Flies, Fly-bottles, and Philosophy

Richard B

“For my own part, I am not at all affronted by the comparison of myself, when I am at grips with a philosophical problem, to an insect that is seemingly entrapped. If I have any grievance against Wittgenstein in this regard, it is that, in my case at least, he fails to achieve his aim; he does not reveal to me the avenue of escape.” From A.J Ayer’s “Wittgenstein”

Ayer is commenting on Wittgenstein’s celebrated gnomic utterances, “What is your aim in philosophy? To show the fly the way out of the fly-bottle.” (PI 309) How does Wittgenstein wish to show the philosopher “out of the fly bottle.” There is an assortment for passages in Philosophical Investigations (PI) that outlines his approach:

PI 7 “….I shall also call the whole, consisting of language and the actions into which it is woven, the “language-game”

PI 23 “…Here the term language-game” is meant to bring into prominence the fact that the speaking of language is part of an activity, or of a form of life.”

PI 124 “Philosophy may in no way interfere with the actual use of language; it can in the end only describe it. For it cannot give it any foundation either. It leaves everything as it is.”

PI 126 “Philosophy simply puts everything before us, and neither explains nor deduces anything. Since everything lies open to view there is nothing to explain. For what is hidden, for example, is of no interest to us.”

PI 130 “Our clear and simple language-games are preparatory studies for future regularization of language--as it were first approximations, ignoring fiction and air-resistance. The language-games are rather set up as objects of comparison which are meant to throw light on the facts of our language by way not only of similarities, but also of dissimilarities.”

However, I believe Wittgenstein overlooks another approach to philosophy that is not just descriptive, but normative. An approach that it shares more with mathematics. Consider the following passage in Remarks on the Foundations of Mathematics (RFM) which uses the same celebrated utterance of a fly in a fly-bottle. In surrounding paragraphs, Wittgenstein provides diagrams where he invites the reader to imagine how someone might be puzzled on how two parallelograms and two triangles can be made into a rectangle, or showing how four triangles can be made into a rectangle. He goes on to say,

RFM Part I 44 “Let us imagine the physical properties of the parts of the puzzle to be such that they can’t come into the desired position. Not, however, that one feels a resistance if one tries to put them in this position; but one simply tries everything else, only not this, and the pieces don’t get into this position by accident either. This position is as it were excluded from space. As if there were e.g. a ‘blind spot’ in our brain here. –And isn’t it like this when I believe I have tried all possible arrangements and have always passed this one by, as if bewitched? Can’t we say: the figure which shows you the solution removes a blindness, or again changes your geometry? It as it were shows you a new dimension space. (As if a fly were shown the way out of the fly-bottle.)”

For Wittgenstein, the mathematician is an inventor not a discoverer, and mathematical proposition are normative.

RFM Part I 32 I might also say as a result of the proof: “From now on an H and a P are called ‘the same in number’” Or: The proof doesn’t explain the essence of the two figures, but it does express what I am going to count as belonging to the essence of the figures from now on.—I deposit what belongs to the essence among the paradigms of language. The mathematician creates essences.”

That said, why should philosophy not have a normative role as well. Maybe we are not discovering what the essence of “Truth”, “Knowledge”, or “Free Will”, but philosophers are inventing what these terms ought to mean.

Shawn

That said, why should philosophy not have a normative role as well. Maybe we are not discovering what the essence of “Truth”, “Knowledge”, or “Free Will”, but philosophers are inventing what these terms ought to mean. — Richard B

Yes, well Wittgenstein was a methodological nominalist, which in regards to common sense we might not be suited to answer the questions you ask about in terms of common sense.

For Wittgenstein, psychology attempts to relate what we experience with physical things, but in fact, we are just relating what we experience with experiences themselves. So, science aside I think we should focus on the problems philosophy ought to or is able to provide a way out for the fly.

180 Proof

[W]hy should philosophy not have a normative role as well. — Richard B

Insofar as one reflectively reasons in order to critique and interpret norms (i.e. rules, criteria, methods, conventions, customs, givens), philosophy is performative. To say, for example, 'one ought to philosophize' does not seem a philosophical statement.

Tarskian

For Wittgenstein, the mathematician is an inventor not a discoverer, and mathematical proposition are normative. — Richard B

Wittgenstein considered his contribution to the philosophy of mathematics to be his chief contribution:

https://en.wikipedia.org/wiki/Ludwig_Wittgenstein%27s_philosophy_of_mathematics

Ludwig Wittgenstein considered his chief contribution to be in the philosophy of mathematics, a topic to which he devoted much of his work between 1929 and 1944.

Wittgenstein has, however, gone into history as someone who does not understand mathematics particularly well:

https://philpapers.org/archive/FLOOSW.pdf

Wittgenstein's remarks on the first incompleteness theorem 1 have often been denounced, and mostly dismissed. Despite indirect historical evidence to the contrary," it is a commonplace that Wittgenstein rejected Godel's proof because he did not, or even could not, understand it.

Wittgenstein's take on the matter was rejected unanimously:

https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems

On their release, Bernays, Dummett, and Kreisel wrote separate reviews on Wittgenstein's remarks, all of which were extremely negative.[38] The unanimity of this criticism caused Wittgenstein's remarks on the incompleteness theorems to have little impact on the logic community.In 1972, Gödel stated: "Has Wittgenstein lost his mind? Does he mean it seriously? He intentionally utters trivially nonsensical statements", and wrote to Karl Menger that Wittgenstein's comments demonstrate a misunderstanding of the incompleteness theorems writing:

It is clear from the passages you cite that Wittgenstein did not understand [the first incompleteness theorem] (or pretended not to understand it). He interpreted it as a kind of logical paradox, while in fact is just the opposite, namely a mathematical theorem within an absolutely uncontroversial part of mathematics (finitary number theory or combinatorics).[39]

In my opinion and based on what he wrote in his "Remarks on the Foundations of Mathematics", Wittgenstein was just confused. In my opinion, Wittgenstein did not understand model theory either:

https://plato.stanford.edu/entries/wittgenstein-mathematics/

From this it follows that all other apparent propositions are pseudo-propositions of various types and that all other uses of ‘true’ and ‘truth’ deviate markedly from the truth-by-correspondence (or agreement) that contingent propositions have in relation to reality. Thus, from the Tractatus to at least 1944, Wittgenstein maintains that “mathematical propositions” are not real propositions and that “mathematical truth” is essentially non-referential and purely syntactical in nature.

For example, the notion of truth in Peano arithmetic theory is defined as correspondence with the set-theoretical structure of the natural numbers. This is an abstract, Platonic reality and not the physical reality, but regardless, truth is still based on correspondence. Hence, arithmetical truth is not syntactical in nature.

Every time I have read something that Wittgenstein has written about mathematics in which he commits himself to a verifiable claim, it turns out to be simply wrong. Hence, Wittgenstein's contribution to the philosophy of mathematics is mostly ... confusion.

Joshs

. This is an abstract, Platonic reality and not the physical reality, but regardless, truth is still based on correspondence — Tarskian

Could you provide your own critique of Platonic explanations of the mathematics, lie that of Goedel, or the correspondence theory of truth? This might shed more light on where you think Wittgenstein went wrong.

Tarskian

Could you provide your own critique of Platonic explanations of the mathematics, lie that of Goedel, or the correspondence theory of truth? This might shed more light on where you think Wittgenstein went wrong. — Joshs

Wittgenstein wrote the following "notorious paragraph" on Gödel's first incompleteness theorem in his "Remarks on the Foundations of Mathematics":

https://en.wikipedia.org/wiki/Remarks_on_the_Foundations_of_Mathematics

Wittgenstein wrote

I imagine someone asking my advice; he says: "I have constructed a proposition (I will use 'P' to designate it) in Russell's symbolism, and by means of certain definitions and transformations it can be so interpreted that it says: 'P is not provable in Russell's system'. Must I not say that this proposition on the one hand is true, and on the other hand unprovable? For suppose it were false; then it is true that it is provable. And that surely cannot be! And if it is proved, then it is proved that it is not provable. Thus it can only be true, but unprovable." Just as we can ask, " 'Provable' in what system?," so we must also ask, "'True' in what system?" "True in Russell's system" means, as was said, proved in Russell's system, and "false" in Russell's system means the opposite has been proved in Russell's system.—Now, what does your "suppose it is false" mean? In the Russell sense it means, "suppose the opposite is proved in Russell's system"; if that is your assumption you will now presumably give up the interpretation that it is unprovable. And by "this interpretation" I understand the translation into this English sentence.—If you assume that the proposition is provable in Russell's system, that means it is true in the Russell sense, and the interpretation "P is not provable" again has to be given up. If you assume that the proposition is true in the Russell sense, the same thing follows. Further: if the proposition is supposed to be false in some other than the Russell sense, then it does not contradict this for it to be proved in Russell's system. (What is called "losing" in chess may constitute winning in another game.)

Wittgenstein mishandled Gödel's witness:

P <-> not provable([P])

By the way, first of all, P could be also be undecidable. We should not simply assume that the problem would necessarily be decidable (true or false). Otherwise, our approach could possibly constitute abuse of the law of the excluded middle.

Next, if P is true then P is not provable.
If P is false then P is provable.

Hence, P is [1] undecidable, or [2] true and not provable, or [3] false and provable.

In fact, we don't know what the actual truth status is of P. That is also not necessary.

Gödel's incompleteness theorem states that there exist in Peano arithmetic (PA) logic sentences that are undecidable, or, true and not provable, or, false and provable. Hence, in constructivist terms, P is indeed a legitimate witness for Gödel's theorem, making his theorem intuitionistically unobjectionable.

Hence, there is nothing wrong with Gödel's witness.

When Wittgenstein wrote:

Just as we can ask, " 'Provable' in what system?," so we must also ask, "'True' in what system?"

Gödel's work is about "provable from PA" and therefore "true in the natural numbers" (as well as all other nonstandard models of arithmetic).

When Wittgenstein wrote:

If you assume that the proposition is provable in Russell's system, that means it is true in the Russell sense, and the interpretation "P is not provable" again has to be given up.

Wittgenstein assumes the soundness of PA ("Russell's system"), i.e. provable implies true.

Gödel's theorem does not assume neither the consistency nor the soundness of PA. The theorem states that "There possibly exist false statements that are provable", i.e. are inconsistent, and also "There possibly exist true statements that are not provable", i.e. are incomplete. So, PA is possibly inconsistent and/or possibly incomplete. The theorem does not say which one it is. It could even be both.

While it is perfectly fine to assume PA's consistency in (ordinary) mathematics, it is not good practice to assume it in metamathematics, where it is often part of the question at hand, such as in Gödel's theorem.

In fact, Gödel proves in his second incompleteness theorem that if PA can prove its own consistency, then PA is necessarily inconsistent. Wittgenstein was clearly also not aware of Gödel's second incompleteness theorem. Consistency was even more the question and not a valid assumption in Gödel's second incompleteness theorem.

In my opinion, Wittgenstein's remarks on Gödel's theorem are confused. He did not point out a problem with Gödel's theorem. What problem in that case? Wittgenstein rather pointed out a problem with his understanding.

Tarskian

This might shed more light on where you think Wittgenstein went wrong. — Joshs

In fact, Gödel's first incompleteness theorem trivially follows from Carnap's diagonal lemma. If you want to attack Gödel's theorem, you can pretty much only do that by pointing out a gap in the proof for the diagonal lemma or by pointing out that the lemma does not apply because isProvable(n) is not a legitimate predicate in PA. Wittgenstein did not do that. Instead, Wittgenstein struggled somewhat with his own flawed interpretation of Gödel's theorem without pointing out a legitimate flaw in the proof.

tim wood

Insofar as one reflectively reasons in order to critique and interpret norms (i.e. rules, criteria, methods, conventions, customs, givens), philosophy is performative. To say, for example, 'one ought to philosophize' does not seem a philosophical statement. — 180 Proof

:100:
Fly in a bottle? An education in a paragraph.

Banno

↪Tarskian

Interesting that you cite the paper in which Floyd argues that Wittgenstein had a much better grasp of Gödel than is often supposed.

The underlying point of Wittgenstein's remarks on Godel is the underlying theme of the later Wittgenstein as a whole: our sentences do not carry their meaning with them intrinsically, or in virtue of something present to the mind ahead of, or apart from, how we give it expression in particular cases. Rather, what we can clearly say about what we mean or think can be made sense of only from within the context of some practice, or ongoing system of use. — Juliet Floyd

And there is this:

A mathematician is bound to be horrified by my mathematical comments, since he has always been trained to avoid indulging in thoughts and doubts of the kind I develop. He has learned to regard them as something contemptible and… he has acquired a revulsion from them as infantile. That is to say, I trot out all the problems that a child learning arithmetic, etc., finds difficult, the problems that education represses without solving. I say to those repressed doubts: you are quite correct, go on asking, demand clarification! (PG 381, 1932) — Quoted in SEP article

I suspect that we might maintain his constructivism, but perhaps rescind his finitism in the light of the considerations of rule-following found in PI.

...philosophers are inventing what these terms ought to mean. — Richard B

Don't these terms - “Truth”, “Knowledge”, or “Free Will” - already have uses and meanings? So to my favourite quote form Austin:

First, words are our tools, and, as a minimum, we should use clean tools: we should know what we mean and what we do not, and we must forearm ourselves against the traps that language sets us. Secondly, words are not (except in their own little corner) facts or things: we need therefore to prise them off the world, to hold them apart from and against it, so that we can realize their inadequacies and arbitrariness, and can re-look at the world without blinkers. Thirdly, and more hopefully, our common stock of words embodies all the distinctions men have found worth drawing, and the connexions they have found worth making, in the lifetimes of many generations: these surely are likely to be more sound, since they have stood up to the long test of the survival of the fittest, and more subtle, at least in all ordinary and reasonably practical matters, than any that you or I are likely to think up in our arm-chairs of an afternoon—the most favoured alternative method. (Austin, J. L. “A Plea for Excuses: The Presidential Address”, Proceedings of the Aristotelian Society, 1957: 181–182)

Austin and Ayer had differing opinions on various topics.

Ludwig V

That said, why should philosophy not have a normative role as well. — Richard B

I don't understand the question. I think that's because norms are not an optional extra in Wittgenstein's philosophy. Perhaps they are not so clearly visible because he is showing through describing.

That collection of quotations was very thought-provoking. Wittgenstein seems to be saying, on one hand, that philosophy is all about description:-

PI 124 “Philosophy may in no way interfere with the actual use of language; it can in the end only describe it. For it cannot give it any foundation either. It leaves everything as it is.”
PI 126 “Philosophy simply puts everything before us, and neither explains nor deduces anything. Since everything lies open to view there is nothing to explain. For what is hidden, for example, is of no interest to us.” — Richard B

But on the other hand:-

What is your aim in philosophy? To show the fly the way out of the fly-bottle. — PI 309

So, a clear description shows the fly the way out. Making a clear description is itself a norm-governed activity, with its own norms and rules and criteria, which philosophy may (rightly) call into question. But these descriptions have a use in the service of the philosophical project - and their contribution to the project is also a criterion by which to judge their success. There is a goal, which (one presumes) is supposed to be a Good Thing. None of these norms are optional extras.
Note that the fly has to do some work as well. It has to grasp what Wittgenstein is showing it for itself. As he says in the introduction to PI:-

I should not like my writing to spare other people the trouble of thinking. But if possible, to stimulate someone to thoughts of his own.

But Wittgenstein has several goes at articulating his project:-

A philosophical problem has the form: ‘I don’t know my way about. — PI §123

There is not a philosophical method, though there are indeed methods, like different therapies. — PI §133

The philosopher treats a question; like an illness. — PI §255

The real discovery is the one which enables me to stop doing philosophy when I want to. The one that gives philosophy peace, so that it is no longer tormented by questions which bring itself into question. — PI § 133

What I give is the morphology of the use of an expression. I show that it has kinds of uses of which you had not dreamed. In philosophy one feels forced to look at a concept in a certain way. What I do is suggest, or even invent, other ways of looking at it. I suggest possibilities of which you had not previously thought. You thought that there was one possibility, or only two at most. But I made you think of others. Furthermore, I made you see that it was absurd to expect the concept to conform to those narrow possibilities. Thus your mental cramp is relieved, and you are free to look around the field of use of the expression and to describe the different kinds of uses of it. — Lectures of 1946 - 1947, as quoted in Ludwig Wittgenstein : A Memoir (1966) by Norman Malcolm, p. 43

He once greeted me with the question: 'Why do people say that it was natural to think that the sun went round the earth rather than that the earth turned on its axis?' I replied: 'I suppose, because it looked as if the sun went round the earth.' 'Well,' he asked, 'what would it have looked like if it had looked as if the earth turned on its axis?' This question brought it out that I had hitherto given no relevant meaning to 'it looks as if' in 'it looks as if the sun goes round the earth'. My reply was to hold out my hands with the palms upward, and raise them from my knees in a circular sweep, at the same time leaning backwards and assuming a dizzy expression. 'Exactly!' he said. In another case, I might have found that I could not supply any meaning other than that suggested by a naive conception, which could be destroyed by a question. The naive conception is really thoughtlessness, but it may take the power of a Copernicus effectively to call it in question. — G. E. M. Anscombe, An Introduction to Wittgenstein's Tractatus, Chap. 12

Whether we are expected to weld these together to form a consistent and complete whole or just regard them as variants on a theme, either to allow different methods for different problems or just to prevent mental cramp, I do not know. Of course, in the end, it is up to us to decide what we will do.

Insofar as one reflectively reasons in order to critique and interpret norms (i.e. rules, criteria, methods, conventions, customs, givens), philosophy is performative. To say, for example, 'one ought to philosophize' does not seem a philosophical statement. — 180 Proof

Does what I have said articulate what you mean here?

PI 130 “Our clear and simple language-games are preparatory studies for future regularization of language--as it were first approximations, ignoring fiction and air-resistance. The language-games are rather set up as objects of comparison which are meant to throw light on the facts of our language by way not only of similarities, but also of dissimilarities.” — Richard B

At first I thought this was inconsistent, harking back to the idea that logic could be the basis of an ideal language, free of all the dross that natural languages carry. But perhaps he doesn't mean re-forming, changing, language, but grasping the order that is already there.

Richard B

Wittgenstein has, however, gone into history as someone who does not understand mathematics particularly well: — Tarskian

Wittgenstein's take on the matter was rejected unanimously: — Tarskian

Based on what I read by your post, I would recommended a much more sympathetic reading of Wittgenstein's views on Godel's Theorem in the book "Godel's Theorem in Focus, Chapter VIII by S.G. Shanker's , "Wittgenstein's Remarks on the Significance of Godel's Theorem. As Wittgenstein said in RFM VII 19 "My task is, no to talk about (e.g.) Godel's proof, but to by-pass it. This is taking a position not as a mathematician, but as a philosopher. Not to critique the proof itself, but the metaphysical assumptions that some would claim, namely the platonists.

I will put forth some passages that I find that present this point:

"In Philosophical Remarks Wittgenstein insisted contra Hilbert that ' In mathematics, we cannot talk about systems in general, but only within systems. They are just what we can't talk about(PR 152). The argument as presented sounds dogmatic, but it follows from the preceding clarification of the meaning of mathematical propositions as determined by intraliguistic rules rather than a connection between language and reality. The point of this normative conception of mathematical propositions and proofs is to clarify that the meaning of a mathematical concept is not an object or 'configuration' but rather, the totality of rules governing the use of that concept in a calculus." Mathematical propostions are not about anything (in a descriptive sense) yet neither are they meaningless: they are norms of representation whose essence is to fix the use of concepts in empirical proposition."

and

"Godel's argument pushes us to accept that there are two versions of the same proportion which is true but unprovable in one system while true and unprovable in another. But the whole point Wittgenstein's argument on the autonomy of mathematics systems is that a mathematical proposition is internally tied to its proof/proof system:"If, then, we ask..."Under what circumstances is a proposition asserted in Russell's game?", the answer is: at the end of his proofs, or as a "fundamental law (Pp). There is no other way this system of employing asserted propositions in Russell's symbolism' (RFM I APP III 6).

and

"In short, the meaning of a mathematical proposition is strictly determined by the rules governing its use in a specific system. If dealing with autonomous calculi then no matter how similar the rules of the two systems might be, as long as they differ - as long as we are dealing with distinct mathematical systems - It make no sense to speak of the same proposition occurring in each. The most that can be concluded is that parallel propositions occur in the two systems which can easily be mapped onto each other. Hence Godel was barred by virtue of the logical grammar of mathematical proposition from claiming that he had constructed identical versions of the same mathematical proposition in two different systems."

Richard B

Don't these terms - “Truth”, “Knowledge”, or “Free Will” - already have uses and meanings? So to my favourite quote form Austin:

First, words are our tools, and, as a minimum, we should use clean tools: we should know what we mean and what we do not, and we must forearm ourselves against the traps that language sets us. Secondly, words are not (except in their own little corner) facts or things: we need therefore to prise them off the world, to hold them apart from and against it, so that we can realize their inadequacies and arbitrariness, and can re-look at the world without blinkers. Thirdly, and more hopefully, our common stock of words embodies all the distinctions men have found worth drawing, and the connexions they have found worth making, in the lifetimes of many generations: these surely are likely to be more sound, since they have stood up to the long test of the survival of the fittest, and more subtle, at least in all ordinary and reasonably practical matters, than any that you or I are likely to think up in our arm-chairs of an afternoon—the most favoured alternative method. (Austin, J. L. “A Plea for Excuses: The Presidential Address”, Proceedings of the Aristotelian Society, 1957: 181–182) — Banno

I see your quote and raise you three:

From Wittgenstein Blue Book "Philosophers very often talk about investigating, analysis, the meaning of words. But let's not forget that word hasn't got a meaning given to it, as it were, by a power independent of us, so that there could be a kind of scientific investigation into what the word really means. A word has the meaning someone has given to it."

From Quine, Word and Object, "There are, however, philosophers who overdo this line of thought, treating ordinary language as sacrosanct. They exalt ordinary language to the exclusion of one of its own traits: its disposition to keep evolving."

And lastly from the arm-chair itself to make my point, from John Rawls "Principles of Justice", "My aim is to present a conception of justice which generalizes and carries to a higher level of abstraction the familiar theory of the social contract as found, say, in Locke, Rousseau, and Kant. In order to do this we are not to think of the original contract as one to enter a particular society or to set up a particular government. Rather, the guiding idea is that the principles of justice for the basic structure of society are the object of the original agreement. They are the principles that free and rationale persons concerned to further their own interests would accept in an initial position of equality defining the fundamental terms of their association."

Tarskian

But the whole point Wittgenstein's argument on the autonomy of mathematics systems is that a mathematical proposition is internally tied to its proof/proof system — Richard B

Wittgenstein's view is not compatible with model theory of which the core understanding is that the provability of a arithmetical proposition is tied to one system (PA) while its truth is tied to another system (ZFC).

https://en.wikipedia.org/wiki/Model_theory

In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the statements of the theory hold).

The model-theoretical approach is necessary in order to discover that there exists arithmetical truth in ZFC that is not tied to a proof in PA. Wittgenstein subscribes to a syntactic notion of truth. Model theory subscribes to the semantic nature of truth.

In fact, Wittgenstein does not properly distinguish between provability and truth.

If dealing with autonomous calculi then no matter how similar the rules of the two systems might be, as long as they differ - as long as we are dealing with distinct mathematical systems - It make no sense to speak of the same proposition occurring in each. The most that can be concluded is that parallel propositions occur in the two systems which can easily be mapped onto each other. — Richard B

Wittgenstein simply rejects the essence of model theory, because mapping propositions in PA to propositions in ZFC is exactly what model theory does. It is exactly about "the same proposition occurring in each".

Hence Godel was barred by virtue of the logical grammar of mathematical proposition from claiming that he had constructed identical versions of the same mathematical proposition in two different systems. — Richard B

In their seminal paper "On interpretations of arithmetic and set theory" of 2006, Richard Kaye and Tin Lok Wong formally proved that PA and ZF-inf are indeed bi-interpretable and therefore that the mapping in model theory -- in and of itself -- does not entail any of the risks that Wittgenstein incorrectly believes to exist.

Bi-interpretability:

- Every logic sentence in PA can be mapped to an equivalent logic sentence in ZF-inf. (Von Neumann)
- Every logic sentence in ZF-inf can be mapped to an equivalent logic sentence in PA. (Ackermann)

https://web.mat.bham.ac.uk/R.W.Kaye/publ/papers/finitesettheory/finitesettheory.pdf

The work described in this article starts with a piece of mathematical ‘folklore’ that is
‘well known’ but for which we know no satisfactory reference.

Folklore Result. The first-order theories Peano arithmetic and ZF set theory with the
axiom of infinity negated are equivalent, in the sense that each is interpretable in the
other and the interpretations are inverse to each other.

Perhaps the first and most obvious conclusion is that statements concerning the equiv-
alence of ‘Peano Arithmetic’ and ‘ZF with the axiom of infinity negated’ require some
care to formulate and prove. It is certainly true that PA and ‘ZF with the axiom of infin-
ity negated’ are equiconsistent for just about any sensible axiomatisation of the latter,
in the sense that interpretations exist in both directions.6 Probably this is the ‘folklore
result’ that most people remember. But for the finer result with interpretations inverse
to each other, careful axiomatisation of the set theory is required. A category theoretic
framework for interpretations is useful to direct attention to these refinements.

It is true that until 2006, model theory had always assumed the bi-interpretability of PA and ZF-inf without formal proof or similar investigation. Gödel never formally proved this mapping either. It was rather being considered self-evident. It would actually have been valid criticism to point out that Gödel assumed bi-interpretability without proof. For Wittgenstein to outright reject bi-interpretability, however, was clearly one bridge too far.

Ludwig V

the meaning of a mathematical concept is not an object or 'configuration' but rather, the totality of rules governing the use of that concept in a calculus." Mathematical propositions are not about anything (in a descriptive sense) yet neither are they meaningless: they are norms of representation whose essence is to fix the use of concepts in empirical proposition. — Richard B

If one thinks about the various developments from, say, to Copernicus to Newton, "fixing the use of concepts in empirical propositions" seems like a more complicated process than this, and it might be thought to violate the purity of mathematical autonomy. The crucial step is the one from "mathematical hypothesis (which the theologians could accept) to description of reality, (which Newton's theory eventually achieved). True, the reality described was modified to accommodate this, but that itself raises questions about the autonomy of systems. I would prefer to say that the application of mathematical propositions to empirical propositions is an extension or development of their theoretical use. How could I rule out other extensions or developments?

Hence Godel was barred by virtue of the logical grammar of mathematical proposition from claiming that he had constructed identical versions of the same mathematical proposition in two different systems. — Richard B

Surely the question whether Godel had or had not achieved that aim is a question for mathematicians. But mathematicians disagree, (don't they?) and perhaps Wittgenstein counts as a mathematician. So the question does not have a determinate answer. That seems to me to be closer to what one might call the truth. I do not rule out the possibility that mathematicians might eventually devise rules for the use of the relevant concepts that would resolve the question. Fortunately, I am barred from attempting the project.

"In Philosophical Remarks Wittgenstein insisted contra Hilbert that ' In mathematics, we cannot talk about systems in general, but only within systems. They are just what we can't talk about(PR 152). The argument as presented sounds dogmatic, but it follows from the preceding clarification of the meaning of mathematical propositions as determined by intraliguistic rules rather than a connection between language and reality. — Richard B

Except when we come to applied mathematics, when that issue becomes central.
________________________

From Wittgenstein Blue Book "Philosophers very often talk about investigating, analysis, the meaning of words. But let's not forget that word hasn't got a meaning given to it, as it were, by a power independent of us, so that there could be a kind of scientific investigation into what the word really means. A word has the meaning someone has given to it." — Richard B

It is better to think that a word has the meaning someone has given to it than to think that the meaning of a word is an eternally existing (subsisting entity floating about in some alternative world. But at face value, for those of us using the words, that is simply false. We learn what words mean - we do not make it up; we discover what they mean (what the rules for its use are), or we do not learn to speak. So there can be a scientific investigation into what the word means - and how its meaning changes. To be sure, sometimes we know who gave a word its meaning, but even if it was coined by someone, its use is the result of a process of dissemination which is rarely documented and we do not altogether understand. But dictionaries often include remarks about it and it could be the object of a "scientific" investigation.

It is fair to say that Austin treats language as a tidy abstract structure, rather than a dynamic and messy collection of language-games, and this undermines his claim to be talking about ordinary language. But still, the resulting arguments are at least worth of serious consideration, so I'm not inclined to get too sniffy about his methodology.

From Quine, Word and Object, "There are, however, philosophers who overdo this line of thought, treating ordinary language as sacrosanct. They exalt ordinary language to the exclusion of one of its own traits: its disposition to keep evolving." — Richard B

"Overdo" is the right word, though whether it applies to specific texts is always going to be debateable. Ryle in "Dilemmas", as I recall, talks about technical and untechnical concepts and concepts that everyone uses whatever technical language they are using, rather than ordinary language.

from John Rawls "Principles of Justice", "My aim is to present a conception of justice which generalizes and carries to a higher level of abstraction the familiar theory of the social contract as found, say, in Locke, Rousseau, and Kant. In order to do this we are not to think of the original contract as one to enter a particular society or to set up a particular government. Rather, the guiding idea is that the principles of justice for the basic structure of society are the object of the original agreement. They are the principles that free and rationale persons concerned to further their own interests would accept in an initial position of equality defining the fundamental terms of their association — Richard B

The difficulty is that I don't trust myself to dispense with all my selfish interests during this imaginative exercise. It is rather easy to say that if I was a slave, I would accept my slavery because those are the rules. It is equally easy to say that if I was a slave, I would do my level best to escape, despite the rules. For my money, it is much better to start where we are. Other people may start in different places. When we disagree, we shall have to have an argument. That's how it works. How can Rawls' exercise help? Back to ordinary language?

Richard B

he difficulty is that I don't trust myself to dispense with all my selfish interests during this imaginative exercise. It is rather easy to say that if I was a slave, I would accept my slavery because those are the rules. It is equally easy to say that if I was a slave, I would do my level best to escape, despite the rules. For my money, it is much better to start where we are. Other people may start in different places. When we disagree, we shall have to have an argument. That's how it works. How can Rawls' exercise help? Back to ordinary language? — Ludwig V

My main point with this example is that Rawls is not looking to the ordinary use of "Just" to come up with his conception of "Justice" nor should he. Could you imagine taking Wittgenstein or Austin recommendation. Look at the ordinary use, look at the language game, the form of life; OK, for example, I live in an environment where "street justice" rules. As a little kid I watch my brother kill someone in front of my eyes and I say to him "Why did you do it" He says, "Because he look at me in a funny way, he disrespected me, so I killed him, it was Just." I understand its use, the action, and the context. Give me that "arm-chair" we can do better. And that goes with any term, maybe it will serve us better if we change it, add to it, subtract from it, etc.. And why can't a philosopher do this, instead of sitting around and describing how the term is actually used.

Ludwig V

↪Richard B

Don't get me wrong. My last question was a question because I don't think that "ordinary language" is the answer. As a matter of fact, I think that the philosophical practice of ordinary language philosophers was at variance with the rhetoric about ordinary language.

It's worth saying that the rhetoric of ordinary language was meant to distinguish their work from the predominately idealist philosophy that was the orthodox of many philosophers at the time, and the analytic and positivist revolutionaries who had emerged in opposition to them. There were good grounds for rejecting both and I would certainly resist returning to either.

And why can't a philosopher do this, instead of sitting around and describing how the term is actually used. — Richard B

There's no reason why not. Nussbaum, Rawls, Russell, and Singer come to mind as stellar examples. It seems to me that WIttgenstein's practice was also at variance from his remarks about just describing. In his case, the business about saying and showing gives some sort of explanation.

My main point with this example is that Rawls is not looking to the ordinary use of "Just" to come up with his conception of "Justice" nor should he. — Richard B

Nor did I mean to imply that he was. Criticizing Rawls doesn't mean that I think we should retreat to describing how the term is actually used. I rather think that the ordinary use of justice would almost certainly lead us to describe it as a term that is the ground of a battlefield, (intellectual and physical) rather than a coherent concept.

Give me that "arm-chair" we can do better. — Richard B

I realise you don't mean that literally, but here's the problem - who is "we"? That's not just a problem for ordinary language philosophy. It's a common usage in philosophy to say "we" say this and that or "we think" this and that.

OK, for example, I live in an environment where "street justice" rules. ........... I understand its use, the action, and the context. — Richard B

It's a very distressing story. It does indeed throws into high relief the simple points that the ordinary is not the same for everyone, and not necessarily justifiable. I have not the slightest inclination to argue against either. If only it were possible to establish an agreement without using force....

Count Timothy von Icarus

↪Ludwig V

It is better to think that a word has the meaning someone has given to it than to think that the meaning of a word is an eternally existing (subsisting entity floating about in some alternative world. But at face value, for those of us using the words, that is simply false. We learn what words mean - we do not make it up; we discover what they mean (what the rules for its use are), or we do not learn to speak. So there can be a scientific investigation into what the word means - and how its meaning changes. To be sure, sometimes we know who gave a word its meaning, but even if it was coined by someone, its use is the result of a process of dissemination which is rarely documented and we do not altogether understand. But dictionaries often include remarks about it and it could be the object of a "scientific" investigatio

What's interesting is that the bolded is true in two senses. First, there is etymological analysis, looking at old texts to determine how some term came to mean what it does. But second, there is looking into the actual physical referents of words to see what they are. So for instance, we know a lot of things about water that we didn't know in 1700. Even grade school kids know that water is H2O.

In the first example, we are talking about scientific inquiry into the history of social practices. In the second we're talking about the natural sciences. The natural sciences in turn often do shift what we even mean by our words. "Water" used to refer generally to any mass of mostly H2O, inclusive of all the stuff dissolved or floating around in it. We still definitely use the word in that sense, but it's also not uncommon today to use it to refer specifically to H2O, and to count anything else as a modifier ("salt water") or different substance. But that only makes sense, science effects how we think of things. Unlike Melville, we don't call whales fish anymore either, but that required knowledge of evolution to pin down decisively. Now preschoolers know "fish" doesn't refer to "whales." The evidence of genetic lineage ended up driving convention.

↪Richard B

And why can't a philosopher do this, instead of sitting around and describing how the term is actually used.

That was Marx's point on Feuerbach: "philosophers have only interpreted the world in various ways; the point is to change it!" -

Joshs

↪Count Timothy von Icarus

What's interesting is that the bolded is true in two senses. First, there is etymological analysis, looking at old texts to determine how some term came to mean what it does. But second, there is looking into the actual physical referents of words to see what they are. So for instance, we know a lot of things about water that we didn't know in 1700. Even grade school kids know that water is H2O. — Count Timothy von Icarus

And what about the etymology of terms like ‘actual’ and ‘physical’? If they undergo as much change as the terms for water , then isn’t a phrase like actual physical referent linguistically self-referential, belonging to the hermeneutic circle along with our changing terms for water, rather than sitting outside of it?

Count Timothy von Icarus

↪Joshs

Sure, but the two don't collapse into one thing. There is still a worthwhile distinction to make between etymology and looking through a microscope. They're phenomenologicaly distinct too. No one mistakes a rock or bee for a word as far as sensory experience goes.

Ludwig V

First, there is etymological analysis, looking at old texts to determine how some term came to mean what it does. But second, there is looking into the actual physical referents of words to see what they are. — Count Timothy von Icarus

Yes. We can discern in both practices what Derrida I believe calls the "wandering signifier". It doesn't half complicate philosophical analysis. We can also discern that "scientific" is not monolithic. We should not presuppose a single "scientific" method.

If they undergo as much change as the terms for water , then isn’t a phrase like actual physical referent linguistically self-referential, belonging to the hermeneutic circle along with our changing terms for water, rather than sitting outside of it? — Joshs

Yes. Terms like "actual physical referent" or "materialism" are increasingly difficult to use in philosophical discussion. That's one reason for doubting how useful the concept of a hermeneutic circle is. Language constantly seems to refer beyond itself, and our practices do not find it difficult to use those terms. Isn't that as good as it gets for defining an outside?

But it's not only science that creates complications. Our changing practices can do so all by themselves. Consider the history of "cash". "Cash" used to mean physical chunks of metal. Then it came to mean physical chunks of metal stamped by authority to certify their physical composition. Then it came to mean physical chunks of metal stamped by authority. Then bank notes..... Nowadays, I discern it coming to mean a credit balance to which I have instant access. It all makes the economy run quicker and more smoothly. Whatever next?

That was Marx's point on Feuerbach: "philosophers have only interpreted the world in various ways; the point is to change it!" - — Count Timothy von Icarus

That's right, of course. The question now is whether one can change the world from one's arm-chair. There's a lot of reason to say that one can. Of course, that might depend on what one regards as meaningful or real change. And yet, one needs a phrase to refer to idle speculation.

Joshs

↪Ludwig V

If they undergo as much change as the terms for water , then isn’t a phrase like actual physical referent linguistically self-referential, belonging to the hermeneutic circle along with our changing terms for water, rather than sitting outside of it?
— Joshs
Yes. Terms like "actual physical referent" or "materialism" are increasingly difficult to use in philosophical discussion. That's one reason for doubting how useful the concept of a hermeneutic circle is. Language constantly seems to refer beyond itself, and our practices do not find it difficult to use those terms. Isn't that as good as it gets for defining an outside? — Ludwig V

I’m not sure we’re understanding ‘hermeneutic circle’ the same way. The circle of discursive practice is not closed in on itself such that it is sealed off from the empirical world. On the contrary, language expresses material practices. For hermeneuticists like Gadamer, the circle indicates that we are neither spinning our wheels in a conceptual void nor at the behest of an independent empirical outside, but involved in a back and forth between lingustic conceptualization and a world that talks back to our concepts within the language that we use , and according to the questions we ask in to make sense of it.’ As Wittgenstein showed, words only have meaning in their actual use , exposed to and altered by fresh contexts of intersubjective situations.

Ludwig V

I’m not sure we’re understanding ‘hermeneutic circle’ the same way. — Joshs

No, we are not. But there are not dissimilar arguments in other quarters about the relationship of Language and Reality, which come to very different conclusions. Perhaps I should not have stuck my nose in. On the other hand, I shall have to look at Gadamer more closely. Thanks.

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