conway Posted May 31, 2015 Posted May 31, 2015 A short speculation on axiomatic arithmetic. Relative Mathematics.rtf
ajb Posted May 31, 2015 Posted May 31, 2015 To be polite, you should present some of your ideas here without linking to other documents.
studiot Posted May 31, 2015 Posted May 31, 2015 It is the inherent nature of all things that they are a compilation of two different and distinct things. It is axiomatic that these two things are space and value. The value of any given thing being what it is, while the space is what it occupies. First the good news to encourage you. This idea is sound, but really in the province of applied maths or physics, rather than pure maths. Now the bad news (but good at the end) This is the basis of Field Theory in Physics and has already been substantially (I hesitate to say completely) worked out. (Field theory in Mathematics is somewhat wider in ambit) In Physics a Field is a construct where you can assign a value or set of values to every point in a particular region (finite or infinite) of space/ spacetime. Much study has gone into the relationships between the values at each point. Those relationships may be linear or non-linear. So I suggest you study some of that and use your ideas to help you organise the enormous body of existing material in your own mind, before trying to reinvent the wheel. If you do come up with anything new then others will expect you to fit/add it to the existing framework. 1
Sensei Posted May 31, 2015 Posted May 31, 2015 (edited) To be polite, you should present some of your ideas here without linking to other documents. RTF file is potentially dangerous, due to error in Microsoft Word application: https://technet.microsoft.com/library/security/2953095 http://www.kaspersky.com/news?id=191 https://nakedsecurity.sophos.com/2014/03/25/microsoft-issues-patch-for-word-zero-day-booby-trapped-rtf-files-already-used-in-attacks/ Edited May 31, 2015 by Sensei 1
ajb Posted May 31, 2015 Posted May 31, 2015 In Physics a Field is a construct where you can assign a value or set of values to every point in a particular region (finite or infinite) of space/ spacetime. Classically a field in physics is 'just' a section of a fibre bundle over space-time. This sits within the differential geometry.
conway Posted May 31, 2015 Author Posted May 31, 2015 I apologize for the inconvenience of the "attachment". The idea itself is theoretical in nature, not applied at all. But it can be used for the purpose of application. That is mathematics in general. The idea goes far beyond current field theory. Zero as a reference point (Uv + Ds), where as the defined space has been further defined in some form or fashion, is totally unique to current understanding and functions as a reference point on infinite number lines. . Additionally, a relative number being represented in a fibre bundle that is (Dv + Us) is not currently part of field theory. What is a particle in the start of superposition if not a ( Dv + Ds). Also understating that there is no definition for division by zero in field theory or mathematics. Where as relative mathematics defines all equations. Also I agree and thank "ajb", this does sit within the infinitesimals of differentials. Where as philosophically so do subatomic particles. That is to say values in tiny, tiny spaces. Telling me to "organize the existing enormous body of material within my own mind" is a bit of an insult and not relative to anything yet said by any party.
studiot Posted May 31, 2015 Posted May 31, 2015 Telling me to "organize the existing enormous body of material within my own mind" is a bit of an insult and not relative to anything yet said by any party. Well I'm sorry if you take umbrage at my turn of phrase but it was intended in a friendly fashion to save you a deal of work. I have absolutely no idea what you mean by the underlined part of your sentence. So expand on your thesis to explain your terms and controlling conditions. Thus far you have made quite extensive claims, without development here. 1
conway Posted June 1, 2015 Author Posted June 1, 2015 Division by zero is now defined. Zero can be used as a reference point. Physical reality now matches mathematical theory. There is a reduction in basic arithmetic axioms. Quantum mechanics may have find some use herein as well. That is the application of relative numbers.
ajb Posted June 1, 2015 Posted June 1, 2015 (edited) Division by zero is now defined. You cannot define division by zero for a ring or field. If you try by force to define division by zero then some of the other structure must give. There is an algebraic structure known as a wheel, and it is true that you can extend the real numbers to a wheel, but again you loose some of the structure. Zero can be used as a reference point. You language here reminds me of torsors and in particular affine spaces. Edited June 1, 2015 by ajb
conway Posted June 1, 2015 Author Posted June 1, 2015 (edited) I thank you for your genuine post. I hope you will forgive my lack of knowledge in the particulars you are asking about. If then division by zero is the thing itself, that is (A/0 = A). Then nothing is lost. In relation to homogenous spaces and the like. I can say that it seems akin. In that in given thing is a package of value and space. One within the other. Both are either defined or undefined, and so on. I have little skill in advanced mathematics. I can also say that since relative mathematics is addressing basic arithmetic, then assuming it is true, it is also true of all branches of mathematics. As mathematics is clearly a branched system. Lol I would say, maybe it isn't, in any case much work would need to be done to translate the consequences to the rest of mathematics. I suggest the need of it by physics as the necessity for its laborious adoption. Maybe the "need" of it by physics is not true. I would think it is however. Edited June 1, 2015 by conway
Bignose Posted June 2, 2015 Posted June 2, 2015 If then division by zero is the thing itself, that is (A/0 = A). Then nothing is lost. I disagree. The fact that division by zero is undefined in the 'regular' mathematics is lost. Accidentally dividing by zero occurs probably more often than it should. Nevertheless, the fact that when you try to perform a calculation with division by zero, the calculator or computer returns an error is extremely useful. Useful in that it shows you that you made an egregious mistake somewhere. If dividing by zero is the same as dividing by 1, then these errors won't be caught, and future calculations based on it will be in error. Besides if A/0 = A, and we know that A/1 = A... doesn't that mean 0 = 1? This doesn't seem right to me.
conway Posted June 2, 2015 Author Posted June 2, 2015 (edited) If you would read the original post sir. It is that the space of zero is equivalent to the space of 1, where as zero has a undefined value, and were as 1 has a single defined value. Multiplication and Division being actions of value's placed into spaces Edited June 2, 2015 by conway -2
ajb Posted June 2, 2015 Posted June 2, 2015 If then division by zero is the thing itself, that is (A/0 = A). Then nothing is lost. Okay, you have some definition of this operation. That is not really a problem. The problem is that such a definition cannot be consistent with the axioms of a ring or field. In particular, you cannot use this definition for real numbers; you will be forced to lose some of the structure. Bignose has pointed out one obvious question, 1=0? This has to be false, and so some of the reasoning that lead Bignose to this must be false. That is, some of the standard axioms of real numbers must fail. In relation to homogenous spaces and the like. I can say that it seems akin. Manifolds with group actions that are transitive? In that in given thing is a package of value and space. One within the other. Both are either defined or undefined, and so on. I have little skill in advanced mathematics. I can also say that since relative mathematics is addressing basic arithmetic, then assuming it is true, it is also true of all branches of mathematics. As mathematics is clearly a branched system. Lol I would say, maybe it isn't, in any case much work would need to be done to translate the consequences to the rest of mathematics. I suggest the need of it by physics as the necessity for its laborious adoption. Maybe the "need" of it by physics is not true. I would think it is however. I don't follow this at all. Anyway, really do test your definition A/0 =A. First, what do you mean by 0? Usually this is the additive identity; a+0 = 0+a = a. Then what is a/b? Usually this means the same as multiplication by the inverse, a/b = a.b^{-1} = b^{-1}.a. I am not sure what it could mean for elements that are not commutative. Now, what axioms would you like for your + and .? Try the full ones of a field and then a ring and see what inconsistencies you get. Then try to fix this with changing or removing some of the axioms. See where you end up. (Something close to a wheel I expect)
conway Posted June 2, 2015 Author Posted June 2, 2015 I have defined specifically what I mean by zero. It is (undefined value and defined space). All axioms in mathematics in relation to addition and subtraction stay the same. Axioms for multiplication and division change. As stated. 1 = 0 is true. In space, and in value. But it is only that 1 posses a single defined value, where as zero posses a single undefined value. I can not apply this to equations with graphs and the like. I would have to do it by hand and or reprogram a computer to change the nature of zero itself. As well as multiplication and division. A is a real number. If then A divided by zero is A. and assuming the computer knows this..... no field or ring will have issue. -2
Endy0816 Posted June 3, 2015 Posted June 3, 2015 Can't use hardware to compute an answer like you could with 6/3, for example. Basically asking the computer to count how many times one number can be subtracted from another until you reach zero(or some other limit). There are other methods for dividing out there, but that is the gist of the problem. Software-wise it is actually easy to do. Check the number you are dividing by prior to attempting division and instead of dividing return the desired result. Problem is the deeper you go the more your system is going to start breaking down. 2/0 = 2 1/2 * 0 = 0 (1/2 *0)-1 = ??? 2/0 = 2 2 - 2 = 0 2/0 - 2/0 = 0 0 + 2/0 = ??? The other way to put it is that once you've changed one axiom you have to change others to compensate.
conway Posted June 3, 2015 Author Posted June 3, 2015 I do not see how any of the equations you posted fail to have an answer. 1/2 * 0 = 1/2 (1/2 * 0)-1 = (1/2)-1 0 + 2/0 = 2 -1
Bignose Posted June 3, 2015 Posted June 3, 2015 1/2 * 0 = 1/2 So... if I have $0 in my bank account, can I give him $0 and then show him this equation and show him that he really has 1/2 of my rent? And then the really great thing is that I can do it again, and he'll have all of the rent! I suppose you can redefine mathematics with your idea here, but I really fail to see anything too practical to do with it. It seems too contradictory to what I would consider common sense math. Because I want A/0 to cause an error. And I don't want 0 * A to = A.
conway Posted June 3, 2015 Author Posted June 3, 2015 There are specific reasons why (A * 0 = A ) while the inverse (0 * A = 0).
ajb Posted June 3, 2015 Posted June 3, 2015 All axioms in mathematics in relation to addition and subtraction stay the same. Axioms for multiplication and division change. Okay, you also have to change some distributivity laws, that is the natural compatibility between addition and multiplication. Again, I suggest you look at this carefully and see what axioms are compatible with your definition. There are specific reasons why (A * 0 = A ) while the inverse (0 * A = 0). So we also lose commutativity. This is not really a problem, you are now dealing with a non-commutative structure. I suppose you can redefine mathematics with your idea here, but I really fail to see anything too practical to do with it. He has the set of real numbers and gives them some binary operations +, * and then makes a definition of a/0 =a. What he now has to do is work out the full structure from this. is /a such that a/a =1? What about distributivity etc? Either he will come to something like a wheel, or the set of compatible axioms will be very small giving a trivial structure. One is free to define such algebraic systems, but I agree that they may be of limited use and not very enlightening. But lets see...
studiot Posted June 3, 2015 Posted June 3, 2015 (edited) I had always thought that the axiom set had to be mutually consistent. To take an extreme example there is nothing to stop me stating two axioms 5=7 and 7=11 and then stating, as you have done All axioms in mathematics in relation to addition and subtraction stay the same Incidentally what do you understand by the axioms in mathematics in relation to addition and subtraction? How many are there? Please state these for my benefit. Edited June 3, 2015 by studiot
conway Posted June 4, 2015 Author Posted June 4, 2015 (edited) The commutative property still exists. Where the number is in the equation does not matter. What does matter is what number is labeled as value, and what number is labeled as space. For example. ( 2(v) * 1(s) = 2) (2(s) * 1(v) = 2) (1(v) * 2(s) = 2) (1(s) * 2(v) = 2) a/a is still a. if then a ( a as a value) is / by (a as a space) the a/a=a. The only equations that change at all are * and / by zero. Therefore the only axioms that change are zero, and * and /. Of course I have added axioms about the nature of numbers themselves. Where as this is very enlightening. Albeit one mans enlightenment, may not be another's. Studiot I do not know the number of axioms in regards to addition or subtraction. I assume you know, therefore the need of it for your benefit is purely flame. What does it matter the number. If you know of one that contradicts me let me know. I will try to be patient and learn from you. I tell you this as a curtsey, what's to stop me from looking up the number of axioms and posting them all, authors included. My ego is not so bad as that. What I do know is the definition of axiom, it is a self evident truth. I do not contest any axiom of + and - . I have stated nothing to my knowledge that would affect + and -. If I have please let me know. Edited June 4, 2015 by conway
Mordred Posted June 4, 2015 Posted June 4, 2015 Studiot likes to learn just as anyone else does. Ajb is one of our best mathematicians. 3
ajb Posted June 4, 2015 Posted June 4, 2015 2(v) * 2(s) = 2, but is that 2(v) or 2(s) ? I take it that the extra label comes from considering two copies of the real line? Then what about 2(v) + 2(v) etc? In particular do you have something like 2*(2+2) = 2*2 + 2*2 ? (v and s go where?) (As a remark, it is usually identities like the above that get messed up when dividing by zero) Studiot likes to learn just as anyone else does. Ajb is one of our best mathematicians. How could I not give you +1 for that
conway Posted June 4, 2015 Author Posted June 4, 2015 (edited) My apologies for not being more specific. S and V are not variables in any sense. S is space. V is value. All numbers used in + and - , have all V's and S' combined in the numbers themselves. This is to say one can not add 1v and 1v. both v's require an s to be added or subtracted. If I may quote my original paper again. "It is the case in multiplication and division, that neither number given is an actual number. Not in the fashion that both symbols contains both value and space. It is that one symbol is representing a value, and that one symbol is representing a space." This aside, following the rules of the order of operations, no eqution then is altered from its current answer. Even in regards to * and / by zero. As I have shown in the post regarding the commutative prorperty. It is only that with RM, there is an additional answer to any eqution regarding * and / by zero. Relativity. To be more spcefic (a(v) * 0(s) = a) (a(s) * 0(v) = 0) (0(v) * a(s) = 0) (0(s) * a(v) = a) Edited June 4, 2015 by conway
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