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May 6

Construction of a Pentagon (a little different.)

The standard construction of a Pentagon more or less starts with a circle and creates a Pentagon inscribed in the circle (choosing one of the corners with a line through the center). What I would like to find is a construction of a Pentagon from a line and a point not on the line so that the point is one of vertices and the opposite side is on the line. (As a bonus if we can get the equivalent construction for when the point not on the line is one of the "other points", neither opposite or one the line).Naraht (talk) 17:27, 6 May 2014 (UTC)[reply]

Well, you can probably use the standard construction to get a line passing through the Point at an appropriate angle so that it meets the first Line in a vertex; that's a start at least. —Tamfang (talk) 06:18, 7 May 2014 (UTC)[reply]

So simply pick a place on the line to do a Pentagon around and then "move" the angle?Naraht (talk) 13:47, 8 May 2014 (UTC)[reply]

This can definitely be done. The important observation to make, is that if you construct a line perpendicular to the given line that goes through the given point, then you have established the length of a 'chord' that goes from the given point the middle of the edge that lies on the given line. This length has a fixed proportion to that of the edge that lies on the given line, which is a proportion that is a constructible number, hence you can construct that appropriate length. The appropriate length is the side length for your pentagon. half this length in either direction from the intersection of the 'chord' and the given line are 2 of the 5 points of your regular pentagon, the remaining 2 are easy to construct. 69.54.133.130 (talk) 06:43, 11 May 2014 (UTC)[reply]

are any axioms true in a physical sense in our universe?

Does it make sense to ask if the axiom of choice (for example) is physically true or false in our universe in a physical sense? If we ask this, what are the possible answers? (e.g. true/false/neither?) What about other axioms? Do any mathematical axioms have a physical value in our universe? (In much the same way that if you put a finite n number of apples together and keep adding one, you will never run out of apples; but if you put a finite n number of apples together and keep removing one, eventually you reach zero. This means there doesn't exist a k such that positive integer n + k = 0;, and for any positive integer n there is a k such that n-k=0. I presume we can actually prove this using a trivial set of axioms; I further presume that these axioms are in some sense true "physically" in our universe.) Please correct me if I am wrong in any of the above assumptions! 212.96.61.236 (talk) 20:43, 6 May 2014 (UTC)[reply]

It's an interesting question, and probably one you can do entire PhD dissertations on (in philosophy departments, not math departments). However, just to get a quick response in, it's not one that mathematicians usually think much about. Mathematical realists, or at least some of them, hold that (at least some) axioms are either really true or really false, but for the most part they are not taking them to describe physical objects, but rather abstract objects that are nevertheless still real and exist independently of our reasoning about them. --Trovatore (talk) 21:06, 6 May 2014 (UTC)[reply]

are you saying that any conceivable mathematical axiom is totally independent of the actual Universe, and in the actual Universe no basic mathematical axiom is 'true' or 'false' but always 'independent'? That even pi does not have an exact value in our universe, but only under a certain set of axioms mathematicians can entertain, which just so happen to be every set of axioms that are of interest. (No set gives a powerful system with a slightly different version of pi.) So even though every set of axioms that is of interest implies a certain value of pi, and even though our universe sure seems like it shows that, still, we can say with absolute certainty that pi does not have an exact value (regardless of whether we could discover let alone prove it) in our universe, but that every single axiom is totally independent in a physical sense? 212.96.61.236 (talk) 22:05, 6 May 2014 (UTC)[reply]

I'm afraid I don't really understand your question. --Trovatore (talk) 22:10, 6 May 2014 (UTC)[reply]

I think you need to take into account that a correspondence between mathematics and our observation of the physical universe is rather loose. Even the correspondence between a formal system (manipulation of symbols according to a set of rules) and mathematics in general is not clearcut, and with the physical universe the correspondence is even less clear. So, what you might consider to be a physical apple does not have a clear definition: where, in the continuous spectrum of transformation from "an apple" to "something other than an apple" do you draw the line? So already you are operating with some convenient idealization of the universe in which apples are back and white (if you'll forgive the mixed metaphor). As such, it is difficult to imagine how to interpret your question as applying to anything at all. —Quondum 22:23, 6 May 2014 (UTC)[reply]

Well take the Banach–Tarski_paradox. In a literal sense is it physically true in our universe? (I am equally happy to hear that it is false.) I'd just like to know whether either those statements, or any other statements, have a literal truth-value? Or is anything nameable, also independent. 212.96.61.236 (talk) 22:59, 6 May 2014 (UTC)[reply]

Sure, it has a literal truth value. It's true — about the Platonic ideal objects it talks about. In our universe? It's true everywhere, including our universe.

However, it doesn't mean you can cut up a physical pea and reassemble it into an equally dense pea the size of the Sun. You can't do that, and Banach–Tarski doesn't say you can, because it isn't talking about peas. --Trovatore (talk) 23:03, 6 May 2014 (UTC)[reply]

With reference to the original question: The Banach-Tarski paradox isn't an axiom; it's a consequence of a set of axioms that treat balls as continuous infinitely finely divisible regions of space. Actual balls are collections of atoms; if you divide them too much they lose their ball-ness, and you have an empty bladder, or a pile of sand, or whatever your ball was made from. So in that practical sense, neither the Banach-Tarski paradox nor the underlying axioms are true of actual space and balls. But the principles that led us to choose those axioms in the first place are derived from our intuitive understanding of ordinary space - so although they're not strictly true, they describe everyday experience very effectively. That's why the Banach-Tarski paradox is a paradox: it's obviously false for real objects, but derives from axioms we invented in order to model real objects.

But in general: an axiom cannot, by definition, be falsified. A physical law, on the other hand, may in principle be falsified, even if we are confident it will not be. Accordingly, no mathematical axioms govern real space. They are different categories of statement. AlexTiefling (talk) 23:09, 6 May 2014 (UTC)[reply]

Correction: It's obviously false for physical objects. It's true for real objects. Just not physical real objects. --Trovatore (talk) 23:13, 6 May 2014 (UTC)[reply]

Do you have an example of a non-physical real object handy with which to illustrate the distinction? AlexTiefling (talk) 23:23, 6 May 2014 (UTC)[reply]

Handy? They're everywhere. --Trovatore (talk) 23:30, 6 May 2014 (UTC)[reply]

I feel like I'm missing something. What are non-physical real objects? AlexTiefling (talk) 23:36, 6 May 2014 (UTC)[reply]

The number 2. God. Your soul. --Trovatore (talk) 23:36, 6 May 2014 (UTC)[reply]

I'd hesitate to call '2' an object; it's a mathematical construct used to enumerate things. And the existence of God, and of souls, are theological questions. Answers like this clearly subvert the intention of the original question. (And for what it's worth, I don't think the Banach-Tarski paradox can be applied to any of the purported non-physical real objects you propose.) AlexTiefling (talk) 23:43, 6 May 2014 (UTC)[reply]

So you're not a Platonist. OK, you're allowed to be wrong. --Trovatore (talk) 23:47, 6 May 2014 (UTC)[reply]

I'm not a Platonist, and I'm also not amused by being called wrong. I especially dislike the way you have treated the unfalsifiable assumptions of your own philosophy as so self-evidently shared that you didn't give me a straight answer when I asked for one. I'd appreciate it if you would apologise for this misleading behaviour, and for the insult. AlexTiefling (talk) 00:01, 7 May 2014 (UTC)[reply]

I did give you a straight answer. I do apologize for the belligerent tone. --Trovatore (talk) 00:03, 7 May 2014 (UTC)[reply]

Saying "Handy? They're everywhere" when I asked for an example cannot reasonably be regarded a straight answer. AlexTiefling (talk) 00:05, 7 May 2014 (UTC)[reply]

I give you credit for being able to figure out what I was getting at. --Trovatore (talk) 00:06, 7 May 2014 (UTC)[reply]

I would say that mathematics models the world, but that statements in mathematics aren't true or false the way statements about the world are. That is not to say that mathematical statements describing the world cannot be true or false: but that is an empirical matter, not an ontological one. Rather, it is to say that mathematics has no real essence that exists apart from its own vocabulary and grammar. When we say that a mathematical statement is "true", we don't mean that it is a true statement about the world, but rather that certain grammatical conditions hold that are internal to the language of mathematics. To most mathematicians, this means that a statement is "true" if it is reducible to some collection of axioms using rules for inference. However, many mathematicians also conflate this notion of "truth" with the notion of "truth of a statement about the world", and thus conclude that mathematics must exist in the world in order to be true or false, independently of any vocabulary: that mathematics is the final vocabulary of the world. This is the essentialist viewpoint of mathematics. Sławomir Biały (talk) 23:55, 6 May 2014 (UTC)[reply]

Well, no, the notion that "true" is the same as "reducible to some collection of axioms using rules" was pretty decisively refuted by Goedel. --Trovatore (talk) 00:00, 7 May 2014 (UTC)[reply]

So what does "true" mean then? Sławomir Biały (talk) 00:13, 7 May 2014 (UTC)[reply]

The objects of discourse of foundationally relevant theories, such as arithmetic or set theory, are well-specified up to a canonical isomorphism by their informal descriptions. What "true" means was explained by Tarski. "The apple is red" is true just in case the apple is red. "Every apple is red" is true just in case, for any apple, it's red. And so on. --Trovatore (talk) 00:15, 7 May 2014 (UTC)[reply]

I don't understand your reply. "Every apple is red" is an empirical statement about the world. "Peano arithmetic is consistent" is not. What does it mean to say that "Peano arithmetic is consistent" is true? Sławomir Biały (talk) 00:20, 7 May 2014 (UTC)[reply]

"Peano arithmetic is consistent" is true just in case there does not exist a formal derivation of 0=1 from Peano arithmetic. --Trovatore (talk) 00:21, 7 May 2014 (UTC)[reply]

So if no one is able to find a derivation of 0=1 in Peano arithmetic, it means that "Peano arithmetic is consistent" is true? Sławomir Biały (talk) 00:25, 7 May 2014 (UTC)[reply]

Oh no. It's evidence for it, yes; how strong the evidence is is open for dispute. But there in principle there could certainly be a derivation that we haven't found. In fact, there could be a derivation so long that it won't fit into the observable universe.

Although one has to be a bit careful about the meaning of the word "could" here. When I say there "could" be a derivation, I certainly don't mean there's a possible world in which there is a derivation. I just mean that our failure to find one does not in itself conclusively settle the issue. --Trovatore (talk) 00:33, 7 May 2014 (UTC)[reply]

But inevitably, the assertion "there exists" should raise the question of "where exists?" (which was the purpose of that bit of sophistry on my part, which I was hoping to clarify just as you replied). Someone might come along with such a derivation, in which case we could say (in an in-the-world sense), "there exists". But when a mathematician says "there exists", it is part of the mathematical language-game. Sławomir Biały (talk) 00:43, 7 May 2014 (UTC)[reply]

If it's all a language game, then there is no explanation for the observed failure to find a contradiction, beyond the mere brute fact that none of the derivations that have actually been tried give 0=1. However, if the natural numbers are real objects, then there is an explanation — namely, PA is consistent because it's true, and truth can't be inconsistent. See for example Quine–Putnam indispensability thesis. --Trovatore (talk) 00:53, 7 May 2014 (UTC)[reply]

Sorry to keep doing this, but I think I can clarify my point a little more. Goedel's incompleteness theorem relies fundamentally on the law of excluded middle. This is a rule of inference in the metalanguage used to model Peano arithmetic.

Also, I don't buy the Quine-Putnam thesis. Doesn't it basically justify any kind of essentialism? Like "the apple has redness"? At what point between Aristotle and the present day did "redness" become no longer an existing essence, replaced by the absorption spectra of phytochemicals in the skin of the apple? Sławomir Biały (talk) 01:14, 7 May 2014 (UTC)[reply]

In what way does incompleteness rely on excluded middle? The proof is perfectly intuitionistically valid. Possibly you can find some use of excluded middle in the interpretation; that wouldn't surprise me. But I'd need to know which one you're talking about.

I'm not focusing on the details of the Quine–Putnam argument. The point is that the ontological claims of realism, or at least some of them, have observable consequences that otherwise have no satisfying explanation. Now, you can say that reality doesn't owe you a satisfying explanation, and things just are the way they are, but that way seems to lie instrumentalism about everything. --Trovatore (talk) 01:29, 7 May 2014 (UTC)[reply]

Sorry, I didn't mean to be saying anything deep about Goedel's theorem or any particular interpretation of it, just that the statement "either Peano arithmetic is consistent or it is inconsistent" obviously uses excluded middle. Sławomir Biały (talk) 01:53, 7 May 2014 (UTC)[reply]

Wait-wait-wait. Where someone questions the law of excluded middle "in-world" -- does that mean that possibly there ARE no actual truth statements in the world? (In a binary sense). For example, it is easy to imagine a dream that doesn't "make sense" and basically has no true or false values in it, just impressions images and feelings. is there a chance the Universe is that way as well, and it is meaningless to ask whether any mathematical axiom is true, simply by virtue of the fact that there is no such thing as truth? (It is a social construct only, and has no meaningful physical interpretation)? 212.96.61.236 (talk) 01:55, 7 May 2014 (UTC)[reply]

It's possible, I suppose, but it's not what anyone here has been trying to say, I don't think. The point I'm making is that statements about the observable world are - at least in principle - testable. That is, they have meaningful truth-values, and the way you get from 'undecided' to 'true' or 'false' is by reference to the actual physical world - never to an axiom. If you had an axiom 'all cows are black', and you found a purple cow, you'd invalidate your system. But if you make a prediction 'all cows are black', and you find a purple cow, you can abandon the old prediction in favour of a new one. Axioms define logical systems - that is, what's true or false in the system is dictated by the axioms. But the real world is real - it is not dictated to by any theoretical construct, but is out there for you to go and check for yourself. It doesn't consist of axioms and postulates, lemmas, theorems, sets and proofs. Those things are used to build abstract models of the way some aspects of the world behave. But the world just is. Truth values of statements about the world derive from the world itself. AlexTiefling (talk) 02:57, 7 May 2014 (UTC)[reply]

That's all very well, but completely talks around whether the world itself in fact has truth-values. Further, you can never learn a truth-value by experiment, just as you cannot prove a mathematical theory by numerical experiments, so it really doesn't matter what we find or don't find. It doesn't affect whether there in fact *are* in fact fundamental arithmetic facts about the universe, for example. There might seem to be (via experiment) but this is irrelevant to whether there are. A dream might certainly seem to obey the laws of physics, but since it is just a dream that you are imagining, it does not in fact obey any physical laws precisely. Likewise the universe can certainly seem rational and consistent, without being such. Do we have any way of knowing for sure that even basic arithmetic is "actually" true in our universe? No real answers given so far. 91.120.14.30 (talk) 09:52, 7 May 2014 (UTC)[reply]

I don't personally believe that the world itself has truth values. Truth can only be assigned to statements about the world, and even the truth of those statements depends on their cultural context (think here of the many different kinds of ontological statements made in different religious contexts). It could be that there is some intrinsic truth that is out there in the world, but I don't think it is useful to talk about such essences. Sławomir Biały (talk) 11:15, 7 May 2014 (UTC)[reply]

I'm putting this below another line, so people can continue the thread above it, if it proves an unhelpful diversion. I've often thought a way into the topic is to look at information as an abstract entity. It's rather hard to argue it exists in a pure form in the real world, because of the possibility of error. I know that errors are treated as noise in information theory, but what I mean is this: You find something that you say has 3 "bits" of information. Now all of that depends on the receiver being originally uninformed, but knowing exactly how much noise is in the signal etc. This could only make sense if you could measure physical states perfectly to begin with, so that you could be perfectly uninformed about those 3 bits, and have perfect randomness for each bit. This is physically impossible, or never found in practice. Random number generation that is perfectly random with a perfectly known distribution is probably impossible, even using quantum experiments. If you have a quantum experiment, you still need to set up the apparatus perfectly, eg. so that each outcome has exactly a 50% chance, and I'm sure that doesn't happen. Now flip the argument around, and remember that if you don't believe in information, all of your knowledge of mathematics came from textbooks and information signals buzzing about in your brain, so tell me it isn't real, and I'll be moved to doubt my Platonism. Let me know if I'm getting anywhere, and better yet, let me know if there's a reference that touches on any of this, and links it to the existing philosophy of mathematics. IBE (talk) 11:32, 7 May 2014 (UTC)[reply]

It is certainly the case that there are some axioms the truth or otherwise of which cannot be meaningfully discussed with respect to the physical universe. The first example you mention, the axiom of choice, is one such, because it only has practical application in the consideration of (in a sense that I carefully leave vague!) sets with an infinite number of elements. Since any set in the universe must have a (possibly astronomically large) finite number of elements this axiom has no meaningful applicability or truth value in the real physical universe. A more general answer to your question first requires some careful consideration of your philosophical position. In my case, I am a (somewhat modified) formalist, and adopt the position that your question is in its general sense meaningless, as any answers that may be provided are not in principle testable. RomanSpa (talk) 12:12, 7 May 2014 (UTC)[reply]

We don't know how big the universe is, only the size of the observable universe. The universe may be infinite for all we know. And whether that is irrelevant or not is another unknown, perhaps the best explanations for what we can see will rest upon an assumption that the axiom of choice is untrue in the universe at large. Seems very unlikely but strange things can happen. Dmcq (talk) 14:02, 7 May 2014 (UTC)[reply]

The universe doesn't have to have only finite quantities in it. For example, the universe itself could be infinite in size (why not.) This does not change whether it is meaningful to ask any questions about the universe (even axiomatic ones) that have or may have a truth-value. Either it is meaningless to ask about whether something is "true" in a physical sense (absolutely, mathematically, in a binary, excluded-middle, axiomatic way) or it is not meaningless to ask this question. Are there any questions for which it is meaningful to talk about the existence of a truth-value in the Universe? (Regardless of whether we can learn the value of that truth-value - probably we can't. But that doesn't mean it can't exist.) 91.120.14.30 (talk) 15:06, 7 May 2014 (UTC)[reply]