Summary Against Modern Thought: God Is Not The Universe

So, Spinoza is wrong. But not because of his own argument is faulty ( contains a logical error) but because he disagrees with Thomas.

Secondly, there certainly are differences in existence. I exist, and the Summa contra Gentiles exist. But I am a material being, and the Summa contra Gentiles definetively is not a material being. You can write it down in a book, but that book is not the Summa contra Gentiles. Because you can have another book also containing the Summa contra Gentiles, which gives you two books, and still one Summa contra Gentiles.

God is not a pipe (after Matisse).

I know pantheists who are convinced that the universe is a part of God.

“This being is the pantheistic deity, if it existed. Obviously, the universe is made of parts,”
Au contraire: then we have only one single entity, just like the Trinity is also three emanations of One God.

“Put another way around, if the universe were God, then nothing could change; things change; therefore the universe is not God. Being cannot alter into new ways of being Being, and nothing can come from nothing.”

Hmm. And what if change was an illusion, and nothing does actually change? Or to put it another way around, all times exist at once, always have, and always will. Nothing has to come from nothing because everything (‘everything’ obviously includes all times) already is, and always has been. Would the argument still be as beautiful? Or would it be even more so?

LOL! I love that little “atheists think like pantheists” dig. Only a religious person would think such a thing. I have news for you. We do not, pretty much by definition.

JMJ

Perhaps only a religious person (whatever that is) has the vocabulary to think such a thing (or to refute such a thing).

what if change was an illusion

Then, in your zeal to dodge You-Know-Who, you have thrown natural science under the bus. Natural science is the close observation of metrical and observable change in material being. If this change is an illusion, the conclusions of natural science are an illusion of an illusion.

the Summa contra Gentiles definetively is not a material being.

A be-ing is anything that has existence; i.e., possesses the property “to be.” Anything material that exists is a synole of matter and form, but the same form is instantiated in different matter, as UGA, the mascot of the University of Georgia, is instantiated in a particular bulldog (Uga IX), in statues, in logos, and so forth. Similarly the form of Contra gentiles exists in various matters– this book, that book, this website, that website — but that does not make these individual existents the same being.

@YOS

A square circle does exist too, if only as a logical impossibility. It might even be different from a round square.

So existing isn’t that hard. Which brings me to the question: what kind of existence does Thomas mean? Having an attribute “exists” being TRUE is clearly not sufficient.

Briggs i can compound the set of logical impossibilities,

“Alice laughed: “There’s no use trying,” she said; “one can’t believe impossible things.”
“I daresay you haven’t had much practice,” said the Queen. “When I was younger, I always did it for half an hour a day. Why, sometimes I’ve believed as many as six impossible things before breakfast.”

“Then, in your zeal to dodge You-Know-Who, you have thrown natural science under the bus. Natural science is the close observation of metrical and observable change in material being. If this change is an illusion, the conclusions of natural science are an illusion of an illusion.”

Where did you get that definition from?

Since relativists often work with the Eternalist ‘block universe’ which is fixed and unchanging, I guess you’re saying relativity isn’t science?

There’s no ‘generation and corruption’ in the block universe, only the parts of a single object near to the past end and near to the future end of the object (and which is an artificial division, anyway).

And if you consider all the different times to exist in superposition – as any time-translate of the universe is equally possible, the sum-over-histories version of QM requires them all – you can even consider them all to exist right now even without using Eternalism. And since the universe is thereby identical to every time-translate of the universe, it is eternal and unchanging.

Modern physics tends to treat time as just another coordinate, and thus ‘change’ is really just geometry. It’s an artifact of a limited/incomplete perspective: ‘illusion’ is as good a word as any for it.

Since relativists often work with the Eternalist ‘block universe’ which is fixed and unchanging

By using Minkowski 4-space, you must consider the universe from outside space and time, so you’d have to postulate a Viewpoint from which you could do that. You would also have to postulate that not only was this sodium atom predestined to combine with that chlorine atom, but they were so predestined even if they were in materials chosen by a chemist from his flasks. But if the scientist had no choice in the experiment, then what confidence can we place in the truth of his conclusions?

Besides, you would have to argue with Heisenberg. I plan to wait until relativity and quantum mechanics are reconciled before deciding that one or the other is actually true-to-life rather than simply instrumentally useful.

There’s no ‘generation and corruption’ in the block universe, only the parts of a single object near to the past end and near to the future end of the object.

But then you would be forced to say that an embryo and a mature adult are the same object, and violence done the one is violence done the other, just as a bash on the head and a stomp on the foot is violence done the same person.

past end and near to the future end of the object (and which is an artificial division, anyway)

It can only be artificial if it is an artifact; so you must postulate an Artificer.

Troubles every way you turn.

@ Briggs

There are no physical square circles, and you cannot draw them either. But you can say that they do cannot be made from matter, and you can say that you cannot draw them.

And that is a kind of existence too. They can be part of a logical expression. If that expression in particular is “a square circle cannot be drawn”‘ then that expression is necessarily true.

Sander, not to labor the point, but you’re engaging in the fallacy of equivocation.

@dover_beach

No, I am wondering about the exact meaning of existence Thomas is using. Trying to coax it out of you people, but again in vain.

@Sander van der Wal:

“No, I am wondering about the exact meaning of existence Thomas is using. Trying to coax it out of you people, but again in vain.”

There is nothing mysterious in the way St. Thomas is using words like “existence” and “being”; these are primitive notions, incapable of further analysis, but everyone understands them, at least everyone with understanding in the first place. That you cannot make the elementary distinction between real beings and beings of reason, is your problem not ours, and certainly not Aquinas. To then go on to say that you are trying to “coax it out of you people, but again in vain” is merely a sign that you are not a credible interlocutor.

That one can slap a name to an idea, say “Santa Claus” to “A chubby red man with a white beard and a red suit, and sponsored by Coca-Cola”, just means that we can slap a label to an idea, and in that sense, “Santa Claus” does have an existence, existence as an idea or intentional object. But just because we can use the name as a logical subject of a predication, say “Santa Claus does not exist” does not allow us to infer thereby that Santa Claus does exist. In common parlance, when one uses such words as “being” and “existence”, one means extra-mental existence and extra-mental beings. It would be very bizarre if Aquinas, when attempting his existential proofs, merely meant to say that God exists as an idea in the mind — a fact that hardly anyone disputes and that raises nary a eyebrow. Scientists do not explicate the meaning of the word existence, yet presumably you have no problems in understanding the plain meaning of the “Higgs boson exists”.

“By using Minkowski 4-space, you must consider the universe from outside space and time, so you’d have to postulate a Viewpoint from which you could do that.”

I don’t see why. I can consider the block universe quite happily from inside it, which is where I am. You don’t have to be outside the spatial universe to imagine all space – why do you have to be outside space and time to imagine all space and time?

” You would also have to postulate that not only was this sodium atom predestined to combine with that chlorine atom, but they were so predestined even if they were in materials chosen by a chemist from his flasks. But if the scientist had no choice in the experiment, then what confidence can we place in the truth of his conclusions?”

In the Everett-Wheeler, sum-over-histories sort of picture, it can be predestined to do both at the same time. The chemist makes both choices, and the outcomes are correlated to his choices, so we may be as sure of his conclusions as we ever were.

The concept of ‘choice’ requires the idea of counterfactuals – things that could have happened but didn’t – to act as alternatives. If there is only one history, it’s rather questionable in what sense counterfactuals ‘exist’. How could you ever know if they were possible? But with Everett-Wheeler, the counterfactuals are as real as the factuals.

“Besides, you would have to argue with Heisenberg. I plan to wait until relativity and quantum mechanics are reconciled before deciding that one or the other is actually true-to-life rather than simply instrumentally useful.”

QM and special relativity (which is all I’m talking about here) were reconciled by Dirac in the 1928. QM and general relativity can be reconciled in a sense, but leave an infinite number of parameters undetermined, meaning you can’t make any predictions. This is now thought to be indicative of QM/relativity being large-scale approximations to a different small-scale physics, most of the details of which cancel out at the longer scales where we can do experiments. It’s like looking at a light bulb on a dark night from a range of a few miles, and observing that the illumination follows an inverse square law. The inconsistency is equivalent to observing that the illumination diverges to infinity as the range approaches zero, and its resolution is to postulate the details of the light-bulb filament, where the inverse-square law breaks down. However, there’s no experiment you can do to observe the shape of the filament from a range of miles.

“But then you would be forced to say that an embryo and a mature adult are the same object, and violence done the one is violence done the other, just as a bash on the head and a stomp on the foot is violence done the same person.”

Yes. Assuming both exist.

In fact, if you take Everett-Wheeler picture to its logical conclusion, all the different elements of the superposition are parts of the same object. So if you choose whether to stomp on a foot, some part of the person always gets stomped on. And it’s important that it be a choice, isn’t it? Or where’s the moral responsibility?

“It can only be artificial if it is an artifact; so you must postulate an Artificer.”

Absolutely. But it’s the division that’s artificial, not the object. There are no perfectly sharp category boundaries in nature – yin and yang, the uncarved block, and all that.

@ NiV

I think you’ve missed the part of YoS’ remark, the one about the conclusions of natural science being an illusion if change is an illusion.
Denying change would cover “psychological” change, including any kind of reasoning, whether deductive or inductive. Given that block-universe is by no means self-evident, and you present it as a conclusion, it being an interpretation of some scientific data, this position is self-undermining. In other words, you cannot -consider- the block universe from inside it, for there’s no consideration at all, no conclusions, no inferences, no predictions, and thus no science (and no you, of course).
I wonder what exactly “time” means when one denies motion/change (time being the measure of change). The fact that modern physicists find it useful to think of time as just another coordinate does not establish that in reality it’s just geometry. If one presupposes that time is analogous to space, of course one can speak of time as analogous to space. And geometry being timeless, of course if “it’s all geometry”, “all’s timeless”. But mathematics being useful doesn’t prove that reality is pure mathematics.

@Georgy Mancz:

“But mathematics being useful doesn’t prove that reality is pure mathematics.”

It better not be! One of the greatest mathematical logicians, the Polish Mostowski, once complained against the (still) prevalent set-theoretic reductionism by remarking that “I am not a set”.

“Denying change would cover “psychological” change, including any kind of reasoning, whether deductive or inductive.”

No it wouldn’t! The entire chain of reasoning can exist simultaneously. Write it down in a book. The entire argument is there on the page, all at once. Turn to the next page to see the next argument. Turn the page back, and you’ll find the original argument is still there. A book is an example of a block universe – you can open the book at just one pair of pages at a time, so you may have the illusion of a progression with a ‘before’ and ‘after’, but in fact the entire story is there all the time, while you are reading it – you just can’t see the rest of it at the moment. When reading a book, you experience the illusion of change and progression, although nothing in the book ever changes.

And you’re not going to tell me that books cannot contain reasoning, whether inductive or deductive!

You could even write a book explaining all this.

“The fact that modern physicists find it useful to think of time as just another coordinate does not establish that in reality it’s just geometry. If one presupposes that time is analogous to space, of course one can speak of time as analogous to space.”

No, time isn’t simply analogous to space, time is space – or more precisely, a particular set of directions in spacetime (the four-dimensional union of time and space). They’re the same thing. It’s like arguing about whether ‘forwards’ is simply analogous to ‘sideways’. If you change the direction you’re facing, ‘forwards’ and ‘sideways’ get mixed up, so what was previously ‘forwards’ is now a bit forwards and a bit sideways. And if you change speeds (and a change of speed is just a rotation in spacetime) then what was previously just ‘time’ (the path of a stationary object) is now a mixture of both time and space (the path of a moving object).

Consider, if you walk ‘forwards’ and your friend walks ‘forwards’ at the same speed but her ‘forwards’ is different to yours because you’re facing a different direction. Then you’ll see her drift sideways and fall behind you. At the same time, she will see you drift sideways and fall behind her. Each of you is going slower than the other in the ‘forwards’ direction, but this works because you’re using different definitions of ‘forwards’. Similarly, if you move through spacetime and you’re moving at different speeds (facing different directions), then she will move through time slower than you, her clocks run slow, as she falls behind, and you will move through time slower than her, and she will see your clocks running slow. This works, because you’re using different directions in spacetime as your ‘time’, and time and space get mixed up in just the same way that ‘forwards’ and ‘sideways’ get mixed up (except with a critical sign change at a certain point in the equations). Time and space are just coordinates in spacetime, and you can rotate your coordinate system and still have everything work.

Walking across the room, turning round and walking back, shifts your personal definition of ‘now’ in the neighboring Andromeda galaxy by about a day. (The speed change is only a very tiny rotation, but it’s a very long lever.) Walk one way and it’s Tuesday there. Walk back and it’s Monday again.

Relativists use the block universe because physics makes no bloody sense if you don’t. Events in the Andromeda galaxy don’t ‘unhappen’ just because you decided to walk the other way. You can slice across spacetime at many different ‘angles’, and physics has to work consistently and according to the same rules in all of them.

Other physicists don’t do it quite as explicitly as the relativists, but they still tend to put time in their equations as just another coordinate. They draw it in graphs and diagrams as just another coordinate. It’s how they think of it. And it’s a very powerful mental tool, too.

>When reading a book, you experience the illusion of change and progression, although nothing in the book ever changes.

And you’re not going to tell me that books cannot contain reasoning, whether inductive or deductive!

You could even write a book explaining all this.

Reading, reasoning, and writing denote three changes.

@ NiV

I might as well tell you that books do not contain reasoning. The text only counts as expressing reasoning for humans. And me reading it (moving my eyes etc.), and going from premises to conclusions involves change, as was noted.

Right, and how does one get (or experience the illusion of getting) from the practice of physicists and useful conceptual tools to the truth of metaphysical four-dimensionalism and change being an illusion?..

“Reading, reasoning, and writing denote three changes.”

That’s just the limitations of the language. The book, and the narrative in it, simply is.

“I might as well tell you that books do not contain reasoning. The text only counts as expressing reasoning for humans.”

Same thing. During the time between the book being written and the book being read, the reasoning must exist, or how could the exact same reasoning arise in the reader as the writer? How could the two be connected, if there is nothing of the nature of reason in the book to connect them? How can anything pass from one place to another without passing through all the points in between?

Books hold meaning independent of the minds that read them. That’s what they’re for: to store knowledge outside our heads.

And anyway, they can express reason for computer software, too.

“And me reading it (moving my eyes etc.), and going from premises to conclusions involves change, as was noted.”

But if you follow the analogy, and consider yourself at each instant of time reading as like a page in the book, or as like a character described in the book’s story, all there at once, then while it is true that on each page the character only remembers the earlier pages and not the later, and only considers one step in the argument per page, all the pages are nevertheless there. You can read and check the steps out of order.

Our language has the illusion of time built in to it. It assumes it at such a deep level that it is extremely difficult – at least without unreadable, strained involutions of language to avoid any use of tenses – to explain what I’m trying to explain. (Diagrams would be easier, but they’re tricky in a comment box.) It does require a little good faith and charity on the part of my audience not to misread an attempt to be more clearly understandable as the final statement of my position. I assume you know what I mean when I say that all the pages of the story are there at once; why not consider the idea as it is clearly intended to be understood?

“Right, and how does one get (or experience the illusion of getting) from the practice of physicists and useful conceptual tools to the truth of metaphysical four-dimensionalism and change being an illusion?”

Through metaphysical four-dimensionalism being the only coherent picture that can make sense of the relativity of simultaneity, and from the four-dimensional picture being unchanging.

I assume you’re asking because you don’t find that obvious. Where do you get the metaphysical falsity of four-dimensionalism from? How do you explain Tuesday rolling back to Monday without it?

“Reading, reasoning, and writing denote three changes.”

That’s just the limitations of the language. The book, and the narrative in it, simply is.

Not at all. Reading, reasoning, and writing denote three changes in the person reading, reasoning or writing. Einstein is no longer with us; another change. Dead leaves are cluttering my sidewalk; more change. None of these changes are illusory or reflect a limitation of language.

“Einstein is no longer with us; another change. “

Only if you turn to the back of the history book. Flip to an earlier page, and there he is!

From the perspective of the characters in a story, time passes. But that’s only because the person at each page can only perceive the page’s past. On each page, a character remembers a different sequence of events as ‘most recent’. Yet looking at the book as a whole we can see that all the pages, and the entire story, are there all the time.

Or if you don’t like the book analogy, consider the alien equivalent of Einstein in the Andromeda galaxy, who died yesterday, from my point of view, but from your point of view is still alive right now since you are on a different point of Earth moving at a different velocity, due to the Earth’s rotation. Both Andromeda-yesterday and Andromeda-today are ‘there’, since they are there in my ‘now’ and your ‘now’ respectively.

From the perspective of the characters in a story, time passes. …. Yet looking at the book as a whole we can see that all the pages, and the entire story, are there all the time.

Yet when the “book analogy,” often conceptualized as “Hamlet,” is used to illustrate the position of God to the world, the whole analogy is dismissed tout court. Of course, it is recognized as being an analogy and not an equivalence. The play “Hamlet” exists, but the prince of Denmark Hamlet “exists” only in an abstract sense.

Both Andromeda-yesterday and Andromeda-today are ‘there’, since they are there in my ‘now’ and your ‘now’ respectively.

But that is a limitation only on our knowledge of Andromeda, not on Andromeda herself.

@ NiV

For the narrative to actually be (considered, read) there must be a book (in this case), it’s a necessary (given the example) condition, but it’s not sufficient. Another necessary condition of reading the narrative is a rational being capable of reading familiar with our conventions. In the text the narrative is present only virtually, but it can be “extracted” by the mind. Computer software is not really different: computers do not think, nor do they reason. Abaci can be complex, but it is us who count and use them for that purpose.

And narratives only work if one knows the (fictional) past – and memory presents things in the past as past. What sense are we to make of “narrative” if there is no change, I wonder?.. Also, things passing from one place to another.
This analogy presupposes the unity/persistence of things, “4-dims” refer to temporal parts (of things) etc. But I don’t see how on four-dimensionalism there can be such a thing. Just lots of instantaneous things (perhaps resembling other, “previous” things) along the coordinate line. What makes them parts of wholes (like the steps of the narrative, or steps of an argument)? The whole, the persistent thing (the character) is really alien to this view, I believe. And this does include arguments, observation, experiment qua reasoning, observation, experiment. These are wholes, “within” and from which you (!) -proceed-.
“Me”-at-t1 (one thing) considering “Moscow is the capital of Russia and capitals are situated in countries they are capitals of” (a different thing) and “me”-at-t2 (a different thing) considering “Moscow is in Russia” (yet another differing thing) are not connected in a way to count as an inference: just two things connected to two other things (‘me’s to things considered, respectively) along the line of coordinates. Not surprising that you’d prefer to use diagrams, geometry, again, being timeless.

Change is evident, observed. You don’t really deny this, for you claim it to be an illusion. But the physical theory (or, more precisely, the philosophical interpretation of it: I’m not aware of anything that makes four-dimensionalism the absolutely mandatory reading of special relativity) you take to entail the irreality of change is dependent on, say, a prediction being made, followed by an experiment, followed by observing the result/testing the prediction, and thus is based on changing things changing. Thus, as YOS put it, ” If this change is an illusion, the conclusions of natural science are an illusion of an illusion”.

To deny change would be to invalidate any basis for such a theory (and therefore of its interpretation). I therefore deny that this view is coherent at all (sorry for a Russianism: I realise that four-dimensionalism entails change being an illusion).
Change is evident and cannot be denied without witnessing the thing denied in the act of it’s denial (whether visible or mental, like typing different words and letters or changing the focus of your attention from one premise to another) or ending up incapable of arguing for the position coherently, and hence four-dimensionalism, like the view of Parmenides, is false. After all, aren’t you trying to -change- our minds? 🙂
So if some interpretations of special relativity entail the irreality of change, well, too bad for them.

“Yet when the “book analogy,” often conceptualized as “Hamlet,” is used to illustrate the position of God to the world, the whole analogy is dismissed tout court.”

Not by me, it isn’t!

Indeed, given that this is an ideal argument to explain how a deity could be omniscient and still allow for free will, I’m puzzled as to why you’re all arguing so vehemently against it. I would have thought you’d like the idea!

“But that is a limitation only on our knowledge of Andromeda, not on Andromeda herself.”

No, it’s actual. The actual ‘now’ there is different for each of us.

“Another necessary condition of reading the narrative is a rational being capable of reading familiar with our conventions.”

Is this like the tree in the forest. Do the pages of a book go blank when there is nobody reading it?

“Computer software is not really different: computers do not think, nor do they reason.”

Computers are not essentially any different to neurons. Do brains ‘think’? If so, can you define the difference without appealing to some form of vitalism?

“Also, things passing from one place to another.
This analogy presupposes the unity/persistence of things,”

So do you deny the persistence of things?

Actually, it assumes the principle of locality: an object is only directly influenced by its immediate surroundings.

“But I don’t see how on four-dimensionalism there can be such a thing. Just lots of instantaneous things (perhaps resembling other, “previous” things) along the coordinate line. What makes them parts of wholes (like the steps of the narrative, or steps of an argument)?”

Does not the same argument apply to space, as well? There cannot be an extended object occupying a non-zero volume of space, only lots of points, resembling neighbouring points along the (spatial) coordinate line. Indeed, the path through spacetime of an atom is more clearly a single continuous object than the agglomerations of many atoms separated by gaps. What makes the atoms of a brick parts of the ‘same’ single object?

“Change is evident, observed. You don’t really deny this, for you claim it to be an illusion.”

Exactly!

“I’m not aware of anything that makes four-dimensionalism the absolutely mandatory reading of special relativity)”

It’s not. You can always construct a sufficiently convoluted alternative model to explain anything. It’s just the simplest and most obvious. I’m assuming we’re willing to apply common sense, here.

“But the physical theory […] you take to entail the irreality of change is dependent on, say, a prediction being made, followed by an experiment, followed by observing the result/testing the prediction, and thus is based on changing things changing. “

No. First, it’s only our knowledge of it that depends on experiment. Second, the experimental process can be re-described in timeless terms: particle worldlines following a specified set of differential equations implement a computational model that represents the state of another set of particles located in the positive time direction from them. etc.

It’s somewhat unnatural to do so, but there’s no logical difficulty in it.

“Change is evident and cannot be denied without witnessing the thing denied in the act of it’s denial”

It depends who you’re talking about. I don’t have any problem doing so, but I agree that it is evident some people do.

“So if some interpretations of special relativity entail the irreality of change, well, too bad for them.”

There are a lot of people who struggle with the ideas of special relativity – the universe doesn’t care.
It is what it is, whether we like it, believe it, or are willing to accept it or not.

@ NiV

Is this like the tree in the forest. Do the pages of a book go blank when there is nobody reading it?

They do not go blank. It’s just that written speech is observer relative, so without the rational person reading it there is no meaning, it’s virtually, not actually there.

Computers are not essentially any different to neurons. Do brains ‘think’? If so, can you define the difference without appealing to some form of vitalism?
No, brains do not think, they are required for human thought. Hylomorphic dualism is not a form of vitalism, so no.

So do you deny the persistence of things?
Actually, it assumes the principle of locality: an object is only directly influenced by its immediate surroundings.

No, what gave you that idea?.. What I deny is four-dimensionalism, which in fact excludes the persistence of things through change (what else?..), for reasons mentioned.

Does not the same argument apply to space, as well? There cannot be an extended object occupying a non-zero volume of space, only lots of points, resembling neighbouring points along the (spatial) coordinate line. Indeed, the path through spacetime of an atom is more clearly a single continuous object than the agglomerations of many atoms separated by gaps. What makes the atoms of a brick parts of the ‘same’ single object?

The trouble here is thinking coordinate lines our abstractions (coordinate lines) as ultimate reality. Sure, they describe it, and given change, space is kind of time-like, because, the things that make up the universe are, well, moving.
Bricks are accidental forms (of substances), and it is substantial forms that determine a material substance’s identity. And with accidental forms, them being reducible to substances ordered in a particular way, identity is always kind of “unstable”, as it were.

It’s not. You can always construct a sufficiently convoluted alternative model to explain anything. It’s just the simplest and most obvious. I’m assuming we’re willing to apply common sense, here.

Right, because denying the reality of change is plain common sense.
I hope you realise that the judgement of the purported qualities of four-dimensionalism is sort of what’s at issue here.

No. First, it’s only our knowledge of it that depends on experiment. Second, the experimental process can be re-described in timeless terms: particle worldlines following a specified set of differential equations implement a computational model that represents the state of another set of particles located in the positive time direction from them. etc.

If our knowledge depends on experiment, which depends on change, which according to you is illusory, then our knowledge is illusory. ///
Oh, naturally (not). Because re-describing something to avoid mentioning change means change is not real (it doesn’t)

It depends who you’re talking about. I don’t have any problem doing so, but I agree that it is evident some people do.

Again, sorry. It should’ve been “coherently deny”. And yes, some people have no problem with being incoherent, no argument here

There are a lot of people who struggle with the ideas of special relativity – the universe doesn’t care.
It is what it is, whether we like it, believe it, or are willing to accept it or not.

I don’t have a problem with special relativity as a physical theory.
What I do have a problem with is an interpretation that denies the reality of change, as I believe I’ve made quite clear.

Sorry, the wrong tag wasn’t intentional.
I need to get some sleep, I suppose.

given that this is an ideal argument to explain how a deity could be omniscient and still allow for free will, I’m puzzled as to why you’re all arguing so vehemently against it. I would have thought you’d like the idea!

An argument is not good simply because it can support some desired conclusion. Minkowsky 4-space is a useful device for doing relativistic calculations, just as epicycles were useful for doing astronomical calculations; but that doesn’t mean they are physically real. The model is not the reality.

However, it is the extension of this mathematical technique to the revival of Zeno’s changeless world that is objectionable. Physicists do distinguish timelike dimensions from spacelike dimensions, there being a very clear distinction between motion through space and motion through time. You can take three steps forward, then three steps back; but you cannot take three seconds forward and then take three back.

“But that is a limitation only on our knowledge of Andromeda, not on Andromeda herself.”

No, it’s actual. The actual ‘now’ there is different for each of us.

Under the hypothesis, what would be an “actual” now?
Taking a smaller scale: Sirius lies 8.6 ly from Sol; Vega lies 25.3 ly from Sol; Vega lies 33.4 ly from Sirius. Thus, if something happened at Vega — let’s say the star suddenly explodes — the Vegans would know immediately, alas; the Terrans would know 25 years later; and the Sirians would know 33 years later.

They do not have different “nows.” Putative residents around each star would simply perceive the event at different time. It is easy to confuse the moment of perception with the moment of occurrence, a consequence of the finiteness of lightspeed. It’s epistemic, not ontological.
++++++

Computers are not essentially any different to neurons. Do brains ‘think’?

Depends. Do hands play the clarinet? Or is the brain the Magic Organ, the only organ that uses the organism rather than vice versa?
One difference between computers and brains is that brains self-assemble from the differentiation of a single original cell.
Another is that computers cannot produce any sentence that is not derivable by steps from its inputs by set rules. Thinking beings, however, can produce true sentences that are not provable within the system (cf. Goedel). Regardless whether brains are computers, minds cannot be. No amount of syntactical rule-processing can ever produce semantic meaning.

“An argument is not good simply because it can support some desired conclusion. Minkowsky 4-space is a useful device for doing relativistic calculations, just as epicycles were useful for doing astronomical calculations; but that doesn’t mean they are physically real. The model is not the reality. “

True. But the Galilean/Newtonian model of time being absolute and the same for everyone, or the Presentist model of there only being a now with the past’s and future’s only existence being what’s recorded in the current state, or predictions from it, is likewise a model useful for calculation. The model is not the reality.

And my point was that Presentism has huge problems dealing with relativity, because ‘now’ is different for different people, and can wiggle backwards and forwards.

“Physicists do distinguish timelike dimensions from spacelike dimensions, there being a very clear distinction between motion through space and motion through time.”

Yes, there is a difference, but it’s not sharply defined. The time-like directions are those in the future light cone. There are lots of them, and any one of them can be used as the time axis for some observer’s coordinate system. The 3-dimensional hyperplane at right angles to this time axis is a set of space-like directions that define the observer’s ‘now’. They are all the events that – for that particular observer – occur at the same time.

However, a different observer, with a different time axis (still in the future light cone of time-like directions) finds that their time axis is at right-angles to a different ‘now’ hyperplane. The planes intersect, but in general, events which one observer considers to be simultaneous the other observer finds happen at different times.

“Thus, if something happened at Vega — let’s say the star suddenly explodes — the Vegans would know immediately, alas; the Terrans would know 25 years later; and the Sirians would know 33 years later.”

No, I’m not talking about propagation delay. Take everything you see and subtract the distance in light years to work out when it happened. The Vegans, Terrans, and Sirians would date the event to the same time. But somebody on a space ship that set off from Vega just before the explosion traveling at 99% of light speed would finally see the explosion, let’s say, just as they arrived at Earth. But they’re moving at relativistic speeds! So they will only have experienced 3.5 years of flight time due to time dilation, and they will see Vega only 3.5 light years behind them (because lengths contract). They will say the event happened 3.5 years ago (just as they remember it), not 25, even though they are in the same location (and subject to the same propagation delay) as people who think it was 25 years ago.

The people on Earth think the fleeing Vegans got it wrong because their clocks are running slow causing them to underestimate the time it took. The fleeing Vegans think the people on Earth got it wrong because their rulers are contracted causing them to overestimate the distance to Vega. The two effects are the same thing seen from different viewpoints. Both are right. Moving clocks don’t really run slow: they’re measuring the local moving time correctly, just as it happens. But the ‘moving frame’ time axis is pointed in a different direction to the ‘stationary frame’ time axis. A moving clock does not measure ‘stationary frame’ time.

‘Forwards’ and ‘sideways’ are different for different people, like ‘time’ and ‘space’ are; nobody thinks it strange if people disagree on how far apart forwards or sideways two events are. Space and time work the same way.

The propagation delay is epistemic, but time dilation, length contraction, and different observers having different versions of ‘simultaneous’ are ontological.

“Thinking beings, however, can produce true sentences that are not provable within the system (cf. Goedel).”

Godel applies equally well to humans. Consider the theorem: “This theorem is not provable by any organic brain.” Can you prove it? If you could, it would be false. So I would suggest that you can’t, and neither can any other organic brain, and therefore the statement is true. You can see that it’s true, but only by making an invalid logical step. A non-organic computer could do so without contradiction, though. Similarly, an algorithm can recognize Godel-type statements about itself as true if they’re allowed to use invalid logic, and there’s no rule to say that they can’t do so.

“No amount of syntactical rule-processing can ever produce semantic meaning.”

Agreed, but computers are not restricted to just syntactic rule-processing.

Presentism has huge problems dealing with relativity, because ‘now’ is different for different people

Relativity of motion was being discussed in the middle ages.

@Nullius in Verba:

“Yes, there is a difference, but it’s not sharply defined. The time-like directions are those in the future light cone. There are lots of them, and any one of them can be used as the time axis for some observer’s coordinate system.”

First, yes there is a sharp separation between a time-like and space-like direction — for starters, it has to do with a pesky minus sign in the Minkowski metric (using signature +++- and sticking to SR for simplicity) and its many consequences in SR. You seem to imply that one can arbitrarily transform a time-like direction into a space-like and vice-versa, but this is simply and utterly false. Besides, for a particle tracing out a geodesic in space-time say, proper-time, not coordinate-time, is an invariant quantity that does not involve anything whatsoever to do with observers, frames etc. Two observers, have different past and future light-cones, an easy observation that does not need any confused talk about “different time-axis”.

And of course, the many things that TOF mentioned.

“The people on Earth think the fleeing Vegans got it wrong because their clocks are running slow causing them to underestimate the time it took.”

If they were dumbasses, they probably would think that. Barring that hypothesis, they would apply the correct transformations to translate the measured results in one reference frame to the other — in fact, Sherlock, that is how you *know* that the dumbass Earthlings get different numerical results from the dumbass Vegans. But the fact that the numbers you get depend on the reference frame is as true of Newtonean mechanics as is of relativity. If you measure something, to know what an observer in a different (inertial) reference frame would measure you have to apply the relevant Galilean coordinate transformation, since the Galilean group is the invariance group of Newtonean mechanics. In SR it is the Poincaré group, and more quantities are reference frame-dependent (e.g. temporal distances).

“Godel applies equally well to humans. Consider the theorem: “This theorem is not provable by any organic brain.” Can you prove it? If you could, it would be false. So I would suggest that you can’t, and neither can any other organic brain, and therefore the statement is true. You can see that it’s true, but only by making an invalid logical step. A non-organic computer could do so without contradiction, though. Similarly, an algorithm can recognize Godel-type statements about itself as true if they’re allowed to use invalid logic, and there’s no rule to say that they can’t do so.”

There is hardly anything that is correct here.

If the sentence in between quotes is true, then it is not provable, and so it is not true in some models of the theory — that thingy you forgot to specify — and therefore it cannot be true in the usual sense, or true in all models. So contrary to what you assert, no one with half a wit would make the obviously “invalid logical step” and take the sentence to be true is you. If it is false, then the sentence “This theorem is provable by any organic brain.” is true, a sentence that looks harmless enough, though also quite meaningless (provable from what?). And why would anyone (and especially TOF) would take the sentence as true? One would only do, if one were committed to say that every sentence is provable or every true sentence is provably true, but absolute provability doesn’t have any meaning. And while such persons may exist for all I know, I think TOF is well furnished with a minimum of good sense to not be one.

And neither could non-organic computers (??) “do so without contradiction”; if you think so, it is simply because you elided the theory in which we are supposed to prove theorems and then jumped on “non-organic” as if *that* made any difference. Newsflash; we are also Turing-complete machines so we can compute and prove as much or as little as computers. Actually, we are not machines, Turing or otherwise, neither computers compute or prove anything, except by an abuse of language. And the fact that the Goedelian sentence is “recognized as true” is not some mysterious “ilogical step”, but is equivalent to the consistency of the background theory (*) — the thingy which you elided. And really, recognizing Goedelian sentences “as true if they’re allowed to use invalid logic”? Too dumb for words.

(*) alternatively, “true” may also mean true in some specified model, say Con(PA) in the standard model of arithmetic.

Furthermore, there are theories that trivially prove whatever Goedelian sentence you can come up, just as it is trivial to prove there are extensions of PA that prove the consistency of PA and other extensions of PA that prove the inconsistency of PA. There are even extensions of PA that prove *all* the arithmetically true statements. The catch is that they are not recursively axiomatizable. And while Goedel proved his theorem using self-referencial sentences, there are proofs that make no use of such. It can go via proof-theoretic considerations, Diophantine equations, Joyal’s reformulation in terms of arithmetic universes, etc.

Now, of course this is all utterly irrelevant to the underlying philosophical issues, issues that, while you plead charitable understanding, you yourself do not understand much less address. Pray tell us, what is this view from nowhere, where supposedly everything that was or will be already is? We (and by we I will tentatively add to my own self the likes of Briggs, TOF, Georgy Mancz and dover_beach) are just waiting for you to spell out the Name of You Know Who. You concede that we, as characters in a book, perceive things temporally, yet you dismiss it as an “illusion”? So how do you explain this ingrained, phenomenal illusion of temporal becoming? Not only that, you dismiss temporal becoming supposedly in the name of some peculiar interpretation of Scientific Theory, theory backed up by, wait for it, sensory evidence, the same sensory evidence that you decry, namely the illusion of temporal becoming. And even if you were hundred per-cent correct, you have presented no argument to prefer growing block theories to eternalist four-dimensional one, and in the former, change still happens. And even if you were hundred and fifty per-cent correct, not *all* arguments for God would be blocked, as can be seen simply by the fact that whatever the amount of misunderstanding of QM and GR you want to throw at it, they neither fix the boundary conditions nor the laws of nature, or even give any account of them.

“Relativity of motion was being discussed in the middle ages.”

True. I think I mentioned Galileo? This is the relativity of simultaneity we’re talking about here.

“First, yes there is a sharp separation between a time-like and space-like direction”

Sorry, I didn’t phrase that very well. The bit I was talking about was “a very clear distinction between motion through space and motion through time”. Time-like motion is not quite the same thing as “motion through time”, since “time” is just one direction but “time-like” describes a whole light-cone of them. Each time-like direction has, in any observer’s reference frame, a time component and three space components, which may be non-zero. Each space-like direction similarly has a time component and three space components, and the time-component may be non-zero.

“You seem to imply that one can arbitrarily transform a time-like direction into a space-like and vice-versa, but this is simply and utterly false.”

I haven’t said or implied that anywhere. You’re making up straw men.

“Besides, for a particle tracing out a geodesic in space-time say, proper-time, not coordinate-time, is an invariant quantity that does not involve anything whatsoever to do with observers, frames etc.”

Excellent! You understand the distinction! Now can you explain to YOS how coordinate time can define different versions of ‘now’ (planes of constant coordinate-time) for different observers. Perhaps you can do it better than I can.

“Two observers, have different past and future light-cones, an easy observation that does not need any confused talk about “different time-axis”.”

That’s crap. Observers at the same position but moving differently have exactly the same past and future light cones, because the null cones are invariant under Lorentzian transformations. They might relabel the vectors inside them with different time and space coordinates, but the cones themselves are the same. And of course, the reason they label them differently is they are using different time coordinate axes.

“If they were dumbasses, they probably would think that.”

Or if they were trying to explain the concept to “dumbasses” who grew up being taught Newtonian physics and need to be led through it gently.

“But the fact that the numbers you get depend on the reference frame is as true of Newtonean mechanics as is of relativity.”

Agreed! Excellent point!

“In SR it is the Poincaré group, and more quantities are reference frame-dependent (e.g. temporal distances).”

Excellent! The fact that temporal distances are reference-frame dependent is exactly the point I’m trying to make. Now, if you could expand on that a bit for YOS…?

“If the sentence in between quotes is true, then it is not provable, and so it is not true in some models of the theory”

Crap. The sentence between the quotes is both true and provable. I even gave the proof: If an organic brain could prove it was true, it would be false, which is a contradiction, therefore an organic brain can’t prove it. QED.

And since you got that bit wrong, the rest of what your say is either irrelevant or nonsense.

“Now, of course this is all utterly irrelevant to the underlying philosophical issues, issues that, while you plead charitable understanding, you yourself do not understand much less address.”

Clearly I understand them better than you lot, since you don’t seem to know what I’ve been talking about for the past couple of days…!

“Pray tell us, what is this view from nowhere, where supposedly everything that was or will be already is?”

The place where “everything that was or will be already is” is called the “block universe” – I think I might have already mentioned it, were you paying attention? – and indeed “the universe” is the subject of this post.

“We […] are just waiting for you to spell out the Name of You Know Who.”

Certainly. “T. H. E. U. N. I. V. E. R. S. E.” Happy?

“You concede that we, as characters in a book, perceive things temporally, yet you dismiss it as an “illusion”? So how do you explain this ingrained, phenomenal illusion of temporal becoming?”

Ah! Good question! The answer – again, I think I might have mentioned it before? – is that Presentism is a useful model for calculation, although the model is not necessarily the reality. More could be said, I’m sure, but I somehow don’t think you’re interested.

“And even if you were hundred per-cent correct, you have presented no argument to prefer growing block theories to eternalist four-dimensional one, and in the former, change still happens.”

What does that even mean? Did you say that right?

“And even if you were hundred and fifty per-cent correct”

Hmm. Something wrong with the math there…
🙂

Despite what you guys seem to think, I’ve got no particular interest one way or the other in proving or disproving the existence of God. It’s not something I care about. I regard it much the same way as D&D players arguing about whether spellcast fireballs work against dragons – it’s not my interest, but I’ve got no objection to how other people harmlessly pass their time. I’ve got no objection to them taking it seriously. It’s none of my business.

But I do find some of the arguments interesting from a philosophical point of view. The subject of the post is Pantheism – a theist position which Aquinas through Briggs is arguing against. Part of one argument for that is that the universe has change while the prime mover does not. Given that Aquinas was using a Presentist conception of the universe, that’s reasonable on his part. My question is whether the argument changes if we instead use an Eternalist picture, as I think is more reasonable given what we now know about time from relativity theory.

In discussing this, I’m temporarily taking the Pantheist position – which you’ll notice is a theist position. That’s partly because it’s easier to make my argument from there, and partly to tweak you guys, to watch you tie yourself in knots arguing against a particular version of God being supported by your own Thomist arguments. (A version that most people would find indistinguishable from your Thomist one – vis-a-vis the Old Testament man-with-beard picture they see in church.) So far, you guys have only attempted to deny the premise, which suggests to me that you fear the argument itself might be applicable.

Personally, I’m more interested in whether some Thomist arguments might be applicable to the debate between different interpretations of quantum mechanics. If the absolute metaphysical reality has to be eternal and timeless, if there’s an actually valid argument for that, then that’s another argument for Eternalist and timeless versions of physics. It seems to me like an idea worth pursuing.

“Relativity of motion was being discussed in the middle ages.”

True. I think I mentioned Galileo? This is the relativity of simultaneity we’re talking about here.

Galileo was not medieval. I was thinking of Witelo, Oresme, and others in the 14th century.
+++

Here’s a thought experiment: The yodelers.
Imagine there are three yodelers up in the Austrian Alps. Each is in a yodeling tower, and these towers are located 340 meters apart, so that it takes one second for a yodel to reach one tower from the previous. The rule relay yodeling is that each yodeler repeats the yodel on the instant he hears it. In this case there is only one intermediate yodeler.

Yodeler A — we’ll call him Alois — calls out his yodel. One second later, Yodeler B (Bernard) hears Alois and adds his yodel. One second after that Yodeler C (Caspar) hears both Bernard and Alois. To Caspar, it seems that Bernard and Alois have yodeled simultaneously with Caspar himself.

But Alois hears Bernard at t(2) and Caspar at t(4), where t(0) is the time Alois himself yodeled. That is the yodel signals arrive at A at intervals, whereas they arrive at C simultaneously. Alien creatures who perceive the world only by means of sonar might go on to develop a sonic theory of relativity, due to the finite speed of propagation of the signals.

Neither Alois nor Caspar would conclude that there is no change.

YOS,

It’s a good thought experiment. And of course, all the yodelers will take the propagation time into account, and subtract the speed of sound times the distance to get the correct time of the event.

Now imagine we add Dieter, who is on a train traveling past Bernard’s tower towards Caspar’s at high speed. From Dieter’s point of view, sitting in the train carriage looking out of the open window, he is stationary and it is the three towers moving past him. He hears the yodel from Alois and notes that Alois is currently 340 metres from him. However, he knows that the yodel must have occurred some time previously, when Alois was in fact closer to Dieter than 340 metres. Let’s say it was only 300 metres. Now Dieter knows the speed of sound is 340 m/s so he knows that Alois’s yodel must have occurred 300/340 = 0.88 seconds previously.

Dieter waits a little while for the yodel to pass along to Caspar and then for Caspar to yodel back. During that time, Dieter moves along the track towards Caspar. He knows that the sound passed him at t = 0, arrived at Caspar’s tower 0.88 seconds later (speed of sound times distance to the event) and returned at the same speed. He would therefore hear the response at 1.76 seconds after hearing Alois’s yodel. If he heard it late, at 2 seconds after the yodel, he’d wonder at that. He’d eventually have to conclude that the speed of sound varied, and was different for different observers, and in different directions. If they sent yodels back and forth often enough, he’d realize that the sound was somehow traveling slower than the speed of sound going from A to C, and faster than the speed of sound going from C to A. One way it was going at 300 m/s, the other way it was doing 380 m/s. That would be very strange!

Sound, as I’m sure you know, does vary in speed in this way. But the speed of light does not! It is the same for all observers, whatever their state of motion. And when time, speed, and distance are strictly related, and speed held constant, then time must vary with distance. Because in Dieter’s moving frame the distance to the yodelers is different to how the yodelers measure it, the time intervals must be different as well. This is simply how the universe works.

@Nullius in Verba:

“That’s crap. Observers at the same position but moving differently have exactly the same past and future light cones, because the null cones are invariant under Lorentzian transformations.”

So it is crap that two different observers have different past and future light cones? And for way of counter-example you present two observers… at the same position. I did not know I had to be pedantic to the point of having to specify that the two (distinct) observers had to be at a non-zero distance (in the relevant metric) from each other, since the concept of two (distinct) observers at the same position… well, it does not seem like a very interesting one.

“Crap. The sentence between the quotes is both true and provable. I even gave the proof: If an organic brain could prove it was true, it would be false, which is a contradiction, therefore an organic brain can’t prove it.”

So:

(1) The sentence between the quotes, call it S, is both true and provable.

And the proof you give is:

(2) If S were provably true, then it would be false, which is a contradiction, therefore it is not provably true. (The rest I fill out for you) Since S says of itself it is not provably true, it therefore is provably true.

Huh huh.

“Clearly I understand them better than you lot, since you don’t seem to know what I’ve been talking about for the past couple of days…!”

Of course you do.

“What does that even mean? Did you say that right?”

Since you asked so politely Growing block universe and references therein.

“I did not know I had to be pedantic to the point of having to specify that the two (distinct) observers had to be at a non-zero distance (in the relevant metric) from each other, since the concept of two (distinct) observers at the same position… well, it does not seem like a very interesting one.”

You’re offering different light cones as an alternative to “confused talk about “different time-axis”.” You’re clearly well-aware that coordinate-time is reference frame dependent, and I’m sure you know that the coordinate axes along which we measure coordinate time in each reference frame are normally passed through the same origin, to simplify the discussion. Using a different origin with parallel axes is irrelevant to the physics we’re interested in, and a frame with parallel axes is usually considered equivalent. To replace the time coordinate axis passing through the origin in each reference frame with the future light cone, it has to be the light cone at the origin. Or the analogy you’re trying to draw falls at the first hurdle. And it’s not hard to come up with scenarios where coincident observers are interesting – the most obvious being asking what an observer would see under two alternative motions.

But I’m sure you know all this, and are just arguing for the sake of it.

“Huh huh.”

Classic!
🙂

“Since you asked so politely Growing block universe and references therein.”

Regarding politeness, I assume by default that Christians would treat me as they would wish to be treated. It’s not normally my way, but I’m OK with it if that’s what you want.

Thanks for the reference to growing block universes. I guessed it might be something like that.

What I was querying was your phrase “you have presented no argument to prefer growing block theories to eternalist four-dimensional one” which I thought was the wrong way round. Surely I ought to be supplying arguments for preferring the Eternalist model I’d presented to the growing alternative?

Assuming that’s what you meant, the argument I gave earlier regarding Andromeda had something very like this in mind. If events ‘happen’ and are added to the block universe in the observer’s ‘now’ (i.e. the ‘coordinate-time zero’ hyperplane), do they ‘unhappen’ when that ‘now’ shifts back and forth?

Or if it’s not happening ‘now’, what shape is this “very thin slice of spacetime, where more of spacetime is continually coming into being”? If it doesn’t coincide with the ‘now’ hyperplane for some observer, it means that there is a spatial boundary, on one side of which is the fixed past, and on the other the undetermined and as yet non-existent future. And if it does so coincide, then the observer only has to change their state of motion slightly for it not to. Thus, we each might stand on the spatial edge of the universe, but we fail to notice the void on our left or right hand because it is being ‘filled in’ faster than the light from there can propagate to us. (We can only ever see our past light cone, in the fixed past, anyway.) Or if the surface is locally convex, we might even be alone in the void, the universe all around us not having come into existence yet.

It’s an odd picture, and requires that we define two independent times: one the internal block time, and the other a meta-time over which the block time gets filled in. Presumably there’s some unobservable physics that implements the filling in process, controlling the shape and progress of the thin slice. How does it work? What’s the mechanism? Clearly, there’s no observable difference from the block universe, so there’s only aesthetics and usefulness on which to decide. It might be so, there’s no experimental test possible to disprove it. However, I feel this is a case for Occam’s razor.

It seems to me this is a highly artificial and arbitrary construct introduced solely because the author doesn’t like determinism, emotionally. Besides retaining indeterminism (which hardly seems like an advantage to a physicist – God does not play dice with the universe) what arguments and advantages are there in its favor? How does it help?

If the ultimate reality has to be eternal and changeless, as Aquinas argues, isn’t that an argument for Eternalism?

@Nullius in Verba:

“”Huh huh.”

Classic!”

If you cannot see what is right in front of your nose, after being laid out in pedantic detail, what can I say? As it is: hopeless.

“If you cannot see what is right in front of your nose, after being laid out in pedantic detail, what can I say?”

Not pedantic enough. You really can’t see the difference between what I wrote and what you wrote?

In case it’s still not clear, let’s try some of that pedantic detail.

Let Pr(X,P) be a predicate equivalent to “Statement X is provable by Prover P”

It has the properties:
[ There exists P such that Pr(X,P) ] implies X
Not-X implies [ For all P, Not-Pr(X,P) ]

i.e. Statements provable by some prover must be true, and untrue statements are not provable by any prover.

So, “This theorem is not provable by any organic brain.” is a self-referential statement equivalent to:
S = [ Not-Pr(S,OB) ]
where OB is the collection of all organic brains, actual and potential.

The proof goes as follows:
Pr(S,OB) implies [ Pr(S,OB) ] ……………… Tautology
Pr(S,OB) implies [ S ] ……………………. ‘Provable implies true’ property of Pr
Pr(S,OB) implies [ Pr(S,OB) and S ] ……….. Combine the above two
Pr(S,OB) implies [ Pr(S,OB) and Not-Pr(S,OB) ] . Definition of S
Pr(S,OB) implies [ False ] ………………… Contradiction
Not-Pr(S,OB) or False …………………….. Definition of implication
Not-Pr(S,OB) …………………………….. Tautology
S ………………………………………. Definition of S

That part’s pretty straightforward. Now there are some more difficult questions I would expect intelligent people to ask, having got this far.

1) Does the self-reference pose a problem?
2) The predicate Pr is defined in English, is that precise enough? Does it exist?
3) You have just given a proof, even though you yourself are an organic brain. Doesn’t this contradict the conclusion?

1) A lot of paradoxes are constructed using self-reference (this proof is modeled after the Curry paradox, for example), to the point where a lot of people regard all form of self-reference as illegitimate. However, there are ways to re-write self-referential statements without self-reference (Y combinators) at the potential cost of generating an infinite tower of definitions, and there are many self-referential definitions that are useful or even essential in mathematics.

So the tricky task is to draw the line somewhere so as to include as many of the useful bits as possible while rejecting all the paradoxes. That’s hard, and different solutions have been proposed. If you reject the legitimacy of all self-reference, or of any of those tricks used to unwind it, then you’re going to reject this one. If you accept some of them, you might – I can’t say. Godel’s cleverness was in finding an indirect way to implement the self-reference via statements of arithmetic or number theory. Rejecting his approach would require rejecting arithmetic, which would be a step too far for virtually any mathematician. But that’s all it was – a way to make the self-reference palatable and to clearly define its limits.

2) To be able to talk about proving and provers, some extra formal machinery is needed. One particular variety of prover is defined by giving a set of rules and axioms, and specifying a process of sequentially exploring every possible sequence of steps looking for the statement it’s trying to prove. Exhibiting a list of such steps is sufficient to demonstrate that the prover would succeed. The prover is restricted: the sequential exploration limits it to a countable space, and often requirements of finite time and memory are imposed. (Or you could add every true statement of arithmetic as an axiom, and get a prover that will complete any proof in one step.)

But that’s not the only way you can do it. You can have provers exploring options simultaneously in quantum superposition, supertask provers, provers with specialised oracles, all sort of things. You can also have provers with tighter restrictions: bounded time and memory, for example.

‘Organic brain’ is not a precisely defined category. Does a brain with non-organic implants count? Should you allow a brain that stores most of its knowledge on the internet? However, for our purposes, all we really require is that you and me are organic brains, and there can be provers that are not organic brains. The first is not in doubt. The second I can imagine you arguing with, but given that most provers in the literature are implemented with some sort of Turing machine, I don’t see that as an obstacle.

For the predicate to be applicable to S, the prover has to be able to reason about provers. It has to be able to work out whether a given prover would be able to prove a given statement. It also ought to know what sort of prover it itself is.

3) And this is where you get the contradiction. The prover attempts to show that it can generate the above proof, but it also has an axiom in its database that says: “I am an organic brain.” (Or even “I might be an organic brain.” It doesn’t necessarily know.) That contradicts the assertion. The failure to check this is the illogical step I mentioned.

The prover can generate the proof by pretending to be a different sort of prover, simulating it. The prover can generate the proof by ignoring the issue, which can be interpreted either as assuming you’re the sort of prover where it doesn’t matter (which is effectively the same as above), or as being an invalid prover, capable of generating ‘proofs’ of false statements.

An organic brain asserting that it has proved S is making a false statement. It knows it is an organic brain, or might be, and is saying something that contradicts this. Proving the statement S is equivalent to saying “I am not an organic brain,” which for some provers might be true but for you and me is false. Organic brains can’t do it; not and be valid provers.

—

In your strawman version, you said: “Since S says of itself it is not provably true, it therefore is provably true.” S does not say of itself that it is not provably true. It says it is not provable by organic brains. This is clearly a different statement.

—

Out of interest, this is the same phenomenon as occurs in the ‘unexpected hanging paradox’. The judge condemns a prisoner to hang on some day during the following week, but specifies that the prisoner is not to be able to deduce the day beforehand. The prisoner uses a simple induction proof to show that this is contradictory, and that he cannot be hung on any day. His hanging therefore comes as a completely unexpected surprise to him, as required.

Again, we’re talking about proofs and deductions where the identity of the prover/deducer is one of the terms of the expression. This makes its provability indexical – it’s provable/deducible for some provers/deducers, and not others.

A lot of people look at Godel’s theorem being applied to algorithmic reasoning, and feel smug and superior that they can see the truth of the statement while the poor algorithm cannot. But it’s a general property of any prover sophisticated enough to represent a prover’s identity included as a term in the expression, as that allows statements that are prover-indexical. A universal Turing machine able to simulate such a prover certainly can. So can humans.