User talk:Tastyummy

Feel free to use this page for correspondence with me. Make a new header above everything else on the page...

like this.

"Conservatism"-- WHY is the alternate and more grammatically-sound term "conservativism" an "incorrect spelling"?

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

"The conservative spelling of this word is 'conservatism.'"

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)[reply]

"Dawkins VS. Popper" ???

Here's something that might interest fellow students of science and/or philosophy, as well as fellow Brights:

Dawkins VS. Popper

(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,

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

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.

Indeterminacy (philosophy)

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[reply]

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 do not have a sockpuppet!

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.

Exceptional Newcomer

The Barnstar of High Culture

Max18well 21:57, 18 August 2006 (UTC)[reply]

Dispute with user Aey

[This is my reply to Tastyummy and Aey at my talk page; I have posted this message at User_talk:Tastyummy and User_talk:Aey also. For convenience, please continue the discussion only at MY page, if you want me to read and contribute to it. Best that it be in one place. - Noetica 13:14, 19 August 2006 (UTC)] Well! Tastyummy, you are very welcome to approach me as you did. I am not an expert on Nietzsche, but Philosophy is an article that I am certainly competent to deal with, and I do give it my attention from time to time. Aey, I am not satisfied that matters have been resolved at that article. I have simply left things for a little while, warning you as I did so that I'd be watching. I have been, and I am tempted simply to revert the text to how it was before your intervention, which I find decidedly substandard. I am just an ordinary editor, not an admin. But I am not impressed with your behaviour around the place. If you do not show more insight into the limitations of your offerings, and more care, consideration, and restraint, I may take the matter to an admin; or if anyone else wants to do that they can rely on my support. I'll continue to monitor the situation, and I'll look more closely at the article Tastyummy refers to. I hope the two of you will feel free to discuss things calmly at my page, with a view to sorting all this out. - Noetica 13:14, 19 August 2006 (UTC)[reply]

A-okay

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)[reply]

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)[reply]

Drop a word here too, if you please: [[1]] Rashad9607 12:31, 29 August 2006 (UTC)[reply]

Re

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)[reply]

Thanks for you kind words! I really appreciate it. Perhaps we'll run-in to one another some other time. —Khoikhoi 10:13, 23 August 2006 (UTC)[reply]

Mathematics, models, reality

Grabbed from the set theory talk page:

Regarding one more thing:

"We can derive almost everything from the formal manipulation of symbols from axioms and rules of inference."

We certainly can, but the axioms and rules of inference of which you speak are not self-evident; they are simply a model and approximation of reality deriving from our continued observation of real phenomena. Nothing is self-evident; hence my inquiry into the origin of even the most basic concepts.

I won't waste any more space here. Thanks again to everyone for helping me with this, and for being patient with my long-winded rants.

-Tastyummy

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)[reply]

Regarding the empty set and zero: You say "zero is a much more tangible concept than the empty set", giving the an example of zero elephants vs the empty set. This is in fact a categorical error. The comparison should in fact be "0 elephants" vs "the set of elephants in my room is empty", not "0 elephants" vs just "{}". You can't see "0" in your room, just as you can't see "{}" in your room, and similarly, you'll never encounter 3/4, -1, i, the identity matrix. *In fact*, the standard construction of the natural numbers from ZF explicitly equates "0" and "{}". I mean, if you can see how sets answer questions about collections of things, how can a non-empty collection be useless? I mean, what is the collection of white swans? A subset of all swans. What is the collection of polkadot swans that have no wings? The empty set. If you can see how "0" is useful in the real world, surely the same can be applied to "{}"?

I do now see that your interest lies in discovering the motivation and mechanics of modelling and abstaction. But there is still much confusion. It seems implicitly somewhere you're implying the usefulness and validity of mathematical concepts is closely correlated with it's ability to model the real world. Perhaps you have some Hume-esque view of the world, where emperical truth and experience is all we can draw upon. I do not (yes, that *is* quite platonic), but what you fail to see is that Fundamental Theorem of Arithmetic, say, (it's the theorem that states that any natural number is uniquely expressible as a product of prime numbers) is valid whether we exist or not. Surely you can see that the formalisation of the natural numbers, as a concept, does not rely on human existence for it's abstract existence. The only thing that humans add is their discovery and naming of said concepts in number theory. Perhaps that *isn't* obvious. But again, you seem a bit of an impiricist, while I will certainly err on the side of platonists. Oh well.

Later on you say "but even the simplest axioms and logic are established as 'true' via their ability to predict real phenomena". This is entirely false. 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. These systems will still be valid mathematical systems. And more concretely, people work in standard systems, but try to prove utterly useless things. Liek the Goldbach conjecture (that every even number is expressible as the sum of 2 primes). What possible use could that be? The answer, actually, is that finding a proof might show us new proof techniques and theorems. The result itself has no application thoug (as far as I know).

As I understand it, you're still confusing provability (the property of being able to derive certain strings of mathematical symbols from axioms using rigid rules of inference) with truth (what interpretation you apply to these symbols, and how that maps to truth values), just as you are confusing mathemtical validity and existence (if some concept X is empirically useless, does it "exist"?) with motivation (why do we abstract things? How is the mind able to do that? Does my system model something in Reality(TM)). As someone who does have more platonic leanings (and some training in formal mathematics), I do separate these quite easily, but I do not claim they are *actually* seprate. I will grant that perhaps our ability to abstract mathematical concepts might be limited in the way we experience the world (could we ever abstract some concept that isn't directly experiential? I actually believe the answer is 'yes'). You obviously believe so. But do not let that belief on the human limitation of the exploration of maths define what maths can or can't be.

Sorry, that was all a bit of a ramble. I don't actually have time to type out a clearer, more concise version! Tez 11:05, 24 August 2006 (UTC)[reply]

Hmmm. The objections you have against the existence of the empty set are in fact also equally valid objections against the existence of the concept of 'zero'. You say:

"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."

If that's what you think, wouldn't it be equally valid to say "there exists no integer whose value is the number of elephants in my room", simply because there aren't any elephants in your room? In other words, any objection you have with the empty set you must equally extend to the number zero. Personally, I don't see any problems with having a empty group, a collection of zero things, and really, I'm still a little confused at what epistimological level you're aiming your argument at. I mean, to answer your question "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.)?" -- well, it does. It is the group of zero elements. The group of elements that don't satisfy some predicate. Or however you want to characterise "zero of something". Out of curiosity, do you actually level these criticisms at 'zero' too?

You then say:

'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'm not sure I used any of these terms. Either I'm not understanding the terms here, or you re again confusing truth and provability. One is a semantic quality (truth), and the other one purely syntactic (formalisation, provability). Again, mathematical truths are already contigent on the truth-value of the interpretation of a theory's axioms. Remember, a theory does not necessarily have any models, and it can have more than one, and (non-tautological) theorems from the same theory will have different truth-values depending on your interpretation. I'm not sure I agree that hypotheticals are irrelevant. A thing that's not observable may be out of the reach of science. But we are talking about numbers. Can you observe '3'? Can you observe '-1', or the imaginary number i? So observability is not really an issue. As for pure hypotheticals, I'm sure it *must* interest you (especially the hypothetical universe where humans don't exist). What we're trying to discuss is the status of existence of mathematical concepts. To discuss whether they are independent of human minds requires us to think about a universe without out our existence. I don't have too many issues with this, despite the apparent paradox.

Later you say "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." I don't understand how you can assert the existence of 'data', especially 'empirical data' without the existence of an observer (and the abstractions inherent in *every* kind of experiment and observation), and then claim that abstraction built on top of them are 'inventions'. Surely then the data "3 zebras" is also an invention? I suppose you allude to that when you say that even language and logic are the product of civilisation. I will say that theories are invented. But then I'd say that the theroems and lemmata that follow from them are indeed discovered. For example, do you believe that every possible chess game doesn't exist, and a series of moves that has never been played before is 'invented', or that every possible chess game *already* exists, and is 'discovered'?

Now for the section where you doubt the existence of mathematcial theormes that don't have an empirical quality (some quotes from your last message to me):

'...the point as that there is no such thing as "pure mathematical truth"'
'"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.'

Here are some:

sum (n : 1 -> inf) 1/n does not exist
sqrt(2) is irrational
the set of algebraic integers is countable
The reals and the rationals are dense in their usual ordering
If there exist functions f:A->B and g:B->A, both injective, then there exists a bijection h:A->B

I have a hard time thinking how these mathematical truths have *any* empirical foundation.

Then you mention:

"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."

Yes! If you are referring to 'noumenon', that is what I was trying to allude to when many posts ago I mentioned that maths is not about phenomenon. Science is about phenomenon. That maths can tie together and model science operational definitions is actually in the realm of science, not maths.

Then you say:

"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."

Well, I talked about this when mentioning euclidean and riemannian geometries. Again, I understand that our models of reality have been shown to be lacking, have been continuously replaced, and will continue to be ammended. Again, this is the realm of science. 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.

Anyway. Whereas I usually *hate* leaving things in a state of "agree to disagree", I think our conception of 'abstraction' and 'formalisation' are just totally different. I can easily see the soundness of the concepts of '3' and '6' (like "3 dogs", and "3 dogs gathering with 3 dogs will result in a group of 6 dogs") without humans. But you are right in the sense that I can only express such concepts 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).

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

I certainly look forward to your response! Tez 09:48, 29 August 2006 (UTC)[reply]

Hi Tastyummy,

To answer the last question first, I suppose I'm using 'abstraction' the way it is used in Abstraction (mathematics) -- ie. the extraction of the underlying essence of some pattern, and the distancing and removal of dependence on real-world objects and phenomena.

As for 'zero' vs the empty set, consider what you write here:

"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."

This rationalisation for zero can be translated directly to a rationalisation for the empty set:

if I must consider the collection of all the elephants in my room, then the collection of them I have is the empty collection".

All I'm trying to convince you of is that the concept of {} is at least as valid as zero or any other mathematical concept. In fact (and I can't remember if I mentioned this already or not), once you study set theory to the point where your textbook/lecturer is constructing the natural numbers from sets, you'll immediately see that the empty set is zero. Further, you seem to be of the view that the empty set is some degenerate case of set. In fact, it is the first set that existence (and uniqueness) can be proved. Every other set is a conbination of the empty set and operations of pairing, union, powerset, etc, etc (see Zermelo-Fraenkel set theory).

As for why everything is a set: well, it is called "set theory". It's the abstraction, formalisation, and rigorous development of the idea of collections of things. What is instructive then is the construction of every other mathematical concept from sets, starting with the naturals, then the rationals, reals, differential and integral calculus, geometry, etc, purely from the idea of sets. In fact, if I were to formalise some theory of collections, I would ask myself "Ok. So where do we start? What is the most obvious collection of things? Collections of naturals? Collections of collections? I know, the collection of nothing. We can then construct other collections by adding things to this starting set". I mean, that does seem the most constructive and least arbitrary starting point. Of course, I'm slightly biased by formal learning in this area :-)

Similarly, have a look at Lambda calculus, where the only thing under consideration is the idea of "function". That's it. And funnily enough, you can also construct the naturals, rationals, reals, etc using only lambda calculus as a foundation.

Mathematical "existence" really means "allowable by the rules of my theory". For example, the Russell-paradox set (see naive set theory) does not exist in the theory ZF since it breaks the axiom of regularity. The chess-game analogy to determining existence of some object is asking whether some arbitrary board position can be derived from the rules and starting positions. For example, any board position with a white pawn on the first rank cannot exist. Mathematical existence does not refer to any outside phenomenon (at least, not once the thoery in which the object is being considered has been formalised to a high enough degree).

I have heard of 'qualia' before. Just as you seem to have a well-informed maths friend (presummably, he did maths or computer science at uni), I have quite a well-informed philosophy friend (maths and philosphy, in fact). He mentions these things to me from time to time, but I don't have the curiosity you seem to have to probe further. Regardless, this would have no bearing on set theory. Equivalence, equality can be determined formally in ZF by the axiom of extensionality. Using the axiom of infinity and the axiom of separation, we can deduce the existence of the empty set, and using the axiom of extensionality, we can deduce its uniqueness. In fact, these deductions can be done in a completely automated way. See automated theorem proving. I will allow that the motivations behind the construction of mathematical theories may be grounded in experience and the Real World (TM). But does the fact that computers can verify maths proofs not indicate further that mathematical existence is completely removed from real-world phenomenon? In fact, we can use the chess analogy again. Automated theorem proving is akin to programming in the rules (rules of inference eg. deduction), the starting position (the axioms) and then enumerating the possible board positions (theorems, proofs).

Again, I do not deny that much of maths is motivated by influences of the real-world. But I can produce a valid mathematical theory that is totally devoid of such motivation, and I think you'd have a hard time working out what real-world phenomenon this is supposed to mirror. Here we go:

Alphabet: "A", "B", "C"
Axiom 1: AABC
Axiom 2: B
Deduction Rule: if you have a string that looks like xBy, you may write down xCCy

Example 1: I have AABC, so I may deduce AACCC (here, x is "AA", y is "C" when using the deduction rule).

Example 2: I have B, so I may deduce CC (where x and y are the empty strings, "", in the deduction rule).

I claim this is a mathematical theory. A rather trivial one, I might add, but a theory nonetheless. I can ask mathsy-type questions like "can I deduce 'BBBB'?", or prove something about all strings producible like "there will always be an even number of 'A's in any string in this theory". So either you can claim that this isn't maths in any way (in which case you'd be wrong), or you can spot some phenomenon this models, or you will have to admit the existence of maths is pretty independent of the real world.

What do you think? Tez 11:13, 30 August 2006 (UTC)[reply]

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)[reply]

Hey Tasty

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)[reply]

"Noone" versus "no one" or "no-one"

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)[reply]

Maths, formalism

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:

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.

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:

"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. My original comment was

"if I must consider the collection of all the elephants in my room, then the collection of them I have is the empty collection"

which was supposed to be analagous to a staement you mentioned to me about your understanding of zero, which was

"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."

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)[reply]

Just a quick PS about axiomatising the empty set: There are several formulations of set theory, and they are provably equivalent (in the sense that theorem T provable in theory A <=> theorem T provable in theory B, roughly speaking). I've seen some axiomatisations of set theory that have the existence of the empty set being an axiom, but in the main, ZF is used, and the empty set is not given by an axiom; it is deducible as a theorem from the axioms. More concretely, in a theory with, say, the axiom of infinity and the axiom of separation, the empty set is deducible.

A quick technical note: In proof theory, a theorem is a sentence that is either an axiom, or the last line in a derivation where every line is an axiom or a statement deducible (by the rules of inference) from the axioms. So to say the existence of the empty set is a theorem means either it's an axiom or it's deducible. Tez 09:45, 10 October 2006 (UTC)[reply]

You say:

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.

Then you say:

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.

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:

Alphabet: "A", "B", "C"
Axiom 1: AABC
Axiom 2: B
Deduction Rule: if you have a string that looks like xBy, you may write down xCCy

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:

And, again, if the axiom of empty set can be proven from other ZF axioms, why is it an axiom?

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)[reply]

More set theory

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:

Firstly, yes, I was referring to things that can be modeled by set theory. But I'm not the only one who does this: <followed by a statement from me asking you to think of 'sets' and 'boxes'>

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.:

if x = { {}, y }, we can say y is a member of x

if x = { {}, {y} }, we say y is not a member of x (where y isn't the empty set).

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.:

if x = { 1, 2, 3 } and y = { 2, 3 }, we can see that y is not a member of x, whereas

if x = { 1, {2, 3} } and y = { 2, 3 }, y is a member of x.

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:

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?

Well. This is simple to counter, if you still believe zero to be a valid number. I will rephrase your objection thus:

To me, an "enumeration of things" minus the whole part about "things" leaves only the idea of "enumeration", "counting", 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?

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:

To me, it seems like people do think that set theory is a "good foundation for everything" ...

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:

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.

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)[reply]

You say:

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.

I'm not quite sure what this means. "If there is a problem with an [underlying axiom]...", for example, could mean a whole host of things. What do you mean by it? If you mean that it might be inconsistent (in other words, that you can prove a proposition P, as well prove its negation, ~P), that is a logic and set-theoretic concern (and will merely imply that we can prove any proposition Q). But this may still model the universe. There is no a priori necessity that the laws of the universe be logically, mathematically, or even "physically" consistent (whatever that may mean).

So instead, if you mean that some aspect of the model doesn't doesn't give a proper interpretation of experimental fact, well, that's not a set-theoretic matter. It is a matter for theoretical and experimental cosmologists. They must examine their operational definitions, their experimental methods and data, and finally the model they've chosen.

So again, 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).

Whether a mathematical framework is internally consistent is a matter of mathematics. Whether the framework models reality is a matter of scientific experiment. Tez 09:37, 24 October 2006 (UTC)[reply]

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

Oh, and some further comments on your views on zero. If you wish to construct the integers without set theory, you can look at the peano axioms (which I'm sure I linked to earlier). With that framework, the emphasis is on the successor relation and induction. Then, you can define addition and multiplication, at which point, it is simple to identify "zero" (say, as the additive identity). While the wikpedia page starts by naming zero as the element without a successor, you can be far more general, and simply name your elements e0, e1, e2, etc, and then point out that e0 has those properties that you'd wish "zero" to have.

You then 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.

Where have you observed zero? Again, all your support for zero can be translated directly to support for the empty set, and all your objections to the empty set can be directly translated to objections about zero. 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"? Tez 11:32, 24 October 2006 (UTC)[reply]

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:

"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..."

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:

"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..."

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:

"If only empirical things exist, then how am I able to talk about them?"

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)[reply]

Laws, "Universal", the Empty Set

Concerning laws, you say:

"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?"

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'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"?"

I didn't say "inconsistencies", I said "exceptions".

Then after the unfalsifiability bit, you say:

"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 (or, in other words, for the Copernican Principle to hold)."

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:

"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."

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:

"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."

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:

A = {A}
R = {S !e S} (where !e means "not a member of") -- Russell's paradox set

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:

"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."

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:

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

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)[reply]

Zero, empty set

You say:

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.)

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:

"Zero can be defined within mathematics without the empty set."

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)[reply]

Constructive sets (as strings of symbols) and structuralist physics

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)[reply]

indeterminacy

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)[reply]

Punctuation

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.

Uh... Hi.

I won't reciprocate your lesson on what is necessary for me with one on what is necessary for you, but I do have a few suggestions:

1. If you'd like to argue that my uses of punctuation marks break "rules", and particularly those observed on Wikipedia, perhaps you'll find the Wikipedia:Manual of Style a useful reference. The following may be found there:

Inside or outside: Punctuation marks are placed inside the quote marks only if the sense of the punctuation is part of the quotation (this system is referred to as logical quotations).

Arthur said that the situation is “deplorable”. (When a sentence fragment is quoted, the period [full stop] is outside.)
Arthur said, “The situation is deplorable.” (The period is part of the quoted text.)
Martha asked, “Are you coming”? (When quoting a question, the question mark is inside because the quoted text itself was a question.)
Did Martha say, “Come with me”? (The very quote is being questioned, so here, the question mark is correctly outside; the period is omitted.)

I'll gladly correct errors if you point them out, and you're welcome to correct them on your own instead of lecturing me. However, I will revert edits that turn grammatically- or stylistically-satisfactory sentences into unsatisfactory ones.

2. The sentences

"When using quotations, the punctuation goes INSIDE the punctuation marks."

and

"It's great that you're so 'verbally competent,' but proper punctuation is also required to appear literate."

evidence the relatively small measure of "verbal competency" in their author's possession. Here's why:

The subject of the first sentence is "the punctuation". "When using quotations" modifies that subject, whose predicate is "goes inside the punctuation marks". Thus, I could write "The punctuation, when using quotations, goes inside the punctuation marks". Do you see why this doesn't make sense?

(And do you see why I didn't have to put the period from your sentence inside the quotation marks I just used? The entire sentence is part of the sentence encompassing it and beginning with "I could write [...]". "Write" is often a transitive verb. One often writes something, and the quoted sentence is, in this example, the thing being written.)

Perhaps you should have written "Punctuate inside of quotation marks", since, I think, you meant to write an imperative sentence.

The second sentence has two subjects with distinct predicates, which itself is fine. The first subject is "It" (from "It's great that [...]") and the second is "punctuation". The "you" from "you're", though, is something else. The first part of the sentence could be replaced with "That you're so 'verbally competent' is great, but [...]", and the second needs the word "you" in it somewhere. Otherwise, it is punctuation that is required to appear literate. ("Hey, punctuation! You know, you have to go over here to look literate!")

3. Grow some balls and sign your posts if they're about my English, or at least study it yourself before making yourself look stupid. (Of course, remember, this is only advice. I wouldn't even pretend to know what you "need" to do.) You might also consider the hypocrisy of your advice about my ego before adding any more.

Have fun,

Tastyummy 20:45, 11 June 2007 (UTC)[reply]

ArbCom elections are now open!

Hi,
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