User talk:Tezh

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Thanks for your response[edit]

Hello Tez,

Thanks for your response on my talk page, and for your advice that I study the foundational principles listed there. I am, in fact, well-aware that in mathematics "the game of axioms" is about how far these axioms can be extended and what proofs derive from this extension, and that maths are not empirical in the sense that the physical sciences are.

What I meant in saying that mathematical truth is not self-evident was not that anyone else was implying that it was; I was simply trying to explain why I had asked such weird questions on the Set Theory discussion page. I didn't mean to imply that mathematical truth can't follow from a given set of axioms and logic; on the contrary, the necessity of proofs given a set of axioms and logic is really quite clear to me. I simply meant that since mathematics derive from human attempts to quantify reality (even numbers themselves are an invention of human beings) I was interested in the origin of the necessity of our approximation of the world through a model in general. As far as I can currently see, mathematics, language in general, et cetera are a product of our ability to represent things symbolically. My interest in set theory comes from my inquiry into the origin of our concepts of similarity and difference: if things are "similar", they can be considered as a set, and if we can identify what makes them similar then we can construct a bounded set of all things similar in that way; however, as I've said, we can continue to find differences between any two things indefinitely, and thus produce two separate sets, each of which includes those properties of each thing not shared by the other. Since no two things are exactly the same, where does the notion of similarity come from? A friend of mine has suggested that it stems from our attempts to predict phenomena by studying history: but, obviously, no science can perfectly predict all phenomena outside of its own models.

An example of the failure of mathematical models to predict real phenomena is prediction of the weather. Since turbulent air is best considered a chaotic system, computation of the next iteration in any iterated model that simulates it becomes increasingly time-consuming-- and more likely to be incorrect-- as the number of iterations increases. Necessary to the accurate prediction of the behavior of turbulent systems would be an extraordinarily large set of data and an extremely-fast computer. It may never be possible on very large scales.

None of this means that I don't believe in the efficacy of mathematical proof: I try to think logically, and mathematical proof is the logical outcome of its premises, or the extended definition of its axioms. This is, in my opinion, philosophy of the highest caliber.

I wouldn't dare be so arrogant as to assume that "Tasty geometry" would ever replace Riemannian geometry, et cetera: I don't know enough about higher mathematics yet even to understand Riemannian geometry. The reason I asserted so bluntly that contradictions existed in set theory wasn't that I was sure I was right; it was because I hoped to see the proof that I was wrong, and I thank everyone who demonstrated it to me.

I don't believe I've confused truth with provability, although I may have interchanged those terms inappropriately; rather, I wasn't familiar enough with the axioms of set theory to realize that, for example, membership in a set is not the same as being a subset of a set. This was my mistake. From misunderstandings like this one, my "finding" of "inherent contradictions" followed. I understand that, given a set of axioms that do not perfectly model reality, proofs still follow from those axioms. Even if a model is flawed in some respects, it can certainly still be useful; Euclidean geometry, with the fifth postulate left intact, is highly useful in everyday life. I realise that "truth" is an intangible and often useless concept.

What troubles me is the seemingly-Platonistic view of mathematical truth that I've heard some of my own more mathematically-talented friends imply. Mathematical provability and "truth" are certainly very different things; this is why I said that I could accept that { { 1 } } is different than { 1 } for the sake of learning more about set theory. But I can't help but ask obtuse questions about things like the necessity of the empty set until I can see that necessity. I don't mean that I am sure that it is unneccessary; but surely you understand that I've got to make sense of it before I can move on, since what I am trying to do in all of my studies of philosophy, science, and mathematics is construct a model that approximates reality. Set theory is very useful in approximating reality in this way; I consider all groups as sets almost naturally. But the empty set, for example, cannot, as far as I know, be used to model any real phenomenon.

I recently visited the article on the empty set and found that it had been equated by some people with the number zero (for the purpose of explanation): I hope you'll agree that zero is a much more tangible concept than the empty set: I have zero elephants in my apartment, but I can't see any empty sets in my apartment, or in anything else, for that matter. Until I can see why it is useful, I can't accept that it "just is" so, even though, as I say, I'm trying to understand why it is useful.

I don't think that there will ever be any "complete" mathematical, scientific, or physical model of the universe. I think that as new theorems are proven, new conjectures will continue to emerge, which will in turn either be proven or eliminated, and so on. I do not think that this is a bad thing: if we ever did have a "perfect" model, life would be boring! Imagine an age in which scientific progress could not continue because science was "finished": wouldn't this be a hell for anyone who likes to think?

Consistency within set theory as I understood it was what I had found lacking: my quotation of the guy who said that since the empty set is an element of every set it is also an element of the complement of every set, and thus both within and without every set, apparently came from my not being familiar with ZF set theory, with which I am now going to attempt to familiarise myself for these very reasons.

Nietzsche spoke of "our falsification of the world through numbers". But he also said that such "falsification" (i.e., the incompleteness of any given model of "reality") was necessary. What he meant was that since in real phenomena perfect equation is impossible, we construct models so that we can describe things to one another, establish observer agreement, et cetera, and that thereby progress is made via increasingly-accurate prediction.

Mathematics are, indeed, not an empirical science: but if we had no empirical data, we would have no maths. They originate from the quantification of non-mathematical objects, and that is why they are useful: if a mathematical "model" didn't model anything, how would it be useful?

Again, thanks for trying to point me in the right direction. I will begin to study each topic you've mentioned. You've been a great help, and I hope you'll allow further questions as I continue to learn about each of them.

Regards,

216.76.78.3 17:21, 23 August 2006 (UTC)[reply]

Sorry I forgot to log in. The above post was, obviously, mine. Tastyummy 17:26, 23 August 2006 (UTC)[reply]

One more thing on my assertion that nothing is self-evident: as I've said, I understand that mathematical proofs are evident from certain axioms and logic; but even the simplest axioms and logic are established as "true" via their ability to predict real phenomena. By using logic and fixed axioms, we can create new technologies, discover new phenomena, and explain and predict the behavior of these phenomena and better predict the behavior of phenomena discovered earlier: the value of even something as fundamental as logic is rooted in its relation to human needs. It does not exist outside of human experience; it appears to do so because we cannot describe anything outside of human experience without making use of it. But no concept derives from something imperceptible or not deductible from perceptible things. This is all I was trying to say.

Tastyummy 20:14, 23 August 2006 (UTC)[reply]

The problem I'm still having here is that I wouldn't, in fact, describe the set of elephants in my room as empty, because I wouldn't describe a set of elephants in my room to begin with. I would say that there exists no set whose members are elephants in my room. I think of sets as groups, or collections, et cetera. How can one have a group of nothing? Why isn't it perfectly logical simply not to call it a group at all?

On the page on the empty set, a philosopher with whom I am not actually familiar, Lowe, is quoted:

"All that we are ever informed about the empty set is that it (1) is a set, (2) has no members, and (3) is unique amongst sets in having no members. However, there are very many things that 'have no members', in the set-theoretical sense—namely, all non-sets. It is perfectly clear why these things have no members, for they are not sets. What is unclear is how there can be, uniquely amongst sets, a set which has no members. We cannot conjure such an entity into existence by mere stipulation."

Why can't we consider some things as sets and others as things other than sets? If a set is a group of elements, and the empty "set" has no elements, how is it a set at all?

You ask whether I can't see that

"[T]he formalisation of the natural numbers, as a concept, does not rely on human existence for it's abstract existence."

This is where we differ: I think that since humans formalised natural numbers, and since only humans can conceive of concepts like "abstract existence", that formalisation does, in fact, rely upon humans. Any given human doesn't necessarily have to understand it for it to be a tenable and valid concept, but if there had never been humans, there would never have been such a formalisation. I've certainly never seen an example of a "concept" not formulated by human beings. Even if humans created artificial intelligence that could do this, that intelligence would be a product of human ingenuity. I am certainly not a Platonist, but I'll respect your views on this unless you'd like to debate on the evidence for the existence of Platonic ideas, forms, and so on (in which case I'd be more than glad to elaborate on my own views on Platonism).

I think that in saying "the fundamental theorem of arithmetic is valid whether we exist or not" you are assuming that I think that, for example, if in some hypothetical universe humans did not exist then the fundamental theorem of arithmetic could not be valid in that universe: this isn't the point I've been trying to make. My point is that such a universe, being purely hypothetical and not observable, is irrelevant to all productive discussion as far as I can tell. What, exactly, is the difference between "recognition of the existence" of mathematical truth and mere "formulation" of it? I will say again that even logic and language are the product of human civilisation, and that they have no existence "in themselves". Such existence is an intangible concept.

If humans had never formulated the most fundamental theorems they have, the raw, empirical data from which they can be established would, indeed, still exist; but theorems aren't "discovered": they are invented. This doesn't take anything away from their value. I am neither criticising their efficacy nor their beauty; I am simply stating my theory on their origins, and I have to say there seems to be more evidence that human beings invented mathematics than there is that mathematics invented human beings. Historical accounts and archaeological evidence can tell us of the earliest systems of counting and we can in a general sense trace the lineage of mathematics to this point. As I say, we use mathematics to describe things that existed before mathematics, but that doesn't mean mathematics have always existed.

"What is the collection of polkadot swans that have no wings?"

There simply and most definitely is no such collection, if a collection is a group. There is no group of wingless polkadot swans, for the simple reason that there are no such swans. There is no need to describe such a concept as a set because it is utterly intangible and useless: why shouldn't sets only contain elements? Why is the empty set so special that it need not satisfy the most commonly-given definitions of a set? (A group of elements, etc.)? "A set is a group of elements-- unless it's the empty set, in which case it isn't." That's my problem. Why not "A set is a group of elements. Things that are not groups of elements are not sets."?

My view of the world is more Hume-esque than it is Platonic, but I disagree with Hume on many points not closely related to this discussion and, indeed, am less familiar with Hume than I am with many other philosophers. There were many other continental empiricists than Hume: Locke, for example. If you're at all interested in precisely what my philosophical positions are, you might consider reading my article, indeterminacy in philosophy. Of course it is completely understandable if you simply don't care; there's no harm in this. But if, as you say, you're (at least roughly) a Platonist, and you're interested at all in the philosophy that relates to this discussion, you might be interested in Immanuel Kant's Critique of Pure Reason (if you haven't already read it), in which he refines Platonistic thought to its logical conclusion: namely, that there must exist a "thing in itself", a "thing whose only property is that it is that thing", a "thing that is and will always be inobservable, indescribable, unquantifiable, et cetera". (I'm not actually quoting Kant but paraphrasing him here.) You might also be interested in modern criticisms of this conclusion, some of which are in the wikipedia article on the thing in itself. Again, you might actually not be interested at all; I'm always trying to interest others in philosophy and it often doesn't work, but that's no problem of yours.

As far as the usefulness of solutions to problems in number theory goes: Are you familiar with asymmetric encryption? The RSA algorithm uses very large primes to generate keys; any number theory problem dealing with the distribution of primes would likely lead to new means of cryptanalysis for asymmetric ciphers. But this is a crude example: the point as that there is no such thing as "pure mathematical truth". There is no hard evidence for it. Mathematical truth is a product of the human mind, which is in turn the product of many other things, on and on, which eventually are best explained in more-or-less "abstract" mathematical terms: but only because it is a human mind that is doing the explaining.

"There exist axiom systems that do not (yet) model any real phenomena, are not intuitive, and are not motivated by the desire to model anything."

They are motivated by a desire to explore their outcomes, no? And whence this desire? I propose that it stems from the fact that they may be useful or interesting to human beings. And even if something only models something very abstract, what it models is still "real" in that we are considering it. Even the most "abstact" thing does, in fact, relate eventually to experience.

"Could we ever abstract some concept that isn't directly experiential? I actually believe the answer is 'yes'"

--Give me a quantifiable example of such a thing and you'll have proven Kant right, which has never yet been done. (I assume that you mean "some concept entirely unrelated to experience"? What does it mean to be directly experiential? What would it mean to be indirectly experiential?) I think you might be referring, as I've mentioned, to the "thing in itself" or "noumenon", the "ground of being" that is separate from all phenomena. There is no scientific evidence that such a thing exists, and there can never be any scientific evidence for it, for the specific reasons discussed in that article: belief in such a thing would be a matter of faith. If you do, in fact, believe that there must exist a thing in itself, then we must agree to disagree, unless you're open to further debate. I, for one, am always open to further debate, but I respect people's right to find me boring :)

And if I've been putting words in your mouth this entire time, then let me make it as clear as possible that this is not my intention: I'm simply wondering whether you'd liken your "concept that isn't directly experiential" to Kant's thing in itself; they sound very like one another to me.

One more thing:

In every age, the thinkers of that age have called their theories "correct", "complete", et cetera: consider how people once talked about Newtonian physics. Inevitably, the thinkers of the next age will exceed us in much the same way. We may get some things right, but if we can use history as evidence, we will be proven wrong on some points. What I mean by "Reality(TM)" is the set of all available data, with which we and others can and will continue both to prove new theories and to disprove old ones as long as there is rational thought.

This conversation's becoming pretty interesting, in my opinion. I hope you'll find the time to continue to respond.

Warm regards,

Tastyummy 12:43, 24 August 2006 (UTC)[reply]

Hello again!

I think I'm beginning to see what you mean in saying that zero is akin to the empty set: Would it be appropriate to state it this way? "If everything must be considered as a set, then there must be some set with no elements"? I still say that zero is a far more tangible concept, for the reason that I see this way of defining zero much more easily: if I must count all of the elephants in my room, then the number I count is zero. The reason for my objection was (and, to some extent, still is, since I'm not entirely sure about this) that while zero can and does model real things-- since the absence of something of which we can conceive is "real" in that we can conceive of it as an absence-- the "empty set" doesn't, as far as I can tell. (I do not level these criticisms at zero): what has helped me the most so far is when someone (I can't remember whether it was you) said that if the intersection of any two sets which do not share members is to be considered as a set, then it must necessarily be the empty set. But I'm still having trouble with why it can't simply be a "non-set". Why, exactly, must everything be considered as a set? It's obvious to me that zero is an integer, but not obvious to me that an empty set is a set. But, as I say, I am trying to come to terms with this. What would help me most would be to see some mathematical structure that necessarily relies upon the empty set in order to model a real phenomenon.

I still say that even sets of mathematical axioms that do not model any known phenomena derive from experiential data, necessarily. But you were right in saying that I had wrongly asserted "the existence of 'data', especially 'empirical data' without the existence of an observer": what I meant was that the best current explanations for the existence of an observer rely upon the existence (or perceptibility, as Berkeley would have said) of other things than the observer. Schopenhauer once wrote an interesting dialogue between two characters, "the subject", (or consciousness, the observer, etc.) and "matter", (or the object, the observed, etc.). In this dialogue he elucidated the interdependency of the observer and the phenomenon. It's really quite an excellent piece of writing, but the problem is that it becomes rather vacuous eventually.

Have you ever heard of qualia? Qualia are, roughly, those elements of human (or any conscious) experience which are indescribable. A (seemingly) very convincing argument was once made by the contemporary philosopher David Chalmers for their existence: it is called the philosophical zombie argument. Here is a rough paraphrasis:

Imagine that you meet two seemingly-identical twins. They both behave in virtually exactly the same way, look alike, share preferences and distastes, et cetera. But one lacks qualia. One of the twins, let's call her A, is conscious and "feels like she is experiencing things", i.e., experiences the illusion of the Cartesian theatre, or the homunculus, et cetera: B, on the other hand, does not. She only appears to be conscious: all of her behaviour is best explained by, or at least correlates extremely strongly with, her being conscious, or, in other words, her being an "observer", but she is, in fact, an automaton of flesh and bone and blood who only behaves as though she is conscious. How could anyone else ever tell which twin was conscious? How, in other words, can one ever know that anyone isn't a "philosophical zombie" like twin B?

This has been called "the hard problem of consciousness" and is still taken seriously by many modern philosophers. I used to take it very seriously indeed, and I would often use it to illustrate to people whom I took to be stupider than me because they didn't think about things just how "useless" empirical studies of behaviour were: it seemed to me that the real, the pressing, the important problem in philosophy for our time was the existence (and, equally, the origin-- I'll come to this later) of qualia! But I had overlooked something very simple:

If the qualia that are present in twin A and not in twin B are completely inobservable, then there is no reason whatsoever to suppose that they "exist"! It is far more productive to study what causes the illusion of qualia then to seriously try to "deduce" their origin, since, by the very definitions of qualia, we can't even be reasonably sure of observer agreement upon their supposed "properties"! No experiment can ever be performed on them; no data can even be gathered from their observation, or from the observation of phenomena influenced by them, since once they influence something else they are descriptible and no longer actual "qualia". This is also true of Kant's Noumenon.

The reason that I spoke before of the "existence" of things independently of an observer is because I am trying to learn about the origin of the observer. I am currently reading a book by Daniel Dennett on the origin of consciousness which purports largely to have "explained consciousness" via an elucidation of (some of) the specific evolutionary processes that led up to what we know as "conscious experience", et cetera. I am not sure of Dennett's conclusions-- I'm not even that far into his book yet-- but I cannot, as yet, refute any of them, and he easily refuted the one last "supernatural" phenomenon that I believed in. (Until then, I wouldn't even admit to myself that qualia were both supernatural and almost totally vacuous!)

The existence, or presence, or perceptibility, et cetera, of any given phenomenon is virtually always explicable in terms of its origins: This is why, while I admit that there are systems of axioms that do not model any observable phenomena, they are derived from observable things. The derivation of something observable from something inobservable is always unprovable, except, as far as I know, for the motion of particles whose positions or momenta are necessarily affected by our observing them in such a way that their original positions or momenta become "inobservable" as a result of the process outlined by Heisenberg's Uncertainty Principle: but even here, we can prove why we can't accurately measure their positions and momenta without altering them, and this leads us to ask "altering what?"-- we now consider their positions and momenta in terms of probability and not definite measured quantities, but this does not mean that they do not exist, since they are still, in some ways, perceptible.

Another thing-- you said that I was "right in the sense that I can only express such concepts [integers, etc.] through language. The difference being, that I don't see 'language' as being necessarily a human-only quality (as an outrageous counter-example: the existence of intelligent aliens)."

In fact, I don't see language as being confined to human use either. In my article, Indeterminacy in philosophy, I state rather early-on that

"Since all communication between sentient beings must be made in some language (be that language human speech or writing, bodily postures, the action of pheromones, or any of a multitude of other types), all scientific and philosophical hypotheses – and indeed all statements in general – are given definition by the 'words' (or whatever other units may be appropriate given which type of language is being used) of that language."

I haven't ruled out the possibility of the sentience of other beings than humans. I don't even necessarily believe that there aren't intelligent alien lifeforms somewhere in the universe (although I do think that the probability that we will ever meet them is "astronomically" low). In fact, I'd say even lower animals can be considered to be sentient to varying degrees: have you ever noticed that a domesticated dog's barks generally correspond to pitches used in the vocalisations of their human owners? This is how humans can tell a "sad" bark from an "excited" bark, and an "angry" or "mean" bark from a "friendly" one, and so on. Humans speak, especially when emotionally excited, in the tones of their musical scales (this is how music conveys emotion), and other animals can pick this up as well. Chimpanzees have been taught sign language with moderate success, and it has been estimated that the intelligence of the average chimpanzee is only slightly below that of a severely-retarded human: and surely even very severely mentally-disabled human beings are sentient!

You were certainly right in insisting that hypotheticals are not utterly irrelevant: we can learn a lot about ourselves by exploring the origins of our strangest hypotheticals. But, as you've said, not all of them model observable phenomena, and I do think that this ability to model reality is almost always what makes mathematics useful. Even seemingly-inobservable concepts, like i, come up in models of real phenomena: i, for example, is useful in electronics and is necessary to the creation of beautiful fractal art. In these and other ways, it is real because its influence is perceptible. Our best models of certain real phenomena necessarily include i, and so we say it "exists", even though there is no "intuitively-real" number which, when squared, equals negative one. Once I find an example of something that relies upon the empty set in this way to model an observable phenomenon, I will thankfully and gladly accept its existence and go on to learn more about set theory. I hope I'm getting closer to finding something like this.

You're also right that you didn't use the term "formulation", although I wasn't implying that you had used the term "recognition of existence" by putting quotation marks around it (I did this because I thought it was a bit of a silly concept and I didn't necessarily want to appear to be implying its possibility). I meant to say "formalisation", as in "the formalisation of the natural numbers, as a concept". My mistake. But my question still stands: what, other than human invention of this formalisation and other humans' perceptions of its influence on other phenomena, constitutes its "existence"? I'll agree in a general sense with Berkeley that existence is perceptibility, but I'll add for the sake of clarity that the "perceptibility" of an object (including a mathematical one) includes perceptibility of its influence. Thus, just because something, like the Big Bang, for example, isn't "directly" observable, I wouldn't call it "imperceptible": we perceive, or can perceive of, its influence on other, "directly" observable, phenomena, like cosmic background radiation (I believe that's the right term).

The "empirical foundation" of the concepts you listed is certain: it is that these concepts are derived from the consideration of other concepts, which are derived from the consideration of other concepts again, and so on, all the way back to the first use of numbers as a model of certain aspects of reality. For instance, we cannot recognise any pattern in the infinite string of digits of a decimal approximation of the square root of two (other than, of course, the fact that each digit we add to this number brings it closer to the "actual" square root of two): but this is, surely, quite empirical! From what are we to try to draw pattern, and thus show it to be "rational"-- and, thus, by constant (and provable) non-recognition of pattern, show it to be "irrational"-- if not empirical data? If humans can discuss numbers with one another, then they are empirically-derived concepts. Every concept has an origin; you can trace the lineage of any concept quite far indeed, (although not to its "true" origin, since humans invented writing-- and this is historical fact evidenced by the utter lack of any writing or even visual representations before what Jared Diamond calls the Great Leap Forward; I believe it was during the Agricultural Revolution).

The original "value" of systems of numeration was their ability to predict: If I count ten dead mammoths outside of my cave, anc I write this number down or remember it, I can predict how many dead mammoths will still be outside of my cave if I go have a cup of lime-water from a stalactite with my cave-wife (unless one is stolen, or miraculously comes back to life). Systems that do not have much value are, in my opinion, those that (seemingly, at least) cannot predict anything: a mathematical system in which the reflexive property is not valid, for example. (I hope I haven't picked a bad example-- I'm very sure, but not completely sure, that this cannot be used to predict anything except the behaviour of other elements of that system-- things true of that system would seem to me to be vacuous.)

In my opinion, we should be asking ourselves: what is the origin of Kant's supposition of the Noumenon? Could it possibly have been the noumenon itself? How do you answer, for example, the scientific critique of Kant's noumenon? I believe that the origin of such concepts is quite well-explained by memetics. A meme is a unit of cultural transmission which replicates via imitation: and what can be more easily "imitated" than something utterly indeterminate? There's less detail to "replicate"-- why do more people have faith than an understanding of the physical properties of their world? I'm not saying I know physics any better than the next amateur: it's a difficult subject, in my opinion: but this is because if I do not imitate my instructors' (be my instructors books, humans, or anything else other than, of course, direct empirical experience, since then this wouldn't be "cultural" transmission) understanding of these principles to a great degree of precision, I have not understood them, to the extent, of course, that these ideas are correct.

Simply "leaping", as Kant put it, (though he wrote in German, obviously), into the "lap of being" gains us nothing, as far as I can see! Isn't it better to explore our world, to change our ideas, to learn more forever, instead of looking for a "final solution"? The biggest "final solution" I ever heard about was Hitler's, and it was one of the most horrific and abominable ideas ever produced by any human being! My conviction that a "final theory" of anything would (at least generally) be a bad thing is, for example, why some portion of my mind remained open to the possibility of a correct argument against qualia even when I was quite sure of their reality, and it is also why I am attempting to find an example of the empty set's usefulness. Zero is clearly useful; i is also useful; perhaps the empty set is also useful. I need to see this for myself, though.

Maths do not purport to be about phenomena, but I will hold to my assertion that all mathematical concepts derive, eventually, from phenomena, and that they are themselves observable phenomena. A mathematical object is still an object; it is just a special kind of object.

Nietzsche once said of "the metaphysicians" (meaning, in particular, Kant) that "From these [their] 'beliefs' they try to acquire their 'knowledge,' to acquire something that will end up solemnly christened as 'the truth'. The fundamental belief of metaphysicians is the belief in oppositions of values. It has not occurred to even the most cautious of them to start doubting right here at the threshold, where it is actually needed the most -- even though they had vowed to themselves 'de omnibus dubitandum' ."

I think (and this is not an opinion that is opposed by many) that science, mathematics, philosophy, history, art, et cetera, are all related and irreversibly intertwined. Historical analysis reveals the influence of each of these fields on each of the others. We should be considering all data, and not simply call ourselves "mathematicians" or "scientists" or "philosophers", et cetera, as though being one of those things precludes being another: I don't know what else to call ourselves: perhaps "students".

Anyhow, I'm glad you're willing to continue this conversation. As I say, I am, (I think,) beginning to understand the need for the empty set, and I sincerely appreciate your kind attention and your long-lived patience with this amateur "mathematician"'s rambling. Please reply any time.

Thanks again and, again, warmest regards,

Tastyummy 06:11, 30 August 2006 (UTC)[reply]

One further question (I forgot to ask this earlier)-- could you elaborate on just what you mean by "abstraction"?

Thanks,

Tastyummy 06:13, 30 August 2006 (UTC)[reply]

Greetings!

I don't have time for a full reply right now, as I'm currently at work, but I'll come back to the rest of your last message later today; I'd like to make a few points, though:

I still say that maths do not exist independently of reality. When I've said that the "usefulness" of maths lies in their ability to model real phenomena, I was referring to the "usefulness" of maths in general, and not to every possible set of axioms, deduction rules, et cetera. Again, the reason I say that maths do not exist independently of "real", observable phenomena is that they derive from them. One of the origins of the mathematical theory you used as an example was you; you can further trace its origins back to experiences you had in learning about mathematics, other influences on your mind, et cetera, and you can trace these back a very long way indeed if you learn about the history of mathematics. And, in fact, that particular set of rules does give results which are observable as mathematical objects. If you couldn't observe them at all, they couldn't be deductible from the axioms and deduction rule. (I haven't even looked at this much yet, but I'll do so later; as I say, I'm short on time right now.)

I'd like to thank you, however, for convincing me of the necessity of the empty set. Would I be correct in saying that I answered my own question by asserting that "If everything must be considered as a set, then there must be some set with no elements"? This is now making a lot more sense to me. You said "if I must consider the collection of all the elephants in my room, then the collection of them I have is the empty collection": I think that's roughly what I said as well. Would you agree? If so, I think I can finally accept this and move on to learning about ZF set theory.

I'll have to look at the article on abstraction, but just offhand I'll say that, as a non-Platonist, I don't really see evidence for the existence of "essences" of any phenomena independently of an observer who imposes these "essences" on things by constantly constructing a model of phenomena. But I'll read the article before I continue to argue against it.

Anyway, I'll read more and post more later. Thanks for your excellent explanation of the empty set; you've been an invaluable teacher. We've really got a lot in common, even if we disagree on a lot of things, at least in my opinion-- we are both interested, to some degree, in fields other than our own.

Have a good day, and I'll write more later--

Tastyummy 16:10, 30 August 2006 (UTC)[reply]

One more thing, and then I've really got to get to work:

You say that

"Just because riemannian geometry has replaced euclidean is the current model of spacetime and gravity does not suddenly make euclidean geometry invalid maths. I'm sure I mentioned this posts ago. The validity of maths is independent of whether it well-models some part of reality. That some esoteric parts of maths do eventually find some model in physics (say) is usually just good luck. Again, see some of the 'truths' above for some theorems that don't have practical applications."

Consider, if you will, the origin of the word "geometry": a more-or-less literal translation of it is "measurement of the earth"-- i.e., of empirical data.

And, I've never said that euclidean geometry was "invalid maths": I said in an earlier post that it was "useful in everyday life" (I think those were more or less my exact words).

Anyway, I haven't even finished reading your post yet, but I'll get to it after work, or perhaps during a break or something.

Tastyummy 16:32, 30 August 2006 (UTC)[reply]

Hello again!

Sorry I never finished responding to your last post. I just found a quotation that I thought might interest you (and with which I happen whole-heartedly to agree):

"Mathematics is the most exact science, and its conclusions are capable of absolute proof. But this is only because mathematics does not attempt to draw absolute conclusions. All mathematical truths are relative, conditional." - Charles Proteus Steinmetz

Steinmetz was, himself, an eminent mathematician, and his statement seems perfectly sensible to me. It doesn't depend upon a thing-in-itself or upon Platonism in general; similarly, science doesn't depend on "absolute certainty", but rather upon falsifiability, in reaching conclusions.

Nietzsche said it well in his Beyond Good and Evil:

"That a theory is refutable is, frankly, not the least of its charms: this is precisely how it attracts the more refined intellects."

This is where I disagree with Richard Dawkins, an intellectual hero of mine: In a post on Bob Seldo's blog, Seldo found that Dawkins (or his editor) had actually misattributed his own statement to Thomas Kuhn in one of his books (A Devil's Chaplain)! I actually quite agreed with the quotation (which was, in fact, penned by Dawkins himself) at the time:

"The best you scientists can hope for is a series of approximations that progressively reduce errors but never eliminate them."

(A further discussion of this topic can be found at Seldo's site.)

I replied to Seldo's original post,

"Isn’t such a 'series of approximations' a perfectly _good_ thing for a scientist to hope for? I’ve actually been of this view lately, as a student of the philosophy that is science. If an 'absolute knowledge' were ever formulated, wouldn’t this be an _end_ of science?

[...]

[Seldo's reply was, roughly, that in Popper's science it is possible to eliminate error, but not to be absolutely certain that you've done so, and that Popper did, in fact, characterise all scientific theories as ultimately conjectural. (I immediately agreed with these points.) Seldo's main problem with Dawkins' treatment of this "quotation" was that it lumped Kuhn in with Popper as a "truth-heckler" when, in fact, Kuhn and Popper's ideas have often fundamentally opposed one another, while mine was that it seemed to me to be a perfectly sensible view.]

[...]

[W]hat I meant in my statement that “a series of approximations that reduce but never eliminate error” was good was, of course, not that I’d ever wish actively to preserve error or to fight against its reduction, but that I think– and I’m basing this only on historical evidence– that there will always be some error left to eliminate, reduce, or otherwise get around– not [necessarily] in every possible well-established theory, but in general– and that there’s nothing bad about this."

A reformulation of the original statement by Dawkins that allows for the possibility of the elimination of error might be

The best that scientists can hope for is a series of theories that progressively eliminate errors but that are nevertheless conjectural in character.

Dawkins has argued against pedantry in the treatment of scientific theories mainly, I believe, because of his opposition to the idea that intelligent design and evolution via natural selection are equally-valid theories of the origin of today's species (and thus of the descent of man). Like Dawkins, I am an opponent of the theory of intelligent design, but, unlike Dawkins (I think), not because a statement of the necessity of the conjecturality of all scientific theories is mere "pedantry" or, as Dawkins has it, "truth-heckling", but because I think, as Dawkins does, that intelligent design is not an evidentially-supported theory.

I think that Dawkins is attempting to address what he sees as a very common and very antiscientific notion: namely, that the conjecturality or falsifiability of scientific conclusions somehow negates their value. Obviously, since Dawkins himself formulated the aforementioned mostly-correct assertion of a scientific epistemology, he probably doesn't himself see falsifiability and conjecturality as being negatory of the value of scientific theories, but he gives the impression that he believes that very-thoroughly-evidenced theories are, in a practical sense, the same as absolute truths. I firmly disagree here: I think that a blind acceptance of the most well-established of theories as "undeniably true", "self-evidentialy true", or "absolutely" or "fundamentally" true inhibits scientific progress. This is historically true, for example, of the common acceptance of the dogma of geocentrism, or-- and this is particularly relevant for Dawkins-- of the common acceptance of creationism as an explanation of the origin of species. Whether or not you believe that evolution via natural selection is a viable explanation of the origin of man and of life as we know it (and your SuggestBot's postings suggest otherwise; please correct me if I've drawn the wrong conclusion here), you can certainly see the point I'm trying to make: that is, that if Darwin had accepted the commonly-held theories of his day about the origins of speciation as fundamentally true without questioning them, regardless of whether they were actually correct, he would never have proposed an alternative to those theories. Similarly, one can approach a well-established theory quite skeptically, look at the evidence that is held to support that theory, and conclude that it is correct independently of any assumptions about the "authoritative" nature of the scientific process via which that theory was originally formulated. Or, one might take a skeptical approach to a given theory and actually disprove it via presenting hard evidence that contradicts it: this, along with a careful reformulation of the newly-disproven theory that accounts for that evidence, is, in fact, how progress in science occurs.

We might be tempted to call it "scientific progress" when, for example, something that isn't necessarily new to science is applied to technology or medicine in a novel way: while this is certainly progress, it is, in my opinion, better-termed social progress, medical progress, technological progress, et cetera, than scientific progress, since it doesn't amount to scientific discovery, or to the productive reformulation of current scientific theories about existing data, et cetera.

I find Dawkins' "truth-heckling" essential to a scientifically-based epistemology; it was, in fact, an almost-identical conclusion, which I reached independently of Dawkins, that led me to consider science as far more valuable than most other philosophy (and I say "other" philosophy because science is, to me, philosophical in character-- science and philosophy needn't oppose one another, even though many modern philosophers are unscientific or even antiscientific). I nevertheless still find good philosophy to be quite useful in modern times, since there are philosophical refutations of unscientific statements made by modern philosophers (which can lead the student of philosophy, as they led me, to a scientific worldview). Dawkins seems to have a "faith in science" which, in my opinion, however well-placed it might be, is not a viable argument for science's efficacy. Scientific theories can predict real phenomena-- that is, there is evidence for their efficacy that precludes the need for any "faith" in that efficacy, in my opinion.

In summary: Dawkins is generally opposed to dogmatism, but a few of his essays seem quite dogmatic. I'm pretty surprised at this. What's your opinion (or do you even really care)?

Regards,

Tastyummy 21:13, 13 September 2006 (UTC)[reply]

Mathematics as Noumenal[edit]

Hello again, and thanks for your response.

I am currently writing a critique of logical atomism, a feature of analytic philosophy that gave rise (in part, at least) to naive set theory via Russell's Principia Mathematica. As far as I can tell (although I could certainly be wrong), logical atomism underlies much of modern mathematical thought; this is evident in such statements as

"[I]t is precisely because theorems in mathematics are syntactically derivable strings from axioms and rules of inference that ideas like falsifiability, testing, etc in general do not apply to mathematics."

That syntactical derivation can constitute proof of anything other than a particular relation between symbols is not a universally-accepted notion. Russel, Wittgenstein, etc. (the logical atomists) asserted that statements made in any language of symbols-- be they mathematical, verbal, or of any other type-- could be reduced to constituent statements, which in turn could similarly be divided, etc., until constituent parts of these statements were met which could not further be reduced to other meaningful statements (these most primitive meaningful statements were the "atoms" of logic). In philosophy, however, and, for that matter, in history, the behaviour of the term "meaning" deserves a closer study than it was given by the logical atomists.

Russell began to formalise his work and produced his highly-influential and masterful Principia, and Wittgenstein degenerated over time into inconsistency: but Wittgenstein recognised the truth of a simple statement that Russell seems not even to have considered: namely, that "of that of which we cannot speak, we must remain silent".

A statement that

"The only reason I say this is because you still refer to mathematical ideas as phenomenon. In fact, they are strictly the opposite. We do not "'experience' infinite series in the world around us. We do not 'experience' the symmetric group S5. These are all noumenon."

is inconsistent with the definition of the noumenon. Noumenal things aren't just abstractions, they are things to which we are not related in any way except in our attempts to describe them; as the article on the noumenon says (and this is not from a "criticisms" section),

"It can be said that for Kant, the noumenal realm is radically unknowable—for when we employ a concept of some type to describe or categorize noumena, we are in fact merely employing a way of describing or categorizing phenomena."

This does, in fact, constitute a thoroughly-modern view of noumenality. What is important here is that the description, categorisation, etc. mentioned above constitute all of the properties of the noumenon. Wittgenstein didn't mean that "of that of which we cannot speak we ought to remain silent", but that we truly are obliged to do so by the very fact that we cannot speak of it: and the noumenon, when taken as it was originally proposed by Kant-- i.e., as fundamentally unknowable-- is something of which we cannot speak except in error: this follows thusly:

1) A description of a thing is an aspect or property of it.

2) (Per Kant,) what can be described is conditioned by the indescriptible.

3) A statement that the descriptibility is conditioned by indescriptibility itself constitutes a description of the "indescriptible".

4) By reduction, we now have "the descriptible is conditioned by the descriptible", an almost-vacuous statement.

I certainly won't call it vacuous; it is, in fact, reiterated by Kant's critics (including me). But very little can be taken from it other than a conclusion that the "meaning" of statements is indeterminate.

But if we have described gauge symmetries, higher-dimensional manifolds, etc., then we have necessarily given them aspect. Our definition, description, and use of them constitute their properties and the ways in which they exist.

"I believe you think that somehow an axiom must be some 'self-evident feature of the universe'[...]"

This is, in fact, the very opposite of what I'm trying to say: I agree that they cannot, in fact, be self-evident, and I further agree that arbitrary assertion of axiomaticity is the only way that we can formalise communication. I am not against mathematics in any way: it's just that I think that mathematical thought can occur without an affirmation of Platonism. In quoting Steinmetz, I hoped to convey, in fact, that I did not think that axioms had to be (or, indeed, could be) "self-evident features of the universe".

My problem is still with what appears to be the inconsistency of various definitions of the "set". A "collection of nothing" can only be considered as a collection if there are various descriptible or quantifiable "bits of nothing" that constitute it. Non-elementality cannot be magically converted into elementality: a set is a collection of elements, and an "empty" collection-- that is, one that is not a collection of elements-- is a contradictio in adjecto.

"[Steinmetz] is not saying that the truths are conditional on any phenomenon. They are conditional on the axioms and rules of inference."

Axioms and inference rules are phenomena. You yourself have seemingly admitted this in arguing against their being "self-evident features of the universe": such features, were they possible, would be noumena, while conditions are phenomena. I do, in fact, understand that in such statements as

If everything must be considered as sets, then non-sets must necessarily be considered as sets

the "truth" of the predicate does not hinge on the truth of the condition, but rather on the "syntax" underlying the construction of statements in general: but my question to epistemology in general, and not only to science, mathematics, religion, philosophy, linguistics, etc., is of the origin of the seeming necessity of the conditionality of every statement and every syntax. It's almost a non-question, but I think it's important because it amounts to a kind of skepticism that is quite rare and quite useful: not a mere questioning of one's own views and the resulting insight, but the questioning of the questioning of one's views, a questioning of the motive of epistemology. I am not asserting anything about "how mathematical epistemology works" as opposed to epistemology in general, but, rather, I am testing propositions about mathematical epistemology as epistemology in general.

This probably doesn't much interest you, and it also has nothing to do with what I would write in the section on the set theory article dealing with objections to set theory that have been made in philosophy (and for sources for these objections, see the page on the empty set, in the section entitled "does it exist or is it necessary?"). I would simply state that, as you say, "the material implication A -> B (in words: 'if A is the case, then B is the case') does not necessarily rely on the 'truth' [...] of A." You helped me to realise that this was what was being said when the existence of the empty set was being asserted; I already knew about the properties of conditional truth, it's just that until I realised that set theory was currently an axiomatic theory I thought that people were saying that the existence of the empty set was a "self-evident truth". This was, indeed, because I am quite undereducated in set theory, and if you think that putting this information on the main article for set theory is not useful because it can be found elsewhere in Wikipedia (which it can), then I will take your advice and leave the article unchanged. (By the way: thanks for that advice! Noone else has said anything about my proposition; I'll probably abandon it. I'll look for some decent source material critical of logical atomism, but I probably won't find much past Derrida, who, unfortunately, reverted eventually to a Hegelian phenomenology that is just as atomistic as anything else out there (i.e., inconsistent with his own arguments that I would be citing); you're right that the prevailing view in today's maths, linguistics, philosophy, etc. does tend toward what I'm arguing against, and that mine is a minority's view of these matters.)

"Theorems are simply contingent on the axioms (which may or may not have an interpretation ie. a truth value). That is what is meant by the Steinmetz quote."

As you already know, they are also contingent on the rules via which they are deduced from their axioms. These rules are just as important as the axioms, no?

One last thing: I've heard from a few people now that the empty set's existence can be proven from the other axioms of ZF set theory: why, then, is it axiomatised? If it is evident from the other axioms, isn't it redundant to include it as a separate axiom? Again, I'm having a problem with what seems to be in need of a shave via Occam's razor.

Anyway, thanks again for your attention and time. I'm glad to hear you're still open to conversation about these topics; I've enjoyed learning from you about these things. As I say, I am thoroughly interested in set theory and in mathematics in general; my background is in philosophy, but philosophy has led me to mathematics; in its economy of thought and expression, mathematics is, in my opinion, the most beautiful of languages.

Warmest regards,

Tastyummy 21:39, 7 October 2006 (UTC)[reply]

P.S. -- My discussion of Dawkins, Popper, etc. above was entirely tangential and not meant as an argument about the validity of anything in maths. I agree that "science must use the language of mathematics". But I do not think that string theory, which has thus far not produced a single experimental result, ought to be considered as "physics" until it is shown to deal with physical bodies: what do you think of this article? I was happy to see what are, more or less, my own independently-reached views printed in the New Yorker, but I do realise that the author of this article is neither a mathematician nor a physicist.

Hello again; I've only got a few minutes right now, so I'll respond to your last post at length later, but I'd like at least to address a couple of what I still see as misunderstandings of what I've been trying to say--

"If everything must be considered as sets, then non-sets must necessarily be considered as sets"

This comment is obvisouly self-inconsistent. How can non-sets be considered a sets? But of course, in fact, you have merely interjected implicitly your assumption that the empty set is not a set.

[...]

I begin with the position that some things are sets and other things are non-sets. I do know that in set theory every object is considered to be a set. (I'm no longer arguing that the article on set theory ought to reflect my views about this; I'm simply trying to clarify my position.) The reason that I take this view is that I am considering the properties of sets in general, i.e., as objects observable and descriptible both within and without axiomatic systems. Here is my objection to the view that every thing-- not every mathematical object, but every object-- can be considered as a set:

If X is an element of every existent object, (and here "existent" means "observable", "demonstrable", etc.) then it is noumenal. No properties of it can be demonstrated, because it cannot be separated from other things: it cannot be measured or compared with anything else. ("To what extent is this composed of X?" "To every extent!" "How many Xes are in Y?" "Infinitely many, since Y, being a set, whose members are necessarily also sets, etc., can always be divided into more objects containing X, which in turn can be divided into objects containing X, etc.") All "unifiers of all things" are vacuous. They cannot predict anything. No descriptible thing is completely disconnected (i.e., totally and utterly unrelated) to any other descriptible thing, since both are descriptible. This seems paradoxical ("descriptibility" and "noumenality" appear, perhaps, to be interchangeable terms), but I can characterise descriptibility quite precisely, e.g., as communicability, symbolic representation, etc. (Even here, a sort of "noumenality" is necessary to language; e.g., "non-(symbolic representation)" can only be expressed as symbolic representation, if you take it that way. But these paradoxes are necessary to definition, and possibly to epistemology in general. Again, see my article on indeterminacy in philosophy for more on this.)

Assuming, then, that non-sets are descriptible, I proceed thus:

1. "Everything" includes both sets and non-sets.

2. "Sets" are "groups", "collections", etc., which necessarily contain members. (Obviously this doesn't agree with the definition of the empty set as a set, but I am taking the same "intuitive" path toward "definition" of the set as do mathematicians who say that "a set is a collection of elements" and that "there exists a set that is not a collection of elements" in a single breath. A "group of zero things" is, emphatically, a contradictio in adjecto. (I'll discuss this later; I apologise for only asserting it for now and realise that this greatly weakens my demonstration. Take it, if you like, as a sort of quasi-axiom for the sake of this example.)

3. If both sets and non-sets must be considered as sets, then contradiction is necessary.

Yes, this does depend on my assertion that non-sets are descriptible, or, alternately (or contingently) on an assertion that the "empty set" is a misnomer; but axiomatisation is a type of assertion. And, again, if the axiom of empty set can be proven from other ZF axioms, why is it an axiom?

I realise that I've presented a very incomplete picture here. I'll get back to this later; I've got to go pick my father up from the airport right now, so, again, I apologise for leaving a "messy" reply, but I will address this further. Thanks very much for your continued help,

Tastyummy 01:07, 12 October 2006 (UTC)[reply]

Hello again[edit]

Here is my objection to the view that every thing-- not every mathematical object, but every object-- can be considered as a set

But no one has claimed that this is the case. Who is saying that everything is a set? More precisely, I think you are referring to things that can be modelled by set theory. But equally, no one is claiming that set theory well-models everything. A very bizarre claim. You claim that this would lead to a contradiction. Well, it does, which is why nobody thinks that set theory is a good foundation for everything.

Firstly, yes, I was referring to things that can be modeled by set theory. But I'm not the only one who does this:

"Since set theory is completely syntactical, and the semantics are defined by a model, try substituting the word 'box' every time you see the word 'set', and see if it makes any more sense."

And, earlier on,

"The language of set theory (again) is first-order logic plus 'is a member of'."

It would be great if that were true; then I'd never have asked the following question on the set theory talk page to begin with:

"Can anyone provide me with three sets of actual numbers A, B, and C such that A is an element of B, and B is an element of C, but A is not an element of C?"

I asked this question because if the only other relation between objects in set theory than first-order logic were membership ("is a member of") then Zakon's statement wouldn't hold:

"[I]f M is a collection of certain sets A, B, C, ..., then these sets are elements of M, i.e., we have A is an element of M, B is an element of M, C is an element of M, ...; but the single elements of A need not be members of M, and the same applies to single elements of B, C, .... Briefly, from p is an element of A and A is an element of M, it does not follow that p is an element of M. This may be illustrated by the following examples.

Let a “nation” be defined as a certain set of individuals, and let the United Nations(U.N.)be regarded as a certain set of nations. Then single persons are elements of the nations, and the nations are members of U.N., but individuals are not members of U.N. Similarly, the Big Ten consists of ten universities, each university contains thousands of students, but no student is one of the Big Ten."

The reason that Zakon was right in his assertion is not evident in his examples about the U.N. and the Big Ten, as I pointed out on the talk page. It was, as you yourself told me, that

"[I had] confused 'is a member of' (which is equivalent to 'is an element of') with 'is a subset of'."

If I could have understood set theory correctly by thinking of sets and boxes, then I wouldn't have been confronted by the following problem:

If box M contains boxes A, B, C, ..., then these boxes are contained within M, i.e., we have box A is in box M, box B is in box M, box C is in box M, ...; but the boxes in box A need not be in box M [...] Briefly, from p is a box in box A and box A is in box M, it does not follow that p is a box in box M.

Of course, this isn't a problem at all if being a subset of something is not equivalent to being a member of it. We are left with "from non-box p is in box A and box A is in box M, it does not follow that p is a box in box M" because p isn't necessarily a box at all. This is why I still don't see the problem with purposefully distinguishing between sets and non-sets: a "non-set" could (in my thinking) be a "member" or "element" of a set (thing that is not a box p can be contained in box M and also be contained in box A which is in M), but not a subset of it (non-box p cannot be a box contained in box M even if it is in box A which is in M), and so it could be substituted for the "p" in Zakon's example unless there's some (unwritten?) law that everything must be considered as a set (box). In other words, I don't see how Zakon's example makes sense without the "single elements" he writes about being non-sets.

By the way: considering sets and subsets but not all elements of sets as boxes makes the empty set possible as an empty box. But as a model, the empty set seems to fail. To me, a "collection of elements" minus the whole part about "elements" leaves only the idea of "collection", "containership", "boundary", "segmentation", "atomism", or any of a host of other things, depending on the context in which one considers it, and some or all of these may (I am not at all sure yet) turn out to be reiterations of the Kantian noumenon. Do you see why? What is fundamental to Platonic ideas is that they are types: there might be an "ideal flower" of which all flowers are imperfect copies; the type, the idea, is "the flower in its most general sense". If we use the language of set theory loosely here, we can say "each flower is a member of the set of all flowers" and "the criteria via which we determine whether thing X is a flower constitute the criteria for its inclusion in the set of all flowers", etc. What is meant, in this sense, by the "empty set"? Our consideration of things as groups without any actual things being considered? To me, this sounds like Kant's consideration of the thing-- that is, the Platonic "idea", "essence", etc.-- "in itself", or, alternately, without consideration of what it is the "essence" of, or what type of thing is being thus categorised. To me, this is a futile exercise in epistemology that is doomed to admit its own unknowables, as did Kant's.

This leads me to your other point:

"[...] I think you are referring to things that can be modelled by set theory. But equally, no one is claiming that set theory well-models everything. A very bizarre claim. You claim that this would lead to a contradiction. Well, it does, which is why nobody thinks that set theory is a good foundation for everything."

To me, it seems like people do think that set theory is a "good foundation for everything": for example: when I try to learn about modern cosmology, I am presented with such notions as topological invariants, whose definitions employ the terminology of set theory, as you can see in that article. If the questions of sciences like physical cosmology aren't as "fundamental to everything" as anything else, given that, to quote that article, "[c]osmology involves itself with studying the motions of the celestial bodies and the first cause", I don't know what is, the vagaries of current theories in those fields regardless.

Since philosophy, too, has often seen attempts at "studying" the "first cause", none of which have yet decisively shown anything. Causality is a type of determinacy, as is implication, which is fundamental to logical atomism, a philosophical cousin of axiomatic logic. Russel was himself a logical atomist. (This leads me to another question: who first used the term "empty set"? When did it first appear; or, alternately, when, in mathematics, was there first considered an object descriptible as "a group/collection/whatever of zero things"? I still can't get past the seemingly sensible position that the only sure criterion for distinguishment between containership and its negation is the actual continence of one or more objects. Problems develop if one uses an object in space, like a box, to explain the concept of the empty set: the intersection of the set of all points in space and the set of points which are neither "contained within" nor "excluded from containment in" the box (i.e., the actual cardboard, or other material, that makes up the box) can be considered as its boundary. (I'm getting this from the wikipedia article on boundary, which says, and in my opinion quite reasonably, that the boundary of a thing is whatever can be approached both from inside and from outside of it.) But the boundary of the empty set is the empty set, according to [pirate.shu.edu/projects/reals/topo/answers/bpoints1.html]. How can anything serve as a "metaphor" for the empty set? (Not that I don't wish something could!)

I'm actually beginning work on a book that will include a critique of logical atomism in general; I will post more on this later.

As for my statement that axiomaticity and assertion were interchangeable terms, you're right; it's not really appropriate. As you point out, and as I know, axioms are not statements of truth: however, they are statements on which hinge the provability of other statements and any given axiomatic system is incomplete (per Goedel as well as per a generalised critique of logical atomism); this is why I equate them with "assertion" when I see them used in definitions in proposed frameworks for explanation of things like causality, etc. The reasons that I can be confident in saying that they are used in such frameworks are plentiful: consider, for example, the following "possible scenario":

Non-causality at the particle level (i.e., quantum indeterminacy) gives rise, via empirically-descriptible (or at least approximable) processes, to the processes via which human beings came to be, which, in turn, give rise over time to language, which gives rise to logical discussion, which leads to attempts at modeling the evolution of language from non-language, and of the illusion of causality from non-causality, etc.

In order to "flesh out" this "brief history of time", one would need a very thorough understanding of physics, which would require familiarity with the language of set theory, which seems to me to involve a logically-atomistic epistemology, which is, as far as I can currently tell, necessarily complete. Logical atomism is a type of axiomatisation of the properties of meaning, whatever that means. (This is, in fact, consistent with Russell's own view. I don't know enough about Wittgenstein, another famous logical atomist, to comment on his work, but I will familiarise myself with it as I write my critique.)

The reason I feel concerned about set theory as a basis for modeling is that it seems to me merely to be an extension and reiteration of Platonic epistemology as a revised logical atomism. If what is provable in an axiomatic system is what is directly derivable from the axioms, deduction rules, formal logic, etc., but any of these is unproven, and the formal definitions of physical topologies, spaces, etc., hinge upon these axiomatic systems, then statements about, for example, cosmology that are composed of statements provable within an axiomatic system are statements provable in an incomplete system; and, more importantly, if any of the axioms is unnecessary or false then its contingent models will include unnecessary or false elements. Yes, this is about model theory: but, to be honest, I have yet to see a demonstrable difference between "model and actual phenomenon", or "signifier and signified", or "idea and representation", or "type and object". Each of these distinctions can be overturned in philosophy via an explanation of how the two "opposing" parts of these divisions are different sides of the same coin in each case.

One question for now: if you find the time, could you set up a proof of the existence of the empty set from other ZF axioms, without the axiom of choice (for the simple reason that I don't fully understand the axiom of choice yet)? I'm just interested to see whether I could follow such a proof. If not, do you know of any text in which the ZF axioms are proven from one another and first-order logic (or however they are supposed to be proven, in case I've asked the wrong question)?

Hello Tezh,

I'm not mixing up different levels of epistemology. I'm writing a book on it, actually.

"[...]after which you list one mathematical topic, and one topic in astrophysics. For topological invariants, please note that the very idea of topological space is defined in the language of set theory (look at the definitions -- they are about the empty set, set intersections, unions, etc etc). To say that topological spaces are "modelled"[sic] by set theory is a bit of a categorical error. This is because topology is already in the language of set theory, and so the model is rather trivial."

"To say that topological spaces are "modelled"[sic] by set theory is a bit of a categorical error."

This is certainly true. But, I already knew that "the very idea of topological space is defined in the language of set theory". I wasn't referring to the topological invariant, but to physical cosmology, as being modeled by set theory. I'm sorry I wasn't more clear about this. That is honestly what I was trying to to.

Cosmology does involve set-theoretical models, to the best of my knowledge, and if there is a problem with an axiom underlying them then cosmology employs them in poverty.

As for zero:

I am currently thinking about attempting a construction of the integers without the empty set. I'm not far yet, because I've been tending toward atomistic logic, from which can be derived mathematics, but which is an epistemological nightmare. I'm trying to connect the ideas of recurrence, replication, explanation, substitution, atomism, noumena, etc. (a host of other weirdness) into a consistent and functioning epistemology. I hope that zero will be easy to define, but since I don't even have a framework within which to work yet, I can't yet say.

But I do "believe in" the number zero. I have observed it. It is a number. Numbers are written and talked about. They definitely exist. I think zero models absence of a given thing quite well. But the empty set is a contradiction in terms.

Sorry for the short post. I'll write more later. Tastyummy 06:24, 24 October 2006 (UTC)[reply]

Hi!

Firstly:

"PS. 'Modelled' is an accepted (though chiefly british) spelling of the past participle of 'model'.

Thanks very much for telling me so; I'm sorry for my mistake. I actually generally make an effort to use the British spelling of a word if and when I know that it differs from a corresponding American spelling because British spellings often convey more information about etymologies than American ones do; I'll be spelling it "modelled" from now on.

Nextly:

"Essentially, as far as the arguments you've put forward so far, existence of zero => the existence of the empty set, and non-existence of the empty set => non-existence of zero. I mean, I can also write down the empty set, and talk about it. In what way is an empty container of things a contradiction of terms?" Again, your interpretation is getting in the way. If you believe that the empty collection negates the idea of "collection", why don't you think that a count of no objects contradicts the idea of "counting", "numerosity"?

Yes, you're right, I suppose it is my interpretation of things that has been getting in the way. I'm still having trouble with it, though: I just can't see why the empty set ought even to be called a container when, to me, being a container means containing something, and the empty set contains no elements. (What does it contain?) The reason that it sounds like a contradiction in terms, at least to me, is that when I ask myself a question like "what is a set?" and get "a collection of elements, unless it is the empty set, which is an exception so special that it has its own axiom in ZF set theory establishing its existence" as an answer (which is certainly not to say that this is what your answers have been like-- rather, this is, roughly, the unsatisfactory picture I've gotten thus far from a miscellany of sources), I can't help that I am not satisfied with such an answer.

It helps to think of sets as containers, and of the empty set as an empty container, only if I can distinguish between a container and a non-container (or between a set and a non-set). If all objects shared any given thing (such as being sets), then that thing couldn't be isolated from the "other" objects, and so statements about it would be trivial to the point of vacuity: but, as you've said, this is a problem of modelling and not of consistency within set theory (I've never yet seen a formal definition of the set, and so I certainly can't claim that because to me, a set must contain a non-zero number of elements ("elements" being anything at all) a theory that does not formally define the set is inconsistent.

"[...]It seems to me that you've conflated two epistemologies; one being the empiricism of the scientific method (and thus the a concern for the philosophy of science), and the other being the formalism of mathematics (a concern for the philosophy of maths)."

To me, empiricism is almost a non-position in epistemology. If only empirical things exist, then how am I able to talk about them? (If everything were empirical, then "being empirical" would be a "thing shared by every object" and, as such, non-information.) At the same time, I cannot escape the assertion that every thing is related, even if only to the very smallest of extents, to every other thing, since if something were totally and utterly disconnected from anything descriptible it would be indescriptible. I'm stuck in the rather annoying state of non-epistemology (i.e., I freely admit that I currently do not have an epistemology) because I haven't seen how to get around this pair of problems yet. It's like a sort of "naiive incompleteness": I don't see how an epistemology can be both consistent and complete. I was inclined toward dumping completeness altogether in favour of consistency, (or trying to do this, anyway), via a quasi-Popperian falsificationism in which incompleteness was to be admitted in each new theoretical refomulation to "make room" for the next, but I realised that for falsification to occur the (consistent) epistemology in which it occurs must contain some elements (such as the possibility for, and definition of, contradiction) in each of its reformulations in order to be considered as a single, connected epistemology. Maybe all epistemology entails contradiction. I'm unsure about this.

Hmmm... I've completely wandered away from set theory, and possibly contradicted myself. I'll just say a little more for now on epistemology/cosmology, and tomorrow I'll hopefully find the time to write more on why zero is more obviously useful to me than is the empty set.

"There is no a priori necessity that the laws of the universe be logically, mathematically, or even "physically" consistent (whatever that may mean)."

This is certainly true; I don't really consider anything to be an a priori necessity. However, the Copernican principle is "applied in cosmology, as the acknowledgement that the Universe is generally homogeneous and isotropic over large scales. These principles are accepted not merely as a philosophical statement but as an acknowledgement that a significant, large-scale deviation from homogeneity and isotropy would be statistically unlikely, and that this acknowledgement has been found to be correct in different contexts in prior observations."

(I bring this up because "physical consistency" would, I guess, mean the inerrancy of a Copernican principle in physics, which, of course, hasn't been established, but which is necessary to a consistent definition of observation. Also: if they're not consistent, what makes them the "laws of the universe" instead of the "laws of certain things within the universe" or the "almost-laws of the universe"?)

Thanks again for your help. I'll be back soon; have a good day (or night).

Tastyummy 06:07, 26 October 2006 (UTC)[reply]

"I'm not really bothered about which spelling you use, just that you (sic)ed me in a quote, so I thought I'd point out the alternative."

I figured as much; sorry about the [sic]; the reason I said what I did about British spelling is that I try to be consistent and have argued elsewhere that it is more informative than American spelling is of etymology, but obviously I wouldn't want to insert [sic] after a correct spelling anyway.

"[Y]ou mention something a little off about the scientific method. You mention quote a description of the Copernican principle from its article page. You say this principle hasn't been established. What do you mean by this? You mention in your very quote that a violation of this "principle" would be statistically unlikely. Doesn't that mean it has been scientifically established?"

I should have said that it hasn't been proven. If "established" is taken as "widely-accepted", then the Copernican principle is indeed scientifically-established. I'll say emphatically that I am not arguing against the Copernican principle! It is completely sensible to assume that "special observers"-- i.e., observers for whom "observation" is something different than what it is for other observers-- do not exist, because to act otherwise is either to use an inconsistent definition of observation or to call observation noumenal.

"[Y]ou make an aesthetic judgment on the form of "laws". A law can easily have exceptions. For example (using some very simplistic mechanics), for an object of mass m, and velocity v, let's say we discovered that the kinetic energy of a mass is 0.5*m*v*v everywhere except right in the centre of Las Vegas, where the energy is 10*m*v*v. We could formulate the law of moving objects thus: Objects, with mass m moving with velocity v, have kinetic energy 0.5*m*v*v, except in Flat 124, 13 Weirdness Drive, Las Vegas, Nevada, where they have energy 10*m*v*v. How is this not a law? It is a law with a well-stated exception which has been accounted for. Surely this qualifies as a universal law."

I don't see how this is an "aesthetic judgment", and I don't really know what "form" means in the above context.

I'll reiterate my question: If they're not consistent, what makes them the "laws of the universe" instead of the "laws of certain things within the universe"?

Wouldn't an utterly universal law be unfalsifiable? If, as in your example, for any given place in the universe an otherwise-universal law doesn't hold, then a corresponding universal law would need to model both the exception and the almost-rule. If Newtonian mechanics didn't hold on Weirdness Drive, then the laws of the universe, which includes both Weirdness Drive and every other place, would need to model kinetic energy in a way that explains events both there and elsewhere. The formula for kinetic energy would have to be something other than 0.5*m*v^2 in order to remain consistent if kinetic energy is to mean the same thing on Weirdness Drive as elsewhere (or, in other words, for the Copernican Principle to hold). When this sort of inconsistency occurs in physics, larger, unifying theories are constructed to explain both the rules and the exceptions of the old ones.

The best current examples of this that I can think of are theories of quantum gravity, in which an attempt is made at reconciling general relativity, which models gravity on large scales, and quantum mechanics, which models the electroweak and strong nuclear forces. But GUTs are, as far as I can see, unfalsifiable. I am not the only person of this view: the following is an excerpt from an article in the New Yorker:

"In their books against string theory, Smolin and Woit view the anthropic approach as a betrayal of science. Both agree with Karl Popper’s dictum that if a theory is to be scientific it must be open to falsification. But string theory, Woit points out, is like Alice’s Restaurant, where, as Arlo Guthrie’s song had it, 'you can get anything you want.' It comes in so many versions that it predicts anything and everything. In that sense, string theory is, in the words of Woit’s title, 'not even wrong.'" [1]

Smolin is an outspoken critic of string theory and is himself an ex-string theorist. I'm planning on reading his book, since I reached the same conclusion independently and for philosophical reasons without any real education in physics. This is to be part of the subject matter of the book I'm beginning to write. This brings me to another question of yours:

"[H]ave you read the Kuhnian view of the scientific method?"

Actually, no, although I am planning on reading Kuhn soon. I actually had a very interesting encounter with what turned out not to be Kuhn's philosophy: I posted the following quotation on this page in the same post as the Steinmetz quote:

"The best you scientists can hope for is a series of approximations that progressively reduce errors but never eliminate them."

I had read this in a book by Dawkins, in which it was attributed to Kuhn. The statement had interested me very much because I had reached an extremely-similar conclusion just days before I read it, and I had searched for it on Google in an attempt to figure out which book by Kuhn it had come from in order to read it. But the only Google results were some weird philosophy journal and this page; the quotation was not from Kuhn or Popper (it was from something Dawkins had written himself), and a rather interesting discussion ensued.

"I'm sure you find it easy distinguishing between cups and non-cups. And something tells me you don't deny the existence of empty cups. Again, the empty set isn't nothing. It is a set that contains nothing."

Yeah, I do find it easy distinguishing between cups and non-cups, whether or not they're full of anything, because of other properties of cups than that they are sometimes full of stuff. I have other criteria for distinguishing between cups and non-cups than that they contain things, but I don't have a better criterion for distinguishing between sets and non-sets right now. However, I have admitted that my problem with the empty set is that I don't think that it can model much of anything, or that I think it ought to be called "the non-set", or something like that. Obviously, if the criterion given is "a set is whatever is defined as such in ZF set theory", then the empty set is a set. But the most sure criterion I can come up with for containership is continence-- in other words, I only call something a container if it definitely contains something; otherwise, it's "something that I could use to contain something else". An "empty container" would, in this view, be interchangeable with a "potential container"; this would work with empty boxes, cups, etc., but not with the empty set, which has no elements.

By the way: what is the complement of the empty set?

I'll be back later. Warm regards, Tastyummy 19:15, 30 October 2006 (UTC)[reply]

Hello!

Firstly, let me apologise for the lengthy interval between this post and the one that preceded it. I've been quite preoccupied lately both with personal issues which you may or may not have seen mentioned on my user page (i.e., that I am attempting to restructure my entire life after having ended a three-year attempt to become insane on purpose, as bizarre as that probably sounds) and with my own investigation of several topics which relate directly to our discussion, and I haven't been on Wikipedia in quite a while. I'd like to resume our discussion of mathematics, and I'll begin with a second look at some of my own questions and arguments about sets.

I have come to realise fully that my own suspicion that the empty set was in some way equivalent to the Kantian noumenon, or to some other inobservable, totally isolated and non-interacting entity, was due to a miscellany of my own misconceptions about its use within mathematics. Firstly, in discussing "sets", I was generally referring to informal "groups", "types", "categories", etc., without any particular commonly-used mathematical definition of sets in mind. The reason for this was my own equation of objects of this sort with Platonic types/forms/ideas/etc., and a subsequent assumption on my own part that something called the "empty set", which, to me, meant simply "the only set for which that which normally distinguishes sets from non-sets (namely, their continence of elements) is not what makes it a set", and, more importantly, an entirely independent and also-incorrect assumption that sets were somehow being taken by some people as unconditionally-atomic objects. I inferred this from statements made by mathematicians in works written, in the main, for the general public on mathematics, theoretical physics, etc. (A particular example that comes to mind is Penrose's The Road to Reality, in which its author does, in fact, construct a tautological ontology in, if I remember correctly, its first chapter, involving a circular determinacy of "the physical world" by "the mathematical (Platonic) world", which is determined by "the mental world", which is determined by "the mathematical world", ad infinitum. Penrose even includes two diagrams illustrating this circular relationship of causes.)

A certain "philosophical belligerence" on my part is to blame for my own unfounded attacks on what seemed to me to be ontological or epistemological assertions which appeared to be inherent to set theory. I had (incorrectly) assumed that, since set theory, implicational causality, etc., were being used fundamental systems of relation between "actual" (modelled) objects (which, I think, in fact, they are, at least by some, given that, to again employ Penrose's work as an example, assertions are made by eminent physicist/mathematicians which amount to this), the set theories employed by those mathematiciant were to blame for the philosophical errors of their proponents. To put it shortly: if the empty set were some fundamental constituent part of the way in which the parts of "Reality(TM)" relate to one another, or if it were elemental to some sort of logical atomism at the heart of modern scientific epistemology or ontology, then I might still stand by my original treatment of the empty set as noumenal within those systems in which descriptible objects must relate to one another only in ways determined by the definitions employed by those systems. I remain an opponent of any theory in which hypothetical objects whose relation to other objects cannot be documented somehow determine the behaviour of observable objects. However, ZF set theory is in no way such a theory, and models employing it needn't reflect such relationships between objects addressed by them, either.

Your statement that "[t]he rules for distinguishing between sets and non-sets are precisely the axioms (and rules of inference) of set theory" was quite enlightening. For me, the entire thing had been a question of terminology in which I had been looking for some term for groups, or sets, or categories, or "somethings" for which the only property common to them and distinguishing them from things of any other sort was that they were comprised of other things ("elements") which were not them. (In retrospect, it seems quite obvious to me that these are not the objects of ZF or any other common set theory; I suppose that the reason for my odd mistake was simply that "set" seemed to me a reasonable term for such objects, but I hadn't known that within formal mathematics, "group", "set", "class", "category", etc., all mean very different things. In all honesty, I didn't know this until I began discussions with you and a few other people who helped me figure this out.)

I had, in fact, already been familiar both with the Quine atom Q={Q} and with Russell's paradox when you mentioned them in your last post. What I had been looking for when I voiced the opinion that the empty set might better be termed a (or the) non-set was, quite precisely, an Urelement. From the article thereon:

"In set theory an ur-element or urelement is something which is not a set, but may itself be an element of a set. That is, if U is an ur-element, it makes no sense to say

X ∈ U,

although

U ∈ X

is perfectly legitimate.

This should not be confused with the empty set where saying

X\in \emptyset

is logically reasonable, but merely false."

I had been considering the empty set as equivalent to what I now know is called an "urelement", and I simply thought it didn't make sense to call it a "set".

I no longer know whether it makes sense or is useful to employ "urelements" in any model at all, but I am going to investigate them, since they are, in fact, employed in some set theories, including those mentioned in their article (New Foundations, NFU, and Kripke-Platek set theory with urelements), although they certainly play no part in ZF. I've been looking into the history and applications of various set theories and formal logics, including the development and properties of first-order logic and "propositional logic"-- which seems to be a basic part of first-order logic and which does, by the way, derive directly from the logical atomism of philosophers like Russell, and which is philosophically unwieldy, since logical atomisms are often identical to Platonic theories of types-- and, along with it, implicational logic (which amounts, it seems, to the properties of necessity, and which, I hope, will answer some of my questions regarding the limits of any description of causality in terms of the properties of definition, which, for me, has been an ongoing experiment), relevance logic, etc. I've been trying to find a copy of Anderson and Belnap's Entailment: the logic of Relevance and Necessity. I am trying to undertake a project in which the relation of entailment is taken as the fundamental relation of determinacy between events and objects, and to study the properties of such a system. I am also trying to figure out what category theory and intuitionism are.

For me, learning about subjects like modern formal symbolic logics and theories of sets, types, groups, categories, etc., means learning about their history and development. To the extent to which there are such things as formal mathematical ontologies and epistemologies (as they have been described and addressed, for example, by mathematical Platonists like Penrose, by nominalists like Quine, and by logical atomists like Russell), there is a direct correspondence between the mathematics based on them, (and here I am treating set theory and propositional logic as possible "bases of mathematics" only because other authors, like Zakon, have done this), and philosophical positions which interest me. Quine, particularly, has attempted to formulate a non-Platonistic theory of types. (I don't know whether he was successful, but, for me, this is worth looking into.)

The last thing I'd like to say for now regards, once again, the empty set.

"'But the most sure criterion I can come up with for containership is continence-- in other words, I only call something a container if it definitely contains something; otherwise, it's 'something that I could use to contain something else'. An 'empty container' would, in this view, be interchangeable with a 'potential container'; this would work with empty boxes, cups, etc., but not with the empty set, which has no elements.'

And again, if you believe in zero, you must admit this reasoning is fallacious, since I may paraphrase it:

But the most sure criterion I can come up with for number is counthood -- in other words, I only call something a number if it definitely counts something; otherwise, it's 'something that I could use to count something else'. A 'zero counthood' would, in this view, be interchangeable with a 'potential number'; this would work with zero elephants, grains of sugar, etc., but not with zero, which has no counthood.'"

I should have said that "[a]n 'empty container' would, in this view, be interchangeable with a 'potential container'; this would work with empty boxes, cups, etc., but not with the empty set, which cannot, by definition, have any elements'", since then zero could, in fact, be defined in terms of "potential counthood" (or something like that) as long as there can be multiple counts of zero. (According to the page on the axiom of empty set, the empty set "exists and is unique" [italics added] in ZF.)

Obviously, I had been thinking, "something 'should only be' a set if it has members". If "[t]he rules for distinguishing between sets and non-sets are precisely the axioms (and rules of inference) of set theory" and the set theory in question is ZF, then my above assertion is obviously false, because of the axiom of empty set. But even in a set theory which does describe objects which are not sets, such as those with "urelements" mentioned above, the naturals, integers, etc. can be constructed. Zero can be defined within mathematics without the empty set. (I don't know whether I'm going to continue to avoid using the empty set or not, since I haven't investigated the alternative theories to any useful degree yet, and am certainly not sure that they make more sense or are in any way superior to ZF, but they are certainly out there. (Not using the empty set in a theory doesn't necessitate the abandonment of zero as a number, or as a particular count of things, etc.

Regards,

Tastyummy 21:58, 12 December 2006 (UTC)[reply]

p.s.-- Merry Christmas, or "happy holidays" (or whatever best applies to you!)