What does your list look like? — Joshs

I'd have to spend some time on Baseball Reference to give a good answer, but my first pick would probably be Greg Maddux. He's currently only #28 on the all-time leaderboard, but I love the way he pitched.

run it through A.I. to highlight the vantage from which each group critiques a previous group — Joshs

You're on a roll tonight.

Can you give me an example of anything other than consciousness and its creations that cannot be explained by physics? — Patterner

Almost everything, depending on how you flesh out "explained by".

Not really a discussion I was looking to have, but this has been really helpful, so thanks!

I've never quite gotten the fascination consciousness has for people around here, why it seems so super special, and it's because we start from very different ideas about—among other things, probably—the unity of science.

Points, though, for the most-to-least advanced list. That gave me a chuckle.

We've met, Josh. I am well aware who's on your fantasy baseball roster.

Okay, I think I get it now. You and @Clarendon believe that all natural science can be reduced to physics, and that all natural phenomena can be explained by physics, with the sole exception of consciousness. Yes?

We know how the properties of the atoms and molecules of living things account for metabolism. — Patterner

"Account for"? Meaning what, exactly? That you could deduce the great variety of living things on earth just from studying carbon and hydrogen and oxygen and so on? Could you instead study electrons and neutrinos and photons and whatnot, and get even better results?

I'll go further than that. The tragedy of Wittgenstein is what was missing from his toolbox.

For instance, an awful lot of Wittgenstein's puzzling over rules and grammar cries out for the sort of game-theoretic analysis David Lewis does later — but Wittgenstein didn't have game theory.

A lot of what he says about concepts and seeing as, the whole midcentury recognition of theory-laden observation and the repudiation of the myth of the given — he's not unique in that, and all of it is stumbling toward what only becomes clear in the Bayesian framework, that evidence is the basis upon which a prior belief is updated, but it is not the basis of belief as such. Ramsey would have gotten there, as "Truth and Probability" shows, but whether he could have dragged Wittgenstein along, who knows?

Wittgenstein turns away from certain old ways of doing philosophy, and he seems to point—so tantalizingly!—toward a destination he never really gets near. It's why he is undeniably vague, inconclusive, difficult to interpret, why he goes over the same issues in subtly different ways for years on end. Having cut loose from the mainland of existing philosophy, he was at sea, and never made landfall. Heroic, in his own way, but tragic.

Pretty sure I'm the only one around here who thinks this.

There is no example of a feature strongly emerging. If you know of one, say. Strong emergence is ruled out a priori by reason, and there is no example of it either to challenge what our reason tells us. — Clarendon

I'm not sure how to proceed here.

Against my better judgment, I glanced at the SEP article on emergent properties to see if I could get a handle on the terminology here, but it's a nightmare, as usual.

Your principle that "you can't get out what wasn't put in" seems much too strong.

Living things grow, they metabolize nutrients and excrete waste, some of them move around, and eventually they die. Their organs and tissues don't do those things on their own, and certainly the chemicals, the molecules, the atoms those components are composed of don't. An engine can give motion to a vehicle it is installed in; the components of an engine cannot do that. One atom and another might be roughly the same size, but when combined with others of their kind, one forms a hard substance, one a liquid, another a gas. Mountains create micro-climates around them, but the dirt and rock they are made of do not, and the plain next door might be made of the same dirt and rock. Any ecosystem is sensitive to changes in its climate or changes in the population of the organisms that in part constitute that ecosystem in a way that no individual or species is. Crowds routinely behave in ways that do not reflect the individual choices of their members. A central bank might lower interest rates with the intent of lowering mortgage rates, but cause the yield on bonds to rise, thus causing mortgage rates to rise — or not, you never can tell. I'm about to use a microwave to heat my coffee, but my microwave manages this not by being made of things that can heat coffee; I cannot get the same effect by removing the glass platter and just sitting my cup on that.

It seems to me everywhere you see more than what was put in, wholes that are not the sum of their parts, unintended consequences.

I expect you'll say all of these are "weak emergence", by which you don't so much seem to mean what your size and shape analogies would suggest, as that you think you understand them. I think consciousness is just like all these, and it is brought about by evolution, which is notably proficient at producing novelty.

I see.

I was wondering if you had other examples of deductions that rely on this principle:

you can't get out what was in no sense put in — Clarendon

With other examples, we could compare the case of consciousness.

Aside from that, this "truth of reason", as you describe it, strikes me as patently false.

wherever there is a gap, God will be inserted, as a kind of explanatory wall filler. — Tom Storm

Reminds me of a nice Wittgenstein aphorism:

A crack begins to appear in the organic unity of the work of art, and so I stuff the crack with straw. But to quiet my conscience, I use only the best straw.

by the same reasoning — Clarendon

Uh huh.

Is it just consciousness?

Are there any other properties of things that, in your judgment, would require strong emergence?

if the parts of that structure wholly lack conciousness, then appealing to structure and complexity just assumes demonstrates that a new kind of property can arise from their arrangement. — Clarendon

FTFY.

almost everything — Srap Tasmaner

Depending on how far down you go. It's obviously everything, if you get to subatomic particles.

Eight hundred leaf-tables and no chairs? You can't sell leaf-tables and no chairs. Chairs, you got a dinette set. No chairs, you got dick!

Examples are not only plentiful, I suspect almost everything, living or nonliving, that everyone on this site has ever interacted with has properties its constituents lack. It is the norm. It is what nature does. Criminy.

Are you saying that I have committed such a fallacy? — Clarendon

@wonderer1 need not say it; you have presented a textbook example.

Read the wiki page he linked. It is educational.

Yes, I think that's right.

In a sense, what the formalism of FOL identifies is that being a member of a domain, or not, and satisfying a predicate, or not, are the same operations for all domains and for all predicates.

In that sense, it is a just a further step along the path Aristotle discovered when he noted the structural similarity of classes of arguments, setting aside the specific contents of the premises and conclusions.

You know, Quine's dictum is a funny thing.

On the one hand, it seems to treat "there exists" as univocal, when discussions like this seem strongly to suggest different sorts of things exist in different ways.

But on the other hand, Quine's dictum does, in its own way, recognize that "is" is "substantive hungry". (Austin's phrase? It's the point that "Alfred is" strikes us as incomplete -- "Alfred is what?") Variables don't float around on their own in classical logic; even when not bound by a quantifier, they only show up governed by predicates.

("What about the domain of discourse? Surely that's just a collection of objects we have assigned names to." But Quine was also inclined to do away with names and use only predicates.)

I think we could follow Quine in saying that, so far as logic is concerned, "there exists" is univocal, while recognizing -- perhaps against his wishes -- that because bound variables are always governed by predicates, there is room for allowing that dogs exist the way dogs exist, numbers the way numbers exist, quarks the way quarks exist, and so on.

(I have complained on several occasions that our logic does not distinguish between predicates and sortals, and this looks like another one of those occasions. But we can similarly recognize that truth functions don't care about that distinction, even if sometimes we do.)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

That's incorrect. — frank

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

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

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

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

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

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

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

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

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

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

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

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

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

Maybe. I'm still mulling it over.

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

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

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

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

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

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

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

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

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

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

Denumerable — frank

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

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

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

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

the real numbers — Banno

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

numerals — Metaphysician Undercover

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

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

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

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

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

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

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

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

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