Hilary Lawson is right to connect the issue of the completeness and consistency of truth with paradoxes of self-reference.
As a kind of summary, consider this story:
It was a dark and stormy night,
and all the Cub Scouts where huddled around their campfire.
One scout looked up to the Scout Master and said:
“Tell us a story.”
And the story went like this:It was a dark and stormy night,
and all the Cub Scouts where huddled around their campfire.
One scout looked up to the Scout Master and said:
“Tell us a story.”
And the story went like this:It was a dark and stormy night,
and all the Cub Scouts where huddled around their campfire.
One scout looked up to the Scout Master and said:
“Tell us a story.”
And the story went like this:It was a dark and stormy night,
and all the Cub Scouts where huddled around their campfire.
One scout looked up to the Scout Master and said:
“Tell us a story.”
And the story went like this:
etc.
In this form, the story obviously exists, but in its implied form, the story cannot be told, because for the story to be “told” is for it to be completed, and it is impossible for it be completed, since it will not be complete until it contains itself, and this cannot happen.
Consider a similar example. You sit in a room at a desk, and decide to draw a picture of the room. You draw the walls. Then you draw yourself and your desk. But then you realize, “there is also a picture in the room. I need to draw the picture.” You draw the picture itself as a tiny image within the image of your desktop, and add tiny details: the walls of the room, your desk and yourself.
Of course, you then realize that your artwork can never be complete, in exactly the same way that the story above cannot be complete.
There is essentially the same problem in these situations as in all the situations we have described which involve self-reference: the paradox of the liar, the liar game, the impossibility of detailed future prediction, the list of all true statements, Gödel’s theorem, and so on.
In two of the above posts, namely on future prediction and Gödel’s theorem, there are discussions of James Chastek’s attempts to use the issue of self-reference to prove that the human mind is not a “mechanism.” I noted in those places that such supposed proofs fail, and at this point it is easy to see that they will fail in general, if they depend on such reasoning. What is possible or impossible here has nothing to do with such things, and everything to do with self-reference. You cannot have a mirror and a camera so perfect that you can get an actually infinite series of images by taking a picture of the mirror with the camera, but there is nothing about such a situation that could not be captured by an image outside the situation, just as a man outside the room could draw everything in the room, including the picture and its details. This does not show that a man outside the room has a superior drawing ability compared with the man in the room. The ability of someone else to say whether the third statement in the liar game is true or false does not prove that the other person does not have a “merely human” mind (analogous to a mere mechanism), despite the fact that you yourself cannot say whether it is true or false.
There is a grain of truth in Chastek’s argument, however. It does follow that if someone says that reality as a whole is a formal system, and adds that we can know what that system is, their position would be absurd, since if we knew such a system we could indeed derive a specific arithmetical truth, namely one that we could state in detail, which would be unprovable from the system, namely from reality, but nonetheless proved to be true by us. And this is logically impossible, since we are a part of reality.
At this point one might be tempted to say, “At this point we have fully understood the situation. So all of these paradoxes and so on don’t prevent us from understanding reality perfectly, even if that was the original appearance.”
But this is similar to one of two things.
First, a man can stand outside the room and draw a picture of everything in it, including the picture, and say, “Behold. A picture of the room and everything in it.” Yes, as long as you are not in the room. But if the room is all of reality, you cannot get outside it, and so you cannot draw such a picture.
Second, the man in the room can draw the room, the desk and himself, and draw a smudge on the center of the picture of the desk, and say, “Behold. A smudged drawing of the room and everything in it, including the drawing.” But one only imagines a picture of the drawing underneath the smudge: there is actually no such drawing in the picture of the room, nor can there be.
In the same way, we can fully understand some local situation, from outside that situation, or we can have a smudged understanding of the whole situation, but there cannot be any detailed understanding of the whole situation underneath the smudge.
I noted that I disagreed with Lawson’s attempt to resolve the question of truth. I did not go into detail, and I will not, as the book is very long and an adequate discussion would be much longer than I am willing to attempt, at least at this time, but I will give some general remarks. He sees, correctly, that there are problems both with saying that “truth exists” and that “truth does not exist,” taken according to the usual concept of truth, but in the end his position amounts to saying that the denial of truth is truer than the affirmation of truth. This seems absurd, and it is, but not quite so much as appears, because he does recognize the incoherence and makes an attempt to get around it. The way of thinking is something like this: we need to avoid the concept of truth. But this means we also need to avoid the concept of asserting something, because if you assert something, you are saying that it is true. So he needs to say, “assertion does not exist,” but without asserting it. Consequently he comes up with the concept of “closure,” which is meant to replace the concept of asserting, and “asserts” things in the new sense. This sense is not intended to assert anything at all in the usual sense. In fact, he concludes that language does not refer to the world at all.
Apart from the evident absurdity, exacerbated by my own realist description of his position, we can see from the general account of self-reference why this is the wrong answer. The man in the room might start out wanting to draw a picture of the room and everything in it, and then come to realize that this project is impossible, at least for someone in his situation. But suppose he concludes: “After all, there is no such thing as a picture. I thought pictures were possible, but they are not. There are just marks on paper.” The conclusion is obviously wrong. The fact that pictures are things themselves does prevent pictures from being exhaustive pictures of themselves, but it does not prevent them from being pictures in general. And in the same way, the fact that we are part of reality prevents us from having an exhaustive understanding of reality, but it does not prevent us from understanding in general.
There is one last temptation in addition to the two ways discussed above of saying that there can be an exhaustive drawing of the room and the picture. The room itself and everything in it, is itself an exhaustive representation of itself and everything in it, someone might say. Apart from being an abuse of the word “representation,” I think this is delusional, but this a story for another time.
Entirely Useless 
> There is a grain of truth in Chastek’s argument, however. It does follow that if someone says that reality as a whole is a formal system, and adds that we can know what that system is, their position would be absurd, since if we knew such a system we could indeed derive a specific arithmetical truth, namely one that we could state in detail, which would be unprovable from the system, namely from reality, but nonetheless proved to be true by us. And this is logically impossible, since we are a part of reality.
I believe this contains a clearly incorrect implicit assumption, which is that any formal system underlying reality must permit some notion of infinity, such as the countable integers. The Halting Problem and Goedel’s incompleteness theorems and all of these other paradoxes depend on self-reference continuing to infinity, but infinity is *obviously* not real.
In fact, there are plenty of formal systems, under “finitism”, that do not permit such infinite constructions. Therefore the argument that knowing the formal system of reality would permit a construction that yields a contradiction does not follow, because reality is finite/bounded.
For your specific example, the arithmetical truth that could not proved might require the number of particles in existence + 1 in order to prove, which of course would be impossible to construct, which is equivalent to “cannot be proved” in such systems. Thus, no contradiction exists, but realism holds.
The infinite self-referring drawing is of the exact same sort. Consider what prevents you from continuing the drawing to ever smaller scales: you cannot draw on an atom (or a quark), and so your self-reference is bounded by the smallest indivisible unit. The assumption of infinite divisibility was at the core of this paradox.
In other words, the solution to realism’s self-reference paradoxes is probably just taking realism seriously.
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Obviously a finite system that does not allow these derivations does not contain arithmetic; so if you are right that reality is finite in this way, then reality does not contain arithmetic at all. So that would be how you avoid the arithmetical contradiction.
I do not think this is obviously true, as you suggest; in fact I think it is obviously false.
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Technically, it isn’t even true that Peano arithmetic (or the weaker minimal system inside it that’s sufficient for incompleteness) permits a notion of infinity at least if by “infinity” we mean “an infinite set of the sort defined by the Axiom of Infinity in ZFC.” The canonical *model* of Peano arithmetic is the set of natural numbers (N) equipped with the appropriate operations and relations, which is itself infinite, but Peano arithmetic can’t talk about N or subsets of N per se, so you can’t prove in PA that there is an infinite subset of N (possibly N itself) in the sense that we do in ZFC. Though of course you can easily prove that there is no greatest element of N in PA, or even that there is no greatest even number, no greatest number divisible by any fixed number “m,” etc., But that’s doesn’t seem to intrinsically involve infinity so much as it is just a property of the way the successor function and order relation are defined and how universal quantification works.
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@wc: By “infinity”, I mean literally anything unbounded. The successor function is where infinity is typically smuggled in. This infinity doesn’t seem too problematic until it’s combined with multiplication. Without the availability of infinite successors, addition and multiplication are unproblematic.
@entirelyusless, I would love to see a proof demonstrating that reality contains unbounded arithmetic. Note that I’m not saying reality doesn’t contain what we might at first blush identify as arithmetical operators, but some truly unbounded natural quantity.
What typically happens is that you end up with bounded by Planck temperature, Planck density, Planck time, Planck length, Planck energy, Planck charge, and gravitational singularities. Quantum theories are our best tested theories, and they are inherently discrete.
You might be tempted to claim that spacetime is unbounded, but we know General Relativity to be false despite its success, because of its singularities. A quantum theory of gravity will shed light on the true nature of spacetime.
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To provide evidence for the claim that reality does not contain unbounded arithmetic, I’ll provide a few citations from working mathematicians and mathematical physicists. From Carlos Rovelli’s book, “Reality is not What It Seems. The Journey to Quantum Gravity”, Chapter 11 aptly titled “The End of Infinity”:
> When we take quantum gravity into account, the infinite compression of the universe into a single, infinitely small point predicted by general relativity at the Big Bang disappears. Quantum gravity is the discovery that no infinitely small point exists. There is a lower limit to the divisibility of space. The universe cannot be smaller than the Planck scale, because nothing exists which is smaller than the Planck scale. […] Quantum gravity places a limit to infinity, and ‘cures’ the pathological singularities of general relativity. […]
>
> There is another case, of a different kind, in which quantum gravity places a limit to the infinite, and it regards forces such as electromagnetism. Quantum field theory, started by Dirac and completed in the 1950s by Feynman and his colleagues, describes these forces well but is full of mathematical absurdities. When we use it to compute physical processes, we often obtain results which are infinite, and mean nothing. They are
called divergences. […]
>
> But the infinities of quantum field theory follow from an assumption at the basis of the theory: the infinite divisibility of space. […] When quantum gravity is taken into account, these infinities also disappear. The reason is clear: space is not infinitely divisible, there are no infinite points; there are no infinite things to add up. The granular discrete structure of space resolves the difficulties of the quantum theory of fields, eliminating the infinities by which it is afflicted. […]
>
> Putting a limit to infinity is a recurrent theme in modern physics. Special relativity may be summarized as the discovery that there exists a maximum velocity for all physical systems. Quantum mechanics can be summarized as the discovery that there exists a maximum of information for each physical system. The minimum length is the Planck length LP, the maximum velocity is the speed of light c, and the total information is determined by the Planck constant h. […]
>
> The identification of these three fundamental constants places a limit towhat seemed to be the infinite possibilities of nature. It suggests that what we call infinite often is nothing more than something which we have not yet counted, or understood. I think this is true in general. ‘Infinite’, ultimately, is the name that we give to what we do not yet know. Nature appears to be telling us that there is nothing truly infinite.
Also worth reading is John Baez’s “Struggle with the Continuum”, https://arxiv.org/abs/1609.01421, where he surveys many unphysical theories of the past that were improved by systematically eliminating infinities, like quantization which led to quantum mechanics. Existing field theories are still plagued by other types of infinities, as Carlos Rovelli describes in the text I quote above, and the ultimate goal is to eliminate them.
Arguably the greatest advance in modern physics was the experimental discovery that reality is discrete in some sense, and scientific progress is only pushing this discretization further and further. Reality is bounded from below by minimum quantities, and bounded from above by gravity, ie. overly large aggregations of information/energy collapse into “singularities”.
Ergo, we have no reason to believe that anything infinite truly exists, and therefore, there can exist no isomorphism to the natural numbers. Thus, unbounded arithmetic is not real.
Of course, bounded forms of arithmetic absolutely do exist in some sense (like modular arithmetic), but these are not subject to the paradoxes described in this article simply because the numbers “run out” before any inconsistent self-referential embedding can succeed.
If you still believe that my claim is obviously false, I look forward to reading your argument!
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(1) Nothing you are pointing to gives a reason to believe the universe is not spatially infinite. Most physicists think that the universe is probably spatially, not that it is infinitely divisible or anything like that. The world can be discrete and still be physically infinite.
There is direct evidence that the universe is spatially infinite: namely that every smaller particular estimate that anyone had in the past has already been proven false.
No one has shown that any real paradoxes result from this possibility.
(2) The above argument is academic, because for arithmetic to be real, it is not required that there be any physical “isomorphism to the natural numbers.” All that is required is for the principles of arithmetic to be true, together with all that logically follows for them.
What is required for your argument is this: some statements that logically follow from the principles of arithmetic are false or meaningless. This is why you cannot derive the conclusion about reality not being a formal system in your system. Because in your system, mathematics is false.
In particular, you are suggesting that it is false that every number has a successor. I say THAT is obviously false, and this does NOT mean “for any collection of physical objects, there is a bigger collection.” Every number can have a successor, and obviously does, even if there is an exact number of “all physical things.”
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> Most physicists think that the universe is probably spatially, not that it is infinitely divisible or anything like that.
I didn’t quote Rovelli on this point since my post was long enough, but here you go:
“There is another infinity which disorientates our thinking: the infinite spatial extension of the cosmos. But as I illustrated in Chapter 3, Einstein has found the way of thinking of a finite cosmos without borders. Current measurements indicate that the size of the cosmos must be larger than 100 billion light years. This is the order of magnitude of the universe we have indirect access to. It is around 10^120 times greater than the Planck length, a number of times which is given by a 1 followed by 120 zeroes. Between the Planck scale and the cosmological one, then, there is the mind-blowing separation of 120 orders of magnitude. Huge. Extraordinarily huge. But *finite*.”
> There is direct evidence that the universe is spatially infinite: namely that every smaller particular estimate that anyone had in the past has already been proven false.
I think it’s bizarre that you consider this direct evidence. By parity of reasoning, the number of grains of sand on Earth must be infinite because every past estimate has been wrong. That’s clearly fallacious.
> The above argument is academic, because for arithmetic to be real, it is not required that there be any physical “isomorphism to the natural numbers.” All that is required is for the principles of arithmetic to be true, together with all that logically follows for them.
That’s what “isomomorphism to the naturals” means. The principles of arithmetic, as commonly understood, are a direct correspondance to the naturals, ie. zero + infinite successors, addition, multiplication, etc.
> Because in your system, mathematics is false.
That’s not correct. I am not necessarily asserting fictionalism. Certain (unqualified) theorems are definitely false, but are true given certain qualifiers. This does not negate all of mathematics.
> Every number can have a successor, and obviously does, even if there is an exact number of “all physical things.”
Only if you assume the conclusion. If you’re not familiar with it, I recommend reading a proper introduction to strict finitism to see why your argument is problematic:
Click to access strict%20finitism.pdf
If you’re a nominalist, then it’s absolutely false that given any N you can construct N+1; numbers may simply not have independent existence, as but one objection. If you’re an intuitionist, then mathematical objects are explicit constructions in the mathematician’s mind, and since every mind is necessarily finite…
Finally, I’m not claiming that the strict finitism presented at that link is necessarily true, but I am asserting that the arguments for the infinite naturals are deeply problematic for many reasons (such as the paradoxes of self-reference). I am further asserting that there is considerable evidence that we not need them, and that the assumptions of infinity underlying them are at the root of many of the problems in mathematics and physics.
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A quote from one individual does not refute the statement that “most” physicists think that the universe is spatially infinite. And they do. Also, the physical space that we inhabit could be finite without it being the whole of reality, and most physicists think this is the case regardless of whether the geometry of this particular space makes it finite or infinite.
The fact that lesser estimates have been refuted is direct evidence. There is nothing bizarre about this; it is simple math. E.g. suppose someone in the past believed:
(1) There is a 10% chance the universe is less than 1 billion miles across
(2) there is a 10% chance the universe is less than 1 trillion miles across
(3) there is 79% chance the universe is finite but more than 1 trillion miles across
(4) there is a 1% chance it is infinite.
If (1) and (2) are proven false, the probabilities of both (3) and (4) go up by defined amounts. So proving 1 and 2 are false is direct evidence for 3 and 4. And it is easy to see that the more estimates you disprove, the more the last estimate (the infinite one) will go up, no matter how low it starts.
If you say that the probability of (4) is 0%, there is no point in discussing it, because by definition that means (if I take seriously someone who says that) that there is even in principle no evidence or argument that could change their mind. See https://entirelyuseless.com/2015/08/08/100/
Nominalism is false, so irrelevant to whether numbers have successors.
Finitism is also false, and in any case would not resolve most paradoxes that result from self-reference, e.g. liar paradoxes.
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> A quote from one individual does not refute the statement that “most” physicists think that the universe is spatially infinite.
First, I don’t see why the consensus fallacy should be convincing. If you suffered from constant headaches and the consensus among all doctors, including pediatricians, obstetricians and family doctors, was that it was benign, but a group of neurologists said it was a brain tumour, whose diagnosis should you believe?
Second, we know GR is incorrect, as I said. Carlo Rovelli and Lee Smolin are specialists in the subject of spacetime and quantum gravity which aims to fix GR’s problems. If you don’t take their position on spacetime seriously, then I don’t know what to tell you.
All the evidence we have suggests that the universe is homogenous and isotropic. This entails neither finite nor infinite extent without additional assumptions being applied.
> If (1) and (2) are proven false, the probabilities of both (3) and (4) go up by defined amounts. So proving 1 and 2 are false is direct evidence for 3 and 4.
First, some pedantry: this does not constitute “direct evidence” of (3) and (4). Direct evidence does not require inference, by definition. Bayes’ theorem is an inference rule. You have direct evidence that (1) and (2) are false, and thus *infer* how the conditional probabilities of (3) and (4) change as a result. That’s technically not “direct evidence” of (3) or (4).
Second, I agree that disconfirming (1) and (2) constitutes evidence for (3) *or* (4), but this is *not* the claim that you made earlier. You said, “there is direct evidence that the universe *is* spatially infinite”. In other words, you claimed that the failures of (1) and (2) was evidence for (4), not for (3) *or* (4) as you now seem to be claiming. Your earlier claim was intended to demonstrate that we have good reason that space was infinite, which it would if that claim were true. This latest argument you’ve does *not* do that, as it provides equal evidence that space is finite.
Third, a GR analysis that results in a number of possible cases one of which is “infinity” would likely assume the conclusion, ie. any such analysis using GR assumes continuous space and then concludes space is infinite. Since we already *know* that GR is wrong, such analyses are somewhat suspect. Which doesn’t necessarily entail that they’re wrong, but infinities in physics always raise suspicion.
> Nominalism is false, so irrelevant to whether numbers have successors.
Where’s the proof?
> Finitism is also false
Where’s the proof?
> and in any case [finitism] would not resolve most paradoxes that result from self-reference, e.g. liar paradoxes.
I have no idea how you have quantified “most”, but sure, every self-reference paradox doesn’t necessarily have only one possible mistake or one possible resolution.
For instance, the Liar Paradox is a problem with the formal interpretation of informal natural language, which Prior already solved, ie. every statement implicitly asserts its own truth, so “this sentence is false” is logically equivalent to “this sentence is true and this sentence is false”, which is false. Nothing to do with infinities or self-reference per se.
But, finitism will certainly resolve some of them because you have to be much more careful with your quantifiers, for instance.
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