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.

Btw:

bijection means that you can take every element from N -- not merely any arbitrary subset of it -- and uniquely pair it with an element from N0. — Magnus Anderson

You have to do it for all of the elements from N.

You don't know if that's possible. You just know that every element from N can be uniquely paired with an element from N0. — Magnus Anderson

Is there also a difference between "all" and "every"? Because you seem to be granting what you denied ...

I see. I would say there's a difference between making a claim about "a subset" and a claim about "any subset"; many of us will treat the former as a "some" claim and the latter as an "all" claim. We similarly take "arbitrary" to imply "all" claims. Perhaps if we simply agreed on how we're using these words, there would be no dispute ...

Substantively, would you accept mathematical induction as showing that the mooted function maps every element of N to an element of N0? The proof is not hard.

Do you see the subtle difference? — Magnus Anderson

What is the cash value of that difference, as you see it?

Let A be { 1, 2, 3, 4 }.

Let B be { 0, 1, 2 }.

Consider the function f: A -> B, f( n ) = n - 1.

Since this is a function, and since functions are relations where every element from the domain is paired with exactly one element from the codomain, this is also a contradiction. Being a function means 1) it must obey the rules, and 2) every element form A must be paired with exactly one element from B. But both are violated. The rules say that the number 4 should be paired with the number 3, but no number 3 exists in B. It being a function means that the number 4 must be paired with exactly one element; otherwise, it is not a function. But it isn't paired with any element. — Magnus Anderson

And is there an element n of N such that n-1 is not a member of N0?

This is a perfectly good argument, but it is not the argument you make about N and N0, which relies on the claim that if B is a proper subset of A, its cardinality must be smaller. Here no mention is made of cardinality.

To argue that f(n)=n-1 does not map every member of N to a member of N0, you must show, as you did here, that there is an n for which f(n) is undefined or not a member of N0.

Isn't it clear that this works, and that there can be no set of objects you could label starting at 0 that you couldn't also label starting at 1?
— Srap Tasmaner

You can't do that. Logic prohibits it. There are more "labels" in N0 than there are in N. — Magnus Anderson

So are you saying there could be a set such that you could label every member of that set starting with 0, but you could not label every member of that set starting at 1? Is that your claim? (And I guess also that "N0" is such a set.)

You seem to be arguing that N must be bigger smaller than N U {0} because, well, 0 is left out. Is that right? (Doofus.)

But try this: instead of thinking of the numbers here as things, think of them as labels. In one set, there is something we have labeled "0"; in the other, there isn't, but suppose there isn't not because the 0-thing isn't there, but because we've labeled it "1" instead.

The functions that have been discussed are instructions for switching from one labeling system to another. Isn't it clear that this works, and that there can be no set of objects you could label starting at 0 that you couldn't also label starting at 1? And vice versa. Or starting at 2 and using only even numbers. Or any number of other ways, so long as you are systematic about it. All these sets of labels are clearly equivalent, and in particular all equivalent to just using N. Now how can that be?

This is yet another thing from the prolific David Lewis, contextualism, the short version of which used to be that we do know things in everyday life that we don't know in the philosophy seminar room.

if JTB can't help us tell the difference between being in a state of knowledge that P, and not being in that state, what good is it? — J

Indeed.

if we can't determine T in some way independent of J, how are we supposed to use JTB as a test for knowledge? — J

Apparently we can't.

Is that what JTB is for?

Take a step back. Is there any prospect for any kind of theory that would pick out all and only true propositions? That would in every case distinguish true beliefs from false ones — or even justified beliefs from unjustified beliefs?

I think we can be skeptical any such theory is possible, either on general grounds of human fallibility or even on logical grounds (the problem of the criterion),

So what are we about?

@Sam26 does seem to want to say, "My claim to know certain things is justified because I used a really good epistemology." I don't think it works that way.

JTB proposes that only true propositions can be known, AND that there is a way to determine truth apart from justifications. — J

I don't think a JTB account is committed to this. You can, and I think this is quite common, simply be a realist (with whatever restriction). That is, it suffices for the proposition to be true or false, whether there is any way to determine its truth value or not.

If I justifiably believe that P, then if P is the case, I am in a state of knowledge that P, and if not then not. Whether anyone knows or can know that I know that P, is a separate issue.

MP, which is a logical "move," not a merely formal property — NotAristotle

It's usually taken as an inference rule, if that's what you mean.

MT is not an instance of MP — NotAristotle

Given MP as an inference rule, can you derive MT? That is, is MT a theorem?

a mode of discursive thought — NotAristotle

I don't know what that is. I was just referring to the form. From ~1 you can next derive ~(1 & 2), so now you have a contradiction.

I don't remember what the point of all this was supposed to be.

1 and 2 then not 1.
1 and 2.
Therefore not 1. — NotAristotle

That's modus ponens.

But JTB is not about what makes something true, but how I can say I know it to be true. [ my bolding, your italics ] — J

I'm not sure this is right.

If we say, a person S knows that P when P is the case, they believe that P, and their belief that P is "justified," in whatever sense we give that word, then what S says or is entitled to say about their possible knowledge that P just doesn't enter into it ― unless you tie justification directly to (even to the point of identifying it with) what S says about how they know that P. That's not crazy, for a lot of reasons, but it's also not forced on us.

And there are cases where we wouldn't want to do that. You might know that the capital of Arkansas is Little Rock, but not even remember how you came to believe it, much less provide an account of that process that would convince a doubter. (Looking it up now in a source trusted by the doubter only proves that you were right, not that you weren't guessing.) There are all sorts of cases.

In general, what you say is going to reflect what you think you know, so sometimes that'll be spot on and sometimes it won't. Just as your confidence is an indicator of the truth of your claims, but a somewhat limited one, so what you offer in the way of reasons and justifications ― what you say ― is likewise only an indicator. Putting too much weight on it will lead us to include cases we oughtn't and exclude cases we oughtn't.

The reasons people say what they do might be somewhat more loosely coupled to what they know and how they know it than philosophers would prefer.

When do I ever know something is true apart from having the right justifications? — J

That's kinda the right question and kinda the wrong question. The J in JTB is supposed to exclude cases of epistemic luck: the truth of your belief, if the belief was not formed in the right way, is not enough for us to count it as knowledge.

The issue isn't whether you know your belief is true — which in most cases amounts to knowing that you know — but whether your reasons for believing something is the case are connected in the right way to its being the case. That's what the J is meant to capture, and it leaves room for epistemic bad luck, where your belief turns out false but anyone would have formed the same belief, and it was a one in a million chance that in this case the evidence misled you.

The point about luck is not incidental: luck is not a strategy. In most cases, a strategy is only likely to produce the result we want, but it's not a guarantee. (Pareto dominance is the exception.) The question is the same with post-hoc justification, whether your strategy was likely to have produced knowledge; the question of whether it did in this case, is different, and must be judged differently.

Doing much better exceeds my ability, I'm afraid.

Bayesian inference is certainly well-suited to formalizing some of these issues, but there are complications I'm not sure how to handle.

Suppose a neighbor calls to say they saw your dog in the road. You'll have to weight that report against your belief that she couldn't get out, and the possibility that it was another dog altogether. (There is a fair amount of work on using Bayes to analyze eyewitness identifications in law enforcement, for example.) Your credence that your dog is out goes up, but not quite to belief, after one report. But if you get another call from another neighbor, your credence probably goes up even more than the first time.

What's crucial here is that the reports be independent. It's no help if Tim calls to tell you that Jane told him she saw your dog, if Jane already called you.

This is why the word "coincidence" is important in Sam's remarks; "coincidence" implies independence. But we know the stories we're evaluating aren't entirely independent. Nancy Rynes mentions thinking, if this is the afterlife, where are my dead relatives? Shouldn't they be here to greet me?

Suppose instead of an escaped dog, we're evaluating UFO reports, or, better still, just reports of some inexplicable object. Tim saw a thing in the woods that he can't identify, but he can describe it; Jane independently gives a very similar description. We might have pretty high confidence that they saw the same thing. That they can't identify it would, I think, actually increase our confidence, and it's worth thinking about why.

Sam argues that the similarity of NDE reports is, in just this way, a point in their favor. Of course, it is, but it's also a problem, because it makes it harder to determine just how independent these reports are, in two senses: first, it strains credulity to claim culture plays no part at all in these stories; second, the claim to know and understand what you experienced means the subject's pre-existing beliefs and concepts play a bigger role than in the case where Tim and Jane give a bare description of something they do not claim to be able to categorize.

In other words, the fact that the experience can be described at all is surprising and therefore troublesome, and that it can be described so well and its meaning be made perfectly clear, that's even worse. (Nor does it help that so much of the afterlife is so much like this reality only better. The place was beautiful, like earth, but more. I felt loved, like people do, but more.) "I came face to face with the ultimate reality, of which what I thought was real is a mere shadow, and I understood exactly what I was experiencing, and I can explain it to you."

If that's so, that in itself is an heroic act of Bayesian updating, one we are asked to reproduce.

Two more little notes on the similarity of stories

Not for nothing, but it's a motif of crime stories, that if two people being interrogated give accounts that seem too similar, especially if they use some of the same specific words or phrases or pick out the same details, they are suspected of colluding.

That point about words and details actually has an academic pedigree: it is a core technique of comparative mythology. If some peculiar detail is repeated in two stories, a character missing a finger, something like that, this is taken to indicate that the stories are related, perhaps one story being a source for the other, or the two sharing a common source, perhaps unknown.

I want to say a little more about this calculation (in which I've corrected a misplaced decimal):

$\begin{align}P(Survival\ \vert\ Observations) & = \frac{P(Survial)P(Observations\ \vert\ Survival)}{P(Observations)}\\0.95 & \approx \frac{(0.01)(0.2)}{(0.01)(0.2) + (0.99)(0.0001)}\end{align}$

That's using Sam's numbers, the most important of which seems to be this:

Chance of Seeing This Evidence If H0 Is True: 0.0001 (0.01%—super low, because if it's all natural, it'd be weird to have so many matching, detailed reports without huge coincidences). — Sam26

You'll note that making this value really small is what makes the posterior probability so high, regardless of how low the prior probability was. It's a ratio: on top is the chance, however low, you assign to consciousness surviving death and people reporting that they experienced this; on the bottom is the total chance of people reporting that they have experienced survival, whether it happened or not (so we add the two cases to get a total).

One way to think of this is as an explanation of how quickly people can update. Consider the characters in a science fiction movie: maybe they don't believe in monsters or aliens, but when one is right in front of them, they might initially resist thinking it's real, but if it demonstrates that it is, they very quickly adjust. Similarly for rare events in real life. You may know for a fact that airplane crashes, church shootings, and tornadoes are rare, but when you're in one, you believe it not quite immediately, but quickly.

This is what Sam is asking of us here. The idea is that something you think unlikely has in fact happened: you never in a million years expected someone to tell you they had experienced the afterlife, but here they are. My prior credence was low; I've gotten the extremely unlikely evidence; now my posterior credence is high. The more unlikely the evidence, the higher my credence will now be. (Hence, Sam above comparing these reports to "coincidences", which raises other issues not addressed here.)

Of course, that is not the view of the skeptic at all. There are two possibilities:

(1) Skeptics believe that these reports are not evidence of an afterlife, and therefore the likelihood of someone offering such a report, having had a near brush with death, just is whatever it is in real life. If five million people last year nearly died but survived, and five thousand of those reported experiencing the afterlife, then the odds of a survivor making such a report are 1000 : 1, and that's it. Whether there's an afterlife doesn't enter into it. Bayes's rule has no use here at all.

(2) Skeptics believe the reports do count, but not so much.

Let's look at how (2) works with an example.

Suppose the chances are 9 in 10 that people will comment favorably on a cute outfit. Suppose further that the chances are 3 in 10 they will comment favorably on an uncute outfit, out of politeness, etc.

How likely are people to say that your outfit is cute? We can't say, because we don't know the base rate ― we don't know how likely your outfit is to actually be cute, so we can't do the calculation. Let's say half your outfits are cute. Out of 20 outfits you wear, 10 of them are cute and you get 9 comments, 10 of them are not cute and you get 3 comments; altogether you get 13 comments out of 20.

Now for the important question: what are the chances that your outfit is cute, given a favorable comment? 9 out of 10? 13 out of 20? Nope. The chances are given by the likelihood ratio of comments on cute outfits to comments on uncute outfits, scaled by the base rate. Given our 50-50 base rate, the chances that your outfit is cute, given a nice comment, are 3 in 4 (because genuine comments are three times more likely). But if only a quarter of your outfits are genuinely cute, a favorable comment makes it only even money that this is one of the cute ones. If only 1 out of 10 of your outfits are cute, the favorable comment gives you only a 1 in 4 chance that this is a cute one.

For our problem, let's say the skeptic considers the odds there's an afterlife a colloquial "million to one". That's the prior. To calculate the posterior odds, we need to know how much more likely we are to get reports of an afterlife, if there is one than if there isn't. It doesn't matter what the odds are, really ― both can be pretty likely or unlikely ― what matters is the ratio. Sam's estimate was that we are 2000 times more likely to get reports if there is an afterlife (0.2 : 0.0001).

Having gotten these reports, what would the skeptic say are the odds there's an afterlife? It's the likelihood ratio scaled by the base bate, in (rounded) odds form:

$\frac{10^6}{1}\times\frac{1}{2000}=\frac{500}{1}$

Still 500 to 1 against.

It's as if the skeptic says, out of a million and one universes, one of them is cute; reports of an afterlife are two thousand times more likely in that universe than in any of the other million; we have those reports, so what are the odds we're in that universe? Bigger than you might think, but still small because the base rate controls. Even if people in the cute universe are dramatically more likely to report an afterlife experience, our chances of being in such a universe ― according to the skeptic ― are so small that they remain small, even when we have those reports.

Sam's skeptic picked a colloquial prior of "a hundred to one", so instead the calculation was (rounding again):

$\frac{100}{1}\times\frac{1}{2000}=\frac{1}{20}$

or 20 chances out of 21, which is about 95%.

So it turns out ― as it almost always does with these kinds of problems ― that the most important estimate Sam gives is not (as I suggested above) the relative likelihood of reports, but the base rate.

If you want to leave open the possibility that we live in a cute universe, you still have to consider:
(a) whether the reports that we do are acceptable as evidence at all;
(b) how much more likely that evidence, if accepted, is in cute universes rather than uncute ones; and
(c) how likely it is that we live in a cute universe.

What will determine whether this evidence controls is the difference between the likelihood ratio of the evidence, in (b), and the base rate of cuteness you give credence to in (c). Is one orders of magnitude bigger than the other? Which one? Sam gets the result he does by treating the evidence as twenty times more likely in the favorable case than the favorable case is unlikely.

(I'm not saying anything about how we might settle on one value or another here. It's just my understanding of the math, particularly for people who found that "95%" somewhat eye-popping. Ignore if you're better at probability than I am.)

the OBE reports often describe a vantage point, like above the body or in the room, that seems spatially anchored. But why assume that's "odd" for something non-physical? — Sam26

Because non-physical entities do not have spatial locations or orientations. "Odd" was perhaps too polite; it's simply a contradiction.

a perspective that's detached but still oriented toward — Sam26

"Perspective", "detached", and "oriented" are all terms describing physical entities.

it's often like a movable viewpoint, not omniscient 360-degree (although 360-degree vision has been reported) god-mode — Sam26

"Movable", "viewpoint", and "360-degree" likewise.

Mystics, when they try to eff the ineffable, frankly admit that they cannot literally describe their experience because it transcends our quotidian, physical vocabulary and concepts.

Your survivors give frankly physical descriptions of physical impossibilities, and then you take that impossibility as evidence of non-physical existence.

"She could not have seen the saw but she did" has to be rescued from contradiction by making two distinctions: "Physical, embodied she could not have physically seen the saw, but disembodied she non-physically did." What I have been pressing you on, is whether you can give any sense to "non-physical seeing" or "non-physical hearing" (and I am passing by whether a disembodied consciousness can be given sense), beyond just positing that they must be a thing because people say they've done it. What exactly is it they've done? What do they mean when they saw they saw these things? In what sense did Pam see the bone saw?

So far, it seems to me the NDE community is satisfied with "exactly like normal seeing but not, you know, physical."

Can you explicitly write out this calculation? — Apustimelogist

I could write it out — Sam26

So could I:

$\begin{align}P(Survival\ \vert\ Observations) & = \frac{P(Survial)P(Observations\ \vert\ Survival)}{P(Observations)}\\0.95 & \approx \frac{(0.01)(0.2)}{(0.01)(0.2) + (0.99)(0.0001)}\end{align}$

French operating-suite amputation (Toulouse)—During surgery under general anesthesia, a patient described rising above the theater and then “looking” into an adjacent operating room where a leg amputation was underway, — Sam26

This is the sort of thing that bothers me, Sam.

(Are the scare quotes around "looking" an acknowledgement of my question about Nancy Rynes looking behind her?)

First, the consciousness that has separated from the body on the operating table seems to have a location in physical space. Doesn't that strike you as odd, for something non-physical?

Second, with or without scare quotes, this consciousness seems to have a definable perspective, a field of view that can be turned this way or that, much the way humans normally see using their front-facing eyes.

Third, this consciousness seems to do one thing and then another thing, meaning it is bound by time. Isn't that also odd, for something non-physical?

I think the significant philosophical question is, why the controversy? — Wayfarer

And your explanation is to look at what you take to be the motivations of the skeptics in your story.

Is that the discussion you want to have? Everyone chooses up sides and then questions the other side's motives while defending themselves as wholesome, open-minded truth-seekers? That's the philosophical approach, in your mind?

As for the postscript argument: "I've got a whole bunch of rocks here; surely a few of them contain mithril."

What other kinds of evidence could there be? — Wayfarer

Take a step back and consider what we're talking about here.

I don't keep up with this stuff, but Wikipedia seems to believe there is still no evidence for extra-sensory perception that is broadly accepted among scientists. So I haven't missed anything.

That's the state of research when you have a definitely living subject in the lab.

So now we're asked to accept that there have been thousands if not millions of cases of indisputable and objectively verified cases of extra-sensory perception, where the perceiver is dead. And on the basis of that evidence, we prove that the perceiver is non-physical.

If anything does, that qualifies as "huge, if true."

There are a great number of interesting issues raised by eyewitness testimony. We've talked about some of them in this thread on and off over the last eight years.

But let's put all that to the side. Why don't you tell me why it turned out to be so much easier to prove there is such a thing as extra-sensory perception when the subject of the perception is dead.

You're trying to make an apple pie with strawberries.

@Hanover gamely pointed out that people can't see without using their eyes, and all of the reports you rely on are of people seeing without their eyes and hearing without their ears. So are you using the words "see" and "hear" the way Hanover and I do, or in some other way?