Wittgenstein vs. Gödel: Are laws of inference superfluous and without sense?

Great Debates
1

What I propose is that we attack and/or defend Wittgenstein’s notion below and what would be the consequences for Gödel’s work on incompleteness:

5.132
W1: If p follows from q, I can make an inference from q to p, deduce p from q.
W2: The nature of the inference can be gathered only from the two propositions.
W3: They themselves are the only possible justification of the inference.
W4: ‘Laws of inference’, which are supposed to justify inferences, as in the works of Frege and Russell, have no sense, and would be superfluous [Gödel’s too?]

Can the sense of logic be gleaned from the study of propositions but not from the study of the [merely corollary] inferences? (Note: I have repeatedly observed something very much like this in numerical physics, has anyone else seen it confirmed in applied research?). If all of this is true then does Gödel’s work on incompleteness, since it seems to be concerned with the laws of inferences, therefore make no sense and is superfluous?

Those who want a quick, opinionated background of the two philosophers please read the remainder. Otherwise, just jump in.

Gödel:

In a nutshell, Gödel disconnected mathematical truth from provability. He “proved” that within formal systems that are consistent (and sufficiently complex) truths exist which can not be proved within that system. His aim, essentially inspired by his love of Platonism, was to show that formalism’s capacity for truth had more reach and power than merely what it could prove. In short he thought he’d discovered something that was strongly suggestive of mathematical realism. To his dismay, his results were quickly hijacked and appropriated by those who basically mounted an all-out assault on objectivity and rationality. The latter context is the most frequent context in which one hears Gödel’s name in serious discussion, but even more so among those who don’t really know anything about formal reasoning. There are some purist formalists who persist that his proofs use techniques which should be off-limits, but they are few, probably because Mathematicians prefer to keep the larger toolbox in the practice of their jobs, and because Gödel can be safely ignored in the using of mathematical tools (as opposed to meta-mathematics, the study of the tools themselves). To the end Gödel felt alienated and isolated, claiming that few understood the true meaning and import of his work. He largely attributed this to the insufficiency of language as a useful basis for communicating truth. One could say that his initial life dream was to build a perfect/pure and therefore common language but the result was more of a nightmare and he believed his work was in the end widely understood in almost exactly the opposite sense of how he had meant it. Whereas in his mind he had extended the domain mathematics, in most minds he had severely curtailed it.

Wittgenstein:

I can only say a few things about Wittgenstein because the more I study him, the less I can say about him without sounding…well…incoherent! His view of formal systems was highly focussed on language, the use of language, and particularly importantly the scope of language, as comprising the fundamental stuff of philosophy and mathematics. If I had to condense Wittgenstein’s work into a slogan I would probably use something like “If it’s important then we can’t talk about it”. As a meta-logician Wittgenstein was dismissive of virtually every school of philosophy virtually all of the time. Yet he was revered by many schools much of the time. The positivists of the Vienna school adopted him as their resident genius guru, and he basically galvanized Bertrand Russell (or…drop forged him?) who was fond of remarking that he didn’t understand Wittgenstein’s work but was pretty certain that it was true! Wiki:

In a letter to Bertrand Russell from 1919, Wittgenstein says of his Tractatus Logico-Philosophicus:
Now I’m afraid you haven’t really got hold of my main contention to which the whole business of logical propositions is only corollary. The main point is the theory of what can be expressed by propositions, i.e., by language (and, which comes to the same thing, what can be thought) and what cannot be expressed by propositions, but only shown; which I believe is the cardinal problem of philosophy.[57]
This corresponds to the Preface where he writes:
The whole sense of the book might be summed up in the following words: what can be said at all can be said clearly, and what we cannot talk about we must pass over in silence.
Those things that cannot be expressed in words make themselves manifest; Wittgenstein calls them the mystical (6.522). They include everything that is the traditional subject matter of philosophy, because what can be said is exhausted by the natural sciences.
4.1 Propositions represent the existence and non-existence of states of affairs.
4.11 The totality of true propositions is the whole of natural science (or the whole corpus of the natural sciences)
4.111 Philosophy is not one of the natural sciences. (The word ‘philosophy’ must mean something whose place is above or below the natural sciences, not beside them.)
So with respect to Frege’s and Russell’s efforts in logic (which is part of philosophy) Wittgenstein responds:
4.121 Propositions cannot represent logical form: it is mirrored in them. What finds its reflection in language, language cannot represent. What expresses itself in language, we cannot express by means of language. Propositions show the logical form of reality. They display it.
5.132 If p follows from q, I can make an inference from q to p, deduce p from q. The nature of the inference can be gathered only from the two propositions. They themselves are the only possible justification of the inference. ‘Laws of inference’, which are supposed to justify inferences, as in the works of Frege and Russell, have no sense, and would be superfluous.
The Vienna Circle, broadly speaking, took this to mean that only empirically verifiable sentences were meaningful, and on these grounds flatly dismissed traditional metaphysical and ethical discourse.
A letter written to Ficker makes Wittgenstein’s own understanding of the scope and goal of the TLP clear:
… I wanted to write that my work consists of two parts: of the one which is here, and of everything which I have not written. And precisely this second part is the important one. For the Ethical is delimited from within, as it were, by my book; and I’m convinced that, strictly speaking, it can ONLY be delimited in this way. In brief, I think: All of that which many are babbling today, I have defined in my book by remaining silent about it.[58]

So with this little bit of background, I would like to discuss the interplay between Gödel’s incompleteness proofs which he thought radically enlarged the scope of Mathematics as a tool for investigating objective reality, and Wittgenstein’s dismissal of Gödel’s work as a dirty trick, perhaps irrefutable if you play by it’s rules, but worthy only of being completely and readily bypassed.

2

“The Tao that can be named is not the true Tao”

I’m with Wittgenstein - Gödel’s theorems are doubtless valid & sound, but they are of dubious utility outside pure mathematics (despite Gödel’s belief), yet have often been misused in exactly that way in other philosophical arguments.

I wouldn’t go so far as to call the theorems themselves a dirty trick, but dragging them into debates willy-nilly (as I’ve seen happen) does nothing but obfuscate. They’re pretty useless in any discussion on consciousness, for instance, yet I’ve argued with people (often dualists) who try and use Gödel to “prove” that you can’t use the reasoning and investigation of the Mind itself to fully explain the workings of the Mind. All this tells me is that they didn’t understand Gödel.

3

It’s odd to me that you’ve chosen to tie that particular passage of Wittgenstein’s to the Goedel incompleteness results as though it was commenting upon them specifically; the former was written more than a decade before the latter. Indeed, I still can’t understand exactly what the connection is you are attempting to draw.

Incidentally, Wittgenstein did later make many (somewhat controversial) comments explicitly about the Goedel incompleteness results, although this was the later Wittgenstein (of what might be called the Philosophical Investigations era), who of course famously retracted and began to argue against many of the views of the earlier Tractarian Wittgenstein, and thus a rather different beast.

4

I’m not sure there’s more to it than just a matter of taste – it’s true that any ‘meta’ level discussion opens your system up to self-reflectivity and paradox, so you can do one of two things: either reject meta-reasoning and curtail the expressiveness of your system, or attempt to deal with the problems that arise if you don’t. I don’t see that either approach is inherently ‘more right’.

5

As for my own tastes, Wittgenstein always throws me for a bit of a loop: if meta-reasoning is meaningless (glossing over the details), then this means that the statement ‘meta-reasoning is meaningless’, being meta-reasoning, is itself meaningless (Wittgenstein was well aware of this; I recall, but can’t seem to find, a quotation of his in which he explains that the reader who truly understands him, will judge the Tractatus itself meaningless).

But doesn’t this conclusion, that meta-reasoning is meaningless, imply that meta-reasoning can’t be meaningless, since otherwise, we couldn’t have drawn that conclusion? This, to me, doesn’t seem any better than the paradoxes that arise through self-reference when one admits meta-level discussion in general, so I’m not sure that there’s anything won when one doesn’t.

6

“My propositions are elucidatory in this way: he who understands me finally recognizes them as senseless, when he has climbed out through them, on them, over them. (He must so to speak throw away the ladder, after he has climbed up on it.)”

Have you read Hume? He makes rather the same point about inference, but in a somewhat less self-referential way. His approach is to say we don’t have any a priori knowledge about particular instances of causation; it all comes from experience. We couldn’t know, ahead of time, that X would lead to Y; we simply observed that X happened, and then observed that Y happened, and after that two-step played out enough times we drew the inference that X leads to Y. Liquid water gets cooler and cooler and cooler until it hits a particular temperature and turns solid; for all we’d known, it would’ve just kept getting cooler, or eventually turned into a puff of smoke. A billiard ball that smacks into another stops as the other one starts moving; for all we’d known, it would’ve bounced off its still-motionless target.

But (a) we still don’t have any special insight into why Y keeps following X; if we’re honest, we merely note that Y keeps following X. In fact, (b) babies and animals can “reason” likewise; they might not flinch from a candleflame that merely looks bright, but after they’ve touched it they soon “conclude” that a painful burning sensation will follow.

What’s more: if (like babies or animals) there’s A-then-B stuff we didn’t actually know before we experienced it, and we still don’t know it after we’ve experienced it twice or twenty times or every day – well, then, it’s not logical inference that powers our conclusions; it’s not some chain of reasoning (which no baby or animal could produce); it’s a fundamentally irrational result of an unproven and unprovable belief that the future will resemble the past, that similar effects will follow similar causes.

Which brings us full circle: how can we prove that the future will keep resembling the past? How can we prove that similar effects will keep following similar causes? Well, we can’t; they have so far, but we don’t know why they do. We’ve only noted that they do.

Now, this certainly flirts with being self-referential – but Hume is, ultimately, just saying that we lack a chain of reasoning that one can produce. For all we’d known prior to experiencing it, snow would taste like salt; for all we’d known prior to experiencing it, similar effects wouldn’t keep following similar causes.

All we can point to is experience – which can’t logically justify relying on experience.

We still can’t rule out the opposite outcome, because we still don’t perceive some necessary connection between the two events; we didn’t see that causal link before the experience played out, and we still don’t see it now. All we can truly do is say, as per Wittgenstein, that such real-world inferences are gathered only from the two observations, that they themselves are the only possible justification of the inference, and that ‘laws of inference’ which are supposed to justify inferences thus have no sense.

(Hume goes on to carve out a mathematician’s exception for consistent-within-a-system factoids that are true by definition: as per Gödel, one can merely declare the properties of a circle so as to make it impossible for one to also be a square, just like there are no married bachelors or whatever.)

7

That’s the puppy, thanks.

Only in the form of secondary sources, but if I’m not misreading you, you’re talking about the invalidity of inferring universal statements from singular ones, i.e. that you can never derive ‘y always follows x’ from finitely many observations of y, in fact, being followed by x – the problem of induction, basically. And that’s very well, but are we, from our inability to formulate the rule that ‘y always follows x’ justified to conclude its nonexistence/senselessness?

Seems to me that this is just a consequence of our limited point of view; if we had, in a manner of speaking, a bird’s eye view of, well, everything, making universal statements would be just as easy as making singular ones – we could directly observe that, for instance, there are no black swans and thus, that all swans are white.

But our situation is more limited – like being trapped in a maze. From the bird’s eye view, finding the exit is trivial, as you can directly observe it and then formulate a strategy to get there – ‘left, left, right, left’, etc. The same is impossible for those inside the maze – being able only to see to the next wall, from their ‘frog’s eye’ perspective, all they can do is to try and see where the next bend will take them; but that doesn’t mean that the ‘optimal route’ out doesn’t exist. And in the end, they can find it, by trying out each path (or adopting some simple algorithm, such as ‘always take the right path, and if that leads to a dead end, backtrack to the last fork and take the next leftmost one’) – proposing the hypothesis that the fork they are taking is part of the ‘optimal route’, falsifying it whenever they hit a dead end.

The same now we can do regarding rules of inference – propose a system which contains, as a universal statement, the rule that ‘y always follows x’, and see whether or not the system works out; that is, whether or not experience falsifies it. (Here, we are justified to rely on experience, since a universal statement can be falsified by a singular one – ‘there are no black swans’ is proven wrong by the discovery of a black swan.)

Thus, it seems to me that our reasoning is logical as long as we work within a consistent system; that we can’t know whether or not that system applies universally doesn’t detract from this.

I’m not sure, by the way, if we haven’t strayed a bit from the OP’s original purpose; but then, I’m not too sure about the OP’s original purpose, anyway. Perhaps I should try to paraphrase what I think is the question under debate: “Call a system Wittgensteinian, when it lacks meta-theoretical aspects; call a system Gödelian, if it contains them. In model-building, is one ever justified to propose Gödelian systems, or are any non-Wittgensteinian systems automatically senseless?” Defero, is that more or less a fair paraphrase, or have I misread you?

8

Yeah, but I’m talking about Hume’s view on causality to come back around to Wittgenstein’s quote about p-follows-from-q inference: it’s not about knowing that each swan happens to be white (which also happens to be beyond our limited point of view, but would be possible if we could bird’s-eye it); it’s about not knowing why each swan happens to be white, which is a different problem.

You don’t know, before counting every swan, whether they’ll all be white or whether you’ll happen to find a black one. But even if you could so count every swan, you still wouldn’t understand why there weren’t any black ones, and so can’t be sure that all future swans will be white; you don’t actually know that, because experience could falsify it tomorrow.

As it happens, you also can’t so count them. But that’s a different kind of limitation; for purposes of causality or induction or whatever, even a full headcount of the present isn’t enough. To make some big fine Law of Inference à la Wittgenstein, we need the ability to make predictions that can’t be wrong; if experience could falsify it tomorrow, then we still lack a real explanation about causality.

I can tell you why you’ll never meet a married bachelor. I can’t tell you why you’ll never find a black swan.

Sure we could. And, as per what Hume wrote about how even infants and animals conclude that “burning” follows when you touch a flame, we do. And in doing so, we’re no different than infants or animals: his point is that performing such an induction therefore doesn’t mean we’ve gained knowledge about some Law of Inference – since, as you suggest, we always have to keep admitting that experience could falsify it tomorrow.

It doesn’t detract from it – but, as per Defero’s quote of Wittgenstein, simply note that the justification isn’t derived from pure logic; there aren’t any known Laws of Inference, because such propositions may well turn out to be incorrect and thus serve merely as superfluous gloss on the only possible justification: Two Events That Have Constantly Accompanied Each Other (a) Thus Far In Our Experience (b) But Might Not Do So Tomorrow.

If that’s all you can say about a p-follows-from-q inference, then you don’t really have a Law. Which brings me to this:

But what, in that framework, is Hume (or Wittgenstein) doing if he makes the following statement:

Is it meta-theoretical to state, as Hume does, that you’re not aware of any such argument? Is it meta-theoretical to state, as Wittgenstein does, that Laws of Inference don’t actually justify the p-follows-q inferences, since the nature of that inference is only gathered from observing that p follows q?

9

I think I missed your point, The Other Waldo Pepper – experiential falsification essentially proceeds via a ‘modus tollens’-deduction, which itself is impossible to be validated through experience; sort of like Carroll’s take on the whole Achilles/Tortoise deal. To that, I don’t think I’ve got an answer.

ETA: Didn’t see your post when I posted this one.

10

You could, if we just resolved to not call an animal that’s black, but otherwise looks like a swan, a swan, but rather something else. Similarly, if we changed the definition of bachelor from ‘an unmarried male’ to ‘either an unmarried male or a married male that doesn’t live with his spouse’, I may still never meet a married bachelor, but the ‘why’ would be much harder to answer. Is there actually more to this ‘why’ than that distinction of linguistic convention?

11

Well, the way Hume puts it: “All the objects of human reason or enquiry may naturally be divided into two kinds, to wit, Relations of Ideas and Matters of Fact. Of the first kind are the sciences of Geometry, Algebra, and Arithmetic; and in short, every affirmation which is either intuitively or demonstratively certain … Propositions of this kind are discoverable by the mere operation of thought, without dependence on what is anywhere existent in the universe … Matters of Fact, which are the second object of human reason, are not ascertained in the same manner; nor is our evidence of their truth, however great, of a like nature with the foregoing. The contrary of every matter of fact is still possible; because it can never imply a contradiction … That the sun will not rise tomorrow is no less intelligible a proposition and implies no more contradiction than the affirmation that it will rise … If we take in our hand any volume; of divinity or school metaphysics, for instance; let us ask, Does it contain any abstract reasoning concerning quantity or number? No. Does it contain any experimental reasoning concerning matter of fact and existence? No. Commit it then to the flames: for it can contain nothing but sophistry and illusion.”

That last bit there is the same problem rephrased from the end of my last post: is it a metaphysical comment because it references metaphysical comments when off-handedly rejecting them in favor of abstract reasoning about relations of ideas and experimental reasoning about matters of fact?

So, coming back around, you can of course make up any definitions you like and apply them consistently within a chosen framework, but you’re not thereby learning anything useful about real-world matters of fact; you’re just playing around (albeit accurately) with relations of ideas – like a geometry teacher who defines a circle as points equidistant from some center point without concluding that any such perfect circles exist in reality, or like some comic-book author who first declares that Green Lantern can use his magic ring to mimic Superman’s various powers and then reasons that GL could fire up telescopic x-ray vision easy as leap tall buildings in a single bound.

12

I’d say it’s metaphysical in the sense that it essentially is reasoning about reasoning, about what one can meaningfully reason about, etc., but that’s kinda dependent on how you want to define metaphysics – personally, I take it literally: if physics is all the thinking you can do about the physical, thinking about how to do the thinking about the physical must be metaphysics. That’s at least how I use this and other ‘meta-’ descriptors (meta-theoretical thus is every theory concerning theories, their formation, for instance; thus, whenever you say ‘theories of this-and-that kind are meaningless, only theories of such-and-such form can be meaningful’, you’re on the meta-level).

But then, that’s all you can really do anyway, isn’t it? There’s nothing (nothing as literally ‘no thing’) but ideas in your head. You don’t have an apple in your thoughts, only the thought of an apple; so whenever you think about an apple, you in fact only think about the thought of an apple.

What this thread is about, I thought, was the question of whether or not it is possible to meaningfully think about thoughts, or about thoughts of thoughts, or if you only ever can think thoughts about things and expect to make sense.

13

Well, let’s maybe push it back a step: would you draw a meaningful distinction between thinking about experience-derived thoughts and thinking about, uh, the other kind?

I mean, okay, let’s say I never actually think about an apple but only ever think about the thought of an apple. So shift my language a little. It used to be that I’d phrase it as “I don’t know whether the next apple I bite will scream in pain and begin to glow, but I do know the next triangle I see will have three sides.” And now I phrase it as “I don’t know whether the next thought-of-an-apple I perceive will appear to scream in pain and seemingly begin to glow during my next purely conceptual experience of possibly hallucinatory ‘biting’ – but I do know the next triangle I see will have three sides.”

Same difference, right? I can still be certain about the next triangle, if there’s a helpful description of “triangle” currently amidst the true-by-definition ideas I’m employing – and I still remain uncertain about the next apple (read: thought of an apple) that I experience in the real world (read: apparent world), since that involves a different kind of idea.

So, yeah, let’s say we redefine the world of mere appearances as the only world we ever actually experience. Does that change anything about Hume’s fork? Or does it merely clarify that said world of mere appearance is, for all practical intents and purposes, the real world, complete with Humean uncertainty regarding matters of fact rather than what Wittgenstein refers to as Laws of Inference – such that any split between “an apple” and “the mere thought of an apple” is, ultimately, irrelevant?

To transition from Wittgenstein and Hume to a Nietzsche quote for a moment:

14

I’m not sure whether we’re not merely talking past each other; however, I do think that I’d disagree with the quoted part: within the conceptual framework I am currently using, I am entirely justified in expecting the next apple not to scream; as much so as I am in expecting the next triangle I encounter (in my mind’s eye, or wherever) to have angles that in sum total 180°. If the apple were to surprise me by screaming, I would have to abandon or amend my conceptual framework, either by calling the screaming thing not an apple, or by allowing for apples to either taste nice and juicy or scream and glow; however, similarly, if my thoughts were to turn to non-euclidean geometry, through whatever processes direct thoughts into some particular direction (which I would argue to be a kind of experience in itself, since the discovery of some new concept within the mind is far more akin to something that happens to us as it is to something we produce) I would have to either amend my framework to include triangles with angle sums different from 180°, or evict those forms with three sides, yet ‘wrong’ angle sums from the category of ‘triangle’, inventing a new one.

True, if I declare a triangle to be a form of three sides, it is impossible for me to encounter, in my thoughts, a triangle with any other number of sides; but similarly, if I declare an apple to be something incapable of screaming, it is just as impossible for me to encounter a screaming apple, as by virtue of its screaming it would show itself to be something different.

15

Well, yeah. But playing with definitions in that way tells us nothing about the next apple-esque thing you encounter; you can, upon preparing to bite into one, declare that apples don’t scream – but remain exactly as uncertain as to whether this item right here (which, to all appearances, thus far seems to be an apple) is about to scream and glow.

Which comes back around to what the OP said. “In a nutshell, Gödel disconnected mathematical truth from provability”, he said. * “I would like to discuss the interplay between Gödel’s incompleteness proofs which he thought radically enlarged the scope of Mathematics as a tool for investigating objective reality, and Wittgenstein’s dismissal of Gödel’s work as a dirty trick, perhaps irrefutable if you play by it’s rules, but worthy only of being completely and readily bypassed”*, he said.

As I read it, any such disconnect leaves the situation exactly as it had been: you can still only make correct-within-a-system conclusions when guaranteedly justifying inferences from definitional truths we merely declare, and you still lack any such guarantee when using p-following-from-q as the only possible justification of real-world inference.

Yes, we can declare some other set of definitional truths at will – for geometry or apples. But, as per the OP, so what? IMHO, that’s “worthy only of being completely and readily bypassed.”

16

So many fantastic points made so far.

Bringing Hume’s skepticism about causation is a great illustration of something akin to Wittgenstein’s skepticism about inference in general.

Indistinguishable said:

It was probably a lot more than 10 years later, and I in no way meant to imply that W. was addressing extant work by G, very sorry if I gave that impression. The connection is that W. argues that ONLY the propositions are meaningful, and even the inferences then ONLY draw meaning from the propositions. Yet Gödel generalizes across all inference irrespective of whatever propositions are their object. Is it then just all nonsense? (I think it may be). Now, where is Gödel really super useful? In arguing against formalists!!

I really don’t like meta-systemic arguments like Gödel’s. For instance the grade-school paradox:
G1. God is an all powerful being.
G2. There is nothing an all powerful being cannot do.
G3. God cannot create an action that he cannot perform.
So there is something God cannot do, so he cannot be all powerful, goes the tautology. But the meaning or sense is perverted. A meta-logic solution would be to say “if God is all powerful then he can alter the laws of inference so that this is not a contradiction.” I you have to admit that defense if you admit Gödel’s incompleteness theorem. It’s similar to Russel’s paradox where he finds problems with the “set of all sets” not being able to be self-containing. It’s really a bit nonsensical. I don’t regard it as a paradox, just an ill-posed question. Wittgenstein’s bypass of Gödel theorem is very elegant but it’s not formal:

So Wittgenstein (I think!) is arguing that you can’t just up and change the sense of something in mid-reasoning any more that you can change the rules of chess in mid-play and then refer back to the original rules. I think he is just calling BS. But on no account does he find Gödel’s work uninteresting. He returns to it often, mentions it often, later on. It may be one of his disposable ladders.

I’ll go out on a limb and apply that (I think!) to Russell’s paradox, which I think may be a bogus paradox. He first defines the set of all sets. Implicit in this assumption is that it’s the biggest set in existence. Then he goes on to say that the set, being a set, must be included WITHIN the set of all sets, and so some larger and different set is newly inferred. Which is ridiculous. I think that the only sense I can make out of it is that you can’t have a set of all sets. And in this case Gödel gives is a tools to PROVE that. Gödel theorem, in my mind, can be read “you can’t ever postulate a system of all systems”. There would have to be a “Gödel system” in there which could never be inferred from any set of systems, yet it would be true were it discovered. Anyone thing I’ve messed that up? I might have! Gödel’s theorem may be true, but I just can’t see how it is real, in any sense. He uses PROPERTIES of a system to argue about MEMBERS of a system. I’m definitely open to having it explained to me how this is OK, but I don’t understand it at this point. I think it’s a lot like saying “Let us assume a system with no rules. Let us see what this tells us about the rules.”

To me, the most commendable thing about Logic, Math, etc. is that it knows when to gracefully bow out. When your calculator displays “E” this is very honest! You won’t see that in most ideologies! I think scope is paramount. I think arguments like Gödel’s should be read as “E”…as in don’t go there! But–and I love Gödel’s work for this reason–if a formalist DOES go there, you can derail them with Gödel a lot of the time.

I’d love to write more but must run…is there anything anyone want to argue over, lol! It seems we may have a shortage of Gödel enthusiasts! I would also love to see how far people can push W.'s work into the “mystical” as he terms it. Also, can someone find his “famous recanting” that was referred to by Indistinguishable? I haven’t read that. Wouldn’t it be funny if W. has already defeated all the points I’m making along the lines of his own work!

17

Except on thing:

Right! But if you don’t define apples as non-screaming, then you are probably better off. If you poke a small hole in an apple and bake it at super high temperature it will scream. A little hotter yet and it will glow after a while. Formalism misses all of these possibilities. In general formalism is very dangerous to apply to the real world. I’ll be back!

18

Well, I remain uncertain of the colour of the apple in my hands until I open my eyes, or even of the fact of whether it’s an apple or an unusually apple-shaped pear until I bite into it; that I have to find out things before I can know them isn’t really that shocking a conclusion – in fact, knowing anything without having experienced it – knowing the colour of the apple without having looked at it, or knowing whether or not it would scream without having bitten into it, if that is the only way to find out – would strike me as far stranger than not knowing.

But basically, I think I agree with everything you said; but what else, other than building models, proposing them with the reservation of their eventual falsification, and holding the inference leading to their falsification to be valid would you have us do?

19

Nothing else. Wasn’t that the point of the Wittgenstein quote in the OP?

So, yeah: don’t do anything else. For one thing, don’t postulate superfluous Laws Of Inference to attempt this or that justification; the fact that p follows from q is all you’ve really got, such that p and q themselves supply the only possible justification of that inference.

Before the experience, you didn’t actually know how it would go down. After the experience, you still don’t actually know whether it’ll do that again next time. Such knowledge would likewise strike me as being far stranger than not knowing; you perform that doglike feat of induction not by recourse to the Laws Of Inference, but just via inferring that the future will resemble the past – which is exactly the sort of knowledge we just ruled out due to unutterable strangeness. And, again, “the inference is not intuitive, neither is it demonstrative. Of what nature is it, then? To say it is experimental, is begging the question. For all inferences from experience suppose, as their foundation, that the future will resemble the past, and that similar powers will be conjoined with similar sensible qualities. If there be any suspicion that the course of nature may change, and that the past may be no rule for the future. all experience becomes useless, and can give rise to no inference or conclusion. It is impossible, therefore, that any arguments from experience can prove this resemblance of the past to the future, since all these arguments are founded on the supposition of that resemblance.”

20

My concern is that you seem to be needlessly and distractingly focusing on Goedel’s incompleteness results when it would appear that much of what you would like to say would apply equally as well (or not as well) to nearly the entire field of mathematical logic. If indeed this is the case, I would prefer to have the discussion at that level of generality, at which I think I could better understand exactly what position is being proposed and what the consequences of it would be, rather than to focus on one particular theorem within mathematical logic with whose details most are rather less immediately familiar anyway.