Because of the First Incompleteness Theorem we know that if S is consistent then G is unprovable in S. Since "G is unprovable in S" is our function G (see above) we can shortcut: If S is consistent then G. Now, we just formulate this statement in S and we know it's provable in S. Now, we assume we could also prove in S that S is consistent. Then by mp G would follow (and thus be proven) in S which is impossible due to the First Incompleteness Theorem. Because of this contradiction our assumption must have been false. — Pippen

Bold emphasis mine.

The bold part is the difficult (or, at least, tedious) part. As I mentioned, we need to show that T proves that Con(T) -> G, which is basically formalizing the first theorem inside T. That's not a "shortcut", since it takes pages and pages of theorems (Gödel himself declined to produce a full proof of it, in part because it was so tedious; the first complete proof was written by Bernays and published in Hilbert & Bernays's Grundlagen)!

1) If a system is complete (as in every possible truth can be proved) then it's necessarily inconsistent (contradictions arise)

2) If a system is consistent (as in there are no contradictions) then it is neccesarily incomplete (some truths can't be proved) — TheMadFool

No, that's not exactly right. The first theorem states that, if T is a theory strong enough to capture all the primitive recursive functions, then, if T is consistent, then T is incomplete. The italicized condition is necessary, since we have examples of arithmetical theories which do not satisfy it and are complete (for instance, the theory of addition or the theory of multiplication in the natural numbers). Going a bit deeper, the reason for this requirement is that the first theorem works because we can construct a predicate inside the theory which captures the provability relation, i.e. the relation prf(x, y) which holds between x and y iff x is a proof of y. This relation, in turn, is primitive recursive, so we need to capture at least as much as the primitive recursive functions in order to prove the first theorem.

The second theorem says that, if T is consistent, strong enough to capture all the primitive recursive functions and some other conditions, then T cannot prove its own consistency. Roughly, the proof of the second theorem works as follows: we start by noting that an inconsistent theory (trivially) proves everything. Hence, by contraposition, if a theory cannot prove everything, then it is consistent (which is desirable: we don't want a theory which proves contradictions!). But this implies that, if the Gödel sentence is true, then the theory is consistent, since the Gödel sentence states that there is a statement which the theory does not prove. More importantly, by showing that the prf(x, y) predicate above satisfies certain conditions, we can actually prove that T proves that G -> Con(T), where Con(T) is a statement which encodes the consistency of the theory (say, it is the statement that T does not prove that 0=1). Using some other conditions (known as the derivability conditions), we can also formalize the first incompleteness theorem inside T, that is, T proves that Con(T) -> G. So we have that T proves that G <-> Con(T). Since G is unprovable, it follows that Con(T) is unprovable.

It is essential that the reasoning above is formalizable in the language of the theory and that T has the necessary resources to prove the various claims being made. That is because consistency is a syntactic property: it says that T does not prove a contradiction. Thus, in order to prove claims about a theories consistency, we have to show that it proves or does not prove something. In other words, we need to actually produce arguments as to why certain proofs are or are not possible starting from T's axioms. That's why it is not sufficient to give mere semantic arguments outside of T: consistency is a stronger property than soundness.

Notice that this does not say that complete theories are inconsistent, since there may be complete theories which do not satisfy the conditions of Gödel's theorems. In fact, that is precisely the case: again, the theory of addition in the natural numbers is complete. Also, note that "complete" does not mean that every truth is provable: it means that every true sentence in the language of the theory is provable.

How would you explain the Second Therorem based on my version with S, G(G is unprovable in S) and my explanation of the Frist Theorem? Maybe that is the best way to show me what you mean, because I will see the difference in my and your version. — Pippen

I already did it, in my last post. What part did you find troubling?

Thank you for taking the time to write that fascinating reply. I see that the a priori / a posteriori distinction is based on the justification of the terms now, - it is an epistemological, not ontological distinction. — Hallucinogen

I'm glad you found my reply useful.

I can't see how natural scientific statements or mathematical statements fall exlusively into synthetic or analytic statements. In order to make scientific explanations, causation has to be involved, ways of referring to time and space, or movement. An example would be "the sun is warming this rock". I don't think synthetic statements are able to do this, they're purely associative right? All they can do is refer to how sets of objects overlap. — Hallucinogen

Well, the whole point of the first Critique is to argue against this idea, that is, to argue that there re synthetic claims which are not grounded merely on empirical association. Kant's argument here (which comprises the entire Transcendental Analytic) is notoriously complex (I myself don't fully understand it---I don't know if anybody does), but the gist of it is that our consciousness of ourselves as abiding (i.e. our consciousness of our own identity throughout time) requires certain conditions which allow us to distinguish between (to use Strawson's turn of phrase) the subjective route of our experiences and the objective world through which it is a route. This distinction in its turn is grounded on certain principles which allows us to distinguish our subjective spatio-temporal order and the objective order of the world (for instance, to use Kant's own example, when I successively experience the different aspects of a house, I hold these to be successive apprehensions of a single object which does not change, whereas if I successively experience the different aspects of a boat going downstream, I hold these to be successive apprehensions of an object in the midst of change, so to speak).

Of course, that does not guarantee that when I judge that something belongs to the objective spatio-temporal order it does in fact belong to that order (unity is never given for Kant, but always produced). Nevertheless, the mere fact that there are such principles which allow me to make this distinction opens up the possibility that some judgments are grounded in these principles themselves, instead of being grounded in my experience of something as this or that. Mathematical judgments, for example, are grounded for Kant in the way we apprehend things as being conceptually identical yet still distinct. And some scientific judgments (Kant thought of his own rather Newtonian Metaphysics of Nature) are grounded in way an objective spatio-temporal order is structured in terms of a community of substances reciprocally acting one upon the other. So for Kant there is a middle term being a judgment being grounded purely in terms of conceptual containment (analytic judgments) and a judgment being grounded merely on empirical association (synthetic a posteriori judgments): some judgments are grounded in principles that make an objective spatio-temporal order possible in the first place (synthetic a priori judgments).

That would be Kant's reply, anyway. For better or worse, it is almost universally rejected today, in part because the supposed principles identified by Kant turned out not to be so necessary---Newtonian physics, for instance, was famously displaced by relativity and quantum mechanics. Some (e.g. Michael Friedman) have attempted to salvage something of the Kantian program by blending it with some variety of positivism: the task of the philosopher would be to identify the conceptual structure that underlies our best scientific theories. Myself, I personally think that the increased power in our logic has given us a much better picture of mathematics, which allows us to defend a variety of (structural) platonism: mathematics describes certain structural features of reality, to which we have epistemic access via proof. On the other hand, I think scientific theories (such as relativity) are not in the business of aiming at truth, but merely of empirical adequacy, of providing nice (generally mathematical) models which save the phenomena, so to speak. These are not merely associations because they turn on certain mathematical or structural features of reality.

5. Since 1.-2. seem true, 3. must be false, and so it follows: LH <-> RH, but that's absurd because it basically says that my left hand can only exist with my right hand and vice versa which is obviously wrong. — Pippen

Bold emphasis mine.

Yes, that is obviously absurd, but it's not what is going on here. The bold part is employing modal reasoning (can only exist), which is not expressible in propositional logic. In the case at hand (sorry!), you are only entitled to conclude that, given that LH and RH are both true, then they have the same truth-value, which is obvious (they both have the value true). Of course, from this you are not entitled to [](LH <-> RH), which would be an adequate formalization of the bold part (with "[]" denoting "It is necessary that...").

Kripke' view that there is a difference between name and descriptions, which challenges the idea that a name that can be replaced by a description and be 'bound by a variable' so without a remainder. But if names do refer to objects, the real-world actual existence of F does not make sense of non-existing objects, such as "the king of France is bald". — TimeLine

But this has nothing to do with the original suggestion, which involves just names and not descriptions. It also does not mention names being "bound by a variable" (whatever that means). Here's the idea. Suppose I have a language, L, and an intended model, say M. Then, instead of saying that a formula such as "exists x such that Fx" being true when there is an object from M which satisfies Fx, we can instead introduce a name for each object in M and say that the formula is true iff there is a name a such that Fa is true. This is called the "method of diagrams" and is one way to avoid talking about satisfaction. Kripke himself proved that this idea can be deployed successfully to entirely avoid objectual quantifiers (see his paper "Is there a problem with substitutional quantification?". where he shows that, given a language L for which truth has already been defined, we can extend L by introducing substitutional quantifiers and this extension is well-defined).

EDIT: Of course, the mere fact that we can define substitutional quantifiers in terms of objectual quantifiers and vice-versa already shows that a facile reading of Quine's maxim is at least problematic...

If truth in arithmetic means provability in ZFC then it is false that every PA formula is either true or false. Thats odd.

What kind of "truth" concept is used in the Gödel and Tarski theorems? — Meta

The first sentence is incorrect. ZFC can formulate a truth predicate for PA in such a way that a formula from PA is true iff ZFC proves that the natural numbers satisfy it. We can also prove that ZFC proves that for this predicate, a formula from PA is either true or false. This does not violate either Gödel's or Tarski's theorems because ZFC is strictly stronger than PA (in fact, it is much stronger), whereas the theorem only applies for theories weaker than PA (including PA itself).

The truth concepts used by Gödel and Tarski in their theorems is the usual model-theoretic one, i.e. a formula from PA is true iff it satisfied in the standard natural numbers.

One problem with construing the quantifiers substitutionally is that you will need a denumerable language if you're working with the natural numbers, and a uncountable language if you're working with the reals. But this seems awkward if you're trying to avoid existence assumptions.

That is not to say that one can't sidestep the problem from the op. One can work inside primitive recursive arithmetic as a metatheory and define ZFC inside it. Using ZFC, one can then formulate a truth-theory for the natural numbers in terms of provability inside ZFC. So everything is (ultimately) syntactic, a kind of formalism, if you will.

What you do in 3. is using the AND-introduction. My question is if I could instead introduce an implication "A -> ~A". I doubt that. I doubt that you can just with two premises P1 and P2 follow P1 -> P2 and vice versa. — Pippen

Actually, that's pretty straightforward:

1. premise: A
2. premise: ~A
| 3. Assumption: A (assumption for ->-intro)
| 4. ~A (p2)
5. A -> ~A (->-intro).

I used | to indicate a sub-derivation.

But this can't be true since it leads to contradictions. — Pippen

Actually, you can derive contradictions in a natural deduction system if you have inconsistent premises. That just shows that you have an inconsistent set of premises, not that the system is unsound. Here's a basic example:

1. premise: A
2. premise: ~A.
3. A & ~A (1, 2, &-intro)

Then why don't I feel ecstatic about someone gifting me $0? — TheMadFool

Suppose I gave you $1. Does that mean I also thereby gave you the number 1?

I thought not having a solution to a mathematical problem is, well, a problem itself. For instance, before zero became a number 2 - 2 had no solution. Zero was invented and now 2 - 2 = 0. Fine. However, 4 ÷ 0 has no solution. So, doesn't this take the punch out of zero's use. It solved some problems but created new ones. — TheMadFool

But (using my previous notation) d(4, 0) is not a "problem", it is a functional term, so it requires no "solution". Suppose I define a function f on the natural numbers as such: f(x) = 1 if x is an even number, otherwise it is undefined. There's no "problem" here about f(1) which would require some kind of "solution".

Also, zero is nothing. And, mathematically, there's no solution to 4 ÷ 0. Put differently, the solution to 4 ÷ 0 is nothing. But nothing in mathematics is, well, zero. So, I shouldn't be completely off the mark in saying 4 ÷ 0 = 0. — TheMadFool

But 0 is not <nothing>, it is something, namely a number, and moreover the identity element in the additive group of integers, just as the identity function f(x)=x is the identity element in the symmetric group of (say) a given set S. Just as the identity function f(x)=x over a given set S is a distinct mathematical object, so is 0.

Goedel's real accomplishment was to formulate the function G in a system that just contains propositional and predicate logic and the natural numbers with addition. — Pippen

The natural with addition and multiplication. The theory of the natural numbers with addition (also known as presburger arithmetic) is actually complete.

Let S be consistent (assumption). Because of the first incompleteness theorem it follows that if S is consistent, then G is unprovable. From this it follows by mp (and is thus proved) that G in S is unprovable. But this is precisely the content of G (see above, the italic style marked one), so that G in S would be proved, which is impossible according to the first incompleteness theorem (there case 1a), so that the consistency assumption must be false. — Pippen

Bold emphasis mine.

The question is: proved where? Your argument proceeds entirely in the metalanguage, so it doesn't suffice to generate the contradiction, because we known (outside S) that G is not provable. What you require is a proof inside S that the consistent statement Con(S) is equivalent to G, whence it follows that Con(S) is also not provable.

I don't see the problem. Let d(x, y) be the result of dividing x by y, i.e. d(x, y) is the unique z such that y*z=x.. Then it's true that d(x, y) is undefined when y=0, as for any x not equal to 0, there is no z such that 0*z=x. But so what? How is this a problem?

The usual arguments for the even parity of 0 are facile, self-serving, and question-begging. There are certain mathematical contexts where it is convenient to assume that 0 has even parity, but it does not follow that 0 MUST have even parity. If it does, it must be proved from set theory, or from the axioms of arithmetic, or better still both. — alan1000

I don't see why they are facile. Given a language L = {+, *, 0, 1} and the normal axioms for the natural numbers, we can define am unary predicate E(x) as (I'll write in prose in order to avoid using LaTeX):

E(x) iff there is a y such that 2*y=x.

It's an easy theorem of number theory that, for every x, 0x=0. So, as previously remarked, 2*0=0, whence (by existential generalization), there is a y such that 2*y=0, so, by definition, E(0).

You seem to be confused by the idea of division. Division (or any other mathematical operation, for the matter) should not be understood too literally, i.e. as a way of breaking down an entity into constituent parts, in such a way that every even number could be somehow broken down into two halves. Rather, as has already been pointed out, one should instead define a relation "x divides y" as holding between two numbers iff there is a (natural) number n such that n*x = y. If you want to define this inside ZFC, you will need first to define what "*" means; this is easily done (for the finite cardinals) as "|A|*|B| = |AxB|", where "AxB" is the cartesian product of the two sets (i.e. the set {<x,y> : x belongs to A and y belongs to B}). Our definition of division can now be established for the finite cardinals, which gives the result that 0 is even (given the definitions). Notice that, given this definition, since the cartesian product of any set with the empty set is also empty, our previous theorem (for every natural number n, n*0 = 0) is now just a special case of a more general theorem, according to which, for every cardinal k, k*0=0.

Finally, notice that it's not entirely uncontroversial that the natural number 0 just is the empty set. We can definitely say that the empty set models the natural number 0 inside ZFC, but it's not entirely clear if it is 0. Some would prefer to identify 0 with a sui generis abstract object (say, by employing Fregean abstraction---cf. neo-Fregean approaches to arithmetic) or with an isomorphism type or a role in a structure, or (...).

A couple of comments:

(a) When reading Kant, it is often useful to take a look at his historical predecessors in order to understand how some of his distinctions are actually tactical maneuvers deployed against positions which he rejected (two books that, incidentally, show the power of such an account for Kant is the classic Kant and the Capacity to Judge, by Longuenesse, and, more relevant to this thread, The poverty of conceptual truth, by R. Lanier Anderson, which is entirely devoted to an elucidation of Kant's distinction between analytic/synthetic). Case in point, Kant considered his analytic/synthetic distinction as a weapon against Wolffian metaphysics. Very roughly, this is the idea:

Wolff and his followers apparently thought that every concept could be positioned in a logical hierarchy, in such a way that immediately above it would be its genus and right below it its species. Thus, we could picture this hierarchy as an upside-down tree, with the most general concept at its root (say, the concept of <object in general>), infinitely branching downward in subdivisions that would systematize the rational order of the world. Importantly, the branches below a given node of this tree had two main characteristics: (i) exhaustiveness, that is, the branches below a node collectively exhaust its species and (ii) exclusiveness, that is, there are no intersections in the path below the branches. This gives us a neat picture of conceptual containment: a concept A is contained in concept B iff A is a node in the path above B.

Wolff wanted to put this picture to use in order to establish a metaphysical system which would mirror the rational structure of the world. The philosopher's task, according to Wolff, was essentially to distill each concept in its component parts (an activity he called "analysis") in order to locate in this hierarchy. This would allow one to ground every (conceptual) truth in this hierarchy. Remarkably, since Wolff actually thought that even our empirical judgments were reducible to conceptual ones, this would mean that every truth would be ultimately grounded in this logical hierarchy. One can thus understand (i) why he gave pride of place to categorical syllogistic inference (because such inferences made explicit precisely relations of conceptual containment, as Arnaul and Nicole had long argued in their logic) and (ii) why he considered the principle of non-contradiction to be a sufficient (!) ground for every truth, including empirical ones (because conceptual containment claims are settled by appeals to the pnc).

Of course, appealing as it is in its simplicity and elegance, Wolff's system is woefully inadequate for the task of describing the structure of the world. In particular, conceptual containment is too crude an instrument to capture even mundane truths such as "The sun is warming this stone" or, to give a more interesting example, "7+5=12". That is the point of Kant's analytic/synthetic distinction: Kant grants to Wolff that conceptual containment does capture some interesting class of truths (namely, the analytic ones); but that class is too meager. In particular, he claims, the B Introduction to the Critique, that (i) Judgments of experience are synthetic; (ii) Mathematical judgments are synthetic (iii) Natural science's judgments are synthetic; (iv) Metaphysical judgments are synthetic. As Anderson makes clear, these four theses are essentially a blunt polemic against the Wolffian paradigm: it is essentially saying that every interesting judgment is outside its scope (hence the title of Anderson's book: conceptual containment has scarce resources to express anything of interest).

Notice that, according to Anderson's analysis, even some logical judgments are synthetic. Kant admitted as distinctive types of inference both hypothetical and disjunctive. But these inferences are grounded in relations between judgments, not concepts, and so a fortiori can't be grounded in conceptual containment relations! So it may be that, perhaps unwittingly, Kant excluded from the class of analytic judgments a whole swath of logical judgments (that this probably wouldn't bother Kant may be gleaned from the fact that, when he announces the pnc as the supreme principle of analytic judgments, he merely says that every analytic judgment is grounded in the pnc, not that every judgment grounded in the pnc is analytic).

In summary, you are right that analytic judgments are a rather impoverished class of judgments. But it was precisely in order to show this that Kant introduced the distinction in the first place.

(b) Note that this way of carving out the distinction between analytic/synthetic is completely different from the current way of appealing to the meaning of the terms involved. For a contemporary treatment of the distinction, you should read David Lewis's outstanding work, especially Conventions andLanguage and Languages.

(c) Finally, regarding the a priori/a posterior distinction, note that this is an epistemic distinction, not a psychological one. In other words, it is a distinction between the grounds of justification for a given proposition, not a distinction about the particular sources for the proposition in question. As Kant himself remarks, "There is no doubt that all our cognition begins with experience (...). But although all our cognition commences with experience, yet it does not on that account all arise from experience". So even though experience may be a necessary condition for me to acquire certain concepts, and hence for me to even formulate certain propositions, that does not mean that experience is a necessary condition for me to justify the propositions in question.

Take, for instance, Fermat's Last Theorem (FLT). Clearly Andrew Wiles needed to go through certain experiences in order even to formulate FLT (in particular, he had to read a certain mathematical textbook when he was a child which explained it to him). But these experiences do not need to be mentioned, and in fact are not mentioned, in his proof of FLT. Only axioms, mathematical definitions, etc., are used in this proof. So it is an a priori, not an a posteriori truth.

Well, I personally think (1) is clearly wrong; there need be no "magical thinking" or "social constructivism" involved in the thought that unfair portrayals can be harmful.

I'd agree with that, but I don't agree that libel is sufficient for harm, either.

On the other hand, "harm" is ambiguous, so we'd need to define it better. — Terrapin Station

I'm not saying that (alleged) libel is sufficient for harm, either. What I am saying is that when it constitutes harm, then there's no magic involved.

As for the ambiguity of harm, I don't think there's any need for a definition, if we can agree that at least some cases of libel are harmful.

I agree that it's almost certainly a Porphyrian Aristotle in the background here, but in truth, I don't think I did justice to Deleuze's reading in the OP. In reality, the engagement with Aristotle in D&R takes place almost exclusively with respect to Aristotle's impositions upon difference. If anything, what is 'selected' for is not where individuals fall under in terms of genera and species (as I put it in the OP), but the kind of difference which is given legitimacy in Aristotle. Aristotle 'selects for' specific difference, while ruling out, as ontologically illegitimate as it were, generic difference - hence the turn to equivocal/analogical Being. — StreetlightX

As I said, I don't think this does justice to the complexity of Aristotle's metaphysics, for a couple of reasons. First, I'm not even sure it makes much sense to talk about specific or generic difference for Aristotle---as mentioned, this seems to be a Porphyrian element extrinsic to the way Aristotle himself thought. What matters is the form of life of an organism, not whether some "specific difference" occurs. Second, this has as an important consequence that Aristotle doesn't "select" for "specific difference" in opposition to "generic difference", since those terms simply don't apply. So, third, that is definitely not the underlying reason for "the turn to equivocal Being" (pace Deleuze's convoluted argument). There is a lot that could be said in regards to this last question, but here I'd note only that (i) the Greek term for "to be" is already equivocal, as can be gathered from Charles Kahn's remarkable studies to this effect (The Verb "Be" in Ancient Greek) and (ii) this linguistic data buttresses, along with Aristotle's logical analysis of the way assertion works and his biological investigations into the diverse forms of life of various organisms, his idea that being is equivocal.

So basically, I think Deleuze may have been to quick in his dismissal of Aristotle, and there may be some problems for his argument for univocity in the first chapter of DR because of this. But I may have already strayed too far from the OP, so I won't pursue this line here.

As I said to Moliere earlier in the thread, the associations of language here might lead us astray, because despite it's 'voluntarist' tenor, 'selection' is anything but voluntary in Delezue, and selection is always the result of an 'encounter' with or 'interference of' a 'question-problem complex' which forces one to creatively engage and fabulate responses as a result (the quoted phrases are Deleuze's). The kind of 'phenomenology' - if we may call it that - of Lewis being 'gripped' by the necessity of imposing the sorts of divisions he does is very much in keeping with the Deleuzian conception of philosophy as involving a 'pedagogy of the concept', where creation - or in this case selection - is very much a matter of imposition, of 'subjective dissolution', if we may put it that way. — StreetlightX

That sounds reasonable.

Libel is a legal term, recall, not a constitution of harm. Courts of law investigate whether a case of alleged libel is unlawful. You don't get to determine that libel would constitute harm. — jkop

But when they do determine that a case of alleged libel is unlawful, they presumably are not engaging in magical thinking.

Some of us aren't in favor of libel laws, by the way. — Terrapin Station

One does not need to be in favor of libel laws in order to recognize that it (libel) constitutes harm.

Why do you rephrase what is open to read? I've said none of those things. Your argument is clearly unsound, and the above is an informal fallacy (loaded question). — jkop

I'm trying to understand your position here. I'll be direct: why do you think that the fact that libel constitutes harm does not require magical thinking, whereas you suggest that to think that some portrayals of women (the objectifying ones) also constitute harm does require magical thinking?

Finally, in Aristotle, it is a matter of 'selecting' what falls under a particular genus and a particular species: Being is 'distributed' according to what categories they fall under, and it is a matter of selecting between what falls where. — StreetlightX

I know this is not the topic of the thread, but I don't agree with this characterization of Aristotle (I think he's a much better reader of Plato). I don't think he was much concerned with "genus" and "species" (these are latter terms), this concern being much more the product of Porphyry (who was a neo-Platonist). Rather, I'd say that he was much more concerned with what is essential and what is inessential; in the case of living organisms, this distinction is rooted in the form of life of that particular organism (so he's very distant from current taxonomical paradigms, which focus more on anatomical features; for Aristotle, anatomical features are something to be explained, not what does the explaining). In other words, essential features of an organism are those which actively contribute to the organism's way of living, whereas inessential features are those which are a mere byproduct of the essential features.

This is not just nitpicking, because Deleuze's argument against Aristotle (in the first chapter of DR) depends on his Porphyrian reading of Aristotle (and here I think he may be operating under the influence of Le Blond), and I'm not sure if it can be patched once Aristotle's subtler points are in view.

Of course, that doesn't detract from your general point, namely that metaphysics is concerned with selection. I think that's an interesting thought, specially since you presumably select something for some purpose, and one could ask what purpose this is (I think Deleuze's reading of Plato is especially nice in this regard). But I'm not sure if I agree. Generally, one would say that one doesn't select one's metaphysical picture, but rather that that picture is somewhat forced upon one. Take, for instance, Lewis's argument for natural properties. Lewis's claims that his division of properties into natural and non-natural properties was almost forced upon him because they did much needed work in a variety of areas (he lists "duplication, supervenience, and divergent worlds; a minimal form of materialism; laws and causation; and the content of language and thought"). So I wonder if it's really about a selection.

It is open to read in my post (e.g. "Granted that some.. portrayals are unfair or misleading...") that here I'm not primarily concerned with the right or wrong of portrayals but the relation in the assumption that one could be diminished or objectified by them. In social constructionism, for instance, it is assumed (incorrectly) that our reality would be constructed by they ways we portray it.

You omit what is said in my post, and instead misuse one of it sentences in a related but different context, libel, which concerns the right and wrong of portrayals. The shift of context makes the sentence appear ironic or irrational, which seems to be your primary concern. But your argument isn't sound, just vengeful sophistry disguised as "logic". — jkop

My primary concern is to point out that your claim is dubious. You asserted that magical thinking is a necessary condition for believing that unfair portrayals could be harmful. I countered that this claim entails the dubious conclusion that libel laws only make sense under the assumption that magical thinking is correct. Notice that libel laws are not simply concerned with the correctness of a given portrayal, but whether the portrayal itself constitutes harm. So my question is: in what sense is the claim that libel is itself harmful different from the claim that some portrayals of women also constitute harm? There must be some difference, if the latter, but not the former, entails magical thinking. But you haven't spelled it out.

I don't deny the validity of your conclusion, but it ain't sound. It is selective and misleading, because my statement, which is selectively used in your argument, is not directed at those who find libel unfair but at those who believe that an unfair portrayal could somehow objectify or diminish what it portrays. It takes magical thinking, social constructionism, or the like, to believe that a mere utterance or depiction could diminish or objectify what it portrays. But one does not have to be a social constructionist to find portrayals unfair or draft libel laws against them. — jkop

I don't see how I'm using your statement "selectively" in my argument (if anything, it seems that you're the one who's using it "selectively" here). You said that it takes magical thinking or "social constructivism" in order to "to believe that a mere utterance or depiction could diminish or objectify what it portrays". But then you go on to say that one does not need that in order to find "portrayals unfair or draft libel laws against them". Now, presumably, the purpose of libel laws is to redress harm. In particular, it redress the harm constituted by an unfair portrayal. So why are feminists selectively singled out as requiring magical thinking when they point out that some portrayals of women constitute harm?

No "postmodern irony" (?), just the logical conclusion of an argument using your premise:

(1) Only "social constructionists" believe that there could be harm from how someone is portrayed in public;

(2) Those who drafted libel laws clearly thought that some harm originated from how someone is portrayed in public;

(C) Therefore, those who drafted libel laws are "social constructionists".

The argument is valid. If you deny the conclusion, you must deny either (1) or (2). I think (2) is obvious. What about you? Do you accept (C), or do you reject (1), (2), or both?

Only social constructionists, or the like, would believe such nonsense; because for them there is no truth beyond our public interaction with words or pictures. As if injustice against women would be caused by how they're portrayed in public. — jkop

Indeed. That's why those people who drafted libel laws are all "social constructionists".

Thanks for the warm welcome!



I don't have the details, but if I'm not mistaken he did own a lot to Frege, as it's clear for anyone who reads the TLP. And I think his Cambridge years helped him immensely to better shape his ideas; to have as conversation partners people like Anscombe, von Wright, Ramsey, and Kreisel was probably essential for the development of the late Wittgenstein. In fact, had he had more formal training in mathematics, for instance, would probably have made him a much better philosopher (his Remarks on Mathematics being infamously weak).

The point is that although Wittgenstein managed to write a substantial treatise without much formal training (though he did have some), his own philosophical outlook vastly improved after he found himself in a more academic setting, in no small part because he was in constant contact with a lot of other brilliant philosophers. So we don't know if lack of formal training was an asset or a hindrance, though we do know that in some cases (mathematics) it was actually a hindrance.

You seem to be on some kind of tirade against classical logic. But this has nothing to do with what I asserted, namely that Gödel's theorems don't assume classical logic and that classical mathematics is not a subset of intuitionistic mathematics (where mathematics A is a subset of mathematics B iff all theorems provable in A are also provable in B). I have no intention of defending classical logic here, nor, for the matter, did Gödel in his incompleteness paper.

Unfortunately, I can't understand how your reply has any bearing on what I said...

Plato, Socrates, many other ancient philosophers, and Wittgenstein..none of them received any formal training. — anonymous66

That's not quite true. Socrates apparently read other philosophers such as Anaxagoras, Gorgias, Parmenides, etc., and Plato, aside from reading those, was also schooled by Socrates himself! That's some pretty good training. Plato himself considered it so good that he went to require that every philosopher should go through it (this can be seen, theoretically, in his discussions of education in, e.g., The Republic, and in practice in the fact that he established a formal school for such training, namely the Academy). As for Wittgenstein, he apparently was tutored by Russell and Frege. Moreover, his time at Cambridge was instrumental for the development of his views, not the least through his contact with Ramsey. Which brings me to my next point.

One thing that I consider invaluable in my time inside my university is the opportunity to meet other academics. Much of my current research has been shaped not so much by the classes I attended, but rather by the people I met, including, crucially, my current supervisor and other logic students (having such good people criticize your work or suggest new research path is an amazing experience). I also benefited greatly from attending congresses and presenting and hearing talks, often about subjects rather distant from my own research area. So I would say that obtaining a degree in philosophy can be very fruitful indeed, for the kind of people you meet, if not for the classes you attend.

I never said it proves classical logic true, merely, that it begs the question of whether it is true or not by assuming the position that it is true. — wuliheron

And I'm saying that no question is begged. If I say "If John is decapitated, then he will die", I'm not "begging the question" as to whether John was decapitated or not!

That's a tricky question and, as I keep saying, I'm not a mathematician and even they don't have the foundations of the mathematics complete as of yet. My own view is with everything being context dependent it depends upon what you mean by provable in any given situation. — wuliheron

Generally, "provable" means roughly follows from the axioms by acceptable rules of inference.

You cannot prove something is true without somehow demonstrating it is true! Conditional reasoning or otherwise, you must assume if nothing else that we can make clear distinctions between true and false! Godel's theorem is based upon the rules of classical logic in that, at the very least, the law of identity and noncontradiction must apply to any proof. You can play around with variations on the excluded middle all you want, but the essential nature of the logic remains the same. — wuliheron

But Gödel's theorems do not state "classical logic is true". They state "if we assume classical logic and some other conditions, then there are some mathematical theories which are incomplete and can't prove their own consistency". In other words, they are of the form "if A, then B". Clearly I don't need to establish "A" in order to prove "If A, then B"; I can show that, if John is decapitated, then he will die, without thereby showing that John was decapitated!

I'm not a mathematician and those that I've read about claimed the foundations are incomplete. That said, subtypes of the overall symmetry will always express a four fold symmetry or supersymmetry that can be expressed as root metaphors or axioms. In physics, a four fold supersymmetry should be expressed in everything observable and can be thought of metaphorically as infinite dimensions or universes all converging and diverging within the singular void and making it impossible for us to perceive anything less than a four fold symmetry in anything clearly discernible. Such a scenario could only be proven statistically by classical standards, but even if it can never be disproved it would mean everything must express four fold symmetry and so you can use eight dimensions and a singularity or 16 or 32 and so on depending on how much accuracy is desired. — wuliheron

That doesn't answer my second question, which I repeat here for the sake of completeness: if A is a subtype of B, does it mean that every theorem provable in A is also provable in B?

The foundations of Intuitionistic mathematics have yet to be fully developed and, as far as I can tell, they first need to be expressed as a systems logic along the lines of what I've described. That mathematicians are beginning to express things like Godel's theorem in Intuitionistic terms merely means they are working on the problem and not that they have left classical logic and mathematics behind as of this date. — wuliheron

Look, here's the fact of the matter: Gödel's theorems do not assume classical logic is true. They are about classical logic. If your logic contains conditional reasoning, then Gödel's theorems will be provable within it.

"Intuitionism is based on the idea that mathematics is a creation of the mind. The truth of a mathematical statement can only be conceived via a mental construction that proves it to be true, and the communication between mathematicians only serves as a means to create the same mental process in different minds."

http://plato.stanford.edu/entries/intuitionism/

Hence, most certainly classical mathematics can be considered a subtype of Intuitionistic mathematics. My own belief is that everything is context dependent making even what is mental or physical a matter of the situation and, for example, the mind and brain have already been demonstrated to substitute for each other at the most fundamental level of their organization for increased efficiency and error correction. They express the particle-wave duality of quantum mechanics which, for me, is simply another way of saying the display extreme context dependence or are "yin and yang". — wuliheron

Question: what is the subtype relation? More to the point, if type A is a subtype of type B, does it follow that every theorem provable in type A is also provable in type B?

My assertion is that Godel's theorem begs the question and is demonstrably useless outside of classical mathematics and limited physical applications. — wuliheron

And my assertion is that the theorem does not beg the question you're saying it begs, namely that classical mathematics is true, because it does not assume classical mathematics; rather, it is about classical mathematics. To put it more forcefully, it's possible to prove the theorem using as a background logic intuitionism, so it obviously doesn't assume any classical theorem. As for being useless outside of classical mathematics and with limited physical applications, yes, obviously, nobody (except maybe Penrose and Hawking) said anything to the contrary.

Categorization is part of the confusion because there is no way to characterize or categorize Indeterminacy. Calling something like quanta random or a joke meaningless or insisting a shadow has no properties is merely another way of saying we can't define them as anything other than false or context dependent. Clearly shadows, for example, exist and calling them false can only have limited usefulness when they can be more broadly defined as context dependent and sharing their identity with photons.

The way around the issue is to use a systems logic where even its own axioms and identity go down the proverbial rabbit hole into Indeterminacy, thus, displaying context dependence in everything which can be established statistically as factual in some contexts and metaphorical or a personal truth in others. — wuliheron

That's nice, but I still don't see how that answers my question. Is classical mathematics a subtype of intuitionist mathematics? Yes or no? If yes, what is the meaning of "subtype", here? Clearly it's not the subset relation, because we know that classical mathematics is not a subset of intuitionist mathematics. So what is it?

Quantum mechanics are noncommutative and you are merely arguing that classical logic and mathematics must be commutative and Godel's theorem is classical. — wuliheron

I quite frankly don't see how you could give this reading to what I said. What does it mean to say that classical mathematics is "commutative"? Some classical theories (Peano Arithmetic) have an axiom stating the commutative of certain operations, others do not (non-abelian groups). So what?

In any case, I repeat: if your problem with Gödel's theorem is that it allegedly claims that every mathematical theory is incomplete, then you have no problem with Gödel's theorem at all, since it does not claim that every mathematical theory is incomplete.

As best I can tell you are confused over the central issue. Classical logic proving internally consistent, yet, contradicting the physical evidence means all classical truths are context dependent and become a jokes in other contexts. The law of identity itself is going down the nearest convenient rabbit hole or toilet of your personal preference and what is classical mathematics or Intuitionistic mathematics also becomes context dependent.

Photons provide a similar example because what appears to be a shadow in a well lit room can become a faint blob of light in a dark one even though it is identical in every other respect other than the changing context. — wuliheron

But how does this answer my question about the inclusion relationship between classical and intuitionist mathematics? Is there any such relationship? If yes, how should we characterize it?

The idea that any theory is demonstrably incomplete is the heart of the matter. For me, a context without significant content or any content without a significantly greater context is an oxymoron along the lines of a statistic of one. What is incomplete defines what is complete just as you cannot have an up without a down, a back without a front. What Godel showed is that it is incomplete by the standards of classical logic and the principles of the excluded middle and noncontradiction. What he did not do is take it that next step further and show how logic itself is context dependent as quantum mechanics suggests. What is a joke and what makes sense is merely a question of the context. — wuliheron

Emphasis mine. If that is the heart of the matter, then it can be quickly be made to rest, since that particular claim is not what Gödel's theorems are about, but only a popular misconception. In fact, Hilbert, Ackermann, Presburger, Tarski and others had shown that many mathematical theories are complete before Gödel proved his theorems, so obviously the latter can't apply to the theories proven complete by those gentlemen (e.g. various weak forms of arithmetic, the theory of real closed fields, the theory of algebraic fields of a given characteristic, etc.). As I mentioned in my last post, Gödel's theorems apply only to recursively axiomatized theory which contain enough arithmetic. By recursively axiomatized, I mean that the set of axioms of the theory should be decidable by an algorithm. By "contain enough arithmetic", it means that the theory should have enough arithmetic to capture the primitive recursive functions (or, as we know nowadays, the theory should contain Robinson's minimal arithmetic). Any theory that fails these two requirements will not be subjected to Gödel's theorems, and thus may be complete (though it's not automatically complete! The theory of groups clearly fails them, but it's incomplete, since it doesn't decide whether a group is abelian or not).

Intuitionistic subtypes are metaphors meaning the subsets of classical logic must also be treated as metaphors if they are to be compatible with the physical evidence and statistically demonstrated to be valid. — wuliheron

Maybe I'm just being dense, but I don't understand what that means or how it answers my question. What you appear to be saying is that a classical theorem should be "compatible with the physical evidence and statistically demonstrated to be valid" before it is accepted as true. But this has nothing to do with relations of inclusion between intuitionistic and classical mathematics. Suppose, for the sake of the argument, that the intermediate value theorem was shown to be "compatible with the physical evidence and statistically demonstrated to be valid". Then we would have to accept a theorem of classical mathematics which is not a theorem of intuitionistic mathematics. On the other hand, suppose that we could somehow show that it is "compatible with the physical evidence and statistically demonstrate to be valid" that every total function from R to R is continuous. Then we would have to accept a theorem from intuitionism that is false in classical mathematics. Either way, though, there wouldn't be any inclusion relation between them, so that none would be a "subtype" of the other.

Kuhn is merely another historian giving his personal interpretation of history in the name of science and philosophy. I'll take experimental evidence over the word of a historian or even the consensus of the scientific community any day. — wuliheron

Actually he was a physicist by formation. In any case, you may do whatever you like, but the point is that scientists don't often proceed in the way Feynman describes, and that's not how science generally progresses.

Godel used classical logic to formulate his theorem and, by the standards he used, if he was not asserting his theorem was true, than he was asserting it was false! — wuliheron

Again, you're misunderstanding the theorems. The theorems are conditional in nature, i.e. they say that "under this and that circumstances, this result follows". In Gödel's case, the circumstances are (i) classical logic, (ii) recursively axiomatized theories which (iii) contain a modicum of arithmetic and (iv) are consistent. So the theorems are, if (i), (ii), (iii), (iv) hold for a given theory, then the theory is incomplete and can't prove its own consistency. There are many theories for which (i)-(iv) don't hold, and the theorem is silent about those (for instance, (ii) fails for the theory of the natural numbers, (iii) fails for Presburger arithmetic, (iv) fails for the inconsistent theory; these theories are all complete, trivially so in the last case). Given that the intuitionists also accept conditional reasoning, it follows that the theorem is valid also in an intuitionist setting.

Mathematicians have already demonstrated that all of classical mathematics and causal physics can be represented using any number of simple metaphors or analogies such as asserting everything is merely composed of bouncing springs, balls of string, or vibrating rubber sheets for all I know. Another study similar concluded they can be fully represented using only two dimensions. In other words, all of causality and causal mathematics are demonstrably based upon what I like to call "Cartoon Logic", that is, the logic of small children who will pick whatever explanation sounds good to them at the time or happens to contradict reality less. The implication is clear that mathematics and logic are merely pragmatic conventions just as quantum mechanics suggest our concepts of reality are. — wuliheron

I don't understand the relevance of the above, since nothing I said contradicts or is even remotely connected to that.

Regardless, I'm still curious about your notion of "subtypes". You said that classical mathematics is a subtype of intuitionistic mathematics. I took that to mean that every theorem of classical mathematics is a theorem of intuitionistic mathematics, i.e. classical mathematics is a (proper?) subset of intuitionistic mathematics. But then that doesn't seem to follow, since, e.g., the intermediate value theorem is a theorem of classical, but not of intuitionistic mathematics. So, is there any other way of understanding this subtype relation?

I would say that that Feynman quotation is incredibly naive in our post-Kuhnian age, but no matter. Gödel didn't assume that classical mathematics was "true"; rather, his result is about classical mathematics. An analogy: Gödel's theorems suppose that the theory in question is recursively axiomatizable. That does not mean that it "begs the question" as to whether all mathematical theories are recursively axiomatizable, which would be plainly false. Rather, it is a theorem about such theories.

As for classical and intuitionistic mathematics, well, classical analysis proves the intermediate value theorem, which is not provable in intuitionistic mathematics. On the other hand, it seems that every total function from R to R in an intuitionistic setting is continuous, something that is clearly false in the classical setting. So one does not seem to be a subset of the other (unless they're inconsistent, in which case they're the same).

I just awarded ModBot the poster of the month award for the shoutbox. She appears to be the most proliferate and productive member over there at the moment.
Sad, sad, sad. :’( — Sir2u

I was wondering about that. Who's posting under ModBot? Paul?