A = x is cognizable, B = x is a noumenon (and ~B = x is an appearance)

1. (A → ~B) v (A & B)
2. (A → ~B) v ~(~A v ~B)
3. (A → ~B) v ~(A → ~B)
True

Not sure what everyone else is doing, or what your P and Q lead to, but the argument is obviously right and I think this is the simplest way to formalize it. (We could do quantifiers and stuff, but it's really simple.)

My way is cleaner than yours because my sentential variables are really pretend predicates, just leaving off the variables. That means we don't stuff quantifiers (like 'all') into the variables.

The main thing is to render 'All F are G' as 'F → G'; that is, read it as 'If something is an F, then it's a G.'

The only other rules used were de Morgan's law to get from (1) to (2), and then the equivalence of P → Q to ~P v Q to get from (2) to (3). Or you can just recognize that if something is both A and B, that means being A doesn't imply not being B.

And then we're done, because (3) is a tautology, so (1) is valid.

2.6 is the section about existence, and it's only a couple pages. I quoted some bits.

One minor point maybe worth going back to is his use of the word 'specifically', also used in our section when recalling 2.6, when he denies that external existence can be taken as something 'specifically different' from our perceptions. I haven't checked, but I strongly suspect that word here means 'different in kind', 'specific' from 'species'. That's how I read his point, but I don't think I mentioned I was relying in part on this particular word.

So the argument begins

(1) Existence, external or not, is not something we have any separate conception of.

But then we get the other part:

(2) The notion of external existence cannot be taken as specifically different from perception.

That sounds like it's telling us what sort of thing something that doesn't exist is.

I think that means (1) is at least a little misleading as I've phrased it.

(2) is part of the larger claim that it's only perceptions in our mind, nothing else.

So if we have an idea of existence, it's that sort of thing, a perception; but (1) had already shown that existence itself is not a perception, but just something part of every perception, so in a way nothing.

(Some of the confusion here is just rhetorical. It's a pretty common move to say something doesn't exist, at least not as the kind of thing you think it is, not if we take it to be what you think it is, but it does exist, just as something completely different. We're keeping the name, but changing the meaning. Like that.)

But there is something else going on here, because Hume says our idea of external existence turns out to be different relations, connections, and durations that we *attribute* to perceptions.

So these are ideas *about* perceptions.

Thus instead of thinking some idea we have is of an object that we also think exists, we will have the perception or idea of the object and then *attribute* to that perception the properties that will be refuted in 4.2, distinct existence and continuity over time.

But attributing, like relating or associating -- these don't sound like perceptions but ways of handling or working with or acting upon perceptions. We can, in addition, have ideas about what we're doing when do this sort of thing, and Hume bundles some of these mental behaviors together and calls them our notion of external existence.

The status of these mental behaviors might be clearer if we look back at Part I where most of the basic machinery is laid out.

On the whole, I find your reading of him to be quite accurate — Manuel

I'm leaning on his actual words too much though.

I'm not absolutely certain I've gotten to the bottom of the arguments in 2.6

I should be clearer. I was only asking if my reading of Hume was plausible, not whether what he says was.

I do want to evaluate the arguments as we come to them, but only once we know what they are!

we do not know — Manuel

That really might be. I'm undecided.

Hume sets out to show that various beliefs we hold cannot be justified either by observation or by reason. But we do, nevertheless, hold these beliefs, and we hold them even once he has done as much as anyone can to demolish them.

Why we would hold reasonable beliefs needs no explanation; why we would give up unfounded beliefs needs no explanation; but why we would hold questionable beliefs and continue to hold them once shown to be groundless — that requires some explanation.

Certainly this talk of nature deciding for us does that. Much of 4.2 is taken up with showing how exactly it works, to make it plausible that this is 'just how we think', willy-nilly.

So perhaps my wondering 'what we get out of it', why nature would so order things, is misplaced. That nature does so order our minds is all Hume is trying to show.

Plausible?

nature has made this issue too important to leave it to us to decide if objects ("body") exist or no — Manuel

Yes, and that's a pregnant suggestion, not really substantiated yet — since I'm less than a page into the exegesis so far.

Nature has not left this [ the principle of the existence of body ] to his [ the sceptic's ] choice, and has doubtless [ ! ] esteemed it an affair of too great importance to be trusted to our uncertain reasonings and speculations.

That last phrase referring to the results of Section I, Of scepticism with regard to reason. There he argues that even doing mathematics is, roughly, a matter of believing you've done it right, and checking your work or having others check it can only raise your confidence that you've done it right, never guarantee it.

Again, all about the psychology. It's not that mathematics can't be trusted absolutely, but that mathematicians can't be! And so it is for all sorts of reasoning. That's how we enter Section II, having established that the sceptic must reason even knowing that he reasons imperfectly. We even get a twin of the point made here:

Nature, by an absolute and uncontroulable [ by us ] necessity has determin'd us to judge as well as to breathe and feel. ... Whoever has taken the pains to refute the cavils of this total scepticism, has really disputed without an antagonist, and endeavor'd by arguments to establish a faculty, which nature has antecedently implanted in the mind, and render'd unavoidable.

If we go back through Part I, we may find some encomium to reason, but I'm not going to bother looking: I think we can just say 'Newton'. Hume may never bother saying in so many words what value to us reason has, because I think in 1739 he would consider the value, indeed the triumph, of mathematics and reason, for which reason nature would instill reason in us, to be perfectly obvious. We may, as he specifically argues, be unable to ground our reliance on reason in reason, but it's clear what we get by this reliance.

Not so for the external existence of objects. There has been nothing yet to explain why nature implanted this habit in us, why the belief in external objects is so necessary. What do we get out of this belief of such great importance that nature implanted it in us?

We will have to be on the lookout as we go through Section II for some clue if not explanation, as to what this belief does for us.

It is curious that he treats reasoning (with the principle example being mathematics) and the belief in distinct, persistent, external objects as separate questions, albeit giving them related answers. In the post-Frege world, we might naturally think these go together. We carve up the world into classifiable objects to make it safe for logic; conversely we analyze the world using the logic of predicates and classes because we have carved it up into distinct objects with properties in common. Logic and objects go together. Without distinct objects, there is nothing for the functions of logic (not the predicates, not the truth functions, quantifiers, or other operators) to be applied to.

tis in vain to ask — Manuel

Good.

I think the warmup is done and now we ought to go right through and discuss the arguments as they arise. I've considered graphing them out, but it'll be more fun to make connections as we go.

I was going to start at the first proper argument, that continued and distinct existence are equivalent, but we should look first at the introductory bit, whence this quote comes, because there's some guidance there.

The sceptic, he says, "must assent to the principle concerning the existence of body ... Nature has not left this to his choice."

We may well ask, What causes induce us to believe in the existence of body? but 'tis vain to ask, Whether there be body or not? That is a point, which we must take for granted in all our reasonings.

Belief in the existence of body is
(1) not supportable by the senses or by reason;
(2) not a matter of choice;
(3) imposed on us by nature;
(4) caused.

Section II, then, takes as given the phenomenon that we believe in body; what we're looking for is a causal explanation of this belief. It will not turn out that we do not so believe, or even that we could, if we chose, not so believe. We do. The only question Hume is addressing is why.

Before explaining some fact, we need to be sure we know what the fact is, and this Hume has already done in Part II Section VI, Of the idea of existence, and of external existence. He will refer to those results almost immediately.

There it is claimed, first, that since to conceive of any thing is to conceive of it as existing, existence is not a distinct idea at all, not a separable perception:

That idea, when conjoin'd with the idea of any object, makes no addition to it.

Specifically what Hume is saying here is that we can have no idea of existence, an idea that we might join to another idea, as a way of having the idea of something existing. It's about our conceptions, not logic, not language. Might Hume have taken another line? Might he, for instance, have said, that to imagine an object differs from imagining it as existing in that the latter is more vivid, or more complete, or something like this? There would still, I think, be no distinct conception of existence. Even if he were to say that conceiving an object as existing is the usual conceiving but accompanied by some particular feeling, that leaves the conception the same, and this is Hume's only point.

But to say that we have no separate conception of existence, which we might add to our conception of an object, is not quite to say that the only existence objects have is in being perceived (esse est percipi). We can form no conception of the existence of an object; that's a fact of psychology, of human nature, the subject of the book, not a metaphysical fact regarding objects and their mode of existence.

But Hume does want to say more, so we get a further argument, which begins:

We may observe, that 'tis universally allow'd by philosophers, and is besides pretty obvious of itself, that nothing is ever really present with the mind but its perceptions or impressions and ideas, and that external objects become known to us only by those perceptions they occasion.

This would all seem to turn on the meaning of the word 'present': I see the book here by me, but the book is not present with my mind, only the perception it occasions; the book remains forever 'out there' beyond the boundary of my mind. A cordon is drawn around my perceptions: within is mind, without is world. The question for Hume is not what's out there or what isn't, but how we may conceive what's 'out there', the psychological question, and perceptions offer him the solid ground of his psychology.

Now since nothing is ever present to the mind but perceptions, and since all ideas are deriv'd from something antecedently present to the mind [ argued earlier ]; it follows, that 'tis impossible for us so much as to conceive or form an idea of any thing specifically different from ideas and impressions.

That is, when we try to conceive the 'external objects' that occasion our perceptions, we have nothing but perceptions to work with — we have no other material with which to construct a conception of 'object', no material that would make such a conception a distinct sort of thing from a perception.

Or: try as you may to conceive, for once, of an external object, itself, you will only produce another idea, an idea derived from previous perceptions, impressions and ideas. It's all your mind can do; there is only one sort of object available to your mind, a perception, and any attempt to bring some other kind of object, whatever it may be, into your mind will fail utterly or substitute an idea of that other kind of object.

There is no claim that perceptions are the only kind of objects, or that all objects are really perceptions, but only that the only kind of object in our minds is perception.

The farthest we can go towards a conception of external objects, when suppos'd specifically different from our perceptions, is to form a relative idea of them, without pretending to comprehend the related objects. Generally speaking, we do not suppose them specifically different; but only attribute to them different relations, connexions and durations. But of this more fully hereafter.

And that's the pointer to our section, Part IV Section II, where he will refer back to this section:

For as to the notion of external existence, when taken for something specifically different from our perceptions, we have already shewn its absurdity.

Note the same language, 'specifically different', meaning some kind of object distinct from the kinds of objects that can be within our minds, perceptions, impressions, ideas.


So how good is Hume's case? Are we convinced by Part II Section VI that external existence is not even an idea?

The introductory bit in 4.2 says we can't raise the question of external objects — because Nature — but we can look for causes of the belief we're stuck with.

But didn't 2.6 say we have, and can have, no such conception of the external existence of objects?

Isn't that a flat contradiction?

There's an out — because in 2.6 he says we don't, you know, really think of objects with external existence (since we can't) but that we do attribute 'different relations, connexions, and durations' to our perceptions.

4.2 is thus entirely about our perceptions, because Hume takes it he has already shown there's no point in trying to talk about anything else — or at least that such talk can be no part of his psychology. Thus, whatever idea we have about objects that exist distinct from our mind and perceptions, will be an idea about perceptions that exist distinct from our perceptions. Big no there. Whatever idea we have about objects that continue to exist when they are not perceived by the senses, will be an idea about a perception that continues to exist when there is no perceiving. Another big no. This is the substance of point (1) above, that neither the senses nor reason take us any way toward the principle concerning the existence of body.

So this is what nature has forced on us, the idea we think of as the external existence of objects, the bizarre belief that some our perceptions continue over time distinct from us the perceivers, since we can only think perceptions not objects. Picturing the tree is picturing the tree existing; picturing the tree existing when no one's looking is picturing the tree. Picturing the tree without the tree being pictured, is not a thing. Picturing yourself picturing the tree for a bit and then stopping, is picturing your picture of the tree waiting for you to come back and resume picturing it. Whatever we may try to think about external objects, this is what we'll end up thinking.

So it is clearly not knowledge or even justified true belief because the J clause fails. — Ludwig V

This is just to deny one of Gettier's premises. And that's fine, of course, but on what grounds?

Gettier is deliberately pretty vague about justification so that his argument applies to various formulations in the literature. He describes the evidence his protagonist has, with the assumption that it's the sort of thing we would usually consider adequate to justify holding a belief. If it's not, in our view, adequate, then we ought to strengthen the evidence until it is.

I take it as obviously true that we can have very strong evidence for a proposition that is false, for the simple reason that evidence is mostly a matter of probability.

In the case at hand, we have a farmer looking out at a field, and he's probably used to the way a cow standing in the field catches just a bit of light so that it's an indistinct bright spot in the otherwise dark field. He's never known anything else to be in the field that had this effect, though obviously a great number of things, including the cloth that turns out to be there, could, so when he sees such an indistinct bright spot he thinks 'cow'. That seems to me a perfectly reasonable belief and he has probably formed a true belief on just such a basis thousands of times before.

There is obviously a gap between the evidence and his conclusion; that gap has never mattered before, but this time reality falls into that gap. C'est la vie. You find the gap too large to allow him to claim to know there is a cow in his field; the point of the Gettier problems is to ask how small the gap has to be before you are willing to allow such a knowledge claim. If there has to be no gap at all, then it's a little unclear how useful the idea of 'justification' is. We may believe inference from empirical evidence can approach demonstrative proof, but we don't generally believe it can actually reach it.

Wherever size gap you choose to accept, even in a single case, that's where your situation can be Gettierized. That's my read. Rational belief is the sort of thing you have evidence for, not knowledge.



I find Lewis's contextualism a pretty obscure doctrine, so I'm not ready to go there yet, and I have more reading to do before taking on contextualism in general. Your view seems to be some sort of hybrid, in which knowledge is still a sort of justified belief, but what counts as justification is context-dependent. (Usually contextualism passes right by justification.)

For this case, Lewis might say that until that fateful evening when the farmer mistook an old shirt for a cow, the possibility of something making the same sort of bright spot in the field as a cow does was irrelevant, but it's not irrelevant for us as the constructors of the hypothetical, so we have to refuse to attribute knowledge to him. But now what? Must the farmer forevermore wonder whether the bright spot is a cow or an old shirt? Because we know he was mistaken once? We might well ask, but Lewis specifically does not make such demands on the farmer, who either will or won't. This is a puzzling theory, that the less imaginative you are the more you know.

What does seem to be the kernel of truth in this story is that — atomism incoming — some states of affairs are relevant to the truth of a proposition P, and some aren't, and some states of affairs are relevant to your knowing that P, and some aren't.

One of the roles of knowledge is to raise our standards of knowledge. I mean cases like this: suppose my son and I haul out the air compressor to inflate his car's tires, and then he's to put the compressor away in the shed. If I later ask him if he got it squared away, his report amounts to a claim to know that he did, and we can itemize that report: he knows to coil up the extension cord; he knows to bleed the hose, else it's too stiff to coil up; he knows to switch off the outlet it's plugged into or unplug it; but does he know to bleed the pressure from the tank? Did I even show him how to do that? If he didn't bleed the tank, then he only believes he put it away properly, but he doesn't know he did because a step has been left out, something relevant to the truth of "I put away the compressor properly."

That sort of relevance analysis, or contextualism, if that's what it is, I'd endorse. It does mean that the more you know, the more cases of putative knowledge might be excluded, because the world is rich with half-assery. But that's nothing like Lewis's acceptance of knowledge going bad when doing epistemology; I'm not imagining that the tank should be bled, I know it. And so it is with those benighted contestants on Deal or No Deal who claim to know the million dollars is in their case: they know no such thing; I know that they don't know not because I'm more imaginative, but because I know how random choice works, and they apparently don't.

Every instance, it seems to me we have a different perception — Manuel

This is another point he makes repeatedly, that we only have these unique, ephemeral perceptions, among which, to be sure, there are resemblances, but that neither reason nor observation justify us explaining these resemblances by positing a constant object they are perceptions of.

One thing notable by its absence in the whole section is conceptualization. Hume allows that one perception resembles another, but shows little interest in, or anxiety about, the sorts of conceptualization we worry about a lot. Hume's system is, to this extent, somewhat more mechanical and less cognitive, than we are used to in these post-Kant, post-Frege days. But it may indeed mean he is closer to mainstream psychology than we usually are; this is a naturalist epistemology.

Good point about the secret springs of nature.

There is something I find fundamentally unnerving about Hume's arguments, that they're hard to categorize as 'empirical' or not.

Will have more later, but I wanted to add what seems like a very strong argument against double existence -- though again it leaves me a little confused which arguments have priority over which.

This is the argument based on his account of our judgments of cause and effect being derived from the experience of constant conjunction. He argues that the claim that some object causes our perceptions cannot be accepted because we never have the opportunity to observe the object, on the one hand, accompanied by the perception, on the other, much less constantly.

That's an awfully strong argument *if* his account of causality is correct.

I have lots of rereading to do, and thinking after that, so I don't have a full picture of the argument yet.

I think the idea is that an impression is something that occurs to us, becomes present to mind, involuntarily, as feelings do, but is otherwise distinguished from an idea only in being more vivid.

In particular, we are counseled not to think we can distinguish impressions from ideas by their source, one external, the other internal. So far as mind is concerned, we are told, they are the same.

If that's right, then indeed the state description is right and the relation description is speculative.

It might seem to be only a formal shift, at one level, from something like (a) stuck-in(arrow, target), a relation, a function from an ordered pair to {0, 1}, to (b) stuck-in-target(arrow), a function from a single object to {0, 1}.

But the 1-place function is just a partially applied 2-place function. The target is just baked in.

Hume seems to think he doesn't need it, that you can coherently say 'impression' and dodge the question, "Impression of what?"

The argument goes round, that the hypothesis of 'double existence' is insupportable, which would be true if impressions are the same as ideas. But is that claim based only on introspection? Or does it arise from a methodological choice not to consider the 'what' that impressions are of? (Here I really have to reread.)

In the latter case, the methodological choice would end up becoming a substantive position, no double existence. Because of course by the time the question is addressed, there's no other answer you can give.

The starting point for Hume's position is that only perceptions (encompassing impressions, ideas, feelings) can be present to the mind.

A second point is that mind can be exhaustively described in terms of such perceptions and their relations (associations, resemblances, and the like).

Which is to say, the mind can be exhaustively described in terms of its states, by which we mean its 'internal' states, there being no others.

Is this plausible?

If a bee is perceiving a flower, isn't the most natural description of such a situation, that there is a relation, perception, that holds between bee and flower?

But Hume will take the bee to be in a perceiving state, full stop, rather than considering the bee as related to another object, the flower, in a particular way.

And the explanation for that move, from relation to state, must be found in Hume's account of the senses.

The question comes down to whether the main character's belief is justified or not; the stories create situations in which it isn't possible to give a straight answer. — Ludwig V

I think that's a pretty common reaction. "No false lemmas" can itself be taken as meaning that the belief in question wasn't really knowledge because it wasn't really justified, or as a fourth condition, separate from justification.

I find the whole approach suspect, as I think justification belongs with rational belief formation, where it's perfectly natural to consider the support offered by evidence as probabilistic, and the beliefs derived as partial. That leaves knowledge nowhere (as some would have it) or as a separate mental state, not belief that's really really justified.

But I'm open to argument that JTB-NFL can be made to work.

If justification and truth run on separate tracks, then justification can sometimes lead, quite reasonably, to falsehood, just as we can sometimes hold true beliefs by luck. (Lotteries provide the clearest examples for both: you can pick the winning number, without justification, and you can only be justified in believing that you didn't, given the odds, but you can't know it.)

"No false lemmas," by stipulating their conjunction, doesn't really address the main issue: either the true, justified lemma is knowledge, or it should face a Gettier case of its own — that is, you will be lucky that your premise is true. (If it's knowledge, then we've taken a step toward Williamson's E = K, the idea that rational beliefs are based on knowledge; but to claim that knowledge must be based on knowledge is either empty — because of course we'll take valid inference to be knowledge-preserving — or circular. If there's a third option, it's pretty subtle, but maybe there is.)

I'll admit, though, that it does seem to help. In Russell's example, checking the time from a clock that's stopped, had you looked a minute earlier or later, you would have formed a false belief, so you were lucky to have looked when you did. Now suppose that the clock was working and had the correct time, but stopped right after you looked; now I think we want to say you do have knowledge even though a minute later you would have formed a false belief. You were genuinely lucky in looking while the clock was still such that it was knowledge imparting.

So what's changed? If you look a minute later, we're exactly in Russell's scenario; a minute earlier, and you're fine. What if we compress things: suppose the clock stopped this time yesterday, briefly surges into life as you approach, just long enough to tell the right time for a minute or so, and then fails again. Now your window of luck is a range of a minute or so — too early or too late is still Russell, but for a brief span, the clock is knowledge imparting. Does that sound right? It sounds a bit dodgier now; you have been nearly as lucky as in Russell's scenario. The clock starting again feels wrong; had it started a minute earlier it would carry on being ahead until it failed, later and it would remain behind. What's missing is the clock actually being set; if a worker had just gotten the clock to work, and set the right time, you would again be acceptably lucky to look while it's keeping the correct time, even if it only did so for a minute before the worker cursed and set to work again.

To say that the clock has been set properly is to say that the time it displays is not only true, but justified, I suppose. But we can keep pushing the problem of luck back into these ceteris paribus conditions, which will grow without bound. Was the worker going by his own watch? What if his watch only happened to have the correct time? We're either going to continue demanding that truth and justification stay conjoined, or we're going to allow them to separate at some point, and that's the point at which Gettier will take hold.

Perhaps though what we're seeing here is that Gettier is the inevitable result of treating beliefs as atomic, and that the revenge cases are indicating that our beliefs never confront reality singly but as a whole, the Quinean view, I guess.

The mistake is to try and classify it as knowledge or not. — Ludwig V

The one thing everyone agrees on is that there is no knowledge here, so I wonder why you think there's a problem saying there is or isn't.

"No false lemmas" is discussed even on the Wikipedia page for the Gettier problem, in a section that begins with this amusing banner:

This article needs additional citations for verification. (October 2021)

Also on the SEP, which says

However, this “no false lemmas” proposal is not successful in general. — SEP

That's at least some places to start if you're sympathetic to the "no false lemmas" response.

He's saying, see, their using JTB and it failed to give knowledge. He's conflating one's claim to knowledge with actually having knowledge. — Sam26

This is not even in the ballpark of the Gettier problem.


For completeness, here's the IEP page on the Gettier problem.

So in this case, we can see that the proposed ideal, is really less than ideal, because the proposed inversion is contaminated by the presence of zero on the number line. — Metaphysician Undercover

Suppose you have a bag of apples you intend to give away.

There are two courses of action open to you:
(1) You can give someone the bag of apples, so that your hands are empty.
(2) You can give out the apples that are in the bag, and keep the empty bag.

There are two perfectly distinct results here: in one you have nothing; in the other, you have an empty bag. In both cases you have no apples.

It's understandable that you might not be inclined to say that a person who has no apples has a certain number of apples, namely 0. What you'd prefer is to say that they do not have any apples. There is no quantity that they have at all, and calling 0 a quantity is an abuse of the idea of quantity. That's understandable. The same with measurement: to say that a person who takes one step to the right has moved that amount is fine, but it is an abuse of the idea of distance to say that a person who has not taken a step at all has moved 0 steps to the right, to the left, whatever direction you like.

What justifies us extending concepts of quantity to include 0?

It's the bag, the difference between not having a bag at all and having a bag with nothing in it. 0 ends up playing a prominent role in positional number systems because the positions in such a number system are like bags laid out on a table into which you can put at most a certain number of items. But the bags are fixed; you do not remove them when they are empty.

Similarly, when we do algebra, we use containers for values, variables, and it may be possible for a variable to hold no value at all, that is, 0. But the mathematical functions we apply to a variable are defined so that they go through even if turns out the variable held a value of 0, or no quantity at all. You just have to follow some rules, so that you don't mistakenly divide by 0, which makes neither mathematical nor intuitive sense, as in this famous 'proof' that 1 = 2:

$\mathrm{Assume}\ a = b.\\\begin{align}a^2 &= ab\\a^2 + a^2 &= a^2 + ab \\2a^2 &= a^2 + ab \\2a^2 - 2ab &= a^2 + ab - 2ab\\2a^2 - 2ab &= a^2 - ab\\2(a^2 - ab) &= 1(a^2 - ab)\\2 &= 1\end{align}$

How does that apply to this poem in particular? — T Clark

To the Danielle Hope poem?

Not sure. I don't have any feel for how she writes.

Perhaps I shouldn't have made my remark about prosody sound so universal. Hebrew poetry, for example, is structured semantically, so there's a sort of rhythm of thoughts, rather than sounds. Or so I understand.

Williams I have some feel for, but the rhythm is the hardest part to analyze or explain. Reading "This is just to say" is like unfolding a bit of origami. He's very tricky about how the syntax is broken up over the lines; you unfold the next bit and it's satisfying but then you're not sure where to tug next and suddenly pop the next fold has come open. By the time you get to the very end and it's all laid out, you're not quite sure how you did it. Some of these little poems of his sound like they're sentences, sound urgently and insistently like sentences, but turn out not to be if you look carefully. Some of that is a commitment to spoken vernacular American, in which syntax can be a bit malleable, but some of it is the way lineation offers a competing structure, and that structure is in part rhythmic.

Which do you think is happening? — frank

No.

If math is self consistent, this is like deciding whether we want to play golf or basketball. No stipulation is taking place. — frank

Here's one way stipulation could enter our play: I don't play golf, but I know roughly how it works. If you know no more than I do, we'll have to make up some rules as we go and agree to them. We'll hope we're getting it roughly right. Our sense of the basic idea isn't enough to get us through an entire round of golf with the sorts of complications that inevitably arise.

Here's another: we could take elements of basketball (teams, a playing area with goals at either end) and elements of golf (small object struck with a special kind of stick) and combine them to make something like hockey or field hockey. Hockey wasn't on your list before so it's not something we can straight up play based only on intuition; we have to make up the rules based on some things we understand from other games.

Prima facie, doing mathematics is not like, say, speaking your native language. Mathematical objects are things we investigate, and make discoveries about. It may resemble playing a complex game like go or chess where you can understand the rules and still not be able to predict what will happen, but the rules of math are only logic and some ideas about counting and shapes and collections that we get, I think, from the real world, so the content isn't exactly arbitrary.

Not at bottom, but we can do things. If you take your ideas about shape and agree not to think about size the way you usually do, to forget that things can be measured, you get a sort of generalization of geometry, and that's topology. You can still talk about types of shapes, and see that there are still some rules about which shapes still count as similar to others, and that these were implicit in the way you did geometry, but by treating shapes in this special way, you get a sort of alternate version of reality in which donuts and coffee cups are the same sort of thing. That's based in our intuitions, but in a selective way. We do the same sorts of thing with numbers, in constructing algebras.

I feel like I'm just not getting the opposition you see here.

I'm saying you might have many intuitions about shapes or counting or collections, and I ask you to rely on only a selection from among those. (They may not even all be consistent.)

Agreement in the selection is effectively agreement about the content precisely because what we're agreeing to select among are the semantic contents of our intuitions.

One way to think about poetry is that it foregrounds elements bedsides the words that shape our understanding of an utterance.

Tiny example. Hugh Kenner tells a story about Eliot, that returning to England on the ferry, someone called his attention to the white cliffs of Dover and remarked that they didn't look real, to which Eliot responded, "Oh they're real enough," a sentence Kenner takes to have four different meanings depending on which of its four words you emphasize.

Prosody matters enormously to the meaning of a poem.

I don't know. What's "real stipulation"? Does that mean "arbitrary"?

If so, no, I don't think the foundations of mathematics are entirely arbitrary. It's not just a game we made up.

But the selection process means not just including but excluding. Think about when you learned to do proofs in geometry. There may be things about a figure you can see are true, must be true, but if you can't show it given only certain premises and inference rules, you can't use it. That's not really much different from your teacher drawing an equilateral triangle on the board and not marking the edges as of equal length. Your intuition is that they're equal, but you're expected to ignore that and treat this triangle, equilateral though it may be, as generic.

There's choice in axioms at least in the sense that we can select which of our intuitions to build on. We don't have to do everything all the time.

So I can say

If an axiom, then based in or captures an intuition.

without being committed to

If an intuition, then captured in an axiom.

See?

But Euclid had axioms.
— Srap Tasmaner

Those aren't a matter of our choices though. They reflect cognitive imperatives. — frank

They clearly are a matter of choice or there wouldn't be non-Euclidean geometry.

no one says negative numbers don't have square roots — Real Gone Cat

No one says it *now*. Wait, actually we don't tell little kids about imaginary numbers, so I guess we do still say it. We don't have to though, because we have the theory in place, and maybe one day we'll teach the complex plane in grade school.

If you're talking about the axioms that protect set theory from paradoxes — frank

No, not just those -- and they're not just to ward off paradox but are an attempt to capture our sense of how collections work, not the most naive sense, of course, but after we've been out on the road with it a while. But Euclid had axioms. Peano gave axioms for natural numbers. Axioms are everywhere in math.

It's known that classical logic is insufficient as a foundation for mathematics, but I remember reading somewhere that all you need to add is Hume's Principle, which is that two sets have the same cardinality if there's a bijection (a one-to-one correspondence) between them.

It's debatable whether math really needs set theory as a foundation, though. — frank

You'd prefer category theory? Nah, you just mean math doesn't need any foundation. And that's obviously true both as an historical and as a practical matter. But there is value in having the foundation. Sets are the lingua franca of mathematics, so they enable making connections, leveraging techniques developed in one context in another, and so on. I think they clarify the use of special axioms too, because there is the common set you can take for granted, and you get your little branch of math by specifying some *additional* constraints. Doing topology, for instance, feels a lot like doing set theory, but with just a handful of other properties in play, and out of that you can develop a rich set of properties and theorems for spaces.

Aren't those things features of how the human mind works? — frank

No, I don't think so. Not exactly.

We have some basic intuitions about collecting and counting, about geometry, and so on, and we build mathematics out of those by making choices, our axioms, and then those axioms have logical consequences.

That means the consequences are implicit in what we made, but not in us as its makers; the properties of buildings are different from the properties of builders.

But really to get there I have to say that I don't think logic is just in our heads, anymore than the physics that underlies structural engineering is. And I don't.

I also don't think our intuitions about counting and geometry are just in our heads, but that doesn't matter for the point I'm making. I think.

Actually 0^0 is called indeterminate and has no value. Any rule you're trying to use to assign a value is not applicable. — Real Gone Cat

Okay. But isn't that just to say either there's no math that defines a value for it or that you're unfamiliar with math that does.

To just say, nope, is like saying negative numbers don't have square roots, or, for that matter, that 2 doesn't.

Math is a big place.

As for 0 carrying a sign, we could have such a system. 0 could be canonically positive and I don't think it would affect anything.

Also consider what happens if you're solving an equation and get all the way down to x = -x. You can see the answer, or you can take another step or two and get x = 0. Which means you can substitute back into the equations you had before, and in particular into x = -x, so that now says 0 = -0. You can make that go away, if you like, but even to do that you have to accept -0 as well-formed. And in fact, I could see having 0 = -0 be a theorem of a system that allowed 0 a sign, or even a definition of zero, that it is the only number for which this is true.

Math is a big place.

find nothing wrong with using mere conceptions to refer — Mww

Wonderful faculty....imagination. Always in use, seldom given its due respect. — Mww

For some cases, the issue is direction of fit. It is one thing to imagine a way of proving Fermat's last theorem, and then spend years actualizing that proof, and another to have written out some mathematics you mistakenly imagine is a proof of Fermat's last theorem. Fermat himself seems to have imagined a proof, which he did not write down, but if he had he would probably have recognized that it was not a proof after all.

Cognitive imperatives? — frank

No, I was thinking more fundamental mathematical principles, or how mathematics as a system works. Things like harmony, symmetry, orthogonality, duality, that kind of stuff. You might want signs just for convenience, but there will have to be a deeper coherence to how a signed number system works, and that's where you'd look to decide whether 0 is +0 or -0.

What is math's rudder? What necessity would inspire us to talk in terms of +0? — frank

Yeah that's a funny thing. Math often allows degenerate cases to pass through for the sake of generality. If you need to say every integer has a sign (for whatever reason) then you'll need 0 to have a sign. Which one? That strikes me as a deep question, in the sense that your reason for giving it a sign is probably not powerful enough to dictate which sign; you'll need some other reason for saying which, and that reason is likely to be "deeper" if you see what I mean.

Something like this problem comes up with 0⁰. We have one rule that would assign it the value 0, and another that would assign it the value 1. Which one should win is an interesting question.

Iambic pentameter, 3 stanzas. Rhymes as follows: ABAB, CDCD, EE — Moliere

That's the Shakespearean sonnet, with the volta coming rather abruptly at the start of line 13. The older form (petrarchan I think) has a group of 8 and then 6, so there's more time after the volta to develop the counterpoint to the first 8.

Trying to make it obvious here how the structure of a poem shapes its meaning.

But I don't think there's much stipulation going on. — frank

Yeah that's what I meant.

I'd forgotten Dennis Ritchie talks about that, but computer scientists (not coders) spend a fair amount of time thinking about semantics. When Jim Backus and his team at IBM invented the first high-level programming language, they had to simultaneously figure out what such a thing would be, and also invented a formal way of specifying its grammar, the Backus-Naur Form still used today.

Assuming you have the semantics of "inverse" to hand, which evidently we don't. In mathematics, it's stipulated. It can be stipulated in other contexts as well.

The trouble comes of what fills the role of stipulation in everyday usage of a natural language.

Santa Claus does refer, and without any annoyance I should think, if the sticker on the gift-wrapped present says “from: Santa”. And if considered from the standard subject/copula/predicate logical propositional format, any conception contained in a predicate, and is thereby the object of it, must refer to its subject. — Mww

Santa Claus is a real fictional character, but not a real person. When the tag on the present is signed "from: Santa" that's supposed to mean it's from the person Santa Claus; no one thinks they're getting a present from a fictional character. Since there is no person Santa Claus, signing a tag that way is pretending that there is such a person.

That's what I mean when I say "reference": an expression that picks out one of the objects in the world. Santa is not one of the objects in the world, so the expression "Santa Claus" does not refer. We pretend it does.

The sense in which a predicate "refers to" its subject, to what it's predicated of, is a matter of syntax not semantics. It's the "of" in "red is true of this ball." Not the same as the expression "this ball" referring, semantically, to some particular object.

I dunno, I don't really feel that way. I find pre-theoretical intuitions interesting and important. No math without 'em.

I read just the other day that a common counting system (among non-literate peoples) is 1, 2, 3, Many. Don't know if it's true, but if so we've come a long way.

non-referring expressions are annoying.
— Srap Tasmaner

As in....infinities with respect to mathematicians, and universals to philosophers? Can we say that which refers to every single thing of a kind is non-referring? — Mww

Kinda. In "This ball is red," "... is red" is a function not an object, the characteristic function of the set of all red things. Of course we also want to quantify over functions, so that means taking them as objects, as in "Red is an easy concept to learn."

That functions don't refer when used as functions shouldn't bother anyone; they're not supposed to refer.

The annoying cases are "Santa Claus", "Sherlock Holmes," that stuff.

Some aid in distinguishing the real from reality, then? — Mww

Clarity for the questions maybe?

We can talk about "the smallest real number greater than 0" but there isn't one, despite our lovely predicate. We get into a muddle if we make that thing and then say it doesn't exist, because non-referring expressions are annoying. But we don't have to do that. We can show that the set determined by such a predicate must be empty. Or we can skip to how we do that, by showing that there is a positive real number smaller than any given positive real number. No non-referring expression needed.

Opposites are commonly — Metaphysician Undercover

And maybe we do it differently for numbers. What's the opposite of 0? If you take it as "none" then its opposite is "some" which is not a number. What's the opposite of one? In many everyday uses, that's "many" or "several", that is, more than one. What's the opposite of 7? Of 94? What?! I mean, you can always just take 'opposite' as 'complement' within the domain, so the opposite of zero is non-zero, the opposite of 94 is {x in Xs | x =/= 94}, given some domain of Xs.

In math we also have inverses, additive and multiplicative. They're opposite-ish, the way equivalence is equal-ish.