Gedanken

Gedanken are literally thought experiments. They have played a role in the advancement of science. Galileo invented an elegant and conclusive thought experiment. He proposed that a cannonball be attached to a light musket ball and then dropped. If the current belief that heavier objects fall faster than less heavy ones were true, the heavier cannonball would be slowed by the musket ball or speeded up because of the combined weight. Conclusion: weight has nothing to do with the speed of fall of objects. Other historical Gedanken are: Newton's bucket, Maxwell's demon, Einstein's elevator, and Schrödinger's cat. Schrödinger's cat is placed in a box with a vial of poison which will be released or not released depending on the "whims" of quantum mechanics. According to the experiment, it is impossible to tell without looking inside the box whether the cat is dead or alive.

But there is a tendency in analytical philosophy to make too much of Gedanken. In Fodor there is an implicit refutation of their usefulness: "The soundness of inference is not impugned by the possibility of equivocating, but only by instances of actual equivocation." (Fodor, The language of thought , 1975)

Certain gedanken are classics in contemporary philosophy and figure in almost any text on philosophy of mind. It appears that researchers feel the need either to use or to refute them. Some of these classic gedanken are:
(1) the Chinese room argument, against artificial intelligence;
(2) Mary, the woman deprived of colour sensations, for qualia (thought up by F. Jackson);
(3) twin worlds, against folk-psycholoy so-called;
(4) the senile woman recalling the assassination of President McKinley,also against folk psychology/propositional attitudes;
(5) the brain-in-a-vat, for functionalism;
(6) brain fission, to explore selfhood
(7) the evil surgeon who switches color perception (maybe Dennett).

Some of these Gedanken cut both ways. Qualia inversion can be an argument for functionalism in the sense that colours are signals for dispositional attitudes. However, it can also be an argument for the existence itself of colours, since it is only in the perception of colours that we can actually adapt to colour inversion.

In the Chinese room argument, formulated by Searle, a person is in a room with a devise which allows him to mechanically turn English words into their Chinese equivalents. From one slot he gets English words which he then dutifully and uncomprehendingly translates into Chinese and slips through an output slot. He is in effect translating but he has not an inkling of what he is doing. This is supposed to illustrate the difference between a machine function and the aware realization of a task. Basically, awareness is recognition, so if there is no recognition there is no awareness. A variation on this argument is that it is possible to record music note by note with a machine. A listener is incapable of such a feat, but he will feel in a way that no machine can.

Kim Sterelny, The Representational Theory of Mind: An Introduction (1990)

"The idea of Twin Earth merely dramatizes the fact that causal origins of psychological states play a role in their representational content. Hence. their failure to supervene on what is inside the head. This fact about representational properties has led some philosophers to urge an irrelevance thesis about representation in psychology: semantic kinds have no places in psychology. Dennett, for example, has written that the mind is a `syntactic engine'; representational notions have in his view real heuristic and predictive value, but they do not have a role in the causal explanation of the mind's inner workings."

"The sadly decrepit Mrs T contrasts with her earlier self in now having a very different (and much impoverished doxastic surround. That's why we no longer count her beliefs as the same. This conclusion is crucial to Stich's ultimate scepticism about belief. For doxastic surround is clearly a similarity notion at best; no two people will ever have the same doxastic pattern."

Gentile

Richard Bellamy on Giovanni Gentile, Opere filosofiche (Edited by Eugenio Garin; Milan: Garzanti). in the TLS, November 8 1992, p. 6

"In his search for a philosophy without presuppositions of any kind--either empirical or a priori--Gentile was driven to make our present activity of reflective consciousness the absolute foundation on which all else depended. The act of thinking became in this way the `pure act' which provided the source of all human experience. Through thought we literally created ourselves and the world around us."

"Thus, each individual act of thought constituted an element within the development of human consciousness as a whole."

This statement defines the imaginary concept of the totality of awareness

German idealism

German or absolute idealism is the post-Kantian belief that, since we can never have the noumenal world, we can assume that what we think is actually what is even to the extent of the determination of history by idealized universal forces that can only be conceived by mind.

Gestalt

See   Psychology

God

The awareness of the ontological chain permits two wide inferences that seem to open significant rational/coherent possibilities. One inference is that the ontological chain is a natural sequence of events devoid of moral or transcendental significance. The rise of awareness in this context is likely to have been an accidental epiphenomenon of matter. In this wide view it would be possible to accomodate in the same perceptual and moral frame the existence of the evil-looking man-eating white shark and the resourceful and efficient SS killer. Another possible inference is that the ontological chain manifests an intelligent and purposeful sequence, precisely the contrary of "might-have-been" or chance. Awareness then is either an insignificant epiphenomenon, in which case we can only be certain of science and technology pointing to absolute nothingness, or it is the means by which one reaches the certainty of a transcendental absolute and the awareness that we are not entirely in the dark and that our means and categories have a transcendental basis. In other words, from the ontological chain we can deduce either a purposeless or a purposeful universe. As between these two wide and conflicting inferences, an appeal to logic or rationality would be futile. From the perspective of a deterministic stance on human behaviour, neither one nor the other can be considered irrational or inconsistent with logic.

The absence of cosmological purpose allows room for the affirmation of purposeless or meaningless meaning. This is the contextual and short-term interpretation of quotidian thought and behaviour that we can infer from a rational overview of our existential situation at any given moment. We know approximately where we stand and where we want to go. From the angle of A-B determinism, we can try to establish rational linkages between premisses and consequences in the ontological chain. Meaningless meaning does not require infallible analysis: only that awareness should have a reasonable understanding of its insertion in the ontological chain. Within the ontological chain it is possible to find meaning in the ordinary course of living, but this sort of meaning involves only short-term existential purpose, hence it is meaningless from the point of view of a transcendental purpose.

But all of this is tantamount to a purposeless universe, because the hypothesis of purpose without the comprehension of purpose contains its own denial. At best, we are back to meaningless meaning complicated by the conundrum of a possibly capricious, unfathombale, even frightening God. In fact, though, this is the deistic paradox, in which it is seen that an abstract God can be said to be no God at all.

Stephen Grover on D. Z. Phillips, Faith after Foundationalism (Routledge, London), in TLS, August 5-11 1988, p.862

"Theological foundationalism can take many forms. For example, it is said that theism is the implicit foundation of religious belief. Others say that religious belief depends on the outcome of historical evidence. Alternatively, it is said that religious belief is grounded in `religious facts' or `religious history'. Again it has been said that religious belief is based on religious experience; an experience given prior to all religious beliefs and theologies. When these forms of foundationalism are subjected to Wittgensteinian criticism, the criticims are often seen as if they were an additional metaphysical theory called Wittgensteinian fideism. In this way, the metaphysical game perpetuates itself."

"Foundationalism holds that a belief can only be rational if is related in appropriate ways to a set of basic beliefs...In the philosophy of religion, foundationalism has generally been taken to imply that belief in God cannot be rational unless supported by evidence, because this belief is not eligible for membership of the foundational set, that privilege being reserved for self-evident truths and reports of sense-experience."

"There is, according to Plantinga, no reason why belief in God should not be held as basic, and so be unsupported by evidence, and yet still count as rational."

"Phillips will have none of this. It is foundationalism itself that has to go, along with the allied metaphysical confusion that there could be some necessity which underpins our practices and determines that they should be as they are."

 In other words, there are no self-evident truths, and especially not the truths of sense-experience

From Phillips

"The reformed epistemologist argues that the religious believer has every right to place their belief in God among his foundational propositions."

 "The classical foundationalist claims that only self-evident propositions and the incorrigible propositions of sense experience are properly foundational. But how does he know this? It is certainly not self-evident that this is so.

"The theist cannot demonstrate his right to place belief in God among his foundational propositions, but the foundationalist is in precisely the same position with respect to the foundational proposition he recognises."

"The reformed epistemologist, unlike the foundationalist, does not claim to know that his foundational propositions reflect reality, but he believes that they do. The Reformers were right: justification is by faith, not by reason.

 "Reformed epistemology, despite its attacks on classical foundationalism, remains within a foundationalist tradition. Both make a final appeal to foundational propositions. For Wittgenstein, basic propositions are not foundational. They enjoy their status within practices where they are held fast by all that surrounds them. Further, foundationalists and Reformed epistemologists regard epistemic practices as though they were descriptions of a reality which lies beyond them. Wittgenstein, far more radically, insists that distinctions between the real and the unreal get their sense within epistemic practices."

John Cottingham on Edward Craig, The Mind of God and the Works of Man (Oxford: Clarendon), in TLS, October 30-November 5 1987, p. 1201

 "In the second half of the book Craig charts the gradual recession of the Image of God doctrine and its replacement by a still dominant world-picture [Weltbild] which he calls the `Agency Theory' or the `Practice Ideal'. No longer a quasi-divine spectator of reality, man becomes `a being that actively creates, or shapes, its own world'...Yet the Agency Theory as conceived by Craig is not simply a matter of seeing man as more actively involved with his environment, but has a deeper metaphysical dimension: the environment comes to be seen as something we not merely encounter but create: `the realities which we meet with are the works of man'. Or again: `As a participant [in the making of reality] Man is autonomous, his creations subject to no controls and no standards other than those which he himself imposes; and with this thesis of man's autonomy comes the corollary image of the surrounding, unresisting but also un-supporting void in which he has henceforth to make his way'."

Anthony Kenny, "The definition of omnipotence", in Thomas V. Morris, ed., The Concept of God, Oxford, 1987

 "Better, however, to say that it cannot be done, rather than that God cannot do it", St Thomas, Summa Theologiae.

"There are advantages, then, in defining omnipotence as the totality of logically possible powers rather than as the power to perform all logically possible actions or to bring about all logically possible states of affairs...The power to change, to sin, and to die are instances of powers which it is logically possible to have--since we human beings have them--and yet which traditional theism denies to God."--p.132 

"Powers such as the power to weaken, sicken, and die will not be part of divine omnipotence since they clash with other divine attributes. What of the power to do evil? Clearly, the actual performance of an evil deed would be incompatible with divine goodness: but some theologians have thought that the mere power to do evil, voluntarily unexercised, is not only compatible with, but actually enhances the splendour of divine beneficence. If so, then the power to do evil, since it is clearly in itself a logically possible power, would be part of divine omnipotence."--p.133 

We cannot define the attributes of God, but we can trust that some of our faculties are of divine origin.

The two schemes: (a) the traditional scheme: from God to meaning in existence; (b) from the limitations of existence to some "despairing" perception of meaning in existence and from there to God.

GÖDEL

The historical sequence which culminates with Gödel is as follows. Frege tried to deduce arithmetic with logic. In order to explain numbers, he had recourse to set theory. Russell discovered a paradox in set theory, which he tried to set right with his theory of types. With his modification of set theory, he too tried to derive mathematics systematically from logic, in PM.

"Russell..showed that most of received mathematics is deducible in set theory...Russell's paradox showed that the unrestricted conception of sets as being determined by all predicates (conditions) was inconsistent. Russell's response was to develop an axiomatized (set) theory in which sets were viewed as being arranged in a hierarchy of types. Types were assigned to any set such that each of its members was of a lower type."

"Formalism. A view pioneered by D. Hilbert (1862-1943) and his followers, in which it was claimed that the only foundation necessary for mathematics is its formalization and the proof that the system produced is consistent...Hilbert's programme was to put mathematics on a sound footing by reducing it (via arithmetic) to consistent axioms and derivation rules, the former being certain series of strokes, the latter ways of manipulating them. Later Gödel showed that the consistency of arithmetic cannot be proved within the system itself, thus demonstrating the impossibility of achieving part of the Hilbert programme." (Flew)

Gödel's proof

 Number theory are the rules of mathematics, and specifically in this case, of arithmetic, i.e., numbers and their rules. The rules of math are logical in the sense either of obeying the principles of logic or of not contravening the principles of logic. However, the rules of math are also learned--not logically derived--because, though conceivably 2x2=4 is an empirical experience in which logical principles are paramount, 1234 times 5679 is not an operation that we can do without the rules for multiplication. It is possible to explain number theory with the use of a formal logical system known as predicate logic, which is the propositional calculus plus the rules for quantification. The exercises we can use to illustrate the predicate calculus are: demonstrations of axioms three and five, illustrations of the rules of specification and generalization, the derivation of axiom two, the derivation of 1x1=1, and the derivation of the commutativity of addition.

 One necessary premise for Gödel's theorem is that although TNT makes provisions for deciding which are wffs and which are not, wffs are not necessarily valid formulæ, that is, they can be either true or false (Hofstadter, Ch.VIII). It is the difference between expressing and representing. Hilbert asked: can predicate logic prove that number theory is both complete (all valid math statements can be derived with predicate logic) and consistent (no derivation of number theory in predicate logic can be the negation of another derivation).

The specific preliminaries that Gödel formulated for his proof are: Gödel numbers, proof-pairs, and the substitution relation. Gödel numbers substitute for all notations in predicate calculus. They establish a one on one relation with predicate logic. Whatever predicate logic can express about number theory can also be expressed in Gödel numbers. Hence, formulas in Gödel numbers can be recursively shown to be valid. Gödel numbers in sum are merely a different set of symbols to be used with predicate logic.

The next specifically Gödelian preliminary to the proof is the concept of proof pair. "Two natural numbers, m and n respectively, form a number theory proof pair if and only if m is the Gödel number of a number theory derivation whose bottom line is the string with Gödel number n."

 The formula "O=O is a theorem of TNT", can be expressed as "TNT proof pair {a, a'}", which can be interpreted as "natural numbers a and a' form a TNT proof pair". It is possible with Gödel numbers to transform this statement into number theory (meta-TNT). Hence, InvEa:{a, a'}. InvEa:{a, O=O}. And the Gödel equivalent: InvEa:{a, 666,111,666}, which translated back to predicate logic means InvEa:{a, 666,111,666 SsO/a'}, where a presumably can be expressed as a huge number of Ss plus O.

[Couldn't the number of SsO which express O=O also express some other combination of Gödel numbers? Presumably not.]

The third and final preliminary to Gödel's proof is the substitution relation involving Gödel numbers, as so:

(a)   a=a                                             262,111,262

(a')  SSO (to substitute the variable)          123,123,666

(a'') substituting                123,123,666,111,123,123,666

The above can be expressed as SUB (a, a', a''), and back from Gödel numbers and into pred-logic as:

SUB (262,111,262 SsO/a, SSO, 123,123,666,111,123,123,666 SsO/a'')

It is possible in the substitution relation to have self-substitution, which means that a is substituted into itself in order to get the third term of the relation.

(a)    ~a=SO                  223,262,111,123,666

(a')   223,262,111,123,666 SsO

(a'')  223,(223,262,111,123,666 copies of S=123),111,123,666

Self-substitution as above is expressed as SUB {a'',a'}, in which a' represents the self-substitution of a.''

With these prelims Gödel derived a formula TNT>G which says of itself that G is not a formula of TNT.

 The formula that Gödel found is:

u=~InvEa:InvEa':{Proof-pair (a,a')^SUB(a'',a')}

Substituting u for the only free variable a'', gives:

G=~InvEa:InvEa':{Proof pair (a,a')^SUB(u/a'',a')}

The above operation expresses a SUB{a'',a'} relation, in which u is the first term, u itself the substitution term, and G the result of self-substitution, hence all of G is the second term in the SUB{a'',a'} relation.

The denial in G means that, since we validly derived the value of a', the variable that is wrong is a.

This is expressed as follows:

InvEa':a' --> ~InvEa:a --> ~InvEa: Proof pair (a,a')

And what this means is that G, which as we saw is a', expresses the denial of a', or G --> ~InvEa':a'^G=a.

Hofstadter throws out the u from which G is derived. He does not explain how Gödel obtained u. [And in particular he does not explain why Gödel numbering is necessary to the logical architecture of G. What Gödel's proof involves is transforming a statement of TNT into a long string of numbers. By self-substitution (substituting this string into a part of itself), the result is an even longer string, which can be interpreted as a denial of itself. Why can't this be done with the symbols of predicate logic? Because with predicate logic it is not possible to make an equation the equal of any of its part, which is the trick that can be done applying the concept of proof pair, which in turn involves the trick of Gödel numbering. Alternatively since TNT is a means to examine number theory, there is a need for a meta-TNT to examine TNT, and this is provided by Gödel numbering.]

Once you have Gödel numbering you can express the concept of proof pair.

Here's an alternative development of Gödel's proof:

InvEa:InvEa':Sub (a,a',a'') [true]

Self Sub (a,a') --> (a'',a')

~InvEa:InvEa':Self sub (a'',a')

What Gödel did was to make the previous statement into a statement about itself using the following equivalence:

Self sub(a'',a')=Proof pair (a,a') [requires Gödel Nos]

~InvEa:InvEa':Proof pair (a,a') ^ Self sub(a'',a')=u

Sub u into u

~InvEa:InvEa':Proof pair (a,a')^Self sub(u/a'',a')=G

InvEa':Self sub(u/a'',a') --> G --> ~InvEa: Proof pair (a,a')

But G=Self sub (u/a'',a') --> InvEa: Proof pair (a,a')

Alternative interpretation. You obtain a long string of numbers. With these you make a longer string. Translating this string you obtain a proposition which is a denial of itself. Why can't this be done directly with the symbols of predicate logic? Because with predicate logic you cannot make the equation the equal of any of its parts whereas applying the concept of proof-pair, which requires Gödel numbers, it is possible to do so.]

The consequences of the argument above are as follows:

if TNT>G, then ~TNT>G.

Suppose then that TNT does not include G.

Since G is the proof that TNT does not include G (and TNT is not a theorem of TNT), then that G is not a theorem of TNT cannot be proven.

TNT is incomplete: there is a true proposition in TNT which TNT cannot prove.

This can be expressed as ~InvEG:G, but it is G --> ~InvEG:G.

The ~G cannot be proven.

This is the incompleteness theorem

"Is G a TNT-theorem? If so, then it must assert a truth. But what in fact does G assert? Its own nontheoremhood. Thus from its theoremhood would follow its nontheoremhood: a contradiction.

"Now what about G being a nontheorem? This is acceptable, in that it doesn't lead to a contradiction. But G's nontheoremhood is what G asserts--hence G asserts a truth. And since G is not a theorem, there exists (at least) one truth which is not a theorem of TNT."

The second Gódel theorem can be derived as follows.

Since G is true, the denial of G must be false. But G says that the denial of G is valid/true. TNT cannot decide between the two propositions. Theoretically, (G^~G) is a wff of TNT [from (P^~P)]. Therefore, TNT is inconsistent. TNT itself cannot prove number theory.

"Since G's interpretation is true, the interpretation of its negation ~G is false. And we know that no false statements are derivable in TNT. Thus neither G nor its negation ~G can be a theorem of TNT. We have found a `hole' in our system--an undecidable proposition. This has a number of ramifications. Here is one curious fact which follows from G's undecidability: although neither G nor ~G is a theorem, the formula <Gv~G> is a theorem, since the rules of Propositional Calculus ensure that all well-formed formulas of the form <Pv~P> are theorems.

    "This is one simple example where an assertion inside the system and assertion about the system seem at odds with each other. It makes one wonder if the system really reflects itself accurately. Does the `reflected metamathematics' which exist inside TNT correspond well to the metamathematics which we do? This was one of the questions which intrigued Gödel when he wrote his paper. In particular, he was interested in whether it was possible, in the `reflected metamathematics', to prove TNT's consistency. Recall that this was a great philosophical dilemma of the day: how to prove a system consistent. Gödel found a simple way to express the statement `TNT is inconsistent' in a TNT formula; and then he showed that this formula (and all the others which express the same idea) are only theorems of TNT under one condition: that TNT is inconsistent...How do you express the statement `TNT is inconsistent' inside TNT? It hinges on this simple fact: that inconsistency means that two formulas, x and ~x, one the negation of the other, are both theorems. But if both x and ~x are theorems, then according to the Propositional Calculus, all well-formed formulas are theorems. Thus, to show TNT's consistency, it would suffice to exhibit one single sentence of TNT which can be proven to be a nontheorem. Therefore, one way to express `TNT is inconsistent' is to say `The formula ~0 = 0 is not a theorem of TNT'...It can be shown, by lengthy but fairly straightforward reasoning, that--as long as TNT is consistent--this oath-of-consistency by TNT is not a theorem of TNT. So TNT's powers of introspection are great when it comes to expressing things, but fairly weak when it comes to proving them. This is quite a provocative result, if one applies it metaphorically to the human problem of self-knowledge." (Hofstadter)

But is TNT w-inconsistent?

G=~InvEa:InvEa':{Proof pair (a, a')^SUB(u/a'',a')}

~InvEa:Sa=O --> InvA:~Sa=O  (interchange)

G'=InvA:~InvEa':{Proof pair (a,a')^SUB(u/a'',a')}, which means that the denial of G entails the denial of G' and the denial of G' entails the denial of G

If we substitute each number in turn for the free variable a, we get that TNT is w-inconsistent. It cannot prove the existence of any number at all. Put another way: no number can form a proof pair with a'. Since we derived a' validly, then it is not possible to derive any number using G. None of the formulas formed with any of the natural numbers is a theorem of TNT. But that is what the formulas assert. They cannot be proven. TNT lets escape infinitely many formulas with any number as a theorem of TNT.

"TNT + ~G is a consistent system under an interpretation which includes supernatural numbers."

Hofstadter (Chapter XV)

Instead of trying to prove ~G through supernatural numbers--which is an arbitrary solution--let us instead increase TNT by tacking on G. This would make it complete and consistent. However, Gödel's proof can be applied to any expression in TNT and therefore to any extension of TNT.

"Is there any reason to expect or hope that TNT + G will be invulnerable to Gödel's proof? Not really. Our new system is just as expressive as TNT. Since Gödel's proof relies primarily on the expressive power of a formal system, we should not be surprised to see our new system succumb too."

"Any system, no matter how complex or tricky it is, can be Gödel-numbered, and then the notion of its proof-pairs can be defined--and this is the petard by which it is hoist. Once a system is well-defined, or `boxed', it becomes vulnerable."

"Here is the point: although theorems of TNT can rule out negative numbers, fractions, irrational numbers, and complex numbers, still there is no way to rule out infinitely large integers...The problem is there is no way even to express the statement `There are no infinite quantities'."

     "This sounds quite strange, at first. Just exactly how big is the number [i.e. I] which makes a TNT-proof-pair with G's Gödel number?...it should be just the size of a number which specifies the structure of a proof of G...."

     "Of course, any theorem of TNT has many different derivations, so you might complain that my characterization of I is non-unique...So you just have to think of I as being some specific of the many possible supernatural numbers which form  TNT-proof-pairs with the arithmoquinification of u."

The way I read this is that you cannot make categorical denials or affirmations with infinite quantities.

 "First of all, nonstandard number theory does not threaten the age-old idea that 2 plus 2 = 4. It differs from ordinary number theory only in the way it deals with the concept of the infinite. After all, every theorem of TNT remains a theorem in any extension of TNT!"

"We haven't yet faced what it means to throw ~G in as an axiom...The point is that ~G has a proof. How can a system survive, when one of its axioms asserts that its own negation has a proof?...As long as we only construct finite proofs, we will never prove G...The supernatural number I won't cause any disaster. However, we will have to get used to the idea that ~G is now the one which asserts a truth (`~G has a proof'), while G asserts a falsity (`G has no proof'). In standard number theory there aren't any supernatural numbers. Notice that a supernatural theorem of TNT--namely G--may assert a falsity, but all natural theorems still assert truths."

 As I read it, since you cannot limit infinity, you cannot pin down an answer except to say that there is one, but that it is non-unique.

Alternatively: to include ~G in TNT implies that G has a proof and that TNT is complete and consistent, but TNT cannot include both G and ~G. This incompatibility evaporates from the infinity premise.

Suppose there is an awareness inside a closed room and a wide world outside, and the awareness wants to make an entry in its diary that says that it is day or night. The awareness reasons this way: it is either day or night, but it cannot be both. I cannot know that it is day, for instance, therefore it is either day or night or it is day and not day, but since it cannot be day and not day, it must be day or night, but it cannot be day and night. The answer that Hofstatdter gives to this dilemma is: write down that outside it is either day or night, and you cannot be wrong.

Another way to view these games is the following: you cannot include in the same system the statements that (a) "white is white" and that (b) "white is black", but you can include the statements that (c) "white is white" and (d) "white is not white", because the possibility that white may not be what it seems is open. In real life, after all, white is not a color.

"When we decided to formalize TNT, we preselected the terms that we would use as interpretation words--for instance, words such as `number', `plus', `times', and so on. By taking the step of formalization, we were committing ourselves to accepting whatever passive meaning these terms might take on...We thought we knew what the true, the real, the only theory of natural numbers was. We didn't know that there would be some questions about numbers which TNT would leave open, and which could therefore be answered ad libitum by extensions of TNT leading off in different directions. Thus, there is no basis on which to say that number theory `really' is this way or that, just as one would be loath to say that the square root of -1 `really' exists, or `really' does not."

"Mathematics only tells you answers to questions in the real world after you have taken the one vital step of choosing which kind of mathematics to apply. Even if there were a rival number theory which used the symbols `2', `3', and `+', and in which a theorem said `2 + 2 = 3', there would be little reason for bankers to choose to use that theory! For that theory does not fit the way money works. You fit your mathematics to the world and not the other way around."

Recursiveness:

(1) x = y; x = a and y = a; therefore, x = y;

(2) "x has a claim on y" can be expressed as x --> y; x = s and y = o; s>t and t>o; therefore, s --> o and x --> y.]

Even though one could argue that what Gödel proved is that x =/= x--as is implied in this quote from Hofstadter: "inconsistency means that two formulas, x and ~x, one the negation of the other, are both theorems"--the claim can be rejected because x = x does not involve self-reference but a relation of identity and therefore x =/= x is not inconsistent but incoherent. Nevertheless, consider the following argument: x = t and ~x = t; :. x = ~x and x =/= x. In other words, it can be argued that what Gödel proved is that a math S will allow the proof of a patently self-contradictory statement. Since self-contradiction necessarily implies self-reference, x =/= x could be a simplified notation of the Gödel conclusion.

 Quine was not satisfied with the self-referential illogicality of the liars paradox. To explain why he wasn't satisfied he turned it into "I am lying" or "this statement is false". This implies recurring to experience: "I am lying when?" and "which statement are we talking about?". So he devised the perfect self-referential sentence. "The problem is to devise a sentence that says of itself that it is false without venturing outside the timeless domain of pure grammar and logic. Here is a solution: "`Does not yield a truth when appended to its own quotation' does not yield a truth when appended to its own quotation."

Incidentally it takes Hofstadter six pages of text to get to what Quine says in one phrase. In part it is because he goes through the process of showing how a phrase can quote itself: "`Is not the title of a book' is not the title of a book", "`is a sentence fragment'" is a sentence fragment", until he finally arrives at Quine's own invention in the version "`yields falsehood when preceded by its quotation' yields falsehood when preceded by its owen quotation." Of course, it is a perfectly nonsensical sentence.

W. V. Quine, Quiddities: An intermittently philosophical dictionary (1987)

"What Gödel proved, then, is that no axiom system or other deductive apparatus can cover all the truths expressible even in that modest notation; any valid proof procedure will let some true statements, indeed infinitely many, slip through its net...Gödel's actual proof...showed how, given any proper proof procedure, to construct a sentence in the notation of elementary number theory that says of itself, in effect, via Gödel numbering, that it cannot be proved. Either it is false and provable, God forbid, or true and not provable; presumably the latter."

Ray Monk, "Strange loops from a reluctant genius", The Higher, September 6 1991, pp. 15 and 19

"Proofs--looked at from a formalistic point of view--can be seen analogously as strings of formulae joined together in accordance with certain inference rules..."Now if we want to say that a certain set of formulas is a proof, we cannot do so in the object language. A proof can be expressed using the object language, but that it is a proof has to be expressed in what we call the `metalanguage', the language we use to talk about a formal system. Thus, `p is unprovable' is not a mathematical proposition but a metamathematical proposition."

From Hilary Putnam, Representation and Reality

"A proof in the ordinary sense (a proof humanly speaking) is a proof in a system which is not just sound, but which a mathematician could, upon reflection, see to be sound, one which a reasonable mathematician would be justified in accepting. `Proof' is an epistemic notion, not a mathematical one."

Paul Davies on John D. Barrow, Theories of Everything: The quest for ultimate explanation (OUP), in THES, 19.4.91, p.18

"The idea that there might exist a complete description of the world in terms of a closed system of logical truths is a beguiling one that dates back to Plato. After all, if the universe is a manifestation of rational order, then we might be able to deduce the nature of the world from `pure thought' alone, without the need for observation or experiment. Most scientists utterly reject this philosophy, of course, hailing the empirical route to knowledge as the only dependable path. But the demands of rationality do impose at least some restrictions on the sort of world that we can know.

    "On the other hand, that same logical structure contains within itself its own paradoxical limitations that ensure we can never grasp the totality of existence from deduction alone. These limitations stem in part from the famous theorem of Kurt Gödel, who in the 1930s demonstrated that a logical scheme based on a finite system of axioms generally cannot be proved to be true from within that system. [What I imagine Davies means is that Gödel discovered a proof, expressed as a theorem, that such was the case, not that the theorem somehow has to do with the limitations of logical systems.] Thus Barrow concludes: `There is no formula that can deliver all truth. No theory of Everything can ever provide total insight. For to see through everything, would leave us seeing nothing at all'."

 "Opposed to Platonism is formalism, which asserts that mathematics is nothing more than an elaborate network of definitions and tautologies invented by mathematicians. According to the formalist position, at rock bottom every mathematical statement reduces to a simple mapping of one set of symbols into another."

 "Can we suppose that the laws of physics share with the mathematics that they manifest an independent Platonic existence, and that for some extraordinary reason Homo sapiens can tap into that Platonic realm? Or are the laws of physics, together with their mathematical expression, merely a human invention, dreamt up to help us make sense of the world?...For unless some trascendant, independently-existing law or principle can bring the universe into existence from nothing, then the origin of the physical world must lie outside the scope of law-like science as we know it."

Philosophical aspects

Despite its rigours, Gödel's theorem holds different lessons for different philosophers. (1) For Ray Monk, a Wittgenstein specialist, it is a validation of his own scepticism about the power of human reason because of the possibility of error. (2) In his philosophical dictionary (Quiddities, Penguin, 1987), Quine affirms that it was a momentous event in the history of human thought, but not because it showed that reason was hopelessly error-prone, but because it proved that "any valid proof procedure will let some true statements, indeed infinitely many, slip through its net". But mostly what it did was to goad him to express a perfected form of the liar's paradox: a very abstract statement that eliminated all contextuality and all reference except to itself. Putnam uses Gödel's conclusions in the epistemological part of his refutation of scientific realism. He assumes a perfectly consistent system and argues that the proof of its consistency means that that there is "a certain number-theoretic version of the proof (a proof of the Gödelian statement `CON S'). Since S was assumed such that every mathematical statement that has a proof has a proof in S, it follows that...`CON S' must be a theorem of S." But since Gödel proved that it was not possible to prove the consistency of any system, the inclusion CON-S in S means that S must be inconsistent. And Putnam extended Gödelian logic to imply that "reason can go beyond whatever reason can formalize". For ourselves, we find that the wider historical as well as the epistemic contexts of Gödel's theorem reinforce the foundational axiom, for its purported proof not only does not invalidate the practical uses of mathematics, but, as Quine exclaims, God forbid that it should serve to prove a falsehood, or as Monk admits, it did nothing to dampen Gödel's "rational optimism".

 You can cut an arithmetical string at any point and it will have meaning. This is discreteness. But you cannot cut linguistic expressions at any arbitrary point and expect to have meaning. This is in part because words are intensional whereas numbers are "extensional". Now, the intensionality of language does not necessarily mean that it is not possible to obtain knowledge through language, because, even though words are indeed intensional, knowledge is a transaction, a give-and-take from which consensus emerges. It is the transactional and consensual character of knowledge that ensures that the intensionality of language does not produce mere babel.

Grammar

Grammar refers to the rules for the correct formulation of linguistic propositions. However, any set of rules for any non-linguistic system of signs can also be called a grammar. Computational languages, e.g., have grammars. There are two antipodal theories on grammar: one, Chomskian, that it is encoded in an innate mental module and that consequently it is an innate, universal feature; and the other, which holds that grammar is learned, that it is contextual, and that it cannot be formalized in an universal sense.

Patricia Greenspan

Patricia S. Greenspan, Emotions and Reason: An Enquiry into Emotional Justification (1989)

 "[Fear]...essentially involves a belief that danger looms...This judgement is partly factual but also partly evaluative...Its detailed factual content typically serves to exhibit an `object' of fear...and thus to distinguihs the emotion from objectless sensation, of the sort that might be held in reaction to the cold. But its negative evaluative content is needed to explain why it amounts to fear rather than some other reaction to an envisioned possibility, such as thrilled anticipation--and why fear amounts to a reasonable reaction in certain situations.

      "I gran these points to `judgementalism', as I shall call this view...But I think they can be incorporated into a borader evaluative view, allowing for propositional attitudes that are weaker than strict belief: states of mind, like imagining that danger looms, that involve entertaining a predicative thought without assent...I hope to lay the foundation for a subtler account, however, by bringing in a noncartesian elemenet of object-directed affect, whose object is an evaluative proposition...Let me propose then that we look at emotions as compounds of two elements: affective states of comfort and discomfort and evaluative propositions spelling out their intentional content...At the outset, bypassing issues of moral justification, I have two corresponding theses in mind, based on the interpretation of emotions as `propositional feelings': (1) although its appropriateness may be explained in terms of belief warrant, the evaluative component of emotions need not rest on reasons adequate for belief. (2) The affective component of emotions gives them a special role to play in rational motivation, as `extrajudgmental' reasons for action...The greater part of this essay will focus on (1), since (2) presupposes the distinction it draws between emotion and belief. But my defense of (2) will eventually bring out my reason for stressing the distinction: it allows emotions a role in rational motivation that is not simply parasitic on that judgment. For `judgmentalism' subsumes emotions under a conventional rational category at the cost of slighting their own role as reasons."

"Rather, the object of my discomfort amounts to the content of the thought: that I face a threat of injury from X. Because the affective component of suspicion is intentionally directed towards this evaluative component on my view, the emotion may be said to have a propositional `internal' object, along with the object given in the proposition, as the source of harm."

"Even if it were `objectless', in fact--if its evaluative component did not pick out a particular source of harm--the emotion would have intentional content; for its affective component must at least be directed towards an indefinite proposition on my analysis...The propositional object of emotional discomfort need not be an object of belief, however."

 [Belief can be false belief, although she may be referring to intensity of belief.]

"My `intuitive' suspicion may be warranted, however, even though I do not take it to be and even though the corresponding belief would not be, under the circumstances."

[This is a philosophical description of so-called feminine intuition.]

"But from my current evidential standpoint the emotion would seem to be best explained by own uneasiness."

 [She's implying that affects arise in and for themselves, that they have their own reasons which language may ignore.]

"If we are to address the question of whether emotions add anything of value to beliefs...it would be well to avoid transferring their intentional content to some attenuated notion of belief. I shall eventually argue...that what emotions add to beliefs depends on their partial justification in extraevidential terms--in terms of practical `adaptiveness', or a kind of instrumental value that is not properly brought to bear on assessments of belief warrant...In general, it seems that belief is just one propositional attitude among others. In the present case we may speak of the subject as `feeling as though' X is untrustworthy--meaning `feeling' in the broad sense indicated earlier, not so easily picked apart from thought and possibly involving belief. Where it amounts to an emotion, though, it also involves an affective state directed towards the corresponding evaluative proposition, which may be held in mind without assent."

"This example should indicate how the special motivational force ascribed to emotions depends on my account of emotional discomfort as object-directed and hence as serving to hold an evaluation in mind more reliably than beliefs and objectless sensations."

"On the view proposed here, the emotion itself serves as a reason for action insofar as it yields discomfort about an action requirement...My discomfort apparently will continue unless and until I satisfy the requirement; so its adds a rationao motive for action to that provided by affect-less thought and desire, even in combination wioth affective emotion symptoms."

In connection with "moral motivation", the author distinguishes between two types of emotional justification: (1) "emotional appropriateness, taken as implying a kind of `backward-looking' justification"; and (2) "practical adaptiveness" as a sort of "`forward-looking' justification, or justification by consequences--in particular by the role of emotions as spurs to action", but then she says" "In fact, though, my account of the notion of appropriateness will turn out to rest on general adaptiveness...For the perceptual situation to which an emotion is appropriate justifies it only as an adequately grounded response. It is one that a subject may quite rationally forgo, if he can--in favor of belief, where belief is warranted, or some propositional attitude short of belief but not involving comfort or discomfort...An evaluation, especially one from which I withhold belief, is unlikely  to have the same grip on my behavior inte absence of negative feeling tone; and it is this fact that lets us complete our justification of the emotion. Appropriateness is not enough to mandate feeling, in short. Where there are no moral "reasons to feel"...we need to bring in practical adaptiveness."

"My discussion of motivational force...will treat the self-interested standpoint as a rational basis for explaining some forms of altruistic motivation via the notion of identificatory emotion."

"Intellectual concern for others may be sufficient in many cases for action on their behalf; but I hope to make out emotional concern, and especially identificatory emotion, as socially adaptive partly because it reinforces moral reasons with self-interest."

"Unless an emotion is practically adaptive in the particular case at hand, that is, there is no reason--apart from moral obligations, and the like--why one ought to feel it instead of simply holding in mind its propositional content without affect. But sometimes there is a `reason to feel'; and the core of my argument here will consist in an attempt to pin it down--to explain how emotions may be justified by their special role in rational motivation. THe argument should serve to counter a long-standing philosophical tendency to dismiss the affective aspect of emotion--and with it, I would say, emotion--as at most a link in some causal chain that leads from belief to action. On the view I shall defend here, the intentional relation of affect and evaluation makes some emotions count as motivating reasons for action, whether or not they stand in the usual relation to belief."

Ground or grounds

"It is frequently alleged that rationalist philosophers like Spinoza failed to distinguish between a cause of an event and a ground for the truth of a proposition." (Dummett)

"The law may indeed be part of the grounds for the explanation, and failure to produce a law may expose one to the charge of having advanced a groundless explanation...But then there are many distinct kinds of ground for any explanation." (Danto)

"Ground" can be understood as:
--the basis for something else to happen;
--a principle, norm, or law for interpretation;
--the proof or argument for a principle, norm, or law;
--a subject of study.
The principle of identity offers grounds for the biconditionality of mind and brain. The grounds of history refers to the principles to grasp and interpret history.

Guilt

Anxiety is the propositional/physical pain associated with a threat to the specific self. Guilt is a type/token of anxiety. There are two forms of guilt. Moral guilt is anxiety over the infringement of social norms. Psychological guilt can have different sources. The most common is the anxiety that arises from the compulsion to achieve perfection. Self-loathing stems from the failure to achieve perfection. Psychological guilt and self-hatred are exactly the same. There is also the transference of guilt.

Neurosis is an internal replication of the impotent perception of paradox. The neurotic child sees love and hatred conjoined, and he does not have the means to penetrate this seeming irrationality, while those who embody the paradox for him are generally not aware of its existence.

The child of Auschwitz does not feel guilt. The specific self despite whatever self-loathing it might harbour wants the protection of secure material surroundings. When it thinks that these are crumbling, it fears for its life and reasons that if it can take the blame for whatever it is that is threatening its security the threat will go away. It takes on the guilt and if the threat is strong and protracted enough the guilt becomes ingrained and makes the guilt anxiety all the more intense.

Regret always accompanies guilt. Psychological guilt self-breeds. It is inherently recurrent. The specific self burdened by psychological guilt, i.e., self-loathing, will be recurrently beset by anxiety. Guilt is the result of subconscious cognitive processes and as such it is inevitable. The inevitability of anxiety produces the sense of impotence and impotence enhances regret and anxiety. Guilt usually inolves a witnessing. What is witnessed in psychological guilt is that the specific self fails when it was in its power to achieve. Paradoxically, impotence, which should mitigate guilt, serves to enhance regret, hence guilt and anxiety.

Logic is complete at birth. The strength of logic is proportional to the growth of memory or experience. There is a phase in the individual history of logic when it is indiscriminate in the acceptance of premises. It is the indiscriminate acceptance of premises that allows psychological guilt--the traits that produce self-loathing--to become part of the specific self.