it's for some reason unacceptable, and offensive to criticize mathematical principles — Metaphysician Undercover

What I apprehend here is that some people take mathematics as a sort of religion. — Metaphysician Undercover

Yes, I attach value to mathematics, but that's like saying I attach value to logic or to language or, you know, to thinking. The basis of mathematics is woven into the way we think, and mathematics itself is primarily a matter of doing that more systematically, more self-consciously, more carefully, more reflectively. The way many on this forum say you can't escape philosophy or metaphysics, I believe you can't escape mathematics, or at least that primordial mathematics of apprehending structure and relation.

When you say you are critiquing mathematical principles, here's what I imagine: you open your math book to page 1; there's a definition there, maybe it strikes you as questionable in some way; you announce that mathematics is built on a faulty foundation and close the book. "It's all rubbish!" You never make it past what you describe as the "principles" which you reject.

So, on the one hand, I think you're simply making a mistake to think that the definition you read on page 1 is the foundation of anything. We are the foundation of mathematics. The definitions and all that, they come later. And, on the other hand, even if mathematics did have the structure you think it does, so that attacking some definition did amount to attacking the entire edifice of mathematics in one blow, I would still disapprove of your failure to engage in the material past page 1. It's childish. Maybe what the adults are doing is foolish, but the evidence for that is not a child, who doesn't understand what they're doing, announcing that it's "dumb."

Recently, one of my supervisors was explaining something to a bunch of us, and she insisted that what she was talking about was true "not theoretically, but mathematically." Put that in your pipe and smoke it.

Defending an idea without understanding it is a sign of a conservative spirit. — frank

You're talking about dogma, I get that, but I think you're missing another possibility.

The other possibility is the sort of thing suggested by Mercier and Sperber in The Enigma of Reason. If you think of reason not primarily as a system a solitary individual would use to deduce one truth from another, that sort of thing, but instead as a tool for critiquing the views of others and supporting your own view against objections raised by others—if, in short, you see it primarily in its social function, then the sort of thing we do around here makes a little more sense.

It's very late in the day, of course, and some people, the sort of people who have devoted some time to systematic thought (logic, mathematics, law, and so on), have been able to internalize the process, and we think of the usage we see there as the norm.

But in its origin, the important thing is the process of communal decision-making and communal understanding. Seen in that light, it's no surprise that we are pretty good at spotting the flaws in the ideas of others and not so good at spotting the flaws in our own ideas. And it also makes sense that logic and argument tend toward dichotomy, black and white, true and false, right and wrong.

Why? Because in the group discussion, each individual is not responsible for figuring it all out on their own; they are responsible for bringing a view to the group and advocating for it, and everyone else does the same. You give reasons to support your view not as an explanation for how you came to hold that view—you probably don't really know that—but to build support among others.

If you start with a view that doesn't hold up, you'll discover that as others critique it, and you begin to see its weakness. But you won't have that experience if you don't bring your idea forward. In hindsight, it might very well look like you were advocating a position you didn't fully understand, but so what? The whole point was to put it to the test. If it failed, so be it, and you're the better for it.

So, no, I don't think it's always just a matter of defending that old time religion, or a conservative mindset. In some cases, it's just playing your part.

but what's most interesting to me is the way people defend it when they don't actually understand it — frank

Gee, I don't know, frank. Isn't that mostly what people do here, no matter what the topic? Or: isn't that the claim of their opponents, should there be an actual debate? @Banno claims not to be a platonist, and @Metaphysician Undercover claims he is anyway—that Banno either doesn't understand his own position or that he doesn't actually hold the position he thinks he does.

And so far as that goes, this is par for the course among real philosophers, not just amateurs like us.

Much like @SophistiCat, there was a time in my life when I could have demonstrated Dedekind cuts for you and proved the Mean Value Theorem on demand. Nowadays, no. Much of the little knowledge of mathematics I once had is gone, along with my undergraduate expertise, but my appreciation of mathematics, the love of mathematics I've had since I was a kid, that remains. Sometimes I like these math threads because it's a chance for me to brush up, blow away some of the cobwebs, and it's a chance to look at math.

I was probably never all that good at math, much as I loved it, but even though I no longer have at my fingertips even the fingertips of the body of mathematical knowledge, I have never stopped looking at the world mathematically. So I enjoy these chances to exercise my math muscles a bit more directly than usual, and I take deep offense at @Metaphysician Undercover's repeated dismissal of mathematics as a tissue of lies, half-truths, and obfuscations.

Yes, we don't always understand everything we're talking about. What else is new? But it's a challenge. I like trying to understand things, and the best way I know of determining whether I do is trying to explain it myself. If I can't, I have some work to do. What else is new? I always have work to do.

Too many participants in too many discussions here evince no such desire to understand. I can take it on faith that they're participating in good faith, but I could not prove from their posts that they are not simply trolling. Maybe some people think the same of me, but I hope not, and if I thought so it would bother me, and I'd rethink how I write. (This is not hypothetical. I have had an analogous experience on the forum.)

By the way, if there's something mathematical you want to know and wikipedia doesn't work for you—some of its mathematics articles are not exactly for the general reader, in my experience—and you can't find another website with a nice explanation, you don't want TPF, you want Stack Exchange. There will be material there that's over your head, sure, but there are also people that know what they're talking about and put a surprising amount of effort into explaining it.

Sorry Srap, I can't see how you make this conclusion. — Metaphysician Undercover

It was a short post, making a single point, which answers exactly this question.

That's incorrect. — frank

It's also an answer to this, I think.

I know you're not a foundations guy, but I for one would appreciate a rap on the knuckles if I get the basics wrong.

(Decades, it's been decades since I did this in school. I could look everything up on wiki, but it's more fun to see if I can piece back together stuff I used to actually know.)

How have you done anything other than described a case of rounding off? — Metaphysician Undercover

It's the difference between saying (1) here is an approximation of the value that is within some tolerance you have specified (or precision, or significant digits, whatever), and (2) here is a value that is within any tolerance you might specify, however small. For (2) to be possible, I must be offering you the actual value.

Maybe this is won't help, but "rounding" is something you do when all you need is an approximation.

It's not that the adjacent members of a sequence become "infinitely close": they become "arbitrarily close", and so the series (in this case, the sum of the members of the sequence) becomes arbitrarily close to — well, that's the thing, to what? And that's your limit.

We are indeed talking about approximation, and therefore approximation of some value. It turns out we can imaginatively construct a "perfect approximation" which just is the value we are approximating. If you can get arbitrarily close to it, you can figure out what you're getting close to.

This is really cool, but I'm not convinced.

The gist of it is that there is a dominant strategy iff the sequence has a limit. If you countenance classical mathematics, you do an existence proof; if you don't, you do a constructive one. And then you have an answer about the game-theoretic application.

I guess what I don't get is that if you want to go the other way—to actually define the limit as a strategy—then you still have to start with an account of how to construct a δ given an ε. You seem to be doing the same thing but saying it's for a different reason.

The limit as strategy is interesting, and it's cool that it can be presented that way, but it looks like you still end up doing exactly the same math (of your preference) you would if you just presented it as a bare question, does this sequence converge? What am I missing?

The Eye of the Heron is another short, preachy one. I'm not sure how she gets away with it, but I think part of it is that she's so smart, you trust her and want to listen. We're used to preaching from people who shouldn't — so there's a sort of double offense — but I just don't seem to mind being preached to by her, which is an odd experience.

Maybe. I'm still mulling it over.

If, as I've suggested earlier, you think of mathematics as the long working out of how to join two sorts of intuitions into a single enterprise (number or count, on the one hand, and something like shape or space, on the other), then Zeno's paradox is a kind of speed bump there, and indicates that this will not be so simple as we might have hoped.

I think that's one reason so many of us grew up thinking calculus somehow "solves" the paradox, or overcomes it, because calculus does represent some kind of completion of a very long process of drawing together various fragments of mathematics.

But something else we might say comes oddly out of the discussion above, about how the reals cannot be counted, and the standard alternative, if we're casting about for a different metaphor, in English anyway, would be that they must be measured. (If you're not a count noun, you're a mass noun.) Somewhere I suggested that "measure" is the first step in joining the two strands of intuition, number and shape, and that's obviously true. But it's also true that a distinction persists. So — to get to it — we don't count distances; we measure them. But the whole structure of Zeno's analysis, despite relying on dividing by 2 and all that, tends toward counting, as if it's an attempt to force distance into the procrustean bed of counting. The whole procedure seems intended to undermine the idea of measuring at all through a perverse insistence on the model of counting. (If that's not clear, I can take another swing at it.)

As for the supertask business, it's the framework that interests me. Zeno insists, apparently reasonably, that to carry out the task of traveling from A to B, you must perform a series of actions — indeed you could say this about anything. It's hard to imagine an alternative, but it's quite an odd thing really. Everything is done by carrying out certain steps one at a time, in order? That's demonstrably false for a great number of things we do. The universe is a concurrent place, and we are concurrent beings. In order to walk, you don't move a leg, then an arm, then the other leg, then the other arm. If you tried to walk by performing a number of actions sequentially, you'd fail.

The most interesting case is probably thinking itself, because we know for a fact that the brain is a massively parallel system, and yet we have put enormous effort down through the generations into shaping that mass of simultaneous activity into something linear and sequential. We get logic that way, and human speech, which is one damned word after another, linearly. We are very proud of our linear triumphs, but it is, so far as I can tell, impossible to say whether that linearity is an illusion.

In short, what interests me about the paradox has less to do with "infinity" and more to do with "sequentiality".

Hyperobjects, that's another hip new member of club.

You left out classes, often in this context called "proper classes," I believe (since the word "class" has many uses), collections that are too big to be a set, for example.

I used to know a lot more of the technical side of this stuff than I do now, but I don't think where people have disagreed it was primarily about technical issues anyway. It looks to me like even our differences regarding mathematics are not primary, but result from broad differences in outlook.

I don't think anyone in this thread had forgotten, or that anyone was confused. Some people reject talking about infinite collections, I think, or reject talking about performing operations on them. We who accept and they who reject disagree, but we all agree on what we're talking about.

Denumerable — frank

Which some authors prefer, but it means what other authors mean by "countable". So long as we know what we mean, "The natural numbers are violet" would do just fine.

Here's another way to look at the difference: the Cartesian product of the natural numbers and the natural numbers is different set, certainly, which you can think of as ordered pairs or as the rational numbers with duplicates, but it's not any bigger and you could still lay them out on a line and you can count them. The Cartesian product of the real numbers and the real numbers is a plane: you go up a whole dimension.

We could express that by saying it appears the set of natural numbers is a subset of the set of reals. — frank

The natural numbers are also a proper subset of the rationals, but they're the same size.

the real numbers — Banno

That's a step in the right direction. You have to switch from a count noun to a mass noun. Water from a fire hose. But even that's not good enough, because with an election microscope you can count individual molecules of water. Maybe the real numbers are closer to something like an electromagnetic field, something where the idea of counting instead of measuring is not just impractical but unthinkable.

Maybe there's no joy there. Still, forcing the unwieldy mass of rational numbers to line up single file to be counted was a master stroke.

I think it's just a coincidence. I used this example because it occurred to me at the time, not because I had read it before. — Ludwig V

That's what I meant. I was very pleased you had the same thought.

letting one sheep through the gate, and one more, let through the next one and so on. — Ludwig V

I had to double-check but I never posted this! A couple times I wrote a post which contained exactly this point. (This post is what was left.) It would have gone something like this:

You can count sheep in a field just by looking but there are a number of challenges. A better way is to force them through a chute into another field or paddock or something, and then counting them as they come through is easy. It's interesting that you needn't care what order they come through in; you have your helper — the dog — start a number of fleeing-toward-the-chute processes that run concurrently, and you count them as they terminate. It doesn't even matter that they interfere with each other.

Zeno insists that we count the sheep — that is, the rational numbers — as we find them, in their natural order. But Cantor showed that there is a way to force them through a chute so that you can count them one-at-a-time. It's interesting that it turns out you cannot do this with the real numbers. (And I'll note again that we might take from Zeno not what we're usually told to, but a clever illustration that the rationals in their natural order do not form a sequence, or as an illustration simply of the reason: Zeno shows us that there is no smallest rational number greater than 0, and so there's no "first step". That was worth learning.)

But it doesn't necessarily tell you what counting actually is.
— Srap Tasmaner
Yes. One would need a demonstration of the written instructions as well. — Ludwig V

I was thinking more of (a) how we individuate objects in our environment, (b) how we consider some of them countable and some not, and especially (c) the idea of associating one list with another. There's quite a little leap in (c), because you have to recognize that two collections have structures that can be treated "isomorphically". In our case, the word "collection" seems a bit out of place, but it's not, because we know what kind of structure the natural numbers have without collecting them all. The rational numbers with that same order (that is, "<") do not have the same structure as the natural numbers, but you can order them differently so that they do. That's (d), the cattle chute, re-ordering a collection (even an open-ended one) so that you can map it onto another, or vice versa. Between (c) and (d) it's hard to say which is the bigger leap in imagination. I lean toward (c). When did shepherds start using notched sticks or knotted strings to count cattle? How on earth did they come up with such an idea? Extraordinary.

But I'm not quite clear what it means to "produce" a number. It's not as if we say to ourselves "I need another number here" and so instigate the procedure. Does your procedure create the numbers it produces from scratch or does it just produce another copy of the number???? — Ludwig V

Eh. A procedure, as I'm using the term here, accepts some input and yields some output. You show me a natural number, and I can show you another.

What I was suggesting was that we can replace our pre-theoretical understanding of counting with this system, consisting of exactly two rules (that 1 is a natural number, and every natural number has a successor), and we will (a) lose nothing, and (b) gain considerably in convenience for doing things that build on counting.

I consider (a) and (b) more or less facts, but there's nothing wrong with examining them closely. Philosophy spends a lot of time doing exactly this sort of thing, but not only philosophy. Linguistics is an easy example quite nearby, where people want to describe a great mass of complex behavior in terms of a smallish set of rules that could account for it. A more or less universal scientific impulse.

So the "axiomatization" of counting here is open to criticism, and I believe it will withstand it.

But it doesn't necessarily tell you what counting actually is.

It might. In a sense, when you come up with a little set of rules like this, if it works pretty well, then what you definitely have is a model of the behavior you want to understand. Whether that model reflects the underlying mechanisms of the behavior, or only simulates the behavior itself, relying on different mechanisms, that's not always perfectly clear. (In one formulation of Chomsky's program, it was of the utmost importance that you have a finite system of rules that can, through recursion, generate an infinite number of sentences, because the system has to be instantiated in a human brain.)

I've been thinking a little, as we've gone along, about the most famous "primitive" counting systems, the "1, 2, 3, many" type. Is "many" a number there? Not exactly. Is it open-ended enough that it might even apply to endless or unbounded sequences? Maybe, maybe not. What I'm trying to say is it might not be quite the same thing as us saying "1, 2, 3, 4 or more" or "1, 2, 3, more than 3", because in our system of counting numbers there is definitely no upper bound.

I suppose I'm bringing that up because we might ask whether people using one counting system are doing something psychologically different from people using another, but we might also ask if there is some difference that philosophy ought to be interested in. The latter, I suppose, would be something about the system itself, and the thoughts that it enables or doesn't, and therefore what would be available as truth, given such a system.

numerals — Metaphysician Undercover

Before we even get to the question of what a numeral refers to, you face an issue of what makes any given numeral count as a 1 (or as a numeral, or as a symbol). If each individual 1 is a token of the type <1>, you have to say what sort of thing the type is. That's not going to work out. A natural move to avoid types as abstract objects is to claim that the various numerals 1 belong to an equivalence class, but that's not so much an explanation as a restatement of our starting point, that each numeral 1 counts as a numeral 1, and it gives you no help actually defining the equivalence class.

I think the tricky bit is that philosophers hear "1 finger and 2 fingers make 3 fingers because 1 + 2 = 3," or even "1 finger and 2 fingers must make 3 fingers, because ..." and this sounds to them like the natural world obeying the "laws of mathematics" or some such. As if the fingers might try to add up some other way, but they would always fail, because there's a law.

But it's actually more like this: if I'm already committed to saying 1 and 2 make 3, then I'm also committed to saying 1 finger and 2 fingers make 3 fingers; if I didn't, I'd be inconsistent. Similarly, I can't say it works with fingers but not with train cars.

Children do have to learn, through trial and error, how much they're supposed to generalize. (Calling cows "doggies" and all that. And learning the difference between count nouns and mass nouns.) And of course what counts as success or failure is determined not by nature alone but also by the adults that mediate a child's understanding of nature.

What's difficult for us, in talking about mathematics, or about language, or about concepts, is that we want to pass over the generation upon generation of practice and refinement, to recreate the primordial scene in which someone, however far back, came up with a way of doing this sort of thing that worked, and we want to identify the features of the environment that enabled it to work, very much as if we expect there would only be one way. Some aspects of our thinking we find relatively easy to change, but some are so deeply embedded that we cannot quite imagine an alternative, so we think this way uniquely fits how the world is.

But it's not just a question of whether other ways of thinking were adequate to "our" needs, but recognizing that there was already adaptive behavior and already learning before there was any conception at all, and even our first conceptual steps were built on that.

Math as we know it piggy-backed the development of money. — frank

Are you saying there could have been a period when people had money, but didn't have amounts of money?

I agree with the spirit of your history lesson, that abstraction was a practical, observable, behavioral thing, but I don't understand the idea that money is the basis of math.

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.

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.

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.

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?

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".

like saying as you count the natural numbers you keep getting closer to the end. Or, as you work out pi to more and more decimal places you keep getting closer to the end. That's false representation. — Metaphysician Undercover

No one ever says either of those things. You're arguing with someone in your head who knows no more about mathematics than you do.

*

Zeno's paradox comes down to this: the rational numbers in their natural order do not form a sequence, unlike the natural numbers.

As it happens, they can made to form a sequence; and as it happens, the real numbers cannot. But I don't think either of those things really matter.

Zeno quite reasonably approaches the problem of moving by attempting to break it into a sequence of "steps" as we call them, for obvious reasons. (A very powerful technique that underlies much of what we do.) But that sequence of actions cannot be mapped onto the rationals in their natural order because that's not a sequence.

Sunday Morning

By Wallace Stevens

I

Complacencies of the peignoir, and late
Coffee and oranges in a sunny chair,
And the green freedom of a cockatoo
Upon a rug mingle to dissipate
The holy hush of ancient sacrifice.
She dreams a little, and she feels the dark
Encroachment of that old catastrophe,
As a calm darkens among water-lights.
The pungent oranges and bright, green wings
Seem things in some procession of the dead,
Winding across wide water, without sound.
The day is like wide water, without sound,
Stilled for the passing of her dreaming feet
Over the seas, to silent Palestine,
Dominion of the blood and sepulchre.


II

Why should she give her bounty to the dead?
What is divinity if it can come
Only in silent shadows and in dreams?
Shall she not find in comforts of the sun,
In pungent fruit and bright, green wings, or else
In any balm or beauty of the earth,
Things to be cherished like the thought of heaven?
Divinity must live within herself:
Passions of rain, or moods in falling snow;
Grievings in loneliness, or unsubdued
Elations when the forest blooms; gusty
Emotions on wet roads on autumn nights;
All pleasures and all pains, remembering
The bough of summer and the winter branch.
These are the measures destined for her soul.


III

Jove in the clouds had his inhuman birth.
No mother suckled him, no sweet land gave
Large-mannered motions to his mythy mind.
He moved among us, as a muttering king,
Magnificent, would move among his hinds,
Until our blood, commingling, virginal,
With heaven, brought such requital to desire
The very hinds discerned it, in a star.
Shall our blood fail? Or shall it come to be
The blood of paradise? And shall the earth
Seem all of paradise that we shall know?
The sky will be much friendlier then than now,
A part of labor and a part of pain,
And next in glory to enduring love,
Not this dividing and indifferent blue.


IV

She says, “I am content when wakened birds,
Before they fly, test the reality
Of misty fields, by their sweet questionings;
But when the birds are gone, and their warm fields
Return no more, where, then, is paradise?”
There is not any haunt of prophecy,
Nor any old chimera of the grave,
Neither the golden underground, nor isle
Melodious, where spirits gat them home,
Nor visionary south, nor cloudy palm
Remote on heaven’s hill, that has endured
As April’s green endures; or will endure
Like her remembrance of awakened birds,
Or her desire for June and evening, tipped
By the consummation of the swallow’s wings.


V

She says, “But in contentment I still feel
The need of some imperishable bliss.”
Death is the mother of beauty; hence from her,
Alone, shall come fulfilment to our dreams
And our desires. Although she strews the leaves
Of sure obliteration on our paths,
The path sick sorrow took, the many paths
Where triumph rang its brassy phrase, or love
Whispered a little out of tenderness,
She makes the willow shiver in the sun
For maidens who were wont to sit and gaze
Upon the grass, relinquished to their feet.
She causes boys to pile new plums and pears
On disregarded plate. The maidens taste
And stray impassioned in the littering leaves.


VI

Is there no change of death in paradise?
Does ripe fruit never fall? Or do the boughs
Hang always heavy in that perfect sky,
Unchanging, yet so like our perishing earth,
With rivers like our own that seek for seas
They never find, the same receding shores
That never touch with inarticulate pang?
Why set the pear upon those river-banks
Or spice the shores with odors of the plum?
Alas, that they should wear our colors there,
The silken weavings of our afternoons,
And pick the strings of our insipid lutes!
Death is the mother of beauty, mystical,
Within whose burning bosom we devise
Our earthly mothers waiting, sleeplessly.


VII

Supple and turbulent, a ring of men
Shall chant in orgy on a summer morn
Their boisterous devotion to the sun,
Not as a god, but as a god might be,
Naked among them, like a savage source.
Their chant shall be a chant of paradise,
Out of their blood, returning to the sky;
And in their chant shall enter, voice by voice,
The windy lake wherein their lord delights,
The trees, like serafin, and echoing hills,
That choir among themselves long afterward.
They shall know well the heavenly fellowship
Of men that perish and of summer morn.
And whence they came and whither they shall go
The dew upon their feet shall manifest.


VIII

She hears, upon that water without sound,
A voice that cries, “The tomb in Palestine
Is not the porch of spirits lingering.
It is the grave of Jesus, where he lay.”
We live in an old chaos of the sun,
Or old dependency of day and night,
Or island solitude, unsponsored, free,
Of that wide water, inescapable.
Deer walk upon our mountains, and the quail
Whistle about us their spontaneous cries;
Sweet berries ripen in the wilderness;
And, in the isolation of the sky,
At evening, casual flocks of pigeons make
Ambiguous undulations as they sink,
Downward to darkness, on extended wings.

source

it's defined as infinite. — Metaphysician Undercover

Maybe for you. For me, that's a theorem.

The point is that a number is not a thing which can be counted — Metaphysician Undercover

There is a very significant error in the idea that a measuring system could measure itself. — Metaphysician Undercover

Then this is nothing to do with infinite sequences, infinite sets, or infinity.

Your position is that you can't count how many numbers there are between 1 and 10.

If I recall correctly, he specifically said "habit" rather than "rule", which suggests naturalizing logic, and indeed I think that's where he was headed.

We do not find numbers in nature, not in the same way we find trees and rocks and clouds. Did we then make it all up, our mathematics? Is it just a game we play with arbitrary rules?

I don't think so. I think naturalizing mathematics means understanding it as behavior, mental behavior. We can interact with rocks carryingly or stackingly or countingly. The concept "rock" is already an abstraction, a mental skill we can apply to objects found in nature. Mathematics, on the other hand, is more like abstraction turned inward, applied to our own mental behavior. The cardinality of a set of rocks is an abstraction of an abstraction of an abstraction. (In many contexts, "idealization" is an even better word than "abstraction". I think Plato was desperate to understand how this works, and if you turn the tools you have, such as idealization, upon thought itself, something like the theory of forms is all but inevitable.)

We never encounter in nature infinitely long lines, endless planes, perfect circles, and so on. But that's fine: those 'objects' are not exactly abstracted from objects we encounter in nature, but from how we think about nature. And it turns out this way of thinking has many properties that are convenient to its smooth functioning that will never be found in nature. And that's fine, because mathematics is not a picture of nature but a tool we use in dealing with it.

I think that it is not necessary for the infinite number of numbers to exist in my mind. All I need to have in my mind is S(n) = n+1. — Ludwig V

This is exactly right, and it is the sort of move I have been trying to hold up as a triumph of human thought. We cannot list them all, but we can give a rule, and a rule we can hold in our heads and work with. (In a similar spirit, Ramsey suggested that universal quantification is actually an inference rule: to say that all F are G is to say, if something is F then it's G.)

I can put it another way: what you cannot calculate, you must deduce.

Infinite sets obviously present a barrier to calculation. So we deduce. Having deduced, we label our results, and then calculation becomes available again. We continually cycle between logic and mathematics, not just here but everywhere.

My claim is that it is definitively impossible to count the numbers. Therefore to represent this as possible is a contradiction. — Metaphysician Undercover

This is to spectacularly miss the point.

Because we can prove what the result would be, we do not have to actually carry out the pairing of every rational number with a natural number. Proof is a further refinement of prediction, beyond even calculation. Of course it's impossible to count the elements of an infinite set as you would the elements of a finite set. But for the results we're interested in here, you do not need to. That is the point. We already know what the result would be if it were in fact possible.

(In my old computability textbook, this was described by having Zeus count all the natural numbers: he could finish, by using half as much time to count each successor. But even Zeus could not count the real numbers, no matter how fast he went.)

a discussion about the nature of measurement — Metaphysician Undercover

That was me.

Now, of course, it's true there are issues with counterfactual definiteness in quantum mechanics, and "experiments which are not performed have no results." Sure.

It is also well-known that those issues do not arise in the same way at the macro scale.

Which is not to claim that acquiring knowledge at the macro scale is easy-peasy and there are no challenges.

Logic and mathematics are mental tools or technologies, habits of mind, that we have developed for dealing with things at the macro scale. And it is of the essence of these forms of thought that we use them to acquire knowledge without messing about with things in the real world.

This is unsurprising since our mental lives consist, to a quite considerable degree, of making predictions. Logic and mathematics enable us to figure out ahead of time whether the bridge we're building can support six trucks at once or only four.

The real world is not always cooperative, of course, and some of our predictions fail. But the link between logic and mathematics, on the one hand, and prediction, on the other, is so strong that it is not implausible that mathematics developed precisely as a refinement and systematization of our pre-existing efforts at prediction.

Which leads, at last, to my point, such as it is: there is something perverse, right out of the gate, about the insistence on "actually carrying it out". It misses an important point about the value of logic and mathematics, that we can check first, using our minds, before committing to an action, and we can calculate instead of risking a perhaps quite expensive or dangerous "experiment". ("If there is no handrail, people are more likely to fall and be injured or killed" -- and therefore handrail, without waiting for someone to fall.)

What's even more perverse is to take the difficulties we find in making good predictions about the natural world, even using logic and mathematics, and conclude not only that there is no way to have knowledge of the natural world ahead of time (which may be true, absolutely, but all we really need are reasonably well-calibrated expectations) but also that we have a similarly absolute inability to deal with our own minds, our own mental tools, that even logic and mathematics are not within our control, as if every time we multiply 5 and 7 the answer might turn out something other than 35.

The natural numbers turn out to go on forever, and we can prove this without somehow conclusively failing to write them all down. We now know that a certain sort of axiomatization of mathematics is not possible, though we once thought it was, and we know this without trying every conceivable way first.

What you get when you turn logic and mathematics upon themselves has a very different flavor than what you get when you try to tame the natural world with them. Mathematics has almost the character of pure thought, like its cousin music.

To see the demonstration that the rational numbers are equinumerous with the natural numbers and complain that it is not conclusive because no one can "actually do them all" is worse than obtuse, it is an affront to human thought.

imagine all of them. Now do you know what I mean? — Metaphysician Undercover

How on earth do you imagine all the natural numbers?

Sorry Srap, it seems you haven't been following the discussion. I suggest you start at the beginning. — Metaphysician Undercover

God forbid you repeat yourself ...

You can list them in a sequence, 1/1,1/2, 1/3, 2/3, 1/4, and so on, and so you can count them - line them up one-to-one with the integers.
— Banno

That's funny. Why do you think that you can line them all up? That seems like an extraordinarily irrational idea to me. You don't honestly believe it, do you?

Do you think anyone can write out all the decimal places to pi? If not, why would you think anyone can line up infinite numbers? — Metaphysician Undercover

The key word in all this seems to be "all". You might as well bold it each time you use it.

Now, it's a known fact that you can line up all the rationals, in the sense of "fact", "can", "all", and even "you" that matters to mathematics. You disagree, and so far as I can tell only because anyone who tried to do this would never finish. Which --

Okay but when you said

Nothing is capable of being put into one-to-one correspondence with all of the positive integers. — Metaphysician Undercover

what are you referring to with this phrase, "all the positive integers"? I know what I would mean by that phrase; I genuinely do not know what you mean.

And a circle contains an uncountably infinite number of points. Oh well, no more analytic geometry.

How do you know that you will be able to produce all of the outputs? — Magnus Anderson

In other words, the problem is that you'll never finish.

Under this view, there are no functions on any infinite set. Not even f(n)=1. No functions on segments of the real line.

You could also demand that to be a set "in the stronger sense" you have to be able to finish listing its elements, and under that definition N cannot be a set.

Which, whatever. It's your sandbox, do as you like.

As usual, I agree with you jgill. Here's the definition you provided: "capable of being put into one-to-one correspondence with the positive integers".

Nothing is capable of being put into one-to-one correspondence with all of the positive integers. We might say [ ... ] — Metaphysician Undercover

I'm just wondering if you think somewhere in the rest of the paragraph (following the bolded sentence) you have provided an argument in its support. Is this the post you will have in mind when someone asks and you claim to have demonstrated that "Nothing is capable of being put into one-to-one correspondence with all of the positive integers"? Because it's just an assertion of incredulity followed by a lot of chitchat. (I think you have in your mind somewhere an issue of conceptual priority, but it's not an actual argument.)

Notice, infinite possibility covers anything possible. — Metaphysician Undercover

Sigh. You can't even pretend to be listing the reals and putting them into a one-to-one correspondence with the naturals. Rather the whole point of this kind of talk about transfinite cardinalities is that they are not all the same.



"Countable" is just a word, of course, and it doesn't bother us that it has been given a technical definition. Maybe "list-orderable" would be clearer.

Not only does none of this bother me, it has all the charm of good mathematics. Cantor's diagonal proof is simple, clear, and convincing. Even better is the zig-zag demonstration that the rationals are countable. ( (I think a more common presentation is just ordering pairs by diagonal after diagonal, but I saw it done first zig-zagging and it's stuck with me.) I think that was even more thrilling for me. In the natural ordering, in between any two, there are an infinite number -- how can they not be bigger than the naturals?! And then you see how they can be rearranged so that there is always a unique next rational. It's brilliant and convincing. People who don't ever see this, or who reject it for semantic reasons, are missing out on some lovely examples of the sort of thinking we should all aspire to.

I think we realize too little how often our arguments are of the form:— A.: "I went to Grantchester this afternoon." B.: "No I didn't." — Frank Ramsey, 1925

Note that to present the point, Ramsey names his philosophers "A" and "B".

Indexicals are very interesting. Their analysis is both interesting and important because everyday speech is riddled with them, so analysing everyday speech requires analysing indexicals.

But I would remind everyone that there is a great liberation that comes with eliminating them from technical discussion.

There is endless discussion here about the centrality of the first-person perspective or even its ineliminability, and so on. To this I say, it wasn't an accident, it wasn't a mistake, it is a step deliberately taken that pays endless dividends.

If you want to know whether A or B went to Grantchester this afternoon, that's a problem you can work on, even if it turns out the evidence is not conclusive. But considering how "I" both did and didn't go is just spinning your wheels. We switch to the third-person on purpose, because it works.