Maybe even think of it this way: you know how to do plus or quus in the way you know how to ride a bike, not in the way you know that Sydney is in Australia. — Banno

Might just leave it here. :smile:

There's no fact regarding which rules. It's a mind bender for sure. — frank

Yes, it can bend your mind. But it doesn't have to. Plus and quus are the same in some instances and not in others. So you can tell which is being followed, provided you consider the full scope of the rule, not just a selected part of it.

There's a classic piece of misdirection going on here. Kripke keeps saying "there's no fact of the matter" and he means that there's no fact of the matter as long as we think only of applications where x,y < 57.

His rule is "x ⊕ y = x + y, if x, y < 57 and = 5 otherwise". It reads rather differently if you write it out in full. Then it is "x ⊕ y = x + y, if x, y < 57" and "x ⊕ y = 5, if x, y => 57".

Don't just pay attention to what conjurers are drawing your attention to. Pay attention to what they are trying to get you to ignore.

There is no fact of the matter if a fact is something we discover. Not if a fact can be something we do. You know how to do plus, as opposed to quus. If you want, you might say that it is a fact that you do 2 plus 2 and not 2 quus 2 — Banno

I don't think we really conflict. I do want to say that it is a fact that someone doing 2+2 is doing something different from someone who is doing 2⊕2. It is true that there is no difference in that application. But if you consider the range of the applications, the full facts of the matter become apparent. To consider that individual case or even a limited range of cases is misdirection.

If you and I are walking down Main Street, and I am going from A to B, but you are going from A to C, our different journeys are not apparent. You have to consider the whole journey to see the difference.

I was writing this while you were writing your comment. I'm happy to leave it here. :smile:

I'm happy to leave it here. — Ludwig V

:up:

↪Joshs Seems we pretty much agree, except that I don't think calling this an "intuition" is at all helpful, since it hints at private mental phenomena. It's not about intuition, it's about action - following a rule is something we do, not a "special sense — Banno

I agree. Intuition isn’t really what I was after. Wittgenstein said it better.

213. "But this initial segment of a series obviously admitted of various interpretations (e.g. by means of algebraic expressions) and so you must first have chosen one such interpretation."—Not at all. A doubt was possible in certain circumstances. But that is not to say that I did doubt, or even could doubt. (There is something to be said, which is connected with this, about the psychological 'atmosphere' of a process.) So it must have been intuition that removed this doubt?—If intuition is an inner voice—how do I know how I am to obey it? And how do I know that it doesn't mislead me? For if it can guide me right, it can also guide me wrong. ((Intuition an unnecessary shuffle.))

… It would almost be more correct to say, not that an intuition was needed at every stage, but that a new decision was needed at every stage.

But I don't think that "invent" is the appropriate description. The story of the irrationals shows that when we set up the rules of a language-game (and that description of numbers is also an idealization), we may find that there are situations (applications of the rules) that surprise us. Hence it is more appropriate to say that we discover these. When these situations arise, we have to decide what to do, in the relevant context - note that there can be no rules, in the normal sense, about what decision we should make, so I would classify these decisions, not as arbtrary or irrational, but as pragmatic and so rational in that sense. — Ludwig V

As for "But I don't think that "invent" is the appropriate description...Hence it is more appropriate to say that we discover these." I guess I disagree, but not strongly. I like "invent" better because it underlines the fact that, as I see it, mathematics is a human invention, a language, and not a fundamental aspect of the universe.

As for the rest of the quoted passage, it seems a like very good description of how mathematics grew from counting to where we find it today. It's much better than the answer I gave @frank.

I like "invent" better because it underlines the fact that, as I see it, mathematics is a human invention, a language, and not a fundamental aspect of the universe. — T Clark

If we're talking about mathematics as a whole, I agree with you. I'm just suggesting that a bit of flexibility in our language within mathematics is helpful. The important point is that when we develop/invent rules and make decisions about how to apply them, we are not totally "in charge". Put it this way - our agreements can lead to undesired consequjences and disagreements, which need to be resolved. We don't invent those - we would much rather they didn't happen, so we don't invent them. We do resolve them. That's not a problem, in itself; it's just part of our practice.

As for the rest of the quoted passage, — T Clark

Thanks.

Can it be that it it is the concept of "beyond our grasp" that is beyond our grasp?
(My old friend Ludvic suggested this to me.)

I agree. Intuition isn’t really what I was after. Wittgenstein said it better. — Joshs

Many of Wittgenstein's contemporaries said it better than Wittgenstein by formally distinguishing assertions from propositions. In particular, Frege introduced turnstile notation to make the distinction between propositions on the one hand, and assertions about propositions that he called judgements on the other.

If P denotes a proposition, then ⊢ P expresses a judgement that P holds true. Judgements can also be conditioned on the hypothetical existence of other judgements, written Q ⊢ P, where Q expresses a hypothetical judgement.

Notably, turnstile expressions don't denote truth values but rather practical or epistemic commitments, and the logical closure of such implications forms bedrocks of reasoning referred to as syntactic consequence. Of course, this does not preclude the possibility of such a collection of judgements from being treated as an object language, thereby allowing such judgements to be analysed, derived or explicated in terms of the finer-grained meta-judgements of a meta-language.

I presume the later Wittgenstein's remarks were not directed towards Frege or Russell - who essentially robbed the turnstile of philosophical significance by automating it, but at his earlier self who argued in the Tractatus that the turnstyle of logical assertion is redundant, due to thinking of propositions as unambiguous pictures of reality whose sense automatically conveyed their truth. But if this Tractatarian notion of the proposition is rejected, thereby leaving a semantic gap between what a proposition asserts and its truth value, then what does the gap signify and how must it be filled?

Evidently Frege was content to leave the gap unfilled and to signify it with a turnstile, and every logician since Russell has been content to build mathematics upon the turnstile by restricting the role of deduction to mapping judgements to judgements.

Logicians generally aren't bothered by the implication of infinite regress when explicating the judgements of object languages in terms of the meta-judgements of meta-languages, as aren't software engineers who often don't rely upon any meta-logical regression (with occasionally horrific consequences). but it apparently took Wittgenstein more time to feel comfortable with the turnstile and to reach a similar pragmatic conclusion.

Many of Wittgenstein's contemporaries said it better than Wittgenstein by formally distinguishing assertions from propositions. In particular, Frege introduced turnstile notation to make the distinction between propositions on the one hand, and assertions about propositions that he called judgements on the other. — sime

I consider the most important and radical implication of Wittgenstein’s later work to be his critique of Frege’s theory of sense as reference. Frege remained mired in a formalistic metaphysics centered on logic, without ever grasping f Wittgenstein’s distinction between the epistemic and the grammatical.

The important point is that when we develop/invent rules and make decisions about how to apply them, we are not totally "in charge". Put it this way - our agreements can lead to undesired consequjences and disagreements, which need to be resolved. We don't invent those - we would much rather they didn't happen, so we don't invent them. We do resolve them. That's not a problem, in itself; it's just part of our practice. — Ludwig V

In what way is the invention of a mathematical rule different from the creation of a language game/form of life? When Moore says ‘this is my hand’, Wittgenstein argues that he confuses an empirical assertion with a grammatical proposition. Moore’s gesture is pointing to the grammar , the rules, of a language game that Moore ‘inherited’ from his entanglement with his culture, but which rules are invisible to him. Moore ‘discovers’ that this is his hand, but doesn’t realize that his discovery only makes sense within the language game. Isnt this form of life an invention, but one that Moore was not ‘in charge of’? Couldn’t we say that scientific paradigms are invented , and the facts that show up within them are discovered?

I had a fascinating response where I was going to argue the arbitrary significance of pi. It was going to be based upon a pi based numeric system, where I would heroically show that decimal based systems would then fall to the same irrationality as pi once we standardized the pi system.

But I found out other folks much smarter than me have shown pi based numeric systems don't work like that. https://math.stackexchange.com/questions/1320248/what-would-a-base-pi-number-system-loosystem.

But it occupied my mind through a boring conference, so there's that.

I consider the most important and radical implication of Wittgenstein’s later work to be his critique of Frege’s theory of sense as reference. — Joshs

A critique of Frege's theory of sense and reference by Wittgenstein isn't possible, because Frege never provided an explicit theory or definition of sense. Frege only demonstrated his semantic category of sense (i.e. modes of presentation) through examples. And he was at pains to point out that sense referred to communicable information that leads from proposition to referent - information that is therefore neither subjective nor psychological. Therefore Fregean sense does not refer to private language - a concept that Frege was first to implicitly refer to and reject - but to sharable linguistic representations that can be used.

The later Wittgenstein's concept of language games, together with his commentary on private language, helps to 'earth' the notion of Fregean sense and to elucidate the mechanics of a generalized version of the concept, as well as to provide hints as to how Frege's conception of sense was unduly limited by the state of logic and formal methods during the time at which Frege wrote.

Frege - the first ordinary language philosopher? ;-)

Frege remained mired in a formalistic metaphysics centered on logic, without ever grasping f Wittgenstein’s distinction between the epistemic and the grammatical. — Joshs

Definitely not, for that makes it sounds like Frege was a dogmatic contrarian as opposed to the innovative and respectable founder of analytic philosophy - apparently the only thinker for whom Wittgenstein expressed admiration. As previously mentioned, Frege had already distinguished the epistemic from the grammatical when he introduced the turnstile. He knew the maxim "garbage in, garbage out".

Yet Frege's perception of propositions having eternal truth suggests that Frege might have been dogmatically wedded to classical logic that has no ability to represent truth dynamics. Indeed, I suspect that the later Wittgenstein's anti-theoretical stance was not a reaction against logic and system-building per-se, but a reaction against the inability of propositional calculus and first-order logic to capture the notion of dynamic truth and intersubjective agreement - a task that requires modern resource sensitive logics such as linear logic, as well as an ability to define intersubjective truth or "winning conditions", as exemplified by Girard's Ludics that breaks free from Tarskian semantics.

A critique of Frege's theory of sense and reference by Wittgenstein isn't possible, because Frege never provided an explicit theory or definition of sense. Frege only demonstrated his semantic category of sense (i.e. modes of presentation) through examples. And he was at pains to point out that sense referred to communicable information that leads from proposition to referent - information that is therefore neither subjective nor psychological — sime

If not subjective nor psychological, then what? Grounded in empirical objectivity? You think Wittgenstein understands sense to be grounded by reference to facts that transcend the normativity of language-games?

I'm just suggesting that a bit of flexibility in our language within mathematics is helpful. The important point is that when we develop/invent rules and make decisions about how to apply them, we are not totally "in charge". Put it this way - our agreements can lead to undesired consequjences and disagreements, which need to be resolved. We don't invent those - we would much rather they didn't happen, so we don't invent them. — Ludwig V

That makes sense.

But it occupied my mind through a boring conference, so there's that. — Hanover

I draw cartoons of the speakers going "blah blah blah."

Cheers.

If not subjective nor psychological, then what? — Joshs

Again, perhaps it's about what we do, how we act as members of a community.

Frege never provided an explicit theory or definition of sense. Frege only demonstrated his semantic category of sense (i.e. modes of presentation) through examples. — sime

Perhaps there was good reason for this - that sense might be shown but not stated, if in being state it ceases to be intensional, becoming extensional.

Amusing and informative article on NY Times about 'Pi Day'. Gift Link.

Can it be that it it is the concept of "beyond our grasp" that is beyond our grasp?
(My old friend Ludvic suggested this to me.) — unenlightened

I think that's about right. To continue the metaphor, though, I'd say that we're grasping for something we sense but do not know where it's at -- such as when we feel a vibration through water of a liferaft being thrown to us. It's just out of grasp and yet we have a sense of where that's at without having a grasp of it.

Second page, and still no pi/pie joke... — Banno

All righty then, I'll give it a go.

There's the pivotal pie scene in the original movie American Pie, for anyone who wants to take a poke.

One could grasp the pie in one sense, physically that is, but in another sense the pie event is un-graspable, in the sense of intelligibility ... thereby making many of us laugh at first seeing the movie.

Then there's the movie Pi. Which can also be grasped and not grasped at the same time. But that one isn't as funny.

For those who haven't seen American Pie:

In what way is the invention of a mathematical rule different from the creation of a language game/form of life? — Joshs

That's a very hard question to answer. My best short answer is, I think, that what I'm saying is meant as a refinement of what Wittgenstein said, not a contradiction. So I'm pretty sure that the distinction between invention and discovery here (in mathematics) can be expected to apply (be useful) wherever we are talking/thinking about rules, language games, practices and forms of life. (Is it forms of life, or ways of life? I'm not sure). More than that, it is reflected in philosophy, as competing theories about mathematics. I've come to the tentative conclusion that neither realism nor constructivism are true, though both have some truth.
The difference is that a language game consists of rules (at least one, and often more), so one can add or modify one of the rules of the game without thereby necessarily creating a new game. I don't pretend that any of the relevant concepts (rule, game, practice, form/way of life) are well-defined. But I'm inclined to think that's a feature, not a bug.

When Moore says ‘this is my hand’, Wittgenstein argues that he confuses an empirical assertion with a grammatical proposition. — Joshs

Yes, but isn't there a rider here, in that W eventually sees the distinction between empirical assertion and a grammatical remark as a matter of what sentences/statements/propositions are used to do - (which, after all, is what meaning means). So "This is red" can be an empirical proposition and an ostensive definition.

Moore’s gesture is pointing to the grammar , the rules, of a language game that Moore ‘inherited’ from his entanglement with his culture, but which rules are invisible to him. Moore ‘discovers’ that this is his hand, but doesn’t realize that his discovery only makes sense within the language game. — Joshs

Well, "discovers" is a bit odd here. What could count as Moore not knowing that that this is his hand? (I can imagine circumstances in which we might not realize that that is his hand, but they are quite special.) However, Moore thinks he is making an empirical statement and that's not wrong. But it seems to leave (does leave) room for sceptical doubt. Wittgenstein wants to eliminate doubt, so I take him to be pointing out that this case, when we attend to it properly, also draws our attention to the conditions for the possibility of doubt.
I sometimes think that Witgenstein was a bit condescending to Moore, though to be condescended to by Wittgenstein is something to be proud of. Moore found a game-changing move against sceptism, even if he didn't have the philosophy to press it home. Nor did Wittgenstein at the time.

Isnt this form of life an invention, but one that Moore was not ‘in charge of’? — Joshs

Philosophers almost always speak as if we are in charge (control) of language - and practices. (I think they hesitate a bit about "forms of life" and that does seem to gesture at something that we are lumbered with, rather than something we invent or are in charge of). But we learn language as something given - how could we not? After we have learnt language we realize, with Humpty-Dumpty's remark in Alice (in Wonderland or through the looking-glass? I don't remember.) that "Words mean what I want them to mean. It's a question of who's in charge." But although in practice we can modify language in some ways, much (most) of what goes on is not under anybody's control. Words don't mean what I, or anybody else, wants them to mean, even though thousands, even millions, of individual decisions make up what goes on.
So it's complicated.

Couldn’t we say that scientific paradigms are invented , and the facts that show up within them are discovered? — Joshs

Mathematics etc. are not quite the same kind of thing as our everyday conceptions of the world. They are more "artificial" than natural language. So I'm happy to agree that we can and we should say exactly that. But I'm after a third category. Our agreement about how to apply a rule defines the rule. So you would think that no difficulty could arise. But sometimes we don't agree, and sometimes our rule throws up peculiar results. (And we can agree when either of those things happen). Negotiation is necessary - changes to the rules, additional rules, etc. These situations do not neatly fit into the usual disctingction between the rules (concepts) and applications of the rules (experience).

I hope at least some of that is helpful or at least not unhelpful.

I'm unsure why this post hasn't gotten any replies, because this gets at the heart of the matter for why pi continues indefinitely.

A perfect circle simply doesn't exist. It can't be made by man, and not by machine. We can get close, but no matter how close we get, it will never be perfect, much like how a digital rendition of an analog signal can also never be perfect.

If we 'zoom in' one pixel (or one decimal) further, the imperfection shows. — Tzeentch

I.E. its a conceptual game of identifying 'gaps' in a systemic series of abstractions with governing rules then giving that 'gap' a new symbol as well as new rules as to how to manipulate these 'gaps'.

@Banno
Isn't this similar to how in ordinary speech there is no problems that we find with talk about holes or absences but philosophers get tied up in knots thinking about them while we use such a concept regardless to great pragmatic benefit?

In principle you could create a nominalism about terms relating to negative notions, absences, or holes but it would be just more clunky. Just as constructive and nominalist approaches (especially finitists) are.

. . . and the kicker. . . THEY ALL AGREE WITH EACH OTHER!

The nominalist, platonist, finitist, constructivist, etc. They will all agree that if we mean a number by a finite writable or computable series of symbolic construction, derivation, or representation then 'real' numbers are in fact NOT real. At least they wouldn't be numbers.

We could, however, call them something different like computational holes. . . then give them a symbol. . . and do some axiomatic derivation as to how holes combine (. . . are manipulated) or whether we get 'actual' numbers out. . . etc. . . etc. . .