As you are probably aware, there are many infinite series and infinite product expansions for pi. In each case, the sum or product can be shown to converge to a unique limit, and you can mathematically guarantee your calculation to always be within a certain range of this limit by taking a sufficiently large number of terms. So just what the bloody hell is wrong with describing pi as that unique number which is approached to arbitrary closeness by such sums and products?
For your info, the existence of \(\sqrt{2}\) isn't taken as an algebraic axiom either, it's defined as the supremum of the set of all real numbers whose squares are less than 2. You're being very picky with your choice of mathematical objects to complain about, probably looking to put together a cheap strawman argument or something.
Try English. Pi is the circumference of a circle of diameter one. Jack, your personal wants don't correspond to whether a description is meaningful or not. \(\sqrt 2\) is irrational and not transcendental. Can you give a description of \(\sqrt 2\) that you "can validate out to infinity that the number found is correct"?
Uh, no. I can do you one better though, here is another of my opinions: You should seriously consider either a) only debate on issues that you have at least a small understanding of, or..... b) get an education before arguing points that you simply do not understand in the least. If you are just trolling, you should bump up your game, as it is weak. If you are serious, you exist in a different spacetime than the rest of us. :blush: This reminds me of that Ohio government fiasco a few years back when they tried to legislate Pi to be defined as no more and no less than "3", period. They said it was to make math easier. Please Register or Log in to view the hidden image! As I recall. Pythagoras went on an extended trip to Sumer and was introduced to the similar triangle theorem. He took that and a few other choice items back to Greece with him when he returned. When the Pythagoreans laid a bisector from one corner of the unit square to the opposing corner, they got 2 congruent triangles. It was when they went to calculate the length of the hypotenuse that they encountered root 2 and the irrationals. They found phi when they compared the lengths of the sides of the top triangle in a pentagram, which is why they chose that emblem for their society. (Likely also why it has come to be associated with 'the devil' and 'magic', early historic anti - science superstitious slander.)
Hasn't the last few weeks taught you to wind your neck in a little? Nice vague criteria there. Its possible to construct algorithms which will be able to tell if any given number is or isn't pi, ie if you give me a number I can tell you with absolutely certainty whether it is or isn't pi. Surely you're aware of the plethora of formulae for pi? We appear to be going down the road of "I don't know about it so it doesn't exist" or "I don't understand it so its wrong". Didn't you learn anything from your experience with special relativity? Your 'I've found something fundamentally wrong with SR' became 'I've failed to understand SR' in pretty short order. And now we've found another area where you prefer to jump to "Everyone else is wrong" rather than "Perhaps its me....".
If you think this, then it is a fact what I am doing is over your head. False, you are stepping on the halting problem See Rogers, Recursion Theory. There is to procedure to calculate PI. This is in the linguistics and logical sense. Folks have not reached the level to understand the cardinality of reality. There is no algebraic equation or language that can determine whether not not PI has been decided. You are simply indicating your ignorance of mathematical logic. This would be the understanding of someone that cannot understand basic logic. Now, exactly how did you did you stop my twins paradox or point I developed? You are a stange person that declares victroy when you have been totally defeated. Othewise, let's go back to my thread you ran from. Now, bring everyone and let's see who wins. I smell fear. I do not think you will do it. If you decide to actually confront me, you will leave a newly trained and defeated person. Let's go.
Wow, it takes much patience. Take your universe and prove PI. You do not get the cardinality of the continuum unless you can prove its origins and existence. You will have no idea what I am saying. Even if I give you transfinite induction, you still will not be able to prove you have PI.
sq rt 2 has an algebraic description. That is describable by language. PI does not have this property. If you can prove it does, then do it. Prove an exact descroption by language so I know the number I have is the correct one. Otherwise, submit to the truth I presented.
Sure, but that's not what I asked, Jack. Can you give a description of \(\sqrt 2\) that you "can validate out to infinity that the number found is correct"? Can you describe a number then prove that its square is equal to 2? Perhaps you can fill in the right hand side of this equation: \(\sqrt 2 = ?\) I just gave you a very clear and concise description of Pi, in the language known as English: Pi is the circumference of a circle of diameter one. Or if you prefer mathematical symbology, here's a simple equation: \(pi = 2.arcsin(1)\)
So the fact I spent several pages walking you through special relativity and basic coordinate geometry, finally getting you to realise you'd made an elementary assumption which was wrong was you 'going over my head'? Yes.... Please Register or Log in to view the hidden image! Given my experience with you I'm going to have to request you be a little more specific in what you're referring to because I heavily suspect that you don't actually know any 'recursion theory' and you're simply throwing out buzzwords so it seems like you know what you're talking about and are providing some kind of retort to what I say. I am not going to assume you know any high school or beyond mathematics so if you can't be precise with what you're referring to I'm going to take it as admission you don't know it. I used to give you the vague benefit of the doubt but you've shown that's too generous an assumption for me to make. This being a case in point. 'Cardinality of reality'? You're throwing out buzzwords which are nonsense. Furthermore your post here would seem to imply you just copied and pasted something from Wiki which you didn't understand. As such unless you can start being a little clearer in what you're saying I'm going to assume you don't understand cardinality. No, you're indicating your ignorance. Firstly, 'decidable' relates to statements. A statement can either be true or false. If it is decidable then you can determine which via some sequence of logic starting at the axioms of the set theory you're working in. If its undecidable you can't tell either way. 'Pi' is not a statement, it is a number so using the label 'decidable' is inappropriate. You can easily make a statement "4 = pi" and it is decidable. As such you can construct a statement which asks where pi is equal to some expression or quantity and it will be decidable because all you need to do is basic arithmetic. For instance, you can define pi as the smallest non-zero number such that \(\sin(x) = 0\). This follows directly from the oscillatory nature of sin and its relation to circles. Now you have a way of telling if a given quantity is pi. If given a quantity y and \(\sin(y) \neq 0\) it can't be equal to pi. If \(\sin(y)=0\) and \(3 < y< 4\) then you have pi. The fact people have come up with many formulae for pi means that they have a way of proving that their complicated summation methods or integrals evaluate to pi. If they didn't they wouldn't be able to say "Here is a formula for pi". The fact this basic bit of logic has passed you by and that you misuse terminology all points to you trying to seem more knowledgable than you are. I explained to you how to do the coordinate transformations, which you were doing incorrectly. I explained that I understood the point you were trying to make and I then explained how what you perceive as a contradiction is not since no causality is violated from any observer's point of view from anywhere in the space-time. I hardly 'ran' from the thread, I patiently explained it several times. Both Pete and I explained things to you again and again and I would imagine it slowly dawned on him, as it did me, that you struggle to understand partly because you don't want to understand. If, despite having several weeks to think about it, you still fail to grasp it then clearly you aren't the whiz at vector calculus you claim to be. You claim you've taught it to undergrads but the fact you struggled with even the formula for spheres, squashed or otherwise, suggests you were fibbing about that too. Ah, you're trying to taunt me. What are you, like 6? You taunted me before, saying I couldn't do the algebra. I demonstrated I could and demonstrated you couldn't. Pete and I have said all that needs to be said. Lorentz transformations do not transform the light cone and all observers in all inertial frames will agree on causal structure. Nothing else is needed. I explained it several times, I suggest you reread it. Failing that enrol in a 1st year special relativity class at your nearest university because obviously the gaps in your understanding are too large to be filled by short exchanges of information on a forum.
\(\sqrt 2 = ?\) Yea, I know if I have this. I use the language x² - 2 = 0. If this number satisifies this equation, I know I have the correct one. Do you have such an equation for PI I can test? Write it does with precision like the one above. Once you find you cannot, you will understand as I said there is no language to represent PI. This means precisely obviously.
There is an outstanding post. Did you want to go there or just talk? It is my memory, you were unable to solve my twins parados that forced you and SR into a contradiction. So, you only walked yourself into a contradiction. It is my understanding, you are still on that path. Yea, I defined a resursive function in the twins paradox using the clock sync method. Now, if you knew what it meant, you would have already discounted SR since the recursive function produces a logically decidable outcome whereas LT calculated two different values that contradicted the clock sync. Again, you simply make baseless assumptions. Tell you what, let me see you construct the real numbers based only on BB and proceed forward that such space achieves the continium cardinality. When you figure out you cannot, then you will understand the cardinality of reality is the finite. Otherwise, stop talking and do it. Let's see now, you are using a trascendental function to decide a transcendental number. That is called vacuous implication. I notice you are an expert at this kind of argument. Good, a formula is in the language of first order logic. Write down the first order formula for PI without the use of a meta language. Yes, I came up with a point under LT such that t' = t. You do not have this ability. You did not even understant the point caused t'=t and I had to show you the math because it was beyond you. Then you write this junk above. I can prove the point has a vector magnitude < c plus v yields a vector magnitude of c. This is pretty funny since the composition of velocity equations cannot produce this. Do you need me to teach you this like I taught you the proof for that point that t'=t?
I didn't discuss the twin paradox in much, if any, detail. I spent most of my discussion with you explaining to you basic coordinate geometry for your 'intersecting spheres' set up. I explained to you several times why there is no contradiction. And I'm not the only one to have spent time explaining it to you. Clearly you have no wish to actually learn because you make it abundantly clear that you simply want to whine about special relativity because you've not managed to get your head around it. Its funny you try to come off as if you've got some amazing understanding yet in this thread and others you repeatedly demonstrate your willingness to use buzzwords in such a way as to demonstrate you don't quite know their meaning. I'm still up for wagering $1000 with you that you can't get your work published in a reputable journal. If you've got a contradiction to SR then you'll win a Nobel Prize. So what's keeping you in the pseudoscience forum? Still rambling about 'decidability' I see. That's what I mean about you using buzzwords yet don't seem to know the meaning. Rather than working through the algebra of special relativity clearly and concisely and get it to say 1=2 you get half way and then blurt out "This is decidable, SR is wrong!". It gives the distinct impression you're trying to wave the 'magic buzzword wand' and hope that by using terminology not familiar to most people you'll convince them you're right. If you could do the algebra you'd not be dancing around crying "decidability!" all the time. Now this is just wandering into meaningless crap. Construct the reals from the big bang?! This is clear proof that you don't grasp the logic and methodology of mathematics (no matter how many times you say 'decidable') because you seem to think that mathematical constructs originate from physical processes or have physical foundations. If you'd ever done the basic analysis needed to get onto stuff like set theory and decidability you'd know that the reals are constructed via logic from a series of base axioms, the simplest method of which is the Peano axioms. The nature of reality has absolutely nothing to do with any of this. For someone touting their supposed grasp of mathematical logic you don't seem to know how to even construct well defined statements. 'Cardinality' is a measure of the size of a set. 'Cardinality of reality' is thus an undefined statement since you haven't specified how you are defining 'cardinality' in this context. Are you counting the number of objects in reality? The number of points in space? The number of angels which can dance on the head of a pin? Logic is all about making clear precise statements which have rigorous proof. Its ironic you try to pretend you know some of it when you come out with such nonsense. Ah, I didn't give you enough credit that when you said 'algebraic expression' you literally meant the space of polynomials. After this shiner it didn't seem right to assume you grasped such concepts. So I'm to write down a formula which is algebraic for a number which is non-algebraic? Nice Catch 22. The fact is that it is possible to prove expressions or quantities are or are not equal to pi. How else would people prove formulae for pi? I pointed this out to you before and you didn't retort it. For instance, \(\sum_{n=1}^{\infty} \frac{1}{n^{2}} = \frac{\pi^{2}}{6}\) was proven by Euler in 6 (or more) different ways centuries ago. Or the Gamma function \(\Gamma(1/2) = \sqrt{\pi}\). Just because we can't describe pi in some ways doesn't mean we can't describe it at all. You can't trisect a general angle, doesn't mean we can't do geometry. And? You found that two different observers in a symmetric set up will have symmetric readings in their respective coordinates. What a shocker! No, hang on.... what an obvious result! Yes, that's it. Person A sees an event happen at time t=T on his watch. Person B sees a symmetric event happen at time t'=T on his watch. So what? There's no contradiction, there's no violation of causality, there's no "We're both younger than each other". You have simply failed to understand it. This is simply a flat out lie. I went through the mathematics myself, using equations of spheres. I had to repeatedly explain to you why the photon sphere hit the spheres on the x axis, a point you repeatedly denied despite it being physically and mathematically clear that you were wrong. You claim to have taught vector calculus to undergraduates yet you couldn't understand the equations of spheres, asking me where their centres are after I'd given you their formulae. I'm now wondering if you've not simply a troll because that's such a pathetically transparent lie that surely you can't honestly believe it'd stick? I went through the geometry, I explained your mistake and I provided repeated explanations for you. Besides, you've been here long enough to know that I am not lying when I say I have taught this stuff to undergrads or that I work in theoretical physics. Or have you missed threads like this or this or this or this. Throwing the "Oh its just because you can't do it" insult really doesn't stick. This seems to be your typical attitude. In a previous post you said to Pete "Otherwise, submit to the truth I presented.". You seem to have trouble with the notion you might be wrong. Obviously experience doesn't teach you much. Do you honestly believe that for 100 years every single person who has done a physics or applied mathematics degree has been unable to multiply matrices together? Because that's what would be needed for you to have managed to have done what you claim. But rather than thinking "I must be wrong, I'll check" you just jump to "Everyone else in the world who has ever done this has gotten it wrong in precisely the same way as one another and I'm the first to do it properly". If you were competent at vector calc you'd realise you're mistaken. If you were competent at group theory you'd know Lorentz transforms are a consistent set of transformations. If you could actually put your money where your mouth is you'd not be dancing around the issue. I love how you keep trotting that out. It's a very pathetic attempt to try to convince the casual reader you schooled me on something. As I said a moment ago, the result that in a symmetric system ttwo observers see particular events at such a time that Person A's clock says t=T and Person B's says t'=T is something I never denied. In fact its an obvious result, the set up is symmetric. This is NOT the same as t=t' because that would imply their clocks are always in sync, there's no time dilation. Two clocks reading the same numerical value for specific events is not the same as them being equal throughout a process. Ironically every time you trot out "t=t'" you demonstrate you are failing to understand this. You repeat t=t' as if its some amazing result. t=T and t'=T for the events in question is obvious. t=t' for all t is wrong. Congratulations, you demonstrate that you simply don't grasp the basic properties of the symmetric system being discussed and that you have no understanding of how SR works. The fact you so firmly believe that the entire population of physicists for the last 100 years have been making exactly the same algebraic mistakes in all theories which include special relativity and that you're the first to find the problem demonstrates how silly you are. Are you so desperate for validation? If you're so right why are you stuck on these forums having to challenge people to discussions in pseudoscience? If I'm so wrong and so thick why waste your time on me when you could be winning a Nobel Prize? Because you know you're just talking shit, its just you're unable to admit it.
uhhhhhh exactly what did you refute in my stuff? Please be specific so your training may continue. Nice cut. I am not here much. But I like to correct you. I used math. You use math to refute me instead of all this met language. Prove this. I see you cannot produce the cardinality of the reals fromn scratch. Just admit that and move on. If you are able to prove you can construct a cardinality in the finite material universe equal to the reals stop talking and do it. Otherwise the cardinality of reality is finite. So your brain has finally figured out what I said. :roflmao: uhhhh you get them sufficiently close in the finite sense. Why do I have to teach you? If you think my twins experiment is flawed and you proved a mistake do it. If you think my twins spheres is flawed prove it. Say it here so folks will laugh at you. You are not able to wipe my feet on this. this rest of your post is junk.
Please Register or Log in to view the hidden image! Sorry, Jack, rearranging the equation adds no more information. Try again. Fill in the right-hand side of this equation: \(\sqrt 2 = ?\) You've been given plenty of equations, Jack. Your problem is that you can't grasp that algebraic descriptions are not the only precise language. What do you think is imprecise about pi=2.arcsin(1) ?
That a contradiction existed because two observers in relative motion disagree on the location of a point. Observers in the rest frames of the two 'physical' spheres disagree on the point which is the centre of the photon sphere and yet there is no contradiction as any two observers will always agree on what region of space-time is inside a light cone and what region is outside. Causality is preserved. You claimed there was a contradiction. I explained there wasn't. Claim refuted. Seriously, do you believe what you're saying or are you simply trying to 'keep up appearances' because you've invested so much into claiming "I'm undeniably right, all of science for 100 years is wrong!" that you simply can't bear to say "Opps, turns out I was wrong". How in any way did this respond to what I said? If you believe yourself so damn sure then put your money where your mouth is. I'm sure we can find a service online which will mediate private bets between people and act as a trusted third party. If you're so sure you're right then submit your work to a journal. I'll even help you do the typesetting so that your work is evaluated on its scientific merit and not its layout. If you're right, why are you still wallowing in pseudoscience? You claim you have taught undergraduates vector calculus. Firstly this is obviously a lie since you struggle to do even basic coordinate geometry. And secondly if you were teaching undergrads it means you've been a postgrad and thus should be familiar with the peer review and journal submission process. As such if you truely believed your claims you'd have submitted your work for review. And yet you're stuck touting it in a pseudoscience forum. Says it all. Someone doing real research wouldn't post their development on a forum. Someone who believes they have something valid wouldn't avoid submitting it to a journal. The people on this forum who are doing research (myself, Ben, Prometheus, Guest and a few others) don't post our results here, we use journals. Because we welcome peer review. You appear to be hiding from it, hoping if you use big words in a pseudoscience forum you can swindle a few people. How can I refute an argument you haven't made? You simply get to t=t' and declare a contradiction by decidability. There's a huge gulf in your logic between t'=t (which I've already commented about) and declaring a contradiction. The very fact I can explain that t=T and t'=T for particular events is entirely within the expected behaviour of SR means that your logic is in need of repair. If there's a contradiction then you can get relativity to say 1=2. By two different methods you can get relativity to say an observer measures two different values for the same thing. You have two different people at different places measuring two different things and getting the same numerical value. t=T and t'=T for a given event doesn't imply t=t' throughout. As such you have failed to show a contradiction. Prove that when someone has a valid and clear justification for their claim they normally provide it? Its self evident. You obviously want to school me, you want to show that I'm wrong and yet rather than doing a clear and precise step by step derivation of SR saying 1=2 (in the manner I just outlined) you get to a result I agree with, that t=T and t'=T for the given events and then jump to "And thus its wrong". If you were able to make your logic explicit you would have. If you're resorting to desperate things like "I don't think you can do this algebra" then its obvious you're out of valid arguments and moving onto ad homs. Excellent strawman. I wasn't trying to. The cardinality of the integers and reals can be constructed from the Peano axioms and though I mentioned them I didn't go into the details. Instead I was commenting that your "Derive the reals from the big bang" is such a ridiculous statement that you have explicitly demonstrated you are not the mathematically well read person you are trying to portray. If you were really up to speed on logic and set theory you'd know that mathematics is an abstract logical construct where a series of axioms are stated and their logical implications derived. Physics has nothing to do with it. Yes physics uses mathematics as a convenient language to describe the universe because we make the assumption the universe obeys the laws of logic but we don't derive mathematics on a fundamental level from physics. Physics might motivate our development of particular areas of mathematics (string theory has motivated enormous amounts of work in differential geometry) but we do not somehow extract the axioms of mathematics from the big bang. A particular set of axioms lead through to the construction of the reals. We can then demonstrate its cardinality is larger than the integers. At no point do we need to consider reality. [edit] Its just struck me, you aren't labouring under the misconception that 'the real numbers' is named because they are supposed to be something real, existing in reality in some way over and above say 'the imaginary numbers'? Because that'd be another nail in the coffin of your supposed mathematical knowledge. It's a classic mistake by the mathematically uninitiated. [/edit] I'm seriously having trouble seeing how you can not possibly grasp this. A pure mathematician needs to know nothing about the real world. Now are you trying to be deliberately obtuse or are you just that stupid? I commented, at length, that your statements like " a cardinality in the finite material universe" are ill defined. If you really know so much logic and set theory you'd know that its essential to clearly define any terminology you use. You talk about 'the cardinality of reality' but this is meaningless. Cardinality is a measure of the number of elements in a set. 'Cardinality of the integers' is well defined because 'integers' are a well defined set. I asked you to say what you mean by 'cardinality of reality', such as number of objects or number of space-time points, and you just ignored me. The question is whether you deliberately ignored me or you're just too thick to understand my point. If you simply make up phrases which you don't then define clearly don't be surprised when people comment you aren't making any sense. Again, you seem to think you score some kind of point when people finally work out what your vague nonsense means. As usual you think the problem is with everyone else when its actually much closer to home. And we're back to just vague nonsense.... If you can't even be bothered to clearly reply to specific things I say and instead just plan to mass quote large sections of my post you don't understand don't bother. Can you retort my comment that clearly people can compare quantities to pi and determine if they are equal as proven by the fact we have a plethora of formulae for pi. If no one could ever work out if a quantity equalled pi how can anyone prove formulae for pi? Its pretty straight forward logic but perhaps its a little too advanced for you. After all, this is the third time I've had to point it out. Because you're under the misapprehension that you understand material you infact do not and harbour the belief that the entirety of physics and mathematics has been wrong for the last century because you're, supposedly, the first person to ever manage to multiply matrices together properly. Actually I said I didn't say much about it but noted that you went from being damn sure about the twin spheres to being damn sure about the train set up, each time ignoring how you'd previously been damn sure on something which turned out to be wrong. I did. We've been over this. Contining the discussion here only serves to allow you to ignore the thread where I've already explained your mistakes. By moving to a new thread you are able to 'plead ignorance' and ignore all the things I've already said to you such that a casual reader might think that I' haven't addressed your points when infact I have. Multiple times. Excellent way to avoid answering direct questions or responding to direct comments. You accused me of not being able to do basic special relativity algebra and I linked you to several threads where I discuss significantly more complex stuff. Unlike you I can put my maths where my mouth is when I need to. You also ignored my comment that for you to honestly believe you're right you need to believe that every person who has done a physics degree (since everyone does basic relativity) for the last century and every person who has done a mathematics degree (since everyone covers group theory on some level) has missed a glaring mistake in material so simply they cover it in their first term of university. And magically they have all made exactly the same mistake every time because all lecture notes, all books, all homework, all examples, all papers, all presentations, all discussions say exactly the same thing. Now any rational person who say that it is pretty silly to claim that millions of people have, independently, made precisely the same algebraic mistake for 100 years (equating to hundreds of millions of man years of work consistently being wrong in the same way) while you, having skimmed over basic calculus and geometry, have been the first person to do it properly. I don't deny that plenty of times I've read results and thought "How in the hell is that the case" but I go away and work through it myself and/or ask knowledgeable people and every time its been a problem in my knowledge. Peer review isn't perfect but what you are effectively saying is not going to happen. You aren't talking about some highly complex obscure result which only 6 people on Earth understand, you're talking about something so basic I sometimes forget there were times I didn't know how to do this stuff. Now while I don't delude myself into thinking I don't come across as arrogant on these (and other) forums but I am at least able to put my maths where my mouth is so the confidence I have in my ability is somewhat justifiable. But even I don't come close to the level of arrogance and self delusion which would be required by you if you honestly believe what you're saying. The fact you aren't submitting your work to journals left, right and centre tells me you don't really. If you believed you have an airtight argument you'd not be here arguing with me. After all, if you believe I'm so thick surely you shouldn't be wasting your time with me when a Nobel Prize awaits! That, after all, is the clincher. Someone who spends considerable time on a forum arguing with other posters about why they have the next big thing is a hypocrite. If they truly believed in themselves they'd submit to a journal.
