understanding is partly a result of the application of Gödel incompleteness to those systems.

The Gödel-Physics Analogy

Despite all the above, the relation between Gödel incompleteness and physics often seems analogical; rather than (strictly speaking) logical.

The incompleteness of physical theories taken individually (or even as groups) has nothing directly - or logically - to do with Gödel incompleteness (which is applied to mathematical systems). The latter is about essential or inherent incompleteness; the former isn't. Or, to put that differently, science isn't about insolubility: it's about incompletablity. (Though it can be said that incompleteness implies - or even entails - insolubility.)

This analogical nature is seen at its most explicit when it comes to scientists and what may be called their scientific humility or modesty.

For example, Stephen Hawking, in his 'Gödel and the End of Physics', said:

“I'm now glad that our search for understanding will never come to an end, and that we will always have the challenge of new discovery. Without it, we would stagnate. Gödel’s theorem ensured there would always be a job for mathematicians. I think M theory will do the same for physicists. I'm sure Dirac would have approved.”

This position is backed up by the words of Freeman Dyson. He wrote:

“Gödel proved that the world of pure mathematics in inexhaustible... I hope that an analogous situation exists in the physical world.... it means that the world of physics is also inexhaustible....”

Stephen Hawking originally believed in the possibility of a/the Theory of Everything. However, he came to realise that Gödel's theorems will be very relevant to this theory. In 2002 he said:

"Some people will be very disappointed if there is not an ultimate theory, that can be formulated as a finite number of principles. I used to belong to that camp, but I have changed my mind."

However, Hawking does seem to be ambivalent on this issue. Specifically when it comes to the analogical nature of Gödel incompleteness and incompleteness in physics.

Stephen Hawking himself uses the word “analogy”; at least within one specific context. That context is “a formulation of M theory that takes account of the black hole information limit”. He then, rather tangentially or loosely, says that

“our experience with supergravity and string theory, and the analogy of Gödel's theorem, suggest that even this formulation will be incomplete”.

Here Hawking isn't talking about Gödel incompleteness. He's simply talking about incompleteness – the incompleteness of a “formulation” of a theory (i.e., M theory). More specifically, it's about incomplete information or incomplete knowledge. Gödel incompleteness certainly isn't about incomplete information or incomplete knowledge.

There's another statement from Hawking that's also really about analogies. (With his use of the word “reminiscent”, Hawking - more or less - says that himself.) Hawking says:

“Maybe it is not possible to formulate the theory of the universe in a finite number of statements. This is very reminiscent of Gödel’s theorem. This says that any finite system of axioms is not sufficient to prove every result in mathematics.”

The question is how reminiscent is reminiscent? Is it vague or strong? Is is substantive or simply analogical? Indeed, on the surface, it's hard to know how to connect the statement that “any finite system of axioms is not sufficient to prove every result in mathematics” to physics generally. Apart form the fact that, yes, physics utilises mathematics and can't survive without it.

Mathematical Models

Hawking himself states a strong relation between Gödel and physics. It comes care-of what he calls the “positivist philosophy of science”. According to such a philosophy of science, “a physical theory is a mathematical model”. That, for one, is a very tight link between physical theory and maths. Hawking says that

“if there are mathematical results that cannot be proved, there are physical problems that cannot be predicted”.

Despite mentioning that tight link, it's a jump from the “mathematical results that cannot be proved” bit to “there are physical problems that can not be predicted” conclusion. The argument must be this:

i) If physical models are mathematical,
and the mathematics used in such models contains elements which can't be proved,
ii) then the predictions which use those models can't be proved either.

That means that mathematical incompleteness (if only in the form of a model in physics) is transferred to the incompleteness of our predictions.

Is “proof” an apposite word when it comes to physical predictions?

Hawking stresses one reason why physics can be tightly connected to mathematics in a way which moves beyond the essential usefulness and descriptive power of maths. He cites the “standard positivist approach” again.

In that approach, “physical theories live rent free in a Platonic heaven of ideal mathematical models”. Thus one (logical) positivist (i.e., Rudolf Carnap) argued that one's theory (or “framework”) determines which objects one “posits”. Similarly, in Hawking's words,

“a [mathematical] model can be arbitrarily detailed and can contain an arbitrary amount of information without affecting the universes they describe”.

This, on the surface, sounds like Hawking is describing an extreme case of constructivism in physics. Or, since Carnap has just be mentioned, is this simply an example of (logical) positivist pragmatism or instrumentalism?

The least that can be said about this stance is that the mathematical model must – at least in a strong sense - come first: then everything else will follow (e..g., which objects exist, etc.). At the most radical, we can say that all we really have are mathematical models in physics. Or, as with ontic structural realists, we can say that all we have is mathematical structures. We don't have objects or “things”.

Hawking doesn't appear to like this extreme constructivist/anti-realist/positivist (take your pick!) pragmatism. Firstly he says that the mathematical modelers “are not [people] who view the universe from the outside”. He also states the interesting (yet strangely obvious) point that “we and our models are both part of the universe we are describing”. Thus, just like the axioms and theorems of a system, even if there are many cross-connections (or acts of self-reference) between them, they're all still part of the same mathematical system. Hence the requirement for metamathematics (or a metalanguage/metatheory in other disciplines).

Finally, all this stuff from Hawking is tied to Gödel himself.

Hawking says that mathematical modelers (as well as their models and “physical theory”) are “self referencing, like in Gödel's theorem”. Then he makes the obvious conclusion:

“One might therefore expect it to be either inconsistent or incomplete.”

Isn't all this is a little like a dog being unable to catch its own tail?

Self-Reference and Paradox

Self-reference and dogs have just been mentioned. Here the problem gets even worse.

Gödel’s metamathematics is primarily about self-reference (or meta-reference). As Hawking puts it:

“Gödel’s theorem is proved using statements that refer to themselves. Such statements can lead to paradoxes. An example is, this statement is false. If the statement is true, it is false. And if the statement is false, it is true.”

Now how can self-referential statements or even paradoxes have anything to do with the world or physical theory? Indeed do the realities/theories of quantum mechanics even impact on this question? (Note Gödel's own position on QM as enunciated in the introduction.) Are there paradoxes in quantum mechanics? Are there cases of self-reference? Yes, there are highly counter-intuitive things (or happenings) in QM; though are they actual paradoxes? I suppose that one thing being in two places at the same time may be seen as being paradoxical. (Isn't that only because we insist on seeing subatomic particles, etc. as J.L. Austin's “medium-sized dry goods” - indeed as particles?) Some theorists, such as David Bohm, thought that QM's paradoxes will be ironed out in time. So too did Einstein.

The ironic thing is that - according to Hawking - Gödel himself tried to iron out the paradoxes from his mathematical theories (or systems). Hawking continues:

“ Gödel went to great lengths to avoid such paradoxes by carefully distinguishing between mathematics, like 2+2 =4, and meta mathematics, or statements about mathematics, such as mathematics is cool, or mathematics is consistent.”

Here again the problem is self-reference. The solution was - and still is - to distinguish mathematics from metamathematics. In the 1930s, Alfred Tarski did the same with his metalanguages and object languages in semantics. Indeed, even before Gödel and Tarski, Bertrand Russell had attempted to do the same within set theory when he distinguished sets from classes (as well as the members of sets from classes) in his “theory of types” (a theory established between 1902 and 1913).

Proof and the Theory of Everything

Wouldn't a/the Theory of Everything be a summing up (as it were) of all physical laws? Thus wouldn't it be partly - and evidently - empirical in nature? Surely that would mean that mathematics couldn't have the last - or the only – word on this.

It can also be argued that a/the Theory of Everything wouldn't demand that every physical truth could be proven in the mathematical/logical sense; even if every physical truth incorporates mathematics.

This is also a case of whether or not proof is as important in physics as it is in mathematics. Indeed, on certain arguments, there can be no (strict) proofs about physical theories.

For example, some have said that the/a Theory of Everything will need to expressed as a proof. Nonetheless, that proof will still be partly observational (or partly empirical – i.e, not fully logical). However, even if only partly observational and largely mathematical, how can it still guarantee a proof? How can there be any kind of proof when a theory includes observations or experimental evidence?

Again, the Theory of Everything would be a final theory which would explain and connect all known physical phenomena. This – to repeat - would be partly empirical in nature. It would also be used to predict the results of future experiments. These predictions would be partly empirical or observational; not (to use a term from semantics) proof-theoretic.

Nature is Not Mathematical

Pythagoras

This piece wouldn’t have been called ‘Who Says Nature is Mathematical?’ if it weren’t for the many other similar titles which I’ve seen. Take these examples: ‘Everything in the Universe Is Made of Math — Including You’, ‘What’s the Universe Made Of? Math, Says Scientist’ and ‘Mathematics — The Language of the Universe’. Indeed from Kurt Gödel to today’s Max Tegmark (who says that the “physical universe is mathematics in a well-defined sense”) and ontic structural realism, mathematical Pythagoreanism (or derivations thereof) seems to be in the air.

The first thing to say is that the claim that “nature is mathematical” hardly makes sense. It’s not even that it’s true or false. Taken literally, it seems almost meaningless. So it must be all about how we should interpret such a claim.

Some of the applications of Gödel’s theorems, etc. to physics, for example, simply don’t seem to make sense. They verge on being Rylian “category mistakes”. This bewilderment is brought about in full awareness of the fact that there would hardly be physics without mathematics. Indeed that’s literally the case with quantum physics and everything which goes with it.

However, when the English theoretical physicist and mathematician John D. Barrow asks us whether or not “the operations of Nature may include such a non-finite system of axioms” (as well as when he replies to his own question by saying that “nature is consistent and complete but cannot be captured by a finite set of axioms”), it can still be a philosophical struggle to see the connection between mathematical systems (or Gödel’s metamathematics) and Nature.

Strictly speaking, Nature isn’t any “mathematical system of axioms” and it doesn’t even “include” such a thing. Mathematics is applied to Nature or it is used to describe Nature. Sure, ontic structural realists and other structural realists (in the philosophy of physics) would say that this distinction (i.e., between mathematics and physics) hardly makes sense when it comes to physics generally and it doesn’t make any sense at all when it comes to quantum physics. However, surely there’s still a distinction to be made here.

Similarly, Nature is neither consistent/complete nor inconsistent/incomplete. It’s what’s applied to — or used to describe — Nature that’s (in)complete/(in)consistent. Again, certain physicists and philosophers of science may think that this distinction is hopelessly naive. Yet surely it’s still a distinction worth making.

This phenomenon is even encountered in contemporary philosophy of logic.

The philosopher Graham Priest, for example, mentions the world (or “reality”) when he talks of consistency and inconsistency. When discussing the virtue of simplicity he asks the following question:

“If there is some reason for supposing that reality is, quite generally, very consistent — say some sort of transcendental argument — then inconsistency is clearly a negative criterion. If not, then perhaps not.”

As it is, how can the world (or Nature) be either inconsistent or consistent?

What we say about the world (whether in science, philosophy, mathematics, logic, fiction, etc.) may well be consistent or inconsistent (we may also say — as with Spinoza later — that the world isn’t “beautiful” or “ugly”). However, surely the world itself can neither be consistent nor inconsistent.

Thus within Graham Priest’s logical and dialetheic context, claims of Nature’s consistency or inconsistency don’t seem to make sense. That must surely also mean that inconsistency is neither a (as Priest puts it) “negative criterion” nor a positive criterion when it comes to Nature itself.

Spinoza vs. Anthropocentrism or Anthropomorphism

What some philosophers of science and physicists are doing seems to contravene Baruch Spinoza’s words of warning about having an anthropocentric or anthropomorphic (though that word is usually applied to non-human animals) view of Nature.

Spinoza’s philosophical point is that Nature can only… well, be. Thus:

“I would warn you that I do not attribute to nature either beauty or deformity, order or confusion. Only in relation to our imagination can things be called beautiful or ugly, well-ordered or confused.”

Spinoza says Nature simply is. All the rest is simply (in contemporary parlance) human psychological projection.

There’s even a temptation to contradict Galileo’s well-known claim about Nature. Thus:

“Nature is written in that great book which ever is before our eyes — I mean the universe — but we cannot understand it if we do not first learn the language and grasp the symbols in which it is written. The book is written in mathematical language, and the symbols are triangles, circles and other geometrical figures, without whose help it is impossible to comprehend a single word of it; without which one wanders in vain through a dark labyrinth.”

Surely we must say that Nature’s book isn’t written in the language of mathematics. We can say that Nature’s book can be written in the language mathematics. Indeed it often is written in the language of mathematics. Though Nature’s book is not itself mathematical because that book — in a strong sense — didn’t even exist until human beings began to write (some of) it.

Yet perhaps I’m doing Galileo a disservice here because he did say that “we cannot understand [Nature] if we do not first learn the language and grasp the symbols in which it is written”. Galileo is talking about understanding Nature here — not just Nature as it is in itself.

Nonetheless, Galileo also says that the the “book is written in mathematical language”. Thus he’s also talking about Nature as it is in itself being mathematical. He’s not even saying that mathematics is required to understand Nature. There is, therefore, an ambivalence between the idea that Nature itself is mathematical and the idea that mathematics is required to understand Nature.

The Mathematical Description of Disorder

Another point worth making is that if mathematics can describe random events or chaotic systems (which it can), then it can also describe just about everything. What I mean by this is that it’s always said that mathematics is perfect for describing (or explaining) the symmetrical, ordered and even “beautiful” aspects of Nature. Yet, at the very same time, if I were to randomly throw an entire pack of cards on the floor, then that mess-of-cards could still be given a mathematical description. The disordered parts of that mess would be just as amenable to mathematical description as its (accidental) symmetries.

Similarly, if I were to improvise “freely” on the piano, all the music I played could be given a mathematical description. Both the chaos and the order would be amenable to a mathematical description and a mathematical explanation. Indeed a black dot in the middle of Sahara desert could be described mathematically; as can highly-probabilistic events at the quantum level. It’s even possible that mathematicians can find different — or even contradictory — symmetries in the same phenomenon.

In a similar way, some of the mathematical studies of Bela Bartok’s late string quartets have found mathematical patterns and symmetries which the composer was almost certainly unaware of. (See this example.) True, Bartok was indeed aware of the golden ratio and other mathematically formalisable aspects of his and other composers’ music. Nonetheless, the analyses I’m referring aren’t really formal in nature. They’re more like micro-analyses of the notes; and they serve, I believe, little purpose. Now there can indeed be interesting formal aspects and symmetries in music which the composers themselves weren’t aware of. Yet, at the same time, a mathematician may still gratuitously apply numbers to specific passages of music in the same way he could do so the same to my mess-of-cards.

Nothing?

The very idea of nothing (or nothingness) is hard - or even impossible - to conceive or imagine. This means that (at least for myself) it fails David Chalmers' idea of conceivability.

David Chalmers (the well-known Australian philosopher) claims that if something is conceivable; then that entails that it's also – metaphysically - possible. The problem with this is we can distinguish conceivability from imaginability. That is, even if we can't construct mental images of nothing (or nothingness), we can still conceive of nothing (or nothingness). I, for one, can't even conceive of nothing (or nothingness).

But can other people conceive of nothing? Do we even have intuitions about nothing or about the notion of nothingness?

So how can we even name or refer to nothing? (We shall see that Parmenides might have had something here.) There's nothing to hold onto. Yet, psychologically speaking, thoughts about nothing can fill people with dread. There's something psychologically (or emotionally) both propelling and appalling about it. And that's why existentialists and other philosophers – with their taste for the dramatic and poetic - found the subject of nothing (or at least nothingness) such a rich philosophical ground to mine. (See if you can wade through Jean-Paul Sartre's Being and Nothingness.)

The very idea of nothing also seems bizarre. It arises at the very beginning of philosophy and religion. After all, how did God create the world "out of nothing"? Did God Himself come from nothing? Indeed what is nothing (or nothingness)?

Not surprisingly, then, Giacomo Casanova (1725–1798) - in conversation with a priest - had this to say on the subject:

“… while the earth, suspended in air, stood firmly at the center of the universe that God had created out of nothingness. When I said to him, and proved to him, that the existence of nothingness was absurd, he cut me short, calling me silly.”

However, John the Scot - or Johannes Scotus Eriugena (c. 815–877) - had previously maneuvered his way around this problem by arguing that God is actually the same thing as nothingness; at least in the context of the question: “How did God create the world out of nothing?” Does this mean, then, that God created the universe out of Himself, not out of nothing?

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Some philosophers use the technical term “non-being” as a virtual synonym for the word “nothing”. (That may be true of the words; though what about the “thing” - nothing?) Having said that, since the notion of nothingness is itself either bizarre or unimaginable, then perhaps the word “nothing” is a technical term too.

Thus the term “not-being” also has its own problems:

i) What is being?

ii) How can there be non-being?

Parmenides

The Greek philosopher Parmenides (5th century BC) based his philosophy of nothingness primarily on logical arguments. Though, as we shall see, this is a prima facie reaction to Parmenides' position.

As soon as the subject was treated scientifically or empirically, however, it can be said that Parmenides' extreme and seemingly absurd position began to fade away.

Parmenides argued that there can be no such thing as nothing for the simple reason that to name it means that it must exist. And nothingness (unlike a stone or a proton) can't exist. This position was resurrected - if in modified form - in the 20th century by philosophers like Bertrand Russell and Willard van Orman Quine. The former obliquely supported it; whereas the latter rejected it. (See later.)

Parmenides' argument is more complete than it may at first seem. Not only is nothing/ness an abstraction to reject; so too is the existence of historical facts or history itself. The possibility of change is similarly rejected.

These are his basic positions (i.e., it's not an argument) on nothing:

i) Nothing doesn't exist.

ii) To speak of a thing, is to speak of a thing which exists.

iii) When one speaks of “nothing”, one speaks of it as if it is something which exists.

In the positions above nothing has been spoken of (it has been named). Therefore, by Parmenides' own light, either nothing must exist or he had no right to speak of it.

What about the events in the past or the past itself? The positions are very similar.

i) If we can't speak of (or name) nothing,

ii) then we can't speak of (or name) things or events of the past.

iii) Such events or things don't exist.

iv) Therefore when we refer to them, we're referring to nothing.

Here again there are references to nothing; which Parmenides warns us against.

What about change, which Parmenides similarly rejects? This rejection of change is strongly connected to his rejection of the past. The argument is this:

ia) If the past doesn't exist,

ib) then only the present exists.

iia) And if only the present exists,

iib) then there can be no change from past to present (or present to future).

iii) Therefore there can be no change at all.

Logical Form and Content

At the beginning of this piece it was mentioned that scientific or empirical philosophers rejected Parmenides's ostensibly pure logical arguments. Aristotle is one example. Indeed he goes further than a mere philosophical rejection. He wrote:

"Although these opinions seem to follow logically in a dialectical discussion, yet to believe them seems next door to madness when one considers the facts."

Nonetheless, Parmenides does seem to be on fairly safe ground. After all, Roy A. Sorenson defines a paradox

“as an argument from incontestable premises to an unacceptable conclusion via an impeccable rule of inference”.

Similarly, Roger Scruton says that paradoxes

“begin from intuitively acceptable premises and derive from them a contradiction – something that cannot be true”.

In other words, it might well have been the case that Parmenides used arguments which are both logically valid and sound. Or, as Aristotle put it, his “opinions seem to follow logically in a dialectical discussion”. It's only when we concern ourselves with semantic (or otherwise) content - rather than logical validity and soundness - that problems arise.

So Parmenides doesn't have it quite so easy. It's also the case that there are logical arguments against his logical arguments. For a start, Parmenides arguments aren't – in actual fact - purely logical in nature. (That is, they aren't purely formal.) This is the case in the simple sense they also involve content. After all, he refers to the “past”, “things”, “change”, the “present” and whatnot. If his arguments had only used variables, propositional letters and other logical symbols (as autonyms), then he'd have been on much safer ground. As it is, his positions - even if they are backed up with logical arguments – are also philosophical (or ontological) in nature.

Leucippus on the Void

One way in which science impacts on Parmenides' position is when it comes to the notion of the void.

Is the void “non-being” or is it something else? Why was the void seen as being “the opposite of being”?

Leucippus (early 5th century BC) - being a naturalist or at least a proto-naturalist - was the first to argue that the void is a thing. Nonetheless, it's a thing without also being a "body with extension" (to use Cartesian terminology).

If the void is non-being, then it throws up many problems. Leucippus , for one, realised that there could be no motion without a void. However, if the void is nothing, then how can something move in it? How can something move in nothing? Or how can some thing move in something which is not a thing?

Leucippus decided that there is no void if it is seen as nothing. Instead we have an “absolute plenum”. This is a space which is filled with matter. And nothing can't be filled with anything – especially not matter. Nonetheless, that didn't solve the problem of motion because the plenum was also seen - in Leucippus's day - as being completely full. Thus how could there be motion within it? Leucippus opted for the solution that there are many plenums; which presumably meant that objects can move from one plenum to another plenum. Democritus (circa 460 BC – 370 BC) seems to have taken this idea of multiple plenums further. He believed that the void exists between things or objects.

Prima facie, the idea of multiple plenums sounds similar to the idea of multiple spaces. However, the idea of a multiplicity of plenums was seemingly contradicted when Isaac Newton propagated the idea of absolute space – as opposed to (relative) spaces (i.e., in the plural).

Science and Empiricism

Aristotle - being a great empiricist and scientist - offered the obvious (in retrospect!) solution to Parmenides's ostensible paradoxes. He simply made a distinction between things which are made of matter and things which aren't made of matter. The latter includes space. In other words, space isn't non-being or even a void. It is, instead, a “receptacle” which acquires objects or in which objects can move.

Bertrand Russell – over two thousand years later - also offers us a good take on this.

Russell - also as an empiricist - started with observed data. He observed motion! From his observation of motion, he then constructed a theory. This is unlike Parmenides; who, when he observed motion, disregarded it for philosophical and logical reasons. In other words, for the Greek philosopher, logic and philosophy trumped observation.

Russell and Quine on Nothing

Bertrand Russell - in his 1918 paper 'Existence and Description' - believed that in order for names to be names, they must name – or refer to - things which exist. Take this remarkable passage:

“The fact that you can discuss the proposition 'God exists' is a proof that 'God', as used in that proposition, is a description not a name. If 'God' were a name, no question as to existence could arise.”

That, clearly, is fairly similar to Parmenides's position on the use of the word “nothing”. Russell's argument, however, is very different. Personally, I don't have much time for it. It seems to have the character of a philosophical stipulation. It's primary purpose is logical and philosophical. Russell, at the time, was reacting to the “ontological slums” (as Quine put it) of Alexius Meinong. However, this semantic philosophy (as I said) simply seems like a stipulation (or a normative position) designed to solve various philosophical problems.

As for Quine, he has no problem with the naming of non-beings or non-existents (though non-being and non-existence aren't the same thing). In his 1948 paper, 'On What There Is', he firstly dismisses Bertrand Russell's position. Quine, however, puts Russell's position in the mouth of McX and uses the word “Pegasus” rather than the word “God”.

Quine wrote:

“He confused the alleged named object Pegasus with the meaning of the word 'Pegasus', therefore concluding that Pegasus must be in order that the word have meaning.”

Put simply, a name can have a “meaning” without it referring to something which exists (or even something which has being). Quine unties meaning from reference; whereas Russell only thought in terms of reference (or, at the least, he tied meaning to reference).

Parmenides, of course, makes similar mistakes (as we've seen). He didn't think that a name could have a meaning without the thing being named also existing or being. However, we can speak of something that doesn't exist because the naming of such an x doesn't imply its existence. Though - in homage to Meinong (as well as, perhaps, to the philosopher David Lewis) - Russell would have asked us what kind of being the named object (or thing) has.

Thus Russell's theory is an attempt to solve that problem by arguing that if a named x doesn't exist (or have being), then that name must be a “disguised description”. (In the case of the name “Pegasus”, the description would be “the fictional horse which has such and such characteristics”.)

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So, as we've seen, nothing (or nothingness) is a difficult notion to grasp. Yet philosophers throughout the ages have had a good stab at it. The problem is (as ever with philosophers) that they've said very different things about it. Then again, nothing (or nothingness) also perplexes physicists and cosmologists; as indeed it does the layperson. Perhaps it's precisely because there's nothing to grasp in the first place that the notion has thrown up so many absurdities and surprises.

Does all this therefore mean that anything goes when it comes to nothing or nothingness?