Feel free to use this page for correspondence with me. Make a new header above everything else on the page...
I'm very interested in an obscure political movement called "conservatism". I've heard of "conservativism" and I know many "conservatives", but I don't know any "conservatists" (proponents of "conservatism"). Why do I keep hearing on the news, on the streets, in books, et cetera, about this thing called "conservatism"?
Someone suggested to me that maybe everyone's just erroneously referring to "conservativism" in writing or saying "conservatism"; this would explain the extreme commonness of the term in political discussions, but not the stupidity of the thousands of people who use the latter term and even promote its necessary correctness over the first:
The fourth Google result for a search for the term "conservativism", an "exposition" of it as error, is utterly lacking in logical foundation: the only evidence it gives for the correctness of the spelling "conservatism" is that
About one year ago I actually wrote an e-mail to the guy who wrote this, asking him whether he had any other reason than the one mentioned above for categorically stating that the spelling "conservativism" is incorrect. He has never replied.
If someone is a proponent of "*ism", he or she is a(n) "*ist", and not a(n) "*ive". For example: someone who practices Buddhism is a Buddhist, and not a "Buddhive". Someone who professes postmodernism is a postmodernist, and not a "postmodernive".
Similarly, if someone or something is "*ive", he, she, or it likely demonstrates some degree of "*ivity" or "*ion" or professes or practices "*ivism". For example: someone who believes that things all are relative to one another, and that nothing isn't relative to something else, is a "relativist". Something related to some kind of relativism (and not "relatism", since we'd be talking about a property of "relatives" if we "reasoned" in the same way that we did in order to "discover" the "correctness" of the term "conservatism") or to something originating from the particular relativism of some particulat relativist, (e.g., to the Einsteinian relativity of mass and energy,) is "relativistic". You may have heard of a couple of his more important theories, namely, those of special and general relativity. And something that is "recursive" exhibits "recursion", but something politically "conservative", at least here in America, often has little, if anything, to do with "conservation", and certainly not with conservation of the resources necessary to the continued existence of Western civilisation! Or perhaps I am mistaken here: perhaps the money-squandering war criminals occupying the Oval Office today are not "conservatives" but "conservatists".
We can call people "liberals" and "conservatives", and we can call the sets of ideologies that characterise "liberal" and "conservative" thought and action "liberalism" and "conservativism", respectively. But the political opposite of "conservatism" is not "liberalism" but "liberism", or something like that-- and I've never even heard of that political movement. What's going on here? Is this some kind of conspiracy? Why won't Paul Brians, the author of the site I mentioned and of a book, Common Errors in English Usage, respond to my question? His book is about common errors-- its very title implies that the criteria he uses in determining whether something is an error must be something more than the mere commonness of a given usage, but his "reasoning" about the incorrectness of the spelling of the word "conservativism" is simply that it is uncommon!
Is he in on it, too?
Tastyummy 05:29, 14 September 2006 (UTC)
Here's something that might interest fellow students of science and/or philosophy, as well as fellow Brights:
(This link works as of 4 September 2006-- leave a note if it isn't working anymore, or search Google for the post's title.)
It seems that either Dawkins, who is a hero of mine, or his editor has misattributed the following quote,
to Thomas Kuhn. I actually quite agreed with the quote, which I read in Dawkins' A Devil's Chaplain, when I read it (and, as I said in my reply to the first post on the page to which I've linked above, I think it's a perfectly good thing for a scientist to hope for!). This is really one of the weirdest things I've ever seen-- not only is the quote not actually from Kuhn, it isn't even apparently a quotation of anyone other than Dawkins himself, who wrote it originally in a Forbes article! Furthermore, Popper himself is labeled in the essay in Chaplain as a "truth-heckler" in what seems to be a rather negative portrayal of his stance on falsifiability-- a stance that is integral to modern science, and rightly so.
Dawkins seems to be trying to make the point that the falsifiability of an argument does not negate its value-- but he does so in so roundabout a way that he seems almost to be arguing against the scientific method in favor of "common-sense truth"! Again, see the link for details and more quotations from the essay in question.
Yeah, I agree with you. I didn't think about checking the capital "P" in philosophy (I found it odd there was a deconstruction page, but no undecidability/indeterminacy page). I'll write one up and put it on.
However, in scanning your page, I'm not really sure as to where it would fit in, exactly. Maybe under a different header, like "Indeterminacy in deconstruction"? -Mordacil 22:25, 1 September 2006 (UTC) PS: If it affects anything, I'm only 17. But I'm hella into philosophy, and have read Derrida and such. Cheers
Nice, also good to know that you're about my age. I've been passively involved in philosophy since my freshman year of high school (2003-2004) because I do debate (specifically, policy debate). I've actively been interested for about a year now, and have read Habermas, Heidegger, Hegel, Derrida, so on and so forth. Mostly postmodern stuffs. I haven't really looked into consciousness from a philosophy perspective, so I can't really give an answer other than what I just think.
I was recently informed by user:Max18well that he was recently accused of being my "sockpuppet". This is silly. Just because he happens to agree with me doesn't mean I made up his account. He was at my house recently because I know him in real life, and this is why he edited from the same IP address that I have. I have just moved and you will see that I am now editing from a new IP address, and that Max18well will rarely, if ever, edit from this address. If anyone can do an IP trace, it will verify that our IPs are not coming from the same gateway or anything. My name is Alex Olsen. I can be contacted at tastyummy@hotmail.com if these accusations continue. My friend's name is Max; he lives in Orlando while I live in Tallahassee, and his e-mail is max18pratt@yahoo.com . Check the IPs; he's been staying here recently because we both grew up here and he wanted to visit, but he is returning to Orlando and an IP trace will confirm this. You'll see an obvious style difference in our writings as well. If there is any further need for proof that we are, indeed, separate people, please let me know how I can do this; I'm not too willing to provide personal info. but I will send a little if it is absolutely necessary. I don't know exactly how to prove that I didn't make a sockpuppet, but I will try to do so however I can. I would never create a second Wikipedia account and impersonate another individual just to edit an article that I can edit anyway. I have no need to do this. If accusations like this continue I will report the accusers to an administrator and a contest of evidence will ensue. Max18well happens to agree with me because we often discuss philosophy together in person.
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The Barnstar of High Culture
Max18well 21:57, 18 August 2006 (UTC) |
Won't you join the Atheism Wikiproject? I've taken the liberty of adding the banner to your userpage. I was hoping you might begin by helping me add to Richard Dawkins Award, currently a stub. Remember to add your name to the list of participants on the Wikiproject page!
I'd love to help with this project. I've signed the page and I will begin to examine and contribute to the Dawkins Award shortly. Thanks also for your contribution to the discussion page for my article on philosophical indeterminacy. It's greatly appreciated as always. Furthermore, if you become aware of any other interesting projects related to atheism, feel free to add their banners to my user page as you've done with this one; I trust your judgment and I will likely agree with any position you take on issues like this.
Tastyummy 21:43, 20 August 2006 (UTC)
There's a discussion which you'd be much suited for than I; should Nihilism be removed from the Atheist category? I think not, but hope to hear your opinion (which you will kindly leave on the linked talk page). Rashad9607 19:39, 28 August 2006 (UTC)
Drop a word here too, if you please: [[1]] Rashad9607 12:31, 29 August 2006 (UTC)
Hi, no problem. :) I'd really rather not get involved in your dispute, however. Try asking someone else. —Khoikhoi 01:45, 21 August 2006 (UTC)
Grabbed from the set theory talk page:
This is not the prevailing view of mathematical epistemology. Maths is not inherently some language that models phenomena. As you are probably aware (you sound familiar with philosophical terms), maths is not an empirical science. It is purely formal and deductive.
Probably the easiest way to allay your doubts is to view theorems of a theory thus: Given a theory T with axioms {A1, A2, A3, ...} and a rule of inference MP (I call it MP because the most popular rule if inference is modus ponens), saying that some proposition P is provable simply means
(A1 and A2 and A3 ...) -> P,
which I think logicians abbreviate with a special symbol |-, and they would write T |- P.
Truth on the other hand, depends deeply on your interpretation (these are very model-theoretic concepts) of your mdel, M, but I guess the idea has an analogous form to provability:
if (A1, A2, A3) are true in my model M then P is true in my model M.
Again mathematical logicians have a shorthand, which I believe looks something like
M(T) |= P
In other words, provability and truth are already known to be contigent on the axioms and model. Informally, theorems that say "P" are really saying "If my axioms are true, then P". There is no need to show somehow that the axioms are "really true" in some way. Maths make no claims to outright truth or provability.
Back to axioms which are "self-evident". This is not how a mathematician has to think. In fact, the more pure the mathematician, the less he has to think like this. Ultimately, the game of maths is "what can I prove with this set of axioms? What can I prove with that set of axioms?" Sure, the theories that model our intuitive ideas of integers, geometry have axioms that are motivated by their ability to well-model basic concepts. But since the C20th formalisation of maths, it has become apparent that this is not necessary, and the sina qua non of maths is usually thought to be consistency (since a theory T being inconsistent implies that every statement generatable by T is also provable).
It is in fact the cosmologist/bioinformatician/chemist/etc that has to worry about whether a particular mathematical construct models his area of study well. For example, it has been shown (and this is the empirical sort of 'shown', not the mathematical 'proved') that spacetime, energy, mass is well-modelled by a Riemannian geometry. If/when the day comes that cosmologists see this only as a rough approximation, instead finding Tasty geometry a better fit, this will in no way reduce Riemann geometry to the status of non-maths, just as Euclidean geometry is still a valid area to study and to prove theorems in. Probably the best course of of reading is everything in foundations (ZF theory, predicate logic) up to Goedel's Incompleteness, and Tarski's theorem of the Undefinability of Truth. But if you do, I have one piece of advice if you do attempt to study mathematical foundations: Do not confuse "truth" with "provability". They are obviously related, but in fact very distinct. Tez 10:19, 23 August 2006 (UTC)
Hmm. Sorry I didn't reply very quickly to your last post on my talk page. I also spotted your post on the set theory talk page.
I still think you're confusing science-based epistemology with mathematical epistemology. One more time: it 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. Perhaps you are confusing the slightly disparate meaning of "theory" when used in science and maths. 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.
You didn't really answer what you thought of my little theory above. If you don't think that that's how maths is done, I'm afraid you'd simply be wrong. If you are aware that that is the essence of much of mathematics, then I don't understand why you are quoting Popper, Kuhn etc, who think and write about a completely different method of epistemology. Their ideas simply do not apply to mathematical derivations and theorems. You can doubt the "existence" of the empty set, of a two-manifold, but in fact, these objects either exist because an axiom says they do, or they are derivable from the axioms.
I believe you think that somehow an axiom must be some "self-evident feature of the universe". This is not the case, as my little theory with the As, Bs, and Cs above demonstrates. They are merely the starting position in a game where the moves are derivations allowed by rules of inference.
I do not really think that a philisophical objections section you mention on the set theory talk page really has much standing, since most of the objections you've raised simply do not apply. I'm willing to have my mind changed if you could show me some source material (which you'd have to do anyway to get such a section into the set theory page).
As for the Steinmetz quote, I think you've taken this exactly the wrong way. You still haven't realised that the material implication A -> B (in words: "if A is the case, then B is the case") does not necessarily rely on the "truth" (whatever you think that means, and it probably doesn't mean in maths what you think it means) of A. He is not saying that the truths are conditional on any phenomenon. They are conditional on the axioms and rules of inference. These axioms aren't things to be believed nor do they need to be "true" (again see my little ABC-theory above) -- they are simply a starting point for the derivation of theorems. 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.
Feel free to make further comments. I'm still open to further conversation. Tez 15:37, 6 October 2006 (UTC)
I'm coming down this weekend, so hold on to your butts...
Read this message Anthony sent me
"I just woke up and saw that you called 8 million times, sorry I was asleep. Here's the deal, you know how Brandi when Brandi talks about Lee she says that after you're around someone all the time everything they do starts to annoy you? Well that's how I feel about being around you and Dan right now. It's nothing personal it's just what happens when I spend way too much time with people. Lately you guys have been getting on my nerves a lot so I kinda just don't wanna hang out with you guys as much anymore. Sorry =.."
My reply you ask? "STFU Noob!"
I then added a ;), so he can't say i'm a total penis-where-my-head-should-be.
Oh, and by the way i'm coming back soon. Max18well 19:41, 30 September 2006 (UTC)
Hi there. Apologies for not replying to your message for a few days. I didn't notice it at the top of talk page (the wikipedia convention is to append new messages to the bottom of the user's talk page, unless you're continuing an existing thread of conversation).
I'm not using "no-one" because of any particular wikipedia style guide, but simply because I thought it was correct, and it's what my spellchecker suggested. However, after looking at a couple of dictionaries (OED and Merriam-Webster), I wonder if "no one" would in fact be the preferred form. In general, hyphenation tends to get dropped from English spellings as time goes on, and my spellchecker seems inclined to suggest hyphens far more often than it should.
My main reasons for disliking "noone" is that when reading it's not automatically obvious how to break up the word. When reading it, I read as far as "noo" (as I would with "noose", "nook", "noodle", "noon", etc), turn that into a syllable and then get stuck with the extraneous "ne". Then I realise this can't be the correct interpretation, backtrack, and see it should be divided up into "no one". Not the end of the world by any means, but a little disconcerting all the same. There are a handful of other words I have this experience with, but I can't think what they are at the moment. Thankfully the same effect doesn't occur with "everyone" or "someone", as "every" and "some" have clear boundaries that are recognised easily.
Cheers, CmdrObot 14:32, 18 September 2006 (UTC)
Yes, my own view of noumenon was hasty -- after actually reading the article, I see now that this really has little to do with the matter at hand.
You say:
This baffles me. None of these things are mathematically defined. "Bits of nothing"? The language of set theory (again) is first-order logic plus "is a member of". Thus, the following is a sentence in set threory: Ax: !(x e y) , where 'A' means 'for all', 'e' means 'is a member of', '!' means 'not', ':' means 'such that', and the other lowercase letters ('x', 'y') are variables ranging over sets. What is (syntactically) provable (ie is a theorem) is that such a 'y' exists (meaning 'is derivable from the axioms') ie: "EyAx: !(x e y)" is a theorem (where 'E' means 'there exists'). That you think this means something about "collections" or "sets" is not of any concern regarding the consistency of set theory. It may relate to what you think these symbols represent, but such an interpretation is not required to do maths, nor derive theorems in set theory. Of course, mathematicians do have the standard mdel in mind (that all these symbols describe a formal structure about "collections of things"), since they mamy or may not help them guide their formal arguments. But this understanding or interpretation, once again, is not strictly necessary.
So, we come to this comment of yours:
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. My original comment was
which was supposed to be analagous to a staement you mentioned to me about your understanding of zero, which was
Right. At this point, I am going to tread very carefully, since you are still confusing the formal, syntactic aspect of this from the interpretation (or the model-theoretic aspect).
You believe zero is a valid number. The reason you think it's a valid number is because you understand how there can be zero elephants in your room. I point out that the same understanding can be transfered to collections of things. The empty collection is simply the collection of no things. I mean, I can easily transfer your objection about the empty set to my own hypothetical objections about zero. On a street with no cars, how can I say there are zero cars? Non-countability cannot be magically converted into countability: a number is a count of things, and a "zero" number-- that is, one that is not a count of things-- is a contradictio in adjecto. Note, this is exactly a paraphrase of your objection quoted above (which can validly be applied to both the empty set and zero). If you believe this is a valid objection against the "existence" of the empty set, you'd have to agree that zero is not a number.
So anyway. All those arguments were not about the validity or consistency of set theory. These are simply very informal attempts to convince you that the standard model (the interpretation most people have when they think about sets) is fine. The empty set is a perfectly good sort of idea: as good an idea as zero, or aleph_0.
But these are not mathematical arguments. Mathematical arguments are simply about whether some statement (string of symbols) are formally derivable from the axioms (somewhere in my previous comment, I elided "and rules of reference", but in fact, there are systems where there are no rules of inference, but simply an equivalent axiom -- this is technical, but not a problem). Of course, since you are not a mathematician (and I'll stress that I'm not one either), you probably are far more interested in interpretations and models rather than the abstract logic of the symbols. This has no bearing on "existence" or "consistency". You might want to look up "soundness" and "validity", but these aren't one-place predicates; they're relationships between a language and some model. Consistency usually means that you can't prove (prove meaning derive a statment of symbols) P while also being able to derive ~P. You can clearly see that your doubts about the "existence" of the empty set (which is derivable from the axioms (and, yes, the rules of inference -- can I eilde this as understood now?)) do not have a bearing on consistency, since you can't derive "the empty set doesn't exist" from the same axioms.
It is quite a rambly post, I'm afraid. The point I'm trying to make are that your objections are in fact about a certain interpretation of the symbols of set theory. The interpretation is not a necessary part of set theory. Different models are equally valid, though probably less useful. Set Theory is really just the syntactic part. To believe me, what you have to notice is that there are several "set theories". There's ZF, ZFC (ZF with the axiom of choice), ZF-C (ZF with the negation of the axiom of choice, which is different from simply omitting the axiom), ZF-I (with the negation of the axiom of infinity), etc etc. No one has any problems with the validities of any of these. Ugh. To be honest, I've actually kinda lost what you or I are arguing for. But I'm having a little fun anyway (today's quite a slow workday :-) Tez 11:41, 9 October 2006 (UTC)
You say:
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.
Then you say:
Axiomatisation is a type of assertion? What? Who says? You seemed to have misread or just simply missed everything I've previously said. Axioms are not assertions of truth. I will fish out the little toy mathematical theory I demonstrated to you not so long ago:
I would like to know what you think Axiom 2 "asserts". Furthermore, you can tell mathematicians don't think axioms are "assertions" from the fact that people work with mutually contradictory axioms. Specifically, someone will prove some theorem in ZF-C (ZF with the negation of the axiom of choice), and some will prove things in ZFC (ZF with the of the axiom of choice). None of these mathematicians are at eachother's throats, or think the "other side" is "wrong", because axiomatic systems don't assert any kind of truth or falsity. I must've repeated this about 6 times now, and you 'still confuse syntax from semantics.
Then this comment:
In the systems where we are given the axiom of infinity and the axiom of separation, we are not (usually) given the existence of the empty set as an axiom (but its existence will be deducible from those two axioms). In other axiomatisations, we may not have, say the axiom of separation, and then we may get the axiom of the empty set. There are several axiomatisations of set theory. The most popular one is ZFC, but there are others. NBG, for example. And there are in fact several different axiomatisations of ZFC, but insofar as one axiomatisation may not have axioms that other axiomatisations do have, those missing axioms will be deducible as theorems. In this sense, different axiomatisations of ZFC are equivalent.
And about your continued doubts about the empty set. 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. Probably you should also replace the phrase "is a member of" by "is directly contained in".
Good luck, chat soon. Tez 17:06, 16 October 2006 (UTC)
Hi Tastyummy,
Sorry, there are lots of snipets I want to reply to, but I will only quote little bits here -- it may seem I'm responding to things out of context, but hopefully things will be clear. You say:
Not the only one who does what? Referring things that can be modelled by set theory? There are many things that can be modelled by set theory. Natural numbers. Groups. But there are things that aren't, either because the model is not well defined (does set theory model clouds? Cats? Business processes?), or because the model is not sound. In your discourse with others, it seems they decided that your level of technical competence with set theory was low. So in many cases, your objections to set theory have been met by explanations at a very informal level.
This has led you to further confusion down the road as you tackle the more abstract topics of language, theory, proof, and models in mathematics. For example, the Zakon example you quote is 'not about the confusion of the membership relation with the subset relation. It was supposed to point out that in set theory, the membership relation is not (necessarily) transitive ie. if x is a memebr of y, and y is a member of z, we cannot in general say that x is a memeber of z (unless the set happens to be constructed as to have that property).
And the point is that the reason this is generelly the way it is with sets is that membership has a syntactic definition that demands that if you're asking whether some set x is a member of some set y, you look and only look "one pair of curly brackets inside", ie.:
The membership/susbet confusion is different, but tells you that when determining whether some set y is a member of some set x, we must not span across commas, ie.:
As to further doubts about the empty set. Let me remind you again that this is a semantic-level criticism, and even worse, we have been talking about it at a very informal level. Your remark was:
Well. This is simple to counter, if you still believe zero to be a valid number. I will rephrase your objection thus:
No, I do not. Are we saying "zero" is an unknowable thing, and thus invalid? Again, you keep coming out with these objections against the empty set that are directly isomorphic to objections about the number zero. If you can understand and know the concept of 'zero', then there can be no problems (with the objections you've raised so far) with the concept of the empty set. Again, so far, the fact that you "like" the number zero is at odds with your "dislike" of the empty set. It seems rather an objection of aethetics (much like the idea of irrational numbers to the greeks) rather than any epistemic or set-theoretic matter. If you want further "motivational" reasons for the existence, consider the idea of closure; consider the integers, and the operation of subtraction. I may take any integer, another integer, substract one from the other, and always end up with an integer. "The integers are closed under subtraction." Similarly, consider a set with and element, say, A. What is {A} minus the A? Does it suddenly become a non-set? No. It becomes the empty set. At a syntactic level, I can write {}. Does that not look like a set? Does it not look like a set with no elements? Would you not call a container with nothing in it "empty"? So it is given the name "empty set". Further, it seems to me your objection is more to do with "emptiness" rather than "set". This is because you think "set" is an abstraction of "something that has members", in which case, emptiness would negate the idea of sets. But it is not. It is an abstraction of (among other things) containership. And I not only believe in, but have seen lots of empty containers.
You then do some appeal to platonism. I'm not sure what relevance this has to our discussion, since properties of well-defined objects in set theory can be proved or disproved (usually), and if we are talking about models, we can usually prove or disprove soundness and completeness. Here's an example anyway. What is the set of four-legged chairs that has three legs? Well, it's the empty set. I have no issues with this. We do not have to appeal to any platonic form. We can simply note that by definition (and we can generalise, or do this case-by-case, depending on our epistemological framework), a three-legged object cannot have four legs, and vice-versa. Thus our set will be the empty set. This happens all the time in maths; you come up with some definition, and the first job will always be to prove existence (ie. that the set of objects that fulfill a definition isn't trivially the empty set).
Then you go on to say:
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" 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. Usually we talk of models between theories with different axiomatic underpinnings, eg. between ZFC and the peano axioms. For physical theories, maths usually only rears its head once we have produced enough operational definitions, and formulated a physical theory in some mathematical language (some group theory, some metric space, some tensor space, etc, etc). Then the idea of the model applies between, say, set theory and the physical model. Whether the physical model is true will depend on the results of experiment.
You then say:
Sigh. Indeed. But you've yet again mixed up different levels of epistemology. You are also using the term "incomplete" very loosely. In a physical universe, the soundness of the model is tested by experiment. Euclidean geometry used to be the model for the addition of velocities. This was shown to be an inaccurate model of reality. But euclidean geometry is (and always was) both consistent and complete. It just does not well-model spacetime. So euclidean geometry contains "false" statements only if you cling on to the belief that euclidean geometry must, by some necessity, model spacetime. But a theory and a model are separate things. As for theories that contain propositions that go "beyond" physical interpretation: yes, sometimes there will be uninterpretable (eg. simultaneous solutions to basic kinematics collisions, where, say, we consider time at negative t), and sometimes, if you think hard enough, these statements will reveal new physical interpretations (negative energy levels of electrons, say).
I will try to find the time to give a formal derivation of the existence of the empty set next time round, though this is something that's eminently googleable (especially the info I've already given to your talk page). You could try to find it yourself, and it would help immensely your understanding of formal mathematical theories. Tez 11:28, 23 October 2006 (UTC)
Hi Tastyummy,
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.
You objection to the empty set in your last message to me was:
Lots of things to say to this. First, it's fine not to "believe" in the empty set. But I will ask again, why don't you apply your reasoning to the number zero? I could say, for example (a paraphrase of your objection), that when I ask myself "what is a number?", I get something like "a count of items, unless there are no items, which is an exception so special, that it has its own axiom in peano arithmetic establishing its existence". So why doesn't your disbelief of the existence of the empty set not apply to zero? It's not so much that your disbelief of the empty set is at odds with my belief in its existence that irks me, it's the obviously contradictory reasoning you apply when you assert the existence of zero while denying the existence of the empty set, using reasoning that directly applies to both. Actually, having just looked at the empty set article, it has a "Commen problems" section that notes that the empty set isn't nothing. It is a set with nothing inside. I doubt you have problems with the exitence of an empty bag, or an empty bin, or an empty box.
I have pointed out that there are axiomatisations of ZFC that don't have the existence of the empty set as an axiom. Further, you should probably note that in axiomatisations with the axiom of the empty set, it is the very first set that we can produce; the existence of all other basic sets (power sets, pairs, all other finite sets) rely on the existence of the empty set; it is the set from which we build all other sets, so the "validity" or "existence" of every other set is only as great as the validity or existence of the empty set. If you have a look at ZFC, this much should be obvious. That you think needing an axiom to exist whereas other sets don't makes the empty set a special case is quite confused. Firstly, we'd need an axiom to assert the existence of some set in order to start building sets. It might be the empty set, but I've pointed out that it can be the axiom of infinity, at which point the empty set is only as deducible as other finite sets, and could not be called a "special case".
You add:
Well, 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. And sets are distinguishable from, say, proper classes in set theory.
About empiricism:
Well, I didn't say that science denies non-empirical things. I was merely pointing out that the gathering of knowledge via the scientific method is an empirical affair. Testing a mathematical (non-empirical) model against Reality(TM) is also an empirical affair. Testing the internal consistency of a mathematical theory or model is a non-empirical affair.
Then you 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? We have gauge symmetry and Noether's theorem, and both have been thoroughly tested. Does that not establish this principle? Perhaps the word "principle" is confusing, because you might think that scientists take this as an assumption. There are no assuptions in science. Everything in a paradigm is always being tested in every experiment. If the universe didn't exhibit large- and small-scale homogeneity and isotropy, we'd have found out by now. So in a sense, we can "assume" it because it has been established scientifically. But it is not an untested assumption or principle.
Finally, you 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.
Oh, also, have you read the Kuhnian view of the scientific method? Rather than naive-falsificationism, he acknowledges that obervations and experiments always take place in a paradigm, and it's possible to continually patch an incomplete theory with ad-hoc stipulations. At this point, Ockham's razor and a desire for succinct statements of universal laws become indispensible when evaluating a scientific theory. This does sound alittle like what you want to lead up to. I think in his view of the scientific method, any/all scientific progressions are an overturning of the prevailing paradigm (usually once that paradigm has so many ad-hoc cases that it's obvious that it's incomplete). Anyway. I'm sure you'll look into it.
Anyway. Talk more later. Tez 11:31, 26 October 2006 (UTC)
Concerning laws, you say:
I'll get to what I mean by 'aesthetic' and 'form' in a moment. You're using the word "universal" ambiguously here. Why does a universal law have to be unfalsifiable? It seems to me you're confusing physical necessity with logical necessity. Here's a universal law: the total amount of momentum of all the things in a closed system will never change (conservation of momentum). It applies to all particles and all interactions in the universe. Why (in theory, or practice, or whatever) would it be unfalsifiable? "Unfalsifiable" doesn't mean "not falsified", so generally the things that are unfalsifiable are true by definition (ie. are tautologous). Is there something in the definitions of "momentum", "closed system", or "change" that make the law tautologous? I can't see that.
Or perhaps your ambiguity is the other way round: you think "universal law" means "a law that applies to all entities in every configuration possible". But nothing we call a "universal law" is like that, since "universal law" is fairly informal, and usually means that with the given restrictions, all other conditions are quantified universally. So for example, the conservation of momentum applies only to things that have a momentum (which is defined operationally). So a photon has a momentum, but a magnetic field does not. Perhaps you think "universal law" should be a statement or expression that encapsulates the truth about everything. But as you have essentially pointed out yourself, such a definition of "universal law" would be as useless as any "law" that fell under that definition.
Following my example using kinetic energy, you say:
I didn't say "inconsistencies", I said "exceptions".
Then after the unfalsifiability bit, you say:
And that's exactly what my example formulation did. Explicitly. Of course, in my example universe, the Copernican Principle is obviously false.
And then you say:
I would still prefer the word 'exception' to 'inconsistency' above, though perhaps you really mean 'non-homogeneity'.
This is what I mean when I referred to form and aesthetics. I would say that a law formulated in a case-by-case manner (somewhat like my kinetic energy example) is a perfectly well-formulated and universal (because it does cover every point in time and space) law. Reality(TM) might actually be full with singularities and exceptions. Having exceptional cases in our description of these laws may be unavoidable. In what way, then would the lack of a "unifying theory" (where I take this to mean a theory with zero or near-zero "exceptions") make these laws any less universal? Of course, it is precisely an appeal to aesthetics that the form of our laws be succinct and apply generally (eg. not just inertial frames, say, but accelerating frames too, in the case of special and general relativity) to as wide an extent as possible. But not having this form does not deny the universality of a law. It is only through a lot of experimental evidence that we're quite confident that there is a formulation that subsumes all current cases and exceptions gracefully ie. that we believe in the Copernican Principle.
When I mentioned inconsistency, what I was referring to is logically inconsistent phenomena in our physical universe. For example, Schroedingers cat, or for an actual experiment that essentially demonstrates "the current is flowing clockwise and the current is flowing anti-clockwise", see [2].
And now, the empty set:
Well, that's because you haven't looked at the links to, say, ZFC that I've pointed to previously. For example, here are some non-sets:
And anyway, sets aren't just collections of things, they are also members of collections (whereas proper classes are collections that can't be members of sets). The rules for distinguishing between sets and non-sets are precisely the axioms (and rules of inference) of set theory.
You further say:
And again, if you believe in zero, you must admit this reasoning is fallacious, since I may paraphrase it:
So, again, fair enough if your interpretation doesn't allow you to see the existence of the empty set, but then the same interpretation must also deny you the existence of zero as an integer or natural number, or whatever.
Finally:
I'm not aware of any unary function "complement" defined in ZFC. Can you cite textbook/website where this is defined? Or perhaps if you define it here (even informally), I can try to work out the answer. Tez 16:23, 31 October 2006 (UTC)
You say:
Not sure what you mean by "multiple counts of zero". Do you know what it means for an entity to be unique? Let's call a predicate that characterises the empty set E(S). That is, if the set S is the empty set, E(S) evaluates to "true", otherwise, it evaluates to "false". Uniqueness means that if we consider sets S1 and S2 such that E(S1) and E(S2) are both "true", we can prove S1=S2. That is, there is only one entity that satisfies the definition of the empty set. In almost all contexts where zero is well-defined (fields, rings, peano set), zero is also provably unique. That is, let's call a predicate charaterising zero Z(N) where N is some number, and Z(N) evaluates to "true" when N is zero, and "false" otherwise. Usually, Z(N) will describe something like zero being the additive identity, or perhaps the element which is not a successor of any other element. Then (again, in most contexts), if Z(N1) is "true" and Z(N2) is also "true", N1 = N2.
Then you say:
But this really has no bearing on any of your objections to the empty set, or my critiques of you objections. To support your point, let us consider a set, call it N, that fulfills the peano axioms. Then, indeed, zero is the unique element of N such it is not the successor to any other member. We haven't used the empty set in the construction, we simply are considering collections of numbers (not sets of sets, as would be the case in "pure" ZFC).
Now, I can still apply your objections to the empty set in set theory (that empty containership is not really containership) to zero in this peano set N (that zero counthood isn't really a real count). I can do this because your objections have nothing to do with the representation or construction of zero (or even the empty set, for that matter), which are formal and syntactic concerns, but instead with your interpretation of what zero and the empty set "mean" (a semantic argument). I can do this because I've informally constructed an isomorphism between natural numbers (whole numbers including zero, constructed in whatever which way you want) and sets, such that for any count of things, there is a set with that many things contained. This isomorphism exists regardless of construction, as long as your numbers and sets have most of the properties that we'd normally consider inherent in numbers and sets.
Therefore, I've setup an equivalence between then meaning of "containership" and the meaning of "counthood" such that for any objection you have about the empty set's existence or "validity" as a set, there is an equally valid objection to zero's existence, or it's "validity" as a number. Similarly, any evidence you have for zero's "existence" or validity as a number is equally evidence for the empty set's validity as a set. Try it yourself: Come up with an objection against the empty set, and simply translate it into an objection about zero. And vice versa. It's pretty easy to do, since the isomorphism is pretty straightforward.
Again, none of this is an argument for the existence of the empty set. It is merely an argument that you cannot consider the empty set any more fictitious or degenerate than zero, nor zero any more real or valid than the empty set. Tez 11:55, 15 December 2006 (UTC)
Hi, maybe the following is of interest to you.
In the above talk, a good amount of semantics is attributed “a priori” to the notion of sets, and their possible application to model (or speak about) the world surrounding us. There is an approach which avoids this, and allows any physical semantics for sets to be recovered (or indeed “added”) later on.
As shown by Bourbaki, set theory may be constructed by adopting rules for constructing, classifying and manipulating strings of symbols from a basic alphabet. This approach has been refined by Edwards and again by Schröter. See e.g. Edwards, R.E., “A formal Background to Mathematics”, Springer 1979.
The result is like a symbolic game with symbols, which mimics (of course: by design!) the “mathematically useful behavior” one expects from classical set theory, which in turn abstracts intuitive notions or expectations. For example, like any other Bourbaki-Edwards (B-E) set, the empty set is simply defined as on particular string of symbols. And lo and behold, when this string is “inserted” into the agreed (defined) string-manipulation schemes for set operations like union, intersection, … it nicely conforms to what we want. Yet the entities (sets) built in this way, are no more than (abbreviations for) strings complying with certain agreed rules, and are at this point devoid of any further meaning.
Note: any abstraction always involves stripping off semantics and reducing, simplifying behavior, on the other hand, any such construction may also bring in some artifacts, so that the construct should never be naively identified with what it’s supposed to model: the model may work only to some extent, if at all.
B-E go on to employ this notion of sets as a basis for mathematical structure-type (or “species of structure”) theories, that cover the bulk of mathematics. Apart from naked sets, they include all sorts of stuff like the natural numbers, other “number sets”, the familiar algebraic structures (groups, fields, rings, vector spaces, algebra’s, …), topological and measure structures, and combinations, extensions and variations thereof (operator algebra’s, manifolds like pseudo-Riemannian spacetimes, etc.).
Let’s keep in mind that, again by design, none of these mathematical concepts carries any interpretation or semantics: everything is but a purely formal game. This is true, regardless of the intuition and heuristics that have guided the choice of the defining axioms for these structures. Once the structure-axioms stand, they are preferably consistent, at best suitable for doing some nice math and with some luck even fun.
Note: the structure-type notion is categorical in nature, but does not go the full length category theory does. It is, in a sense, more cautious, which seems appropriate when probing deeper philosophical questions as to the how and why, scope, ontology etc. of such a tremendously powerful human art as is physics.
This is as far as B-E mathematics goes (which is pretty far). So what about physics? A general way how physical interpretation may be appended to all this was proposed by Ludwig. This is one of the “structuralist” programs for theoretical physics, as listed in Stanford Encycopedia.
According to Ludwig, at the core of any Physical Theory (PT) resides a B-E Mathematical Theory in the above sense, which is to serve as a model for some excerpt of reality. Observe that for each (attempted) PT, its MT is chosen or “proposed”, postulated, if you like; it is never formally inferred.
Next, one goes on to specify the “known inputs” for the PT. These consist first of appropriate templates for “observational statements”, which are accepted as relevant for the intended PT. In general, any concrete (experimental) observation may be formulated in set-theoretic language as “the constant a is an element of the set B”. Each observation leads to a formal such sentence, which is appended to the list of axioms of MT, thus extending it.
Actually, in order to make the MT into a PT, one also has to specify which “basis sets” in MT may appear in the observational statements. This convention, together with the “input templates” are of course additions to the formal scheme, outside of the MT considered. As such, they are “meta” notions. Together, they constitute the PC’s “mapping scheme” or “mapping principles”.
What we also have to do, is to describe in “natural language”, which parts of nature are adopted as “known inputs” for the intended PT. This is referred to as the “(fundamental) domain” of the PT.
In other words, the basic ingredients of a physical theory PT are given by the triple (domain, mapping scheme, MT). For the most elaborate and in-depth development, see Schröter, J., “Zur Meta-Theorie der Physik”, W. De Gruyter, 1996. A nice summary is given on Martin Ziegler’s page.
Ideally, the structural axioms of the MT used, have an immediate physical interpretation. This is for example not the case for the Hilbert space axioms, as used in quantum mechanics. This indicates that Hilbert space is only an ancillary mathematical structure, possibly fine for practical calculations, less so for proper understanding. (In fact, Hilbert space is just the carrier of a – very practical - mathematical representation of other MT’s that do permit more direct physical interpretation.)
Another interesting footnote is that to the extent that the Ludwig program works (as is essentially established for physical models like General Relativity and non-relativisitic Quantum Mechanics), it also shows that traditional binary logic is sufficient for physics (as all B-E mathematical theories are based on it).
Sorry, this has become a bit more lengthy than intended. Just discard if deemed inappropriate.
--Marc Goossens 10:20, 9 March 2007 (UTC)
Hello. Can you help clean up the unfortunate situation described at Talk:Indeterminacy (Philosophy)? Two articles indeterminacy (philosophy) (with a correctly lower-case initial "p") and indeterminacy (Philosophy) (with an incorrectly capitalized initial "P") now exist and should get merged. Michael Hardy 21:23, 5 April 2007 (UTC)
Mr. SAT Man,
You need to study your punctuation rules some more. When using quotations, the punctuation goes INSIDE the punctuation marks. Example: It's great that you're so "verbally competent," but proper punctuation is also required to appear literate. Additionally, you were right about some people thinking you a prick. Lose the ego, son; it's unbecoming.
Hi,
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