Monday, 24 July 2017

Deflating Gödelised Physics: The Basic Argument (2)


Gödel and Einstein.
[This introduction is cut and pasted from part 1. Skip if you've read it before.]

This piece, of course, isn't about deflating Kurt Gödel's metamathematics or even deflating his own comments about 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.

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First we have two statements:

1) Mathematics systems contain unprovable statements. (Or: Gödel's theorems apply to all mathematics.)
2) Physics uses mathematics.

This is the argument:

ia) If physics utilises mathematical systems,
ib) and Gödel's discoveries apply to such mathematical systems,
ii) then Gödel's discoveries also apply to physics.

Thus a physical theory is either complete and inconsistent or consistent and not complete. Either way, it is said, physics/science looses... Or does it?

Freeman Dyson, for one, did see a strong link between Gödel incompleteness and physics. He wrote:

... no finite set of axioms and rules of inference can ever encompass the whole of mathematics; given any set of axioms, we can find meaningful mathematical questions which the axioms leave unanswered. I hope that an analogous situation exists in the physical world. If my view of the future is correct, it means that the world of physics and astronomy is also inexhaustible; no matter how far we go into the future, there will always be new things happening, new information coming in, new worlds to explore, a constantly expanding domain of life, consciousness, and memory.”

The above should be classed, however, as scientific incompletability, not scientific insolubility. In other words, it not about a Gödelian lack of proof within a system - or even within all systems. It's about the “inexhaustible” (to use Dyson's own word) nature of “physics and astronomy”. Yes, the words “inexhaustible” and “incomplete” are near-synonyms; though this still isn't Gödel incompleteness.

Besides which, Dyson himself says that the link between Gödel and physics any amounts to an “analogous” - not a logical - link.

On the other hand, John D. Barrow puts the case against a thoroughly Gödelised physics in the following:

.... it is by no means obvious that Gödel places any straightforward limit upon the overall scope of physics to understand the nature of the Universe just because physics makes use of mathematics. The mathematics that Nature makes use of may be smaller and simpler than is needed for incompleteness and undecidable to rear their heads.”

Here we can highlight the word “understand”. To put what I think is Barrow's position, scientific understanding doesn't require Gödel completeness. It may simply mean that an understanding of a physical theory - even a full understanding of a physical theory - isn't affected by Gödel's theorems. That, of course, begs the question as what scientific understanding is. It may mean that we can describe and explain a physical theory (or nature itself) without the notion of completeness having any substantive affect on that description or explanation. Perhaps that's because the completeness would only be relevant to the mathematics required to understand or describe a physical theory (or nature).

On that last point. Barrow also makes the technical point that even taking into consideration the necessary and vital role maths plays in physics, it may still be the case that, as Barrow puts it, the “mathematics Nature makes use of may be smaller and simpler than is needed for incompleteness and undecidable to rear their heads”. That's a technical point that I - as neither a mathematician nor a physicist - find hard to comment upon. It's clearly, nonetheless, a statement that Gödel incompleteness doesn't apply to all maths. And that mathematical remainder may be all that's required for mathematical physics.

Thus, to sum up. Does it follow that Gödel incompleteness is automatically a negative conclusion for physics? Doesn't it all depend on a whole host of other factors? As just stated, physics may not require the entirety of mathematics. Moreover, it may require only those parts of maths that aren't affected by Gödel's theorems. And even if Gödel's theorems do somehow affect the maths employed in physics, that may not be to the detriment of physics in any substantive way.

Succinctly in three statements:

1) Physics may not (or does not) utilise the entirety of mathematics.
2) Physics may utilise only the parts of mathematics which aren't affected by Gödel's theorems.
3) Physics may survive (or not be substantively touched) even if physics is affected by Gödel's theorems.

To put all that another way. Only certain aspects of mathematics are applied to physical reality. Those aspects are decidable or computable.

This is an alternative argument which is nonetheless related:

i) Physics doesn't need (or have) strict proofs.
ii) Gödel's theorems are primarily partly about proof.
iii) Therefore the most important aspect of Gödel's theorems may not be directly applicable to physics.

More specifically, one formulation of Gödel incompleteness to physics doesn't seem to work. Say that the (weak) argument is put this way:

i) Mathematical systems contain unprovable statements.
ii) Physics is based on mathematics.
iii) Therefore physics won't be able to discover everything that is true.

As stated, physics doesn't - strictly speaking - have proofs. (Though the mathematics included it it may need proofs.) Neither does it require proofs. Secondly (and relatedly), discovering everything isn't the same thing as proving everything. And what sort of claim or aim is it anyway to “discover everything”?

Tangentially, there is a sense in which the word “incompleteness” can indeed be applied to problems in physics. Is this Gödel incompleteness? Not really.

One kind of incompleteness in physics simply applies to the situation in which new observations can't be accounted for by older theories. Thus the older theories must be incomplete. Again, this has no direct connection to Gödel incompleteness.

Physical Laws as Axioms?

Perhaps taking the laws of physics as axioms is at the root of the problem. After all, if one takes physical laws as axioms; then, somewhere along the line, there may be Gödel incompleteness or inconsistency.

Yet physical laws both are and aren't axioms. For one, they aren't self-evident or intuitively acceptable. One reason for that is that physical laws are things which couldn't - even in principle - by intuitively obvious because intuitions can't apply (in any strict sense) to laws which govern things which lie outside experience (at the cosmological or the quantum scale, for example). Added to that, if physical laws are axioms, and what we derive from laws are theorems, then what about the unpredictable consequences or predictions which we derive from our axiomatic laws?

Take a purely formal logical deduction or argument. In such a thing, we move “from an incontestable premises to an acceptable conclusion via an impeccable rule of inference”... Can all that - in any way whatsoever - be applied to axiomatic physical laws and there theorems? Indeed is it correct to use the word “axiom” at all in physics?

In addition, can any law of physics ever be as simple and as pure as an axiom in a logical or mathematical system?

There is a direct consequence of this way of thinking.

Gödel's theorems require that the axioms of a theory or system be listable. Can it be said that all the laws of physics are (or could be) listable? And even if they were listable, would the theorems which we derive from physical laws bear a strong resemblance to the theorems which are a derived from the axioms of a logical or mathematical system? In other words, do we have entailment (or strict deduction) from physical axioms/laws to physical theorems? Do we have either metaphysical or logical entailment or deduction when it comes to axioms/laws and the theorems derived from them?



Thursday, 20 July 2017

Deflating Gödelised Physics, With Stephen Hawking (1)



This piece, of course, isn't about deflating Kurt Gödel's metamathematics or even deflating his own comments about 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 sometimes simply suggests - 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 Every 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 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.

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 (to an audience):

"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. (Then again, 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 even 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 can not be proved, there are physical problems that can not 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 with 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) positivistic 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 constructivism/anti-realism/positivist pragmatism (take your pick!). 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 all part of the same mathematical system. Hence the requirement for metamathematics (or a metalanguage or 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 there 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 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. Alfred Tarski, in the 1930s, did the same with 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 the members of sets and classes) in his “theory of types” (a theory established between 1902 and 1913).

Proof and the/a 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.



Sunday, 9 July 2017

Nature isn't Mathematical!



This piece wouldn't have been called 'Nature isn't Mathematical' if it weren't for the many other titles which I'd seen, such as: 'Everything in the Universe Is Made of Math – Including You', 'What's the Universe Made Of? Math, Says Scientist', 'Mathematics - The Language of the Universe', etc. Indeed from Kurt Gödel to today's Max Tegmark (“the physical universe is mathematics in a well-defined sense”) and ontic structural realism, mathematical Platonism (or derivations thereof) seem to be in the air. Though that's if being a realist about mathematics (or numbers) is the same as being a realist about the mathematical reality of... reality.

The first thing to say is that the claim - i.e., 'Nature is mathematical' - hardly make sense. It's not even that it's true or false. Taken literally, it's meaningless. So perhaps it's all about how we should interpret such a claim.

Some of the applications of Gödel's theorems 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 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 Godel's metamathematics and reality/the world.

Strictly speaking, Nature isn't any “mathematical system of axioms” and it doesn't even “include” such things. Mathematics is applied to Nature or it's 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 maths 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 not consistent/complete. It's what's applied to - or used to describe - nature that's complete/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 of logic and logician 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 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 - with Spinoza later - that the world is “beautiful” or “ugly”). However, surely the world itself can be neither 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 “negative criterion” (as Priest puts it) 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 this 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.

Thus 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 wrote (some of) it.

Yet perhaps I'm doing Galileo a disservice here because he does 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 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 explain or describe random events or chaotic systems (which it can), then it can also explain or 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 the (accidental) symmetries.

Similarly, if I were to improvise “freely” on the piano, all the music I played could be given a mathematical description. That is, both the chaos and the order would be amenable to a mathematical description and even a mathematical explanation. Indeed a black dot in the Sahara desert could be described mathematically; as can probabilistic events at the quantum level. It's 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; though it's not the one I'm familiar with.) 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 composers weren't themselves 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.


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*) Kitty Ferguson, in her book The Fire in the Equations, writes:


"It is very surprising to find, then, that the chorales Bach wrote himself contain quite a few parallel fifths. The rule forbidding them is a rule which Bach himself violated without the least compunction; in fact, it's doubtful whether he was even aware such a rule might exist." (238)





Friday, 30 June 2017

Nothing?




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

David Chalmers claims that if something is conceivable; then that entails that it's also – metaphysically - possible. The problem with this is that Chalmers distinguishes conceivability from imagination. 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).

In addition I have no intuitions about nothing or even about the idea of nothing.

Then 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, a nothing-thought about nothing can fill one 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.

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 manoeuvred 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 that mean, then, that God created the universe out of Himself, not out of nothing? Why is that idea any less “silly” (to use the priest's term) than the former one?

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Some philosophers use the technical term “not-being” as a virtual synonym for the word “nothing”. (That may be true of the words; though what about the thing - nothing/ness?) Having said that, since the notion of nothingness is itself either bizarre or unimaginable, then perhaps the word “nothing” is a technical term too.

The term “not-being” also has its own problems. One: What is this being? Two: How can there be non-being?

Parmenides

The early 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 his 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 resurrected itself - if in modified form - in the 20th century with philosophers like Bertrand Russell and Quine. The former obliquely supported it; whereas the latter rejected it. (See later.)

Parmenides' argument is more complete that it may at first seem. Not only is nothing an abstraction or a thing (?) 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 (not 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's something which exists.

Having said that, 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 past or things or events in the past? 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 argument/s. 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.

However, 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. 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 simply 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're backed up with logical arguments – are philosophical (or ontological).

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.

If the void is non-being 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 such a thing 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 contradicted when Isaac Newton propagated the idea of absolute space – as opposed to (relative) spaces (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” (i.e., that which receives) 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, must have disregarded it for philosophical and logical reasons. In other words, for the Greek philosopher, logic and philosophy must have 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, 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”.