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.

Once the super-stardom of neurotransmitters fades, I think we'll see support more in line with the bigger theories — Isaac

That sounds plausible. Also wouldn't be surprising if the role of neurotransmitters was overestimated because we got to them first, because there were tractable problems about their role we were able to address before we could make any headway on neuronal networks. That means there will be a step coming at which neurotransmitters will likely be given too steep a discount, before the pendulum swings back to finding a harder-to-reach role for them.

This is all just science journalism though. One thing I always have in the back of my mind is that science can cheerfully proceed this way, iterating and refining, and at multiple levels — individual experiments can be repeated but done better, theories can be replaced by other theories within the same program or paradigm, programs and paradigms can be replaced by others. The latter shifts can be difficult to explain, but should engender, always, more and better science. A scientist can expect her field to move over the course of a career, and must expect to say, "When I was in grad school thirty years ago, we all thought ..., but now ..." Philosophy moves, but not quite like this. For what sorts of values for X would a philosopher say, if X, then we'll all have to start thinking about Y differently? It happens, but much less often, so there must be a different mechanism here. (Assuming it's something besides fashion.)

there is also a big chunk of the theory that simply not disputable - not because of the weight of evidence, but because, like mathematics, it's just re-arranged the equations to say something interesting. You can dispute that it's interesting ... — Isaac

Right, and that has to do with interpretation, but this is so complicated, because there's science-engendering generality and interpretation, and there's public-facing, also general (at least because detail-poor) interpretation, which is in some ways close to application. Scientists who can be very clear about 'what this means' for the field, can be very wrong about 'what this means' for non-scientific purposes. Down in the valley of the nitty-gritty, the generality of the program is still a constraint, but non-scientific generality is worse than useless; in philosophy, those two sorts of generality should be nearly the same (because of "saving the appearances"), and when they aren't philosophers say the same sorts of things scientists say: you think you have knowledge but here's what's really happening when you think you know something (philosophy); you think you see things but here's what's really happening when you think you see something (science).

It's more a description of what it means to avoid entropy (remain organised) than a modelling assumption, in that sense. — Isaac

And maybe in that sense no more than a reworking of Kant, who knew enough to expect the subject of knowledge to have a sensorium, but not enough to expect the subject of knowledge to be self-organizing, with all that entails. Such an enriched Kantianism might be interesting, but it's not really comparable to the original. You could as well say that the subject of knowledge must have arisen through evolution by natural selection.

It's just not at all clear — and @apokrisis is right about this — at what point we are really passing from a priori to a posteriori. Kant's conditionals are supposed to be awfully strict, meaning they are intended to rely to the greatest extent possible only on logic and not on how the physical world happens to be.

Point being, it is interesting to know how an organism might acquire knowledge, not least because we are organisms. It is less clear that only an organism can be the subject of knowledge, but if you're a biologist that's exactly what you're going to assume because you only study the natural world, not the possible world. Kant's concern was knowledge, not the knowledge of organisms.

But there is this grey zone, and I take it this is where apo's grand synthesis lives, in which we consider what a possible natural world could be, and that means, to begin with, showing that the actual natural world can be described without remainder as such a world. It's just not clear to me what else this is: it's definitely not science, because you can only do science with the actual natural world , and this level of generality is somewhere above what usually defines a research program; but it's not just interpretation either because there's more here than the science of the actual natural world.

Thank you both for forthright answers.

@Joshs, your describe something that sounds like a research program a la Lakatos, which seems pretty reasonable. My grasp of the philosophy of science is just strong enough to find that plausible but not strong enough to quibble. — I do wonder though, whether we are the right audience for your cheerleading, not being scientists who may do research developed around the core of your suggested program. Why us?

I don't know what you're describing, @apokrisis, but it sounds somewhat more — I'm not sure how to put this — messianic.

That would be like asking house painters what they thought about abstract expressionism. — apokrisis

I'm sure this is meant to suggest what we benighted AP folks call a "category mistake," but it sounds a lot like contempt.

It is about a willingness to stand apart from the herd. — apokrisis

At least there is, it seems to me, a natural audience for what Josh has to say; it's just not us. You, on the other hand, are a voice crying in the wilderness. No point talking to the herd; they won't listen and wouldn't understand if they did. So why us?

But how can you assess the evidence — apokrisis

Wouldn't dream of it. I'm not competent to.

But if, on my next day off, I wandered over to the Life Sciences building at the local state university, and asked everyone I met there about biosemiosis and Friston and Salthe and all the rest, they would all assure me that it is universally accepted — except perhaps for a handful of dinosaurs on the verge of retirement — and as well-supported as, say, evolution.

Is that what you're telling me?

Why isn't there a -0?



That question can be rephrased: why does -0 = 0?

The usual mathematical approach to such a question is to try negating it and seeing what happens.

You should try that. What happens if you have a -0 unequal to 0?

We now have a testable theory of the modelling relation that accounts for ... — apokrisis

Is it "accounts for ..." or is it "may be able to account for ..."?

You and @Joshs (but @Isaac I think to a lesser degree) are always talking as if this is all a done deal, as if all we needed was the theory, as if having a theory you can imagine testing is the same as testing and confirming it, as if saying you could in principle fill in the details of an account is the same as actually filling in those details rather than the giant IOU it actually is.

All well and good if you're laying the groundwork for a research program, but am I wrong to think there's rather little in the way of observation to support all this theory? I'm not denying the elegance of the theory, or at least the bit of it I understand, but does it have anywhere near the body of support that, say, QM or evolution by natural selection has?

To be clear: not critiquing whatever science there is here, which you are vastly more competent to judge than I am. I am questioning whether you are in so secure a position that you are entitled to be as dismissive of doubt as you generally are. Insofar as people ask questions in order to better understand what you're pitching, of course you should be answering, 'yeah that's it' and 'no that's way off' — you're the world's leading authority on your own position.

But this forum is not exclusively dedicated to your views, so insofar as you or @Joshs answer the sorts of questions philosophers talk about with "Shiny new theory says X" without assuring us that shiny new theory has much claim to truth, why should we listen to you? You guys have preferences among theories, good for you; let us know when you have overwhelming evidence.

A computer can easily calculate all the moves of chess — introbert

Well no.

The number of possible chess games is so large that — here I'm guessing — we'd need quantum computers to actually decide chess. As of now, it is common knowledge that white has some advantage; the statistics have been clear for a long time. What isn't clear, and no computer has determined yet, is whether that advantage is sufficient to win. Which just tells you that the concept of "advantage" is still, even with computer chess, a little tricky.

Point being: chess being still undecided, computers are not in a qualitatively different position from us; they just have infallible memories, fewer biases (nowadays, not at the beginning), and can calculate very much faster.

But there are also techniques to achieving such successes. Robert Kowalski, a key early figure in logic programming (especially the development of Prolog) and thus early AI, has suggested that humans might consider — instead of trying only to get them to think like us — learning to think a bit more like them.

The set of things to be explained exists but is empty.
— Srap Tasmaner

Can a set of things to be explained be empty? The set of explanations of things that exist may be empty, but that thing to be explained must be a member of the set of all things. — Mww

I only meant that we can talk about the set of statistical anomalies that the Bermuda Triangle is thought to be a member of and then discover that it is not.

There's a story, probably apocryphal, that Frederick the Great once gathered his court scientists and philosophers together and asked them to explain why a dead fish weighs more than a live one. They went around in turn each offering a theory, and once they had all offered their explanations, he pointed out that it does not.

An idea that seems well thought out to one person based on their (his and her etc) knowledge and capacities may be a very weak game to someone else more experienced or better able. — introbert

That's right. In making a move, you put your ideas to the test, but it's not generally a dispositive test, only what another fallible player like yourself could come up with under the same constraints as you. These days, if you really want to know the truth, you'll ask Stockfish, but in the old days, you had to do your own analysis. A writer might give their analysis of a famous game between top-ranked players, only to be contradicted later by another writer who came up with some ideas the first writer overlooked.

And this is another way in which chess has, before the computer era at least, resembled philosophy, or the sciences: it is cumulative. There is voluminous accumulated knowledge on openings and endings, middlegame strategies and combinational patterns. The first really serious use of computers was in completing, and in some cases correcting, our knowledge of fundamental endings. Now they just do everything better than us.

None of which is really a surprise, because, as John von Neumann remarked, chess is not a game but a form a calculation. Of course it can be turned over to computers.

Whether there is some reason philosophy cannot be, is an open question.

Discussion of anything presupposes its being real or possibly real enough to discuss. — Mww

Except when that's clearly false? You and I, discussing whether the Bermuda Triangle is a thing, with a mysterious ship- and plane-eating property, cannot be assuming that it is real: that is the question we are addressing.

(1) Is it possible that the Bermuda Triangle is a real thing? Is the idea consistent with the laws of nature as we understand them, for instance? Is there some suitably naturalist explanation for the disappearance of ships and planes thereabouts?
(1a) Our understanding of nature may be correct to the extent of ruling out Bermuda Triangles.
(1b) Our understanding of nature may be incorrect at least in ruling out Bermuda Triangles.

(2) If the existence of a Bermuda Triangle is consistent with our understanding of nature, or if our understanding of nature incorrectly rules it out, then the question remains whether we live in a world that has a Bermuda Triangle. It may be possible and thus real somewhere, just not here.

But now consider the actual Bermuda Triangle, marked off as a region of the Atlantic ocean through which ships and planes pass, and within which ships and planes are lost at roughly the same rate as any similarly heavily trafficked coastal region anywhere in the world.

With that in mind, to say that the Bermuda Triangle is not real, is to say that there is not something to be explained, but nothing, there being no statistical anomaly in need of explanation. The set of things to be explained exists but is empty.

the conception of universals is prior to the apprehension of particulars. — Metaphysician Undercover

But there are very good reasons people think it goes the other way.

For most people, for most concepts, acquaintance with instances of the concept precede, in time, the possession of the concept, and exposure to those particulars is instrumental in acquiring the universal they fall under. That's the argument from ontogeny: you are acquainted with moving, barking, licking particulars before you know that they are dogs. And there is a related argument from phylogeny: modern humans have a great many concepts that they were taught, often through the use of exemplars, but it stands to reason that not every human being was taught: there must have been at least one person who passed from not having to having a concept unaided. In essence, we imagine that person somehow teaching themselves a concept through the use of exemplars, and we imagine that process proceeding as we do when analyzing a population of objects, looking for commonalities.

In thinking about this thread, I was reminded of the Sesame Street approach to teaching about classes, an approach presumably backed by research, probably the most famous educational bit in Sesame Street:



(The irony of this song, "One of these things is not like the others. One of these things doesn't belong," in a show teaching inclusion and tolerance, was not lost on the makers of the show, and the bit was largely retired in favor of "three of these things go together," which is not much of an improvement.)

What's of interest here is that resemblance is not only relative, but comparative: resemblance is a three-way relation, a given object resembles another more, or less, than it resembles a third.

uncertainty avoidance — Isaac

Most people avoid tense situations. Repo man spends his life getting into tense situations.

I see. Static bad; dynamic good.

Serves me right for needlessly poking Isaac.

It is a point of interest how much you can do with a preference for order and predictability — any order, however arbitrary.

Sellars has that just-so story in "Philosophy and the Scientific Image of Man" in which he derives the idea of natural law from the observation that some of the persons in nature (old man river, old man mountain, that sort of thing) are set in their ways, the way people get, and thus predictable, the way some people are. (Big Lake is freezing over again, like he always does this time of year.) He suggests we recognized the efficacy of habit first and derived the idea of mechanical determination from that. (A sort of corollary to the 'theory' that we derive the idea of force from our own efficacious action.)

DishBrain is able to identify habits or tendencies in the "ball" and to develop matching habits or tendencies or propensities. For what purpose? In an earlier age, we might have heard this described as a manifestation of the death drive, the will to become mechanical, but maybe Freud was on the right track in seeing life as paradoxically trying always to reduce irritation and excitation, or to predict it well enough that it ceases to be experienced as surprise. (See, @Isaac, I do listen. Did you know you're a closet Freudian?)

Insofar as the practice of philosophy is largely a sort of conversation, there are obvious analogies to dance.

The issue of imaginary numbers is different though. It is an issue of there being two distinct conventions, yet each convention is correct in its own field of application. In the one case there is no square root of a negative number, in the other case there is. — Metaphysician Undercover

I don't think so. It remains true that negatives do not have *real* square roots, and that's the same as saying that if your domain is discourse is restricted to real numbers they have *no* square roots. The complex plane is a perfectly natural extension of the real line.

It is how the negative are conceived to relate to the positive, that creates the problem, i.e. it is not a straight forward inversion due to the role that zero plays. — Metaphysician Undercover

Not following this at all.

This is a terrible idea.

Chess is illuminating because it presents questions that may be decidable in principle but are not, for humans, in practice. The alternating reliance on calculation and heuristics, with the goal of grounding a decision under uncertainty, is very reminiscent of philosophy, which rarely gives opportunities for decisive arguments and must content itself with persuasion. And you still calculate whenever you can.

I usually don't see what we do here on the forum as competition. — T Clark

But don't forget that chess is also cooperative. Takes two to play a game.

Given a single ball being dropped into a Dalton Box, can you tell me where the ball will finish? — Banno

I will read. Shouldn't have neglected her.

In the meantime, and acknowledging that I may be putting my foot in it, this is a slightly bizarre way to talk about Galton boxes, the point of which is that even if chance is real, and not just a consequence of our non-omniscience, nature rather makes a point of capturing chance and turning it to the creation of order. The path of any single ball on a Galton box is at least effectively, for us, random, if not genuinely random, but the result of thousands of balls flooding onto the board is a perfectly predictable gaussian distribution. The actual shape of the distribution will vary from run to run, and the amount of variance is also predictable. Nature seems to believe in statistics.

<offtopic>

I'll consider adding "useful for us to believe" to the OP I want to write about "context dependent" and "purpose relative."

Maybe I'll save it for the one about "from our human perspective."

</offtopic>