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

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

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

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.

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.

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]

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.

. 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. — Joshs

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.)

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

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.

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

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

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

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

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

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)

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

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

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

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.

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

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

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

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

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

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

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).

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

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

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.

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

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 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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. — Joshs