I agree with everything you say. But it is not easy to say it clearly.

A rule can fix the standards for correctness without implying that the entire infinite list exists as a finished thing. We often feel “it’s already there” because the rule is firm, but what’s “already there” is the method, not a completed infinite inventory. — Sam26

What bothers me is that we seem driven to talk about processes in connection with infinity, as you do in the first sentence. But does such a concept make sense in the context of mathematics? Or does it mean that constructivism must be true, at least in the context of infinity?

Well, I can't say I understand exactly what you are proposing, but it seems like you are saying the question of the medium is secondary, but then you explain why it must be primary. — Metaphysician Undercover

No, it is simpler than that. We are using "medium" is different ways. I think. For me, empty space is not a mediium. A medium is substance that fills a space. Space is a co-ordinate system, which defines the possibilities where certain kinds of object may be. Objects are distinct from mediums because the latter are found everywhere, but objects have a locating within space.

What bothers me is that we seem driven to talk about processes in connection with infinity, as you do in the first sentence. But does such a concept make sense in the context of mathematics? Or does it mean that constructivism must be true, at least in the context of infinity? — Ludwig V

In math, process doesn’t have to mean a thing happening in time. It may just mean a rule, a precise recipe that tells you how to get the next step, or how to compute the nth term. Infinity shows up because the rule has no final step.

That doesn't itself prove constructivism. Math is just comfortable saying this exists even when you don’t have a method to build it. Constructivism demands the method, at least that's my take.

Wittgenstein’s point is to be careful not to treat the infinite as a finished object sitting out there. What we really have is a rule and the proofs we proceed with. That leans constructive in spirit, but it isn’t a knockdown argument that constructivism must be true.

For me, empty space is not a mediium. — Ludwig V

Of course it's not empty space, or else it wouldn't qualify as a medium. That's the point I was making. There is no such thing as empty space between objects. So to make a co-ordinate system which shows the positions which an object could have requires knowing the type of object and the type of medium.

Space is a co-ordinate system, which defines the possibilities where certain kinds of object may be. Objects are distinct from mediums because the latter are found everywhere, but objects have a locating within space. — Ludwig V

So "space" here is completely conceptual. And the point I was making is that it needs to be conceptualized according to the objects which are to be mapped and the medium between the objects. If we make a co-ordinate system which allows any objects to be anywhere (infinite possibility) that produces Zeno paradoxes. It's the faulty conception of space which allows for infinite possibility that creates Zeno type paradoxes.

In math, process doesn’t have to mean a thing happening in time. It may just mean a rule, a precise recipe that tells you how to get the next step, or how to compute the nth term. Infinity shows up because the rule has no final step. — Sam26

How could "the next step" not imply "a thing happening in time"?

How could "the next step" not imply "a thing happening in time"? — Metaphysician Undercover

Good point. Does a typical mathematical sequence imply motion in time?

I made it far enough in mathematics, before getting too ornery, to know that you have to do multiplication and division before you do subtraction and addition.

How could "the next step" not imply "a thing happening in time"? — Metaphysician Undercover

Because next can mean two different things.

1) Next in the definition (logical next).
In mathematics, next often just means “the item with the next label in the sequence.” It’s part of how the rule is set up, so if you tell me where you are, the rule tells you what counts as the next one. That doesn’t require anything to be happening in time.

2) Next in our activity (temporal next).
When you or I actually work it out on paper, then there really is a next moment: first this line, then the next line, etc.

So, the word next doesn’t automatically imply time. It can be about the structure of the sequence, or it can be about our act of calculating it.

How could "the next step" not imply "a thing happening in time"? — Metaphysician Undercover

Because it doesn't mean that.

"Next" here implies a relation, and mathematics is the study of the relations between its "objects," which it is happy to treat as effectively undefined. That's why "What is a number really, and do numbers actually exist?" is not a question mathematicians are much interested in, though non-mathematicians of all sorts, even philosophers, are.

Does a typical mathematical sequence imply motion in time? — jgill

You know it doesn't, unless you mean something pretty subtle by "imply".

Wittgenstein’s point is to be careful not to treat the infinite as a finished object sitting out there. What we really have is a rule and the proofs we proceed with. — Sam26

I wouldn't argue Wittgenstein's point, though doesn't that point us firmly in the direction of the Aristotelian distinction between actual and potential infinity? Which itself leans heavily on our actions in relation to infinity. The second sentence is true if we are talking about our activity in relation to mathematical formulae.

That leans constructive in spirit, but it isn’t a knockdown argument that constructivism must be true. — Sam26

Fair comment. I used to think that constructivism was the way to go. No longer. Now, I'm seriously bewildered and working things out. I have noticed how time and process show up so often in talk about infinity and am wondering how deeply rooted it is.
Perhaps it is a metaphor. Perhaps it is an application of terms in a new, stretched, language game. Notice, though that your talk of the infinite as unfinished implies a process.

A rule can fix the standards for correctness without implying that the entire infinite list exists as a finished thing. We often feel “it’s already there” because the rule is firm, but what’s “already there” is the method, not a completed infinite inventory. — Sam26

Aren't you leaning here on an idea of what exists and/or is real? Isn't it that idea that leads us into difficulties about the status of the sequence. In one way, you are right. In another, you seem to be saying that there are natural numbers that don't exist or aren't real (non-mathematical sense of real). Aristotelian talk of potential numbers tries to find a half-way house, though I think it is a most unhelpful concept.

"Next" here implies a relation, and mathematics is the study of the relations between its "objects," which it is happy to treat as effectively undefined. — Srap Tasmaner

Are you happy to defend an interpretation which regard S(n)=n+1 as a remark about the relations between numbers? It must be that, unless you are thinking of the number line, which is a spatial metaphor. But if is just a remark about the relations between numbers, it seems more like a generalization that a rule.

There is no such thing as empty space between objects. — Metaphysician Undercover

Empirically, that may be true - especially if you regard a field (gravity, magnetism) as a medium. But setting up a set of co-ordinates does not require a medium in addition, so far as I can see.

I think you’re right about how quickly the language of infinity goes into time and process. But “process” can be doing two things, viz., it reports our activity, calculating and checking, and it can mark a feature of the rule, that there is no last step built into it. Calling it “unfinished” need not mean a temporality is at work, it can mean the grammar contains no stopping point.

On your Aristotelian comment, Wittgenstein might ask what “actual” and “potential” are doing in our language, and whether they clarify the use of symbols or just swap one picture for another.

And on existence, I am not denying that numbers exist. I’m blocking a slide in what “exist” means here. In mathematics, “exists” is governed by proof and use, not by the idea of a completed infinite inventory sitting somewhere. So, the rule can be firm without that extra picture.

The philosophical problem isn’t infinity; it’s the pictures our words seem to imply when we remove them from the practice that gives them sense. When we keep the use fixed, the mystery largely disappears.

Calling it “unfinished” need not mean a temporality is at work, it can mean the grammar contains no stopping point. — Sam26

OK.

On your Aristotelian comment, Wittgenstein might ask what “actual” and “potential” are doing in our language, and whether they clarify the use of symbols or just swap one picture for another. — Sam26

As I said, I don't think the Aristotelian account clarifies anything much. If anything, it deepens the mystery.

And on existence, I am not denying that numbers exist. I’m blocking a slide in what “exist” means here. In mathematics, “exists” is governed by proof and use, not by the idea of a completed infinite inventory sitting somewhere. So, the rule can be firm without that extra picture. — Sam26

Perish the thought of denying that numbers exist!

The philosophical problem isn’t infinity; it’s the pictures our words seem to imply when we remove them from the practice that gives them sense. When we keep the use fixed, the mystery largely disappears. — Sam26

Yes, but here, we need to deal with the adaptation of terms that already have a use in some contexts, but need adaptation for this specific context.

I’m blocking a slide in what “exist” means here — Sam26

What are you blocking it with? Sentiment?

Because next can mean two different things.

1) Next in the definition (logical next).
In mathematics, next often just means “the item with the next label in the sequence.” It’s part of how the rule is set up, so if you tell me where you are, the rule tells you what counts as the next one. That doesn’t require anything to be happening in time. — Sam26

The "logical next" is next in time in this context. The only other option is "beside" in space, and this is clearly not the case. "The item with the next label in the sequence" is the one which comes after the other. Therefore the sequence is temporal. Without the separation of before and after, there is no sequence. The rule tells you "what counts as the next one", but unless you follow the rule, and produce "the next one", then the next one never comes. And following that rule is a temporal process. Therefore the sequence is a temporal process.

One might argue, that the order of such mathematical things simply exists, as eternal platonic objects, and that "the rule" is a description of that platonicly existing order. Then we'd have the nontemporal order, without having to fulfil the process of following the rule. But platonism is clearly wrong here. the rule is clearly not descriptive, because the proposed platonic objects cannot be observed to be described. They have no spatial/temporal existence. Therefore the rule is a prescriptive rule, and the sequence only comes into existence by following the process temporally.

"Next" here implies a relation, and mathematics is the study of the relations between its "objects," which it is happy to treat as effectively undefined. — Srap Tasmaner

Yes, "next" implies a relation, as you say. It implies a temporal relation. "Next" has two distinct meanings, a spatial relation, or a temporal relation. In this case it is not a spatial relation, therefore it must be a temporal relation.

You may insist that mathematics keeps "objects" as undefined, But mathematics would be useless if it cannot define its relations. And this is a serious consequence of having "object" as undefined. If we cannot identify an object, how can we formulate relations? In other words, we cannot unequivocally understand the proposed relations between objects if we do not know what an object is.

So, you are asserting that "next" implies a relation. Do you think you could explain what "next" means in the context of a mathematical sequence, without describing it as either temporal or spatial? Otherwise you are simply making an unjustifiable claim.

Empirically, that may be true - especially if you regard a field (gravity, magnetism) as a medium. But setting up a set of co-ordinates does not require a medium in addition, so far as I can see. — Ludwig V

I know we can do that, and that's the point I was making. We can, and do set up sets of co-ordinates without reference to the medium. That is a universal conception of "space", which allows in principle, for infinite positioning. But it is conceptual only. And if one sets up such a universal set of co-ordinates, with infinite possibility, and applies it to a real medium, it is a false representation as the primary premise in the representation which will follow. That false premise is what creates Zeno's paradoxes.

The point being that we can, and do set up such co-ordinate systems, I'm not arguing against that. What I am saying is that when we apply them they are applied as false premises, As such, they produce unsound conclusions as demonstrated by Zeno. Zeno concluded that motion cannot be real.

I don't have much more to say on the subject. Thanks.

Are you happy to defend an interpretation which regard S(n)=n+1 as a remark about the relations between numbers? It must be that, unless you are thinking of the number line, which is a spatial metaphor. But if is just a remark about the relations between numbers, it seems more like a generalization that a rule. — Ludwig V

With regard to the number line, I'll say first that the intuitions most of us have, formed in school days, can be a bit misleading, because we are on the far side of a great many developments in mathematics, which bring together the numerical and spatial through measure. The "purely spatial" without any sense of measure gives you not geometry, not the number line, but topology. In short, I wouldn't agree that the number line is purely spatial.

But I think I understand what you had in mind. You can talk about one number coming later or earlier than another in a temporal sequence, or you can talk about a number being to the left or to the right of another, as they are laid out in space. And I'm saying those are much more the same thing than you might think at first, because a 1-manifold of 0 curvature doesn't have any numbers on it at all.

The number line of grade school is neither Euclid's "breadthless length" nor the 1-manifold of topology. It's an axis ripped out of a Cartesian coordinate system because they intend eventually to teach you analytic geometry and calculus.

Now there is a question about whether our mathematics is built upon one sort of fundamental intuition or two: is it all numbers (and collections and so on) or is it also shape and space? There's a pretty strong case for saying that the spatial intuition is distinct, and that much of mathematics has been occupied with somehow bringing together the two sorts of intuition (as in the number line).

But if they can be brought together, what enables that? Doesn't that indicate these are two different ways of looking at the same thing? Maybe. It's at least clear that the ways of doing things with our numerical and our spatial intuitions are closely related, so we can generalize at least enough to say something about that, and that's why we say mathematics is the study of systematic relations among things, be those things numbers or shapes, integers or angles of polygons, or what have you. (The proof of Fermat's last theorem, the statement of which looks like the barest number theory, takes a very long detour through algebraic geometry, if I recall correctly, and falls out as a special case. Part of the interest of that series of results, as I remember it, was how many fields were brought together in those proofs.)

Finally, you ask whether we're talking about a generalization or a rule, which sounds quite a bit like asking me if mathematics is discovered or invented. It's an unavoidable issue, and I've suggested before where my intuitions lie, which of course involves answering "neither". I'll only add that I think too often we think we can fruitfully approach this issue by staring really hard at the natural numbers or at triangles and circles to figure out what they really are and where they came from, when we would do better to look at the practice of mathematics to see what's going on there. It is empirically false that mathematics is all working out the consequences of arbitrarily chosen rules.

I can give a small example, not very good, but maybe it'll indicate what I have in mind. I was recently asked to look at a bit of statistical analysis someone had done of sales in several stores. There were all these numbers and percentages calculated, the usual stuff, but it didn't actually mean what they were saying it meant. There were no errors in the calculations, but the numbers they were comparing just shouldn't be compared, and certainly not in the way they were doing it. Why not? I couldn't really explain why, except to say that I had never seen it done, it had never occurred to me to do it, and I knew in my bones that it shouldn't be. I suggested that someone smarter than me and higher up the pay scale might be able to explain why we don't do this, but I could only give hunches. Still, I knew intuitively that it was gibberish.

I think you can see the same sort of thing among mathematicians. There are certain ways of developing the field that feel like mathematics. If you're doing something quite odd like inventing non-Euclidean geometry, you might get some pushback, but the way you'll win over the naysayers is by getting them to dig into it enough that they get a feel for it and can see that it is not arbitrary, not chaotic or random or meaningless, but still recognizably mathematics. There are other things you might try that just feel off, or feel wrong, that just aren't mathematics.

(You can see exactly the same thing in chess: there are legal moves that are, in effect, meaningless, because they don't address the position; there are also the obvious moves, but sometimes there are moves that don't make sense at first but once you understand them, they address the position even more deeply than the obvious moves, which come off looking superficial. Really playing chess is something different from just following the rules.)

Is any of this in the neighborhood of what you were asking?

One additional thought. We've alluded to the spatial and temporal metaphors we often use talking about mathematics, but another very common metaphor in mathematics (and in mathematics-adjacent discourse) is the tree. Trees are interesting because the main thing we want out of them is the parent-child relation, which suggests numerical change over time, but that relation is also naturally related to thoughts of growth, or spatial change over time.

That false premise is what creates Zeno's paradoxes. — Metaphysician Undercover

Do you mean the premiss that space can be infinitely divided, not merely conceptually, but also physically?
I think most people would accept some version of that. But a physical limit to the process of division doesn't undermine the conceptual description. The physical limit will allows the conceptual division to continue.
Zeno produces an paradoxical analysis of the race. We can brush it aside and stick with the conventional analysis. There is an alternative, which is not paradoxical. Simple arithmetic and the definition of speed and (distance/time) tells us when Achilles will overtake the tortoise. So it is only a question of how you look at it. But still, people get hung up on the paradox. However, I think the real problems emerge in the analysis, for example, of circles and ellipses, which are not so easily dealt with in that way.

I don't have much more to say on the subject. Thanks. — Sam26

That's fair enough. Thank you for your comments.

:up:

So we have :

A. Field structure (algebraic axioms)
B. Order axioms
C. Completeness (least upper bound property) — Banno

Now the field structure and the order axioms are the rules that @Sam26 and @Ludwig V have been discussing, that set up the sequence of numbers in order.

We've already left Meta behind, since he has claimed numbers are not ordered...

The completeness axiom is a second-order statement (because of the quantification over subsets S), and it expresses completeness of ℝ.

∀ S ⊆ ℝ, (S ≠ ∅ ∧ ∃ M ∈ ℝ: ∀ s ∈ S, s ≤ M) ⇒ ∃ L ∈ ℝ: (∀ s ∈ S, s ≤ L) ∧ (∀ L' < L, ∃ s ∈ S, L' < s)

∀ S ⊆ ℝ says the axiom quantifies over subsets of ℝ, and does so without specifying which subsets.

(S ≠ ∅ ∧ ∃ M ∈ ℝ: ∀ s ∈ S, s ≤ M) is the antecedent in the axiom. "S ≠ ∅" discounts an empty domain. ∃ M ∈ ℝ: ∀ s ∈ S, s ≤ M specifies that there be numbers bigger than or equal to those in S; mathematically, S is bounded above.

So the antecedent is "if there is a non-empty set of real numbers with some upper bound..."

∃ L ∈ ℝ: (∀ s ∈ S, s ≤ L) says that there is some number that is larger than or equal to every number in S.

and

(∀ L' < L, ∃ s ∈ S, L' < s) says that there is also always some number that is less than that larger number, but still a part of S. That is, we have an L such that we can't lower L even slightly and still have the upper bound, and yet anything smaller than L fails to give us that upper bound.

Putting it together, For every non-empty set of real numbers that is bounded above, there exists a real number which is the smallest number greater than or equal to every element of the set.

By way of an example, If S were {.9, 0.99, 0.999...}. And S is not empty; S is bounded above by 1; and so by the completeness axiom, there is a real number which is the smallest number greater than or equal to every element of the set. Again, int his case, 1.

With S: {.9, 0.99, 0.999...}, we would have a process, "keep adding another 9", that would be acceptable to a Wittgensteinian - a rule that allows arbitrarily many interruptions. But the true-blue Wittgensteinian would deny that we thereby have a whole set; we have a rule for constructing an arbitrarily long string, and nothing more. To get to the compete set we need S={x∈R∣x=1−10−n for some n∈N}, which presumes ℝ and so presumes already that we can talk about infinite sets.

From a modal perspective, S is a subset of ℝ with 1 as its supremum. From a Wittgensteinian perspective the rule “add another 9” never produces 1, “approaching 1” is not “being bounded by an element”, and the talk of a completed S is a projection of grammar. So the statement "The supremum of S is 1” is treated as a useful way of talking, not a statement reporting a fact about a completed domain.

Now what I would maintain is that the two are for all intents and purposes the same. That is, the ellipsis as it stands does not tell us how to continue on, and so falls to the sort of view expressed by Kripke; but we dissolve this by insisting that there is a correct way to carry on, given by the model theoretical account.

And I would add that this amounts to no more than following more rules. We have the definiendum ℝ on the left, and on the right we have a rule setting out what counts as ℝ.

And all this by way of showing that some rules are not procedural at all; they are constitutive norms.

______________
My apologies for that post, it's sloppy, and under argued, and I moved from limits to ℝ as I went through the argument. The whole needs reworking, but I'll let it stand because it sets out the direction of my thinking in response to the last page or so from @Sam26, @Ludwig V, @Srap Tasmaner and @frank; that we can legitimately reify a procedure with a "...counts as..." constitutive rule. In this case the axioms count as setting up ℝ.

One follow on question is the extent to which this is a reflection of what Wittgenstein is getting at in PI  §201. @Sam26 may well insist that Wittgenstein had no such thing in mind. I'm not so sure. While he didn't use the "...counts as..." terminology, it seems to me implicit in his continuation of the account.

One follow on question is the extent to which this is a reflection of what Wittgenstein is getting at in PI  §201. Sam26 may well insist that Wittgenstein had no such thing in mind. I'm not so sure. — Banno

Okay, I guess I do have something more to say. I can see why @Banno would connect it to PI 201, at least as an analogy. PI 201 is about the gap between a rule and its application; the worry is that any finite formulation can be made to fit different continuations unless our practice fixes what counts as going on correctly, which is close in spirit to the worry about ellipses and “…” in the infinity case.

But I'd be careful about saying Wittgenstein had the real numbers or completeness axioms in mind there. In 201 he is not doing foundations of analysis; he is diagnosing a philosophical temptation: the idea that the rule must contain its own application as rails laid to infinity. His answer is that correctness is in the shared practice of following a rule, not in a ghostly extra fact.

If the link is: “reifying a procedure with a counts as norm is one way of making our practice explicit,” then yes, that resonates with 201. But if the link is: “201 is really about completeness or model theory,” I think that is a stretch. The Wittgensteinian moral is still that a rule does not interpret itself, and neither does an ellipsis. What settles the continuation is the rule plus the practice that gives it grip.

he is diagnosing a philosophical temptation — Sam26

Yep, and that diagnosis applies to the foundations of maths - the area in which he thought he had made the greatest contribution.

A rule does not interpret itself. Yet we have rules that set up novel interpretations. Following a rule can involve treating something as if it were something more. The move is essentially to build a new language game on the back of another. And something like this seems implicit in a form of life. The whole remains embedded in human activity, in a form of life.

some rules are not procedural at all; they are constitutive norms — Banno

I was thinking some days ago that, though I'm not sure what the favored way to do this is, if pressed to define the natural numbers I would just construct them: 1 is a natural number, and if n is a natural number then so is n+1. I would define them in exactly the same way we set up mathematical induction. (Which is why I commented to @Metaphysician Undercover that the natural numbers "being infinite" is not part of their definition, as I see it, but a dead easy theorem.)

And this will be handy later when we want to prove things by mathematical induction because our definition of the natural numbers is ideally suited to just that use.

Is this the sort of thing you're getting at? I have a procedure for producing one natural number from another, but more to the point is that the natural numbers just are what you get when you do that. It's the definition. It doesn't "turn out" that adding 1 to a natural number gives you another. That's not something we discover. It's part of what we mean by "the natural numbers".

On the other hand, it seems you could easily prove that adding 1 to an integer must produce an integer. The question is, what would you be doing in that proof? I think it would amount to showing that the definition you started with is good enough, that is, not self-contradictory in some sneaky way, and that it's all you need to generate the objects you want.

I guess that last sentence points to the fact that even here, we're talking about coming up with rules that give a complete account of a pre-theoretical practice of counting. So there's something a little disingenuous about saying I'm "defining" the natural numbers. (Famously, the Big Guy did that.) But I think we can still say that such a definition is an adequate account of our practice, so in that sense it's not quite the norm itself, but a usable form of it -- because having a definition in hand allows us to do all sorts of clever things.

A rule does not interpret itself. Yet we have rules that set up novel interpretations. Following a rule can involve treating something as if it were something more. The move is essentially to build a new language game on the back of another. And something like this seems implicit in a form of life. The whole remains embedded in human activity, in a form of life. — Banno

I agree with most of that.

Wittgenstein does think his approach bears on the foundations of mathematics: of course, the temptation is to imagine that the rule, or the proof, carries its own application and its own interpretation independent of what we actually do. A rule doesn't interpret itself, it's not an aside, it's aimed at that picture.

At the same time, youa'e right that we can introduce further rules that effectively stabilize new ways of speaking. We can take an earlier practice and add a counts as norm that extends it. In this sense, following a rule can include treating a construction as if it were something more, because we have adopted criteria that make that treatment correct within the extended game.

But the Wittgensteinian idea is that this isn't a metaphysical ascent to a realm of completed entities. It's a reworking of our practice (what we do), still embedded in human activity and a form of life. The novelty comes from what we now allow as a correct move, not from discovering a new kind of object behind the calculus.

Is this the sort of thing you're getting at? — Srap Tasmaner

Pretty much. So we have "Any number has a subsequent number", a procedure - if something is a number, then there is a subsequent number. But we need another step - "1 counts as a number" - to get the procedure moving.

Calling on procedure alone is insufficient. We need there to be stuff to perform the procedure on.

And I just don't suppose that Wittgenstein, a clever chap, had missed this point as was saying that all we need in maths is procedures.

We need there to be stuff to perform the procedure on. — Banno

I keep thinking about how we teach basic arithmetic with applications, and it's a very subtle thing. We say, "If I hold up 1 finger, and then 2 more, I'm holding up 3 fingers" and the important thing is getting the child to say that this is because 1 + 2 = 3. That "because" is very interesting.

But the Wittgensteinian idea is that this isn't a metaphysical ascent to a realm of completed entities. It's a reworking of our practice (what we do), still embedded in human activity and a form of life. The novelty comes from what we now allow as a correct move, not from discovering a new kind of object behind the calculus. — Sam26

Yep.

It's not platonic.

My own view is different, but I think that's Wittgenstein's take as I interpret it.