Is the number pi beyond our grasp?
What do we do with numbers like pi that go on forever? I can't deny that I live in a world where there are such shenanigans: numbers that can't be completed.
It's definitely not an aspect of counting, because I can't count to pi. I could say it's just a matter of arithmetic, but but what about that endless thing going on?
"The Philosophy of Pi
The allure of pi extends beyond the concrete realms of mathematics, science, and engineering, spilling over into the realm of philosophy. The infinite, non-repeating decimal expansion of pi provides a tantalizing metaphor for the endless pursuit of knowledge. It invites us to grapple with concepts of infinity, perfection, and the limitations of our understanding.
"The mysterious nature of pi is emblematic of the paradoxes inherent in the human condition. Just as we strive to know pi to ever-increasing decimal places without ever reaching an end, we humans are driven by an insatiable curiosity to understand our world, while fully aware that absolute understanding is likely beyond our grasp." --here
It's definitely not an aspect of counting, because I can't count to pi. I could say it's just a matter of arithmetic, but but what about that endless thing going on?
"The Philosophy of Pi
The allure of pi extends beyond the concrete realms of mathematics, science, and engineering, spilling over into the realm of philosophy. The infinite, non-repeating decimal expansion of pi provides a tantalizing metaphor for the endless pursuit of knowledge. It invites us to grapple with concepts of infinity, perfection, and the limitations of our understanding.
"The mysterious nature of pi is emblematic of the paradoxes inherent in the human condition. Just as we strive to know pi to ever-increasing decimal places without ever reaching an end, we humans are driven by an insatiable curiosity to understand our world, while fully aware that absolute understanding is likely beyond our grasp." --here
Comments (111)
Quoting frank
It's an irrational number after all.
Pi = the ratio of a circle's circumference to its diameter.
My mind tells me one of the main revelations of pi is the picture of the straight line of the diameter surrounded by the encircling circumference. This juxtaposition shows concisely that the rectilinearity (straight-lining) of science is only partially commensurable with the curvilinearity (curving) of nature.
The straight lines infinitesimal of the analysis of calculus can only approximate nature's reality.
Science is nature-adjacent rather than natural.
As technology diminishes and displaces nature, humanity rejiggers itself out of mysterious existence into self-reflection. The trick of AI and SAI is baking in a component of mystery and a component of error. Mystery and error support otherness, a component essential to forestalling the cognitive suffocation of an enclosing self-reflection.
Intentional mystery and error preserve the irrationality pictured by pi.
We must pull on and push against the idea our natural world is full mystery and error because some prior race of sentients understood the essential importance of forestalling cognitive suffocation. Having original sin in the mix is better than the damnation of perfection.
Against utopia!
Well, here we are, talking about ? - so, no, it is not beyond our grasp...
At least for some of us.
And what that AI describes as "the philosophy of Pi", isn't - any more than are the outbreaks of verse that sometimes litter these fora. Fluffy nonsense, like knowing the millionth digit of Pi. (5, according to Wolfram Alpha).
Just as the index finger can grasp the coffee cup by the little handle, the mind grasps pi by the bits we know.
I didn't realize that quote in the OP came from an AI. :lol: Don't worry, it was just a thought that popped up while I was cleaning house.
That's what I first thought -- and not just talking about pi, but knowing what we're talking about in saying pi.
Quoting ucarr
What else could "grasping" consist in such that we don't grasp pi in the manner @ucarr says above?
There could be something deep in there --
Quoting frank
I'd suggest we stop at the point we are satisfied, while knowing that the procedure can carry on.
Quoting Moliere
Yep.
Quoting Moliere
Stop which - the calculation, or the thread?
Well, it must be the calculation since the thread will never satisfy -- given how often we go past the point of explanation here :D
I'm guessing we all pretty much get the joke?
To keep my annoying persona, I must say both-and :D
I didn't mean that up front though. And find:
Quoting DifferentiatingEgg
A refreshing change of pace.
Quoting Banno
I'll take as the original joke. Not beyond our grasp, though there are some of us... -- it's a rimshot joke.
Though if forced I'd say that the litter of outbreaks in verse on these fora are closer to philosophy than the nonsense of the AI bots.
It's not a joke. It goes on forever, so you can never know it completely.
I didn't get that memo. Why not?
Yeah we can. the ratio of the radius to the circumference of a circle; that is it exactly and entirely. There are other ways to say the same thing, such as the aforementioned mentioned smallest positive number where the sine function is equal to zero or ?=ln(?1)/i from Euler's identity or Cd/2LP for Buffons Needle or any number of other neat-o calculations.
You are completely correct...
or not.
Well, that's another joke, because he sure fooled you and your homeboy Russel...
Funny thing is, if I'd started a thread that said we can know pi in its entirety, you would have said that ridiculous. :confused:
:lol:
True, that.
There's no way to know all the digits that go on it. I think you can express the concept by just saying "pi."
Interesting thing is that while we cannot know everything, there is (arguably) nothing in particular that we could not know.
Why not?
True. Nothing that we know about anyway.
:wink:
You asserted P
When I asked for your justification, you said
"Why not P?"
Does that sound rational to you?
It probably doesn't matter because, as you say, and as @Banno said, we grasp the concept.
Yep. It's an extension of "the world is all that is the case".
The thought that came to mind was how if the thread was posited this or the other way @Banno would say his bit, and I was thinking how it'd be right to say it -- whether it be against big knowledge claims or for small knowledge claims, it'd be right to point out those difficulties in relation to a philosophical question.
That's just blatant idealism.
Quoting Banno
Doesn't realize that shame/guilt doesn't work on someone who understands Nietzsche.
How rude.
:smile:
I agree, but feel like I shouldn't...
Welcome, brother! :D
To my circle of thinking that ends in . .. circles... of thinking......
Appropriate, given the topic...
Second page, and still no pi/pie joke...
There are infinitely more irrational numbers then there are rational ones, so it's not just pi. They're just regular old Joe Sixpack numbers. I guess we should get used to dealing with them. It is my understanding that all mathematics is based on counting, but there are many, many instances where it has gone beyond it.
Ah, "Fecal Freakal," I understand.
The Left Rights - Take a Shit
Oh, far more than just that... :nerd:
How did that happen? If it's based on counting, how did it give rise to things that can't be counted?
To reverse the usual formulation: God may be sufficient, but not necessary, for madness.
How did we get zero? How did we get negative numbers from natural numbers? How did we get rational numbers from integers? How did we get real numbers from rational numbers? How did we get complex numbers from real numbers? Humans invented them.
I can't see an alternative to saying that the numbers are based on counting (apart from some platonic story about how they always already exist, though not in this world).
Quoting T Clark
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.
Certainly, the eventual decision to simply incorporate the irrational numbers into the system of real numbers (I may not have expressed this quite correctly) was, in some sense, perfectly reasonable. In one way, order was restored in the world. In another way, the problem was simply labelled, rather than restored.
There is not one answer to your questions. We just need to pay attention to the actual, historical processes in each case, and give a more detailed description of what went on.
They were first discovered by the Pythagoreans. They were horrified by them though. They aggressively suppressed the knowledge of irrational numbers per legend. If it was an invention, it was not a welcome one. How do we explain that?
Quoting T Clark
This is a fascinating story involving the transcription of Babylonian abacus results.
This thread puts on display the way people try to escape from wonder. They assume a conclusion when they don't actually know any facts that support it. Psychic protection strategy?
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.
One third of 1 is 0.33333...........continuing to infinity.
If we altered our numbering system, such that we replaced 1 by 3, then one third of 3 is 1. This avoids any problem of infinity.
This suggests that the problem of infinity is an artificial problem of our numbering system.
Similarly with pi.
I don't understand how we could replace 1 by 3. That doesn't make any sense. But with the new numbering system, 1/3 would be 1/5.
If there is one object in the world, dividing it into three parts does not involve infinities. In our numbering system, dividing 1 by 3 does involve infinities.
This suggests that infinity is an artificial problem of our numbering system. Perhaps a different numbering system would avoid the problem of infinity altogether.
I was fascinated by this, but I couldn't find anything specifically on it, although there are many versions available. On the other hand, this version does refer to accountancy, which does seem to me a practical application that is bound to trip over both 0 and negative numbers. (Both are needed to represent the critical difference between debit and credit and neither.)
Scientific American - Zero
Quoting RussellA
I can see your point. but the ancient Greeks did not need the decimal system to prove that the square root of 2 or pi is irrational.
But, more fundamentally, if you define the numbers by reference to the operation "n+1", you already have infinity. Similarly "divide by 2" will also produce an infinite series, no matter what number system you have.
If the rules of a language game make rational numbers intelligible, then isnt it a new set of rules that make irrationals intelligible? In other words, dont we have to invent irrationals as well as rationals?
It's a fiction that meaning arises from rule-following. There's no fact of the matter regarding what rules you've followed up til now.
Check out Zero: The Biography of a Dangerous Idea by Charles Siefe. It's pretty good.
Quoting frank
If were talking about Wittgenstein on rule-following here, then there is no intelligible meaning without rules, criteria, forms of life. But at the same time, in applying those concepts, criteria and rules, we dont simply refer to them as a picture determining in advance how to go on. The rules underdetermine what to do in each new situation. There is an element of invention in following rules.
The Private Language argument indicates that there's no way for you to know what rules you've been following up till now. Check out Wittgenstein on Rules and Private Language by Saul Kripke.
Or better, there's no fact of the matter about what rules you've been following.
The Pythagoreans denied their existence for a long time
Quoting frank
Quoting frank
The Pythagoreans denied their existence for a long time after they realized the problem. No doubt they were working on arguments to establish that. They failed. It seems odd to describe that process as "inventing the irrationals". I don't know enough history to even comment on whether the rationals were invented or discovered. The number
Quoting frank
That's true, in a sense. But not the whole story.
Quoting Joshs
You state the problem nicely, but don't mention Wittgenstein's solution.
Quoting frank
The PLA (insofar as it is an argument) establishes, IMO, that there is no way for you to know what rules you have been following up to now, if they are private rules. "Private" means that your say-so determines what is correct and what is not. So "correct" and "incorrect" have no application - no meaning.
What gives meaning to rules is human agreement in the context of human life. Think of how the fact that we agree on how to use words is enough to make them words. (This fact is, perhaps, not a fact of the matter, but it is a fact nonetheless.) What often gets left out of this is that we sometimes find that we don't agree on how to apply our rules; so we have to make a decision about how to go on.
There's just nothing you can point to and say, "See, this is the rule I've been following for the use of this phrase."
Quoting frank
Funny that this came up here just after I had used it in another thread.
Kripke misunderstood Wittgenstein's answer, found in PI §201
It's what we do that is of import. If Kripke were correct, you would not know how to count, yet you do know what it is to count, and by twos and threes as well as by ones. You understand what it is to carry on in the same way, and while you cannot say what this is, you cna show it by counting. This is the import behind the now cliched appeal: "Don't look to meaning, look to use".
Quoting Joshs
Don't look for an abstract thing called "the meaning". Look instead at what one is doing as a participant in the various activities that make up our daily lives. Then at least you will have a better idea of what Wittgenstein said.
Quoting Ludwig V
Quoting Ludwig V
Its not human agreement , as though each individual voices their opinion and then the group arrives at a consensus. Socially normative meanings function prior to and already within individual experiences of rules and criteria of action. At the same time that such social norms allow us to make sense of our own perspective within them, we can differ among one another within shared language games as to how to proceed. And whether or not we agree on how to apply our rules, those rules never are enough to tell us how to go on. It is only within the actual context of the situation that we intuit the specific sense and use of a rule. This intuitive knowing is the solution, not waiting for a consensus from a group.
But then I reject such a phenomenological approach.
I agree with every word of that, except the word "intuit". But it's just a fancy name for the fact that we agree and usually, but not always, can resolve our disagreements on the basis of reasons, which, again, are reasons only because we are persuaded by them.
I would comment, though, that social norms, in this context, are not norms because they tell us what to do - that would make them just more rules; they are norms in the sense that we do in fact follow them for the most part. When we don't follow them, they cease to be norms.
Quoting frank
If one could, it would just be another rule, and so not explain anything.
Yet, there are things we can point to and say "See, this is the rule I've been following". But that's because we have learnt the human practice of following rules, not because the rule tells us anything - apart from the words we put into its mouth.
Exactly, especially about Kripke.
In the end, it comes down to "This is what I do".
This shows a misunderstanding of Kripke's point. There's no denying that we do things with words, and that we do on occasion follow rules. There's just no fact regarding what rules you've been following up till now. If you think you know the rules you've been following, you need to take a closer look at the PLA.
I doubt it
I doubt it.
Quoting frank
Sure, if what you mean is that the rule cannot be stated. But that is irrelevant, since the rule can be enacted.
Perhaps you need to take a closer look at the PLA.
Added: I'll fill that in a bit, rather than leave the implied but unintended offence. Kripke has his own idiosyncratic version of the private language argument. it is not generally accepted as what Wittgenstein argued for.
There's no fact regarding which rules you've been enacting.
Where does "fact" fit here? What is a "fact"? And how does being or not being a "fact" fit in to enacting a rule?
If a fact is something we discover, find out abut, then it would be odd to think of what we might choose to do as being a fact... odd, for example, for you to say that you discovered that you had responded to my post. You didn't discover that response, it's just what you did. Sure, it's a fact you responded, but that's after the act. See the difference in direction of fit here? Following a rule is changing how things are to fit how you want them to be. Setting out a fact is changing what you say so that it matches how things are.
There's no fact regarding which rules. It's a mind bender for sure. It took me a good while to digest the implications.
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.
And if you don't know which you are doing, then there's perhaps not much more to be said here, since I, and others, do understand what it is to follow plus and quus and to choose which to enact.
Can confirm.
That you cannot state what you do to ride a bike does not imply that you cannot ride a bike.
Might just leave it here. :smile:
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.
Quoting 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:
:up:
Quoting Banno
I agree. Intuition isnt really what I was after. Wittgenstein said it better.
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.
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.
Quoting T Clark Thanks.
(My old friend Ludvic suggested this to me.)
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.
Quoting sime
I consider the most important and radical implication of Wittgensteins later work to be his critique of Freges theory of sense as reference. Frege remained mired in a formalistic metaphysics centered on logic, without ever grasping f Wittgensteins distinction between the epistemic and the grammatical.
Quoting 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. Moores 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 doesnt 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? Couldnt we say that scientific paradigms are invented , and the facts that show up within them are discovered?
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.
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? ;-)
Quoting 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.
Quoting 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?
That makes sense.
I draw cartoons of the speakers going "blah blah blah."
Quoting Joshs
Again, perhaps it's about what we do, how we act as members of a community.
Quoting 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.
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.
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:
[hide="Reveal"]Quoting https://en.wikipedia.org/wiki/American_Pie_(film)#Plot.[/hide]
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.
Quoting 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.
Quoting 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.
Quoting 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.
Quoting 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.
@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. . .