Anyone have a link for the confusion in pure mathematics about the rings part? Did this get resolved since or something?

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Also relevant: Philosophy, Science, and Method-- a new blog dedicated to promoting philosophy that maintains significant contact with mathematics or science.

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While I agree with the sentiment about the typical philosopher's over-reliance on intuition and the need for formalization and calculation when possible, nonetheless many sub-disciplines of philosophy, such as epistemology, philosophy of science and (der) logic use formal methods regularly.

I apologize for going all laundry list on you guys, but one has just to look at the the many philosophical journals (The British Journal for the Philosophy of Science, Erkenntnis, Journal of Symbolic Logic, etc.), conferences (e.g. Formal Epistemology Workshop) professional associations (The Society for Exact Philosophy, Association for Symbolic Logic, etc.) and books that feature formal treatments of philosophical problems. Moreover, examples abound of philosophers using, and better, contributing to, Bayesian networks (Williamson, Glymour, Bayesian statistics(Seidenfeld), formal learning theory (Putnam, Kelly), set theory (Dana Scott and many others), and more formal subjects. Hell, there is even a move toward experimental philosophy for the more naturalistically inclined.

See this online copy of Patric Suppes' Representation and Invariance of Scientific Structures for one approach to formal philosophy. Suppes is an excellent example of a formally-minded philosopher that is responsible for many innovations in fields outside the discipline.

Best Regards.

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OK, I just checked my email and I see it was rejected as too long and insufficiently relevant. Very well. If anyone wants to see what I said (Caledonian and Unknown, I suppose this means you), mail me at mitchtemporarily@hotmail.com.

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I made a lengthy reply here yesterday. I even saw it go through. Half of it was about the real issue (the quantum worlds), half of it about these conceptual tangents. It's gone now and I didn't save a copy. I'll wait a few hours to see if anyone happens to have preserved it, and then I'll do it again from scratch.

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Before I get lost in these semantic and epistemic complexities, I will say once again what the problem is.

We are endeavoring to interpret the wavefunctions or state vectors of quantum mechanics: to form a hypothesis about the reality they describe. The hypothesis is: before decoherence there is one "quantum world", after decoherence there are many quantum worlds. As the difference between "one world" and "more than one world" is discontinuous, but the process of decoherence is continuous, with no sharp boundary between before and after, I asked exactly where the transition from one world to more than one world occurs. The reply was that that is not an issue, since the answer would make no difference to the argument in the papers. I conceded that it makes no difference to this particular argument, but the issue itself must be faced; the existence of these worlds, if they are to be taken seriously, must be an objective matter.

Somehow, having attempted to argue for that last proposition, I find myself being asked to define what I mean by "existence", to accept that someone's existence can be "vague", and who knows what else is going to come up. I accept the desirability of trying to elucidate fundamental concepts as thoroughly as possible. But can I first ask: If a person said that according to their theory of the universe, at one time you have one of something, and later on you have many copies of that same thing, but there's no particular moment in time when the one becomes the many, and that doesn't matter because the something only has a vague, fuzzy existence... wouldn't you think that the theory might have a few problems, or at least be missing a part?

Everything I have said about worlds, and observers in worlds, and about the certainty of one's own existence as an individual observer, has been meant to drive that home. That chain of relationships is the detailed reason why it is unacceptable to have a blase attitude towards the conditions of existence of quantum worlds. They must be regarded as existing or not existing, in a completely objective, absolute, non-relative way, or the concept becomes a nonsense, because worlds must play the role of hosting entities whose existence is definitely not vague or relative, namely, us.

Does no-one understand or sympathize with this line of thought?

I will get on with the philosophy in a moment. But I ask that those of you who may find yourselves in a protracted debate with me over these tangential questions, please consider anew the foregoing argument and ask yourself whether it is desirable or even possible to settle for a vague notion of "world", given the theoretical burden it has to bear.

Caledonian asks what I mean by "existence". I confess that I am unable to define it without using a synonym, which is not much of a definition. There may be quite a few similar basic indefinables, which we nonetheless manage to talk about; "negation" may be another example. It seems that all I can do is talk about it, and hope some recognition dawns. I know I already have a disagreement with Caledonian in this matter, because earlier this month he wrote here that existence is relative and depends on the possibility of interaction, something I would never say, because it confuses existence per se with something like knowability - the epistemic grounds whereby one observer may assert of one thing that it does indeed exist. We who live now, our existence was not knowable to anyone who lived a thousand years ago. Nonetheless, we do exist, here and now, and it is a fallacy to relativize our existence, and say "we exist for each other, but we don't exist for those people in the past". It is a basic confusion of knowability with reality.

Now, Unknown, what am I to do with you? Your line is that existence is a vague concept because I cannot define it without being circular, or that I cannot define it in a way which offers a clear decision procedure for existence. My line would be that we all know perfectly well what "existence" refers to - the property of being there, the property of being a part of reality, the property of not being nonexistent - but that the metaphysical depths of its nature are not so obvious. Again, one is constantly making implicit judgements about what does and does not exist. Does al Qaeda exist? Does Xenu exist? Does the special discount on milk at the corner store still exist? Existential judgements are ubiquitous in human thought. We all possess a basic facility with the concept. Does the inability to crisply define it or place it in an ontological scheme mean that one only has a vague concept of existence? I don't think so, because I think the criterion of vagueness in a concept is that its referent, the specific thing which it designates, is underdetermined (i.e. there are several different things it might be referring to), not that the nature of the referent remains incompletely specified. I think the particular referent of the human concept of existence is unambiguously known, but the nature of that referent may be obscure to the human mind. But this is a complicated matter.

And one more time: this metaphysics is an interesting and even vital topic, but it is somewhat of a tangent from the main issue, which is the meaning of "world" in many worlds.

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Mitchell, I have no doubt that I exist. I also have no doubt that certain people are bald. You are confusing doubtful claims with vague claims. A claim can be vague but certain, or a claim can be distinct (to a certain degree; my point is that absolute distinctness is impossible) but doubtful.

When I say that someone is bald, I may be certain or uncertain. Either way, the claim is vague.

When I say that I exist, I am very certain. In this case, according to me, the claim is certainly also very vague.

Once again, if you say that this claim is distinct, please offer an exact definition of existence that clearly manifests those and only those things which are included in existence. You have not yet done this, saying that it is not easy to do without being circular. If so, this reveals that the idea of existence is extremely vague, even if very certain.

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mitchell porter, can you explain what it is that you mean by 'existence'?

When you say things like:

So let me first try, one more time, to say it very simply: You exist. You know you exist. What is it that you're asserting Unknown knows?

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Unknown, I feel like I really shouldn't have to go through all these contortions and byways to make a very simple point. So let me first try, one more time, to say it very simply: You exist. You know you exist. According to many worlds, you are part of a "world". Since you unquestionably do exist, the world that contains you must also exist just as absolutely as you do. Therefore, a theory which shrugs its shoulders rather than be exact about the existence of worlds is, at best, incomplete (and at worst, it is entirely wrong, along with the whole many-worlds notion).

Now back to details...

You bring up a paradigm example of a vague property, baldness. But there is a big difference between attributing a vague property to something whose overall existence is not in doubt, and saying that the very existence of a thing "is vague". I'm not even sure what you mean by the latter. Are you saying that some of your properties are vague? But that's not what this is about. Let's distinguish between thatness and whatness; the fact of your existence, and the nature of your existence. That you are, that you exist, should not be in doubt. What you are, your nature as an existing being, the collection of all your properties, is much more in doubt. It's complicated, open-ended, there is plenty of scope to attribute to you properties which are vague because they are complex and underspecified, as I wrote earlier. Baldness is very definitely a property of that kind.

Now what does it mean to talk about the existence of a vaguely defined object? It means that for some cases, the existence or nonexistence of the object will vary across specific refinements of the definition (refinement to the point of exactness, i.e. to the point of offering an exact yes-or-no answer to the question 'Is it there?' in all possible worlds). Already this means that there must be some definition of "you" which is not vague, because there is at least one sense in which you unquestionably exist, namely, the sense in which you are a locus of awareness. (For the sake of argument I assume I am not talking with a zombie or a hallucination.) Your existence, in that sense, cannot be dependent on a definition. Your existence as a locus of awareness is a certainty, regardless of what uncertainties there may be regarding other aspects of your ultimate nature.

But this also applies to all your copies. If they exist, they exist unequivocally, in definite quantity. And since there is one world per copy - since a copy owes its existence to being part of a "quantum world" - then worlds too must exist just as unequivocally. And so an ultimate theory cannot still treat the concept of world as a merely heuristic concept that can remain vague. It must become "sharp", exact, in the final reckoning.

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One area where no formal models can even capture all the platitudes is in the theory of truth. Every meaningful sentence is either true or false (but not both); we can express meaningful sentences containing the words "true" and "false"; and it's certainly possible for a card to have the following two sentences written on opposite sides: "the sentence on the opposite side of this card is true" and "the sentence on the opposite side of this card is false". There has been a lot of work done in recent years to improve formal models of this sort of stuff (look at the work of JC Beall, Kevin Scharp, Hartry Field, Anil Gupta, Graham Priest, and others) but none of them really work yet.

Even in areas where the formal models are very well developed and seem to do a good job of capturing the ordinary notions are still generally full of problems - there's a recent paper by David Baker pointing out that recent derivations of the Born rule from the many-worlds interpretation are circular in the way they presuppose decoherence; there's lots of work in Bayesian epistemology pointing out the ways in which particular models assume all sorts of things, like logical omniscience and omniscience about one's own mind.

Someone earlier in the thread said that words are apt to change meaning, while formal symbolism is not. I would say that in fact that in many cases, the situation for formal symbolism is far worse - there are far too many distinct axiomatic theories that aim to discuss the same concept, and use very similar notation, so that it's very easy to interpret a formula in one system, when it's meant to be interpreted in the other. Even consider the confusion caused in pure mathematics by different conventions of whether rings are commutative and have identity, or whether topological spaces are compact or Hausdorff. The important thing is to be really careful and clear about what you're talking about, and to do this right, we often need to use an appropriate combination of formal methods and informal language. We can get a very precise grip on something by using formal methods, but informal language is generally much closer to the thing that we actually want to talk about. Our aim is to somehow bridge the gap, so that we can get a very precise grip on the thing we actually want to talk about.

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Mitchell, your claim "any theory which makes your existence a matter of convention is already such a theory" is not evident and needs to be established.

If I see someone who is bald, saying that the word "bald" is vague and cannot be precisely defined is not saying that the bald person does not exist, nor that he is not bald. It is not saying that "there is no such fact." It it simply saying that he is in fact bald, but his baldness is something vague.

In the same way, there is no reason not to admit that I exist, but my existence is something vague.

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Unknown, above all I am saying that realities are not vague. There is nothing at all mysterious about a concept whose vagueness arises from its being about highly complicated entities. To exactly define the scope of such a concept will itself typically be a highly complicated matter, because there are so many cases to be dealt with.

But "existence" is not in any way a vague concept. Why would you think it is - because it applies to everything? (Everything that exists, at least.) Because it has no properties of its own? A vague concept is one whose reference is underdetermined. But intensionally speaking, the scope of "existence" should be crystal clear, even if it is hard to define in a noncircular way: X exists if and only if X is part of reality - reality being everything that exists. If I say there's an elephant outside your window, you don't puzzle yourself over what I mean by the "is" in "There is an elephant...", you proceed immediately to judge the truth of my utterance by referring to reality itself, perhaps by looking out the window.

In any case, as interesting as the discussion about the nature of existence is (is it a property, is it something else, how do we know of it), it is not necessary to have that discussion to make my point. As an individual observer, just about the most elementary thing you can know about reality is that it is there. That something exists is the most epistemically basic fact there is - followed by the fact that it's you (whatever you are) that knows this. A theory of your existence which implicitly asserts that there is no such fact would be wrong, but any theory which makes your existence a matter of convention is already such a theory, and yet that is what we have if the existence of quantum worlds is held to be determined by arbitrary convention.

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Mitchell, all possible concepts are vague, and the vaguest of all is "existence." If you disagree, show some examples of well-defined concepts. (Mathematics and logic do not provide examples, as you will see after thinking about it for a bit.)

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I find that I said this all already, five years ago.

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Robin: there are only arbitrary ways to define the exact moment when you are born or die

Oh, come on! First of all, it's the period in between that we're concerned with. That's when you're a functioning "observer", and that is the part where it would be nonsense to say, "There's not actually any fact about the number of copies of me that exist in the multiverse. It depends on where you set an arbitrary cut-off." Again, according to the hypothesis, you right now are one of those copies, and because you definitely exist here and now, your existence and the existence of this world cannot simply be a matter of definitional convention, dependent on where we set a decoherence threshold parameter.

I also think you underestimate the objectivity with which the beginning and the end of your existence in time can be defined. The principle is the same: if you are aware of these words, then at that moment there is some sense in which you exist, categorically and absolutely. This in turn implies that there is some quite objective criterion of personal existence available. Consider the idea of a "neural correlate of consciousness". One might say: there's a person there, if and only if that neural correlate is there. I concede that even here, multiple definitions will be possible: one would be, the person exists if and only if the neural correlate is there and it is in a state of consciousness; another would be counterfactual, the person exists if the neural correlate could become conscious. Personhood is a complex concept, and unless its ontological basis proves to be much simpler than one would presently expect, there will be multiple objective thresholds at which one might draw the line between person and not-a-person, just as there are degrees of consciousness, a variety of mental faculties which may or may not be possessed, and so forth.

But I bring up all those complications only to demonstrate that there is no excuse anywhere for treating things as elemental as "existence" and "awareness of existing" as being on a par with those fuzzy, conventionally defined concepts. If your theory is fuzzy about them, it means the theory isn't complete - that's all.

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Dr. Hanson,

When you claim " they seem too enamored of words over more formal notation, and they seem to trust their intuitions way too much," would you include even Drs. Stalnaker, Williamson, Bostrom, van Fraasen, Tennant, etc., here?

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