The discussion centers on a new mathematical framework proposed by Lama, which aims to redefine fundamental concepts such as tautology, sets, and the real line. Key axioms include the independence of points and segments, the duality of elements, and the completeness of collections. The framework emphasizes the relationship between absolute and relative properties, suggesting that the real line is a fractal system with invariant cardinality across various scales. Critics question the validity of the proposed definitions and their equivalence to established mathematical constructs like Dedekind cuts and Cauchy sequences. The conversation highlights a clash between traditional mathematics and Lama's innovative approach, which seeks to address complexities overlooked by conventional methods.
#401
Lama
466
0
Hurkyl said:
I merely mean that you can prove they exist, and have certain properties.
Thank you dear Hurkyl for your reply, but I am afraid that you did not see the rest of my axiomatic system and what I can do with it, so for a better picture of it please read all of it until the end of it, at lease twice before you reply about my axiomatic system, that can be found here:
Nowhere have you shown any understanding of the actual meaning of tautology in the sense used by logicians.
I know perfectly well what a tautology is; you don't seem to understand it.
#404
Lama
466
0
arildno said:
Nowhere have you shown any understanding of the actual meaning of tautology in the sense used by logicians.
I know perfectly well what a tautology is; you don't seem to understand it.
Please read at least 3 times my questions about where the premise and conclusion is on your "tautology", read at least 5 times what wikipedia actually says about tautology, post a reponse explaining what the premise and conclusion is in your "tautology", proofread it 2 times, then present it for us to read.
Only then can anyone make meaningful remarks on your "work." THat is, you have to actually respond to what people ask, rather than give reading assignments or reposting the same stuff over and over.
Lama:
As yet, you have not convinced anyone that your first axiom is not total gibberish.
Instead of wasting virtual space with tons of cooked-up definitions from your own fancy, please focus only on elucidation of your first axiom.
If you think it cannot be "understood" out of context with the rest, then it is not an axiom at all, but just gibberish.
#407
oreopoj
7
1
How do complex numbers fit into your new theory? I see only talk of real numbers here. I have read through all 28 pages of this thread and I have had time to look at all of your files on your website. Please explain i=sqrt(-1) in terms of your new notation. Please do not refer me to one of your pdf files, as I know that there is no reference to the complex numbers.
#408
Lama
466
0
Dear oreopoj,
When we have {},{.}_AND_{._.},{__} then we can construct any information form that we like.
Complex numbers are based on R part + Z part, so all they are is no more then a technical extension that help us to solve equations where numbers like √-1 are involved.
My research about the language of Mathematics is exactly in the opposite direction, which means:
Instead of searching unsolved problems within the standard framework (based on 'How' questions) I went back to the most fundamental concepts of this beautiful language and used most of the time 'Why' questions, that helped me to develop my non-standard framework.
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#409
Lama
466
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ex-xian said:
Please read at least 3 times my questions about where the premise and conclusion is on your "tautology", read at least 5 times what wikipedia actually says about tautology, post a reponse explaining what the premise and conclusion is in your "tautology", proofread it 2 times, then present it for us to read.
Only then can anyone make meaningful remarks on your "work." THat is, you have to actually respond to what people ask, rather than give reading assignments or reposting the same stuff over and over.
All you need is to read and (I hope) understand my work.
Good luck.
#410
Lama
466
0
arildno said:
Lama:
As yet, you have not convinced anyone that your first axiom is not total gibberish.
Instead of wasting virtual space with tons of cooked-up definitions from your own fancy, please focus only on elucidation of your first axiom.
If you think it cannot be "understood" out of context with the rest, then it is not an axiom at all, but just gibberish.
Why do you need more people around you to be sure that my work is (by you) just gibberish?
At least respect yourself and say it clear: "I, arildno, think that your work is nothing but a gibberish!".
Believe me it will look much better then using the "not convinced anyone" style.
Because my framework is a paradigm-shift in the Langauge of Mathematics,
you have no choice but to understand it by the most fine intuition/reasoning
interactions abilities that you have.
Naturally well educated mathematicians are the first persons that have the biggest problems to understand a paradigm-shift because of more then a one reason, for example:
1) The foundation of their own work can be changed for better but also for worse, but in both cases they have to reexamine their work according the paradigm-shift, which is a very unpleasant situation, for persons who afraid from deep changes.
2) They are already trained by another 'school of thought', therefore it is hard for them to see fundamental things from a different point of view.
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#411
hello3719
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0
Lama said:
Naturally well educated mathematicians are the first persons that have the biggest problems to understand a paradigm-shift because of more then a one reason, for example:
1) The foundation of their own work can be changed for better but also for worse, but in both cases they have to reexamine their work according the paradigm-shift, which is a very unpleasant situation, for persons who afraid from deep changes.
2) They are already trained by another 'school of thought', therefore it is hard for them to see fundamental things from a different point of view.
Mathematicians are the first people that would be happy for some deep changes. You just don't explain your definitions well. Every new term you introduce in your axioms should be explained using the most fundamental concepts (ex. < ,> = ...)
#412
Lama
466
0
Dear hello3719,
I am glad to know that you have no problems with fundamental changes.
(ex. < ,> = ...) are simply and clealy explaind in:
Lama, it isn't fair for you to criticize anyone for not understanding what you're doing, nor is it fair to say that the fact that virtually everyone has panned what you've done.
You've failed to explain what the premise(s) and the conclusion(s) are in your tautology. You've invoked something called "number," which you've not defined, in order to explain your first two axioms. This isn't the way math, any math, is done. You can't just pull concepts out of the air.
If you want to be taken seriously, you need to present your definitions, then your axioms. Your axioms need to be understandable and explainable only by appealing to the definitions or previous axioms. When you have to pull out mathematical words that you haven't defined in order to explain your axioms, you're not being at all productive.
I'm trying my best to understand you and to give you the benifit of the doubt. Here's what we have so far:
Set: A set is a finite or infinite collection of objects in which order has no significance, and multiplicity is also ignored.
1) Since your building up the numbers from scratch you need to give definitions for "finite" and "infinite."
2) Also, do you have commonplace definitions in mind for "order" and "multiplicity?" That is, is {x, x, y} the same set as {y, x}?
Tautology: x implies x.
1) Since you refernced wikipedia, then any mention of a tautology necessarily means you have a proposition in mind.
2) That is, "x implies" is the same as "if x, then x."
A definition for a point:
A singleton set p that can be defined only by tautology ('='), where p has no internal parts.
1) What is a singleton set? Since you're building the numbers up from scratch, you either have to admit to the standard way of constructing the counting numbers or explain "singleton set" w/o referencing numbers.
2) What are internal parts? Remember, you have to define this only with previously defined terms.
3) You say that a point can be defined only by tautology, which you equate with the operator "=". When one writes "x = x" is this the same as saying "if x, then x"?
See if you can take care of these questions, please?
#414
Lama
466
0
Dear ex-xian,
There comes a time where you have no choice but to use your own finest internal intuition/reasoning interactions abilities to understand something.
From this moment you are in your own and nobody, included me, can help you to understand this thing.
In short, I did my best to represent my ideas in the simplest and clearest ways that I can.
From this point I can only answer to questions that shows that the person who asks them already made his paradigm-shift in his mind and leaped by using his own abilities to my new framework.
for example, you wrote:
ex-xian said:
1) Since your building up the numbers from scratch you need to give definitions for "finite" and "infinite."
All you need is to understand how the axiom of completeness
(The axiom of completeness:
A collection is complete if an only if both lowest and highest bounds are included in it and it has a finite quantity of scale levels.)
is related to the rest of my work, because one of the important things in my system is, that you have to understand the whole of it in order to understand a part of it, where what I call 'the whole of it' is the simplest level of understanding, which stands in the basis of my work, and cannot fully shared with others (by using written axioms) that are not already share within them this common source of the simplest state, which is beyond any definition.
And this is exactly the deep meaning of the words ‘paradigm-shift’.
There is always an unclosed gap that has to be closed by an intimate-private journey that a person does in the deep and fine silence of his both heart and mind.
Good Luck.
Lama
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#415
moshek
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oreopoj said:
How do complex numbers fit into your new theory? I see only talk of real numbers here. I have read through all 28 pages of this thread and I have had time to look at all of your files on your website. Please explain i=sqrt(-1) in terms of your new notation. Please do not refer me to one of your pdf files, as I know that there is no reference to the complex numbers.
The very well known mathematision Michael Atiya, gave a very interesting lecture about a year ago at the conference about "The unity of mathematics". He said that we are waiting now to a new-Newton that will break the enigma of mathematics and Physics.
He gave the example of I ( root of -1) to how new concept bring high resistant when they are appearing.
Well, I hope that you can see the analogy to this complitly
new theory about mathematics as one whole - M(i).
Moshek
#416
oreopoj
7
1
What in the world has this forum become?! Lama, what is the point of repeatedly posting links to your pdf files and saying the same thing over and over again to show the community your brilliant new way of doing math if you say:
"nobody, includ[ing] me, can help you to understand this thing".
Asking people to abandon their current way of thinking about math is not reasonable. If you want to share the wonderfulness of what you've found, you must prove that it works to others in a way that is comprehensible to people. Why else would you have tried to share these ideas, producing 28 pages of a single thread. This is what forums and discussion are for. That is why you are here. If you just want to advertise your new ideas without accepting criticism, then please just use your website.
Lama, please try to understand that the reason some people have demanded that you demonstrate some kind of knowledge of the current system of math is because there hundreds of years of theoretical and applied research to back up its use. Math is taught in schools and university the way it is because it works and has been put to use in so many real world applications; ie, computers, physics, economics, etc. That's not to say that math is a static thing, certainly not. It, as you have said, evolves through research and new ideas by bright minds, but the great thing is, each time a new breakthrough in mathematics is made, it ENCOMPASSES and is CONSISTENT WITH all the old ideas and methodologies that came before.
I have read through all of the previous messages in this thread and have gone through every single pdf file on your website. I am not convinced that you have discovered a new branch of mathematics in which the old mathematics is merely the "shadow" of yours.
Why not attempt to re-express your ideas in a new way that is compatible with the way it is done normally. I think you should start by defining every single unique term in your axioms in your own words and not simply use the definitions found on Wolfram's website. If your definitions and following axioms are inconsistent in some way as some people have pointed out so far, go back rework them further until they are consistent.
And don't be so defensive when someone points out a mistake in your reasoning! It's annoying. Please do not say, "I have clearly shown [such and such] on this file at my website." You sound like a broken record when do this. Everyone has probably looked at the link you're referring to already. We need a new and clearer explanation of what you're trying to say in order to understand you.
Lama, you can brand me as a "bodyguard of math". Yes, there are bodyguards of math, and for a good reason. Research in science and math is a vigorous process involving the constant judgment of one's submitted ideas by the other peers of the research community. A new idea cannot be accepted as truth without first being questioned and scrutinized. This is the power of the research process. Imagine how crazy and disorganized science and math would be if every single idea was immediately incorporated into the current body of knowledge. Your ideas should be able to predict and explain the behavior of a certain model or system, and, if this is successful, then those same ideas should be able to do the same on a different system or model. If the new idea fails in the respect just once, then it must be rejected in favor of the old ideas that can already correctly explain some phenomenon. At that point, it's time to go back to the drawing board. I can tell you from experience that less than 5% of any research that is done in anyone research group or laboratory in an academic setting ever goes somewhere significant. Don't be surprised if someone shows that you are wrong. A good mathematician or scientist is able to admit when they are wrong.
And back to my original beef with what has been going on here in this thread. I’ll quote you again:
"nobody, includ[ing] me, can help you to understand this thing".
Let us suppose for the moment that you actually have come up with some new way of doing math that will lead to tremendous advancements in the field and in science. What a shame it would be if that idea were to die with you some day then. You must be able to convince people who study math that you are correct in order for that advancement to take place. If you cannot make us understand, then it is all for nothing and you should not be wasting our and your time on this forum. If you cannot do this, go and learn how to; go and share your ideas with a professor in mathematics that is near you. Maybe he/she can help you express your ideas in a way that most people can understand. Believe me, I think it’s good that people like you want to explore and discover new things in mathematics. Mathematics is a beautiful thing as you have said before and is a very noble pursuit; but it is not one to be taken alone. If you want us to embrace your ideas, you have to be willing to embrace our thoughts and ideas as well.
#417
sparkster
153
0
Well, I guess that settles it. You're not seriously trying to do math after all. Your trying to do mysticism and smear math words over it.
If you want to be a mystic, that's fine and good. But at least be honest about it.
All you need is to understand how the axiom of completeness
(The axiom of completeness:
A collection is complete if an only if both lowest and highest bounds are included in it and it has a finite quantity of scale levels.)
Then you need to give definitions for lowest bound, highest bound, finite, and scale. And you have to define them all w/o using numbers, since you're building everything from scratch. Either that or concede the standard way of defining finite sets--which would concede the standard way of building the natural numbers.
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#418
oreopoj
7
1
I agree, sometimes it sounds more like mysticism to me.
#419
sparkster
153
0
oreopoj said:
I agree, sometimes it sounds more like mysticism to me.
And I actually don't have a problem with mysticism per se, but I wish more of them subscribed to Wittgenstein's philosophy of mysticism.
"Of that which we cannot speak, we must pass over in silence."
#420
oreopoj
7
1
I with you on that one. I would prefer mysticism to religion any day of the week. But look at how many times people in history have looked silly for trying to mix mysticism/religion with science; for example, Kepler tried to explain the orbits of the planets in terms of known 3D polyhedrons of the time because they were "perfect" or "divine". That's garbage. It was only after he let go of the idea of "perfect" shapes that considered using elliptical orbits and sucessfully formulated his laws.
I with you on that one. I would prefer mysticism to religion any day of the week. But look at how many times people in history have looked silly for trying to mix mysticism/religion with science; for example, Kepler tried to explain the orbits of the planets in terms of known 3D polyhedrons of the time because they were "perfect" or "divine". That's garbage. It was only after he let go of the idea of "perfect" shapes that considered using elliptical orbits and sucessfully formulated his laws.
Perhaps the most blatant example of misplaced mysticism may be found in the philosopher Hegel.
He became so enthused by the idea that he'd found an overarching system(which "integrated" religion/morality/society/science/nature) that his "philosophy of nature" is just about the worst crank writing imaginable.
However, his earlier, less connected works (in which he didn't suffer from his unifying obsession to the same degree as later on) does contain bits an pieces of rather poignant and subtle social analysis.
If he had had less than the crank nature in him, he might well have developed a truly interesting social theory (placing him in the company of Max Weber, and others)
#422
sparkster
153
0
arildno said:
Perhaps the most blatant example of misplaced mysticism may be found in the philosopher Hegel.
He became so enthused by the idea that he'd found an overarching system(which "integrated" religion/morality/society/science/nature) that his "philosophy of nature" is just about the worst crank writing imaginable.
However, his earlier, less connected works (in which he didn't suffer from his unifying obsession to the same degree as later on) does contain bits an pieces of rather poignant and subtle social analysis.
If he had had less than the crank nature in him, he might well have developed a truly interesting social theory (placing him in the company of Max Weber, and others)
Yes, Hegel is a good example. My most favorite of his statements is when he claimed that the scientists were wasting their time searching for heavenly bodies--if they would just study philosophy they would realize that since 7 is the number of perfection, there can only be 7 and no more.
Incidentally, this brings me back to Lama/Dorian/Organic/etc. In E.T. Bells' Men of Mathematics, he is discussing the immortal Gauss, and what Gauss had to say about people like Hegel. He states that "those who wish to peck away at the foundations of mathematics would do well to sharpen their dull beaks on some hard mathematics first."
So instead of being able to give a tautology you've now had to remove it from your definition. Does this mean you're going to remove all your pointless posts where you refused to give a tautology and just reposted the same silly definition.
Now, if you could just remove direction from your new definition, which you've not defined then you're getting somewhere. Note, it appears that you're presuming the real numbers exist, but what are they in your system? You've just presumed they exist when they don't necessarily since they are just a construct in our system that you reject. Also note that you've admitted something we've been telling you for a while that you've ignored: that the organic numbers which you state are BASED on N, cannot therefore be more basic than them since you;ve used N in its definition.
#426
Lama
466
0
So instead of being able to give a tautology you've now had to remove it from your definition
Thank you very much for your open-hearted replies.
1) I do not start every single concept from scratch, so words (for example) like, finite, and scale have their standard meaning.
2) Lowest bound or Highest bound are simply the edge values of a non-empty ordered set.
3) Scale factor is determined by the ratio of any R member to the entire R members.
As I wrote to ex-xian, I cannot do the whole work alone to convince someone about my point of view, if a person choose not to move from his spot, where things looks fundamentally different.
In short, you cannot stand in some place and say: “Please convince me to move to your place by describing what do you see from your place”.
I can do my best to describe (define) what I see, but the best advice that I can say is:
Please come to my place and see it by yourself, and only then we can argue about our different interpretation about what we see from this place.
Again, no one of you did this simple step, which is: To come to my place and see things by using his own eyes.
And this is the reason why I said that I have found that the most persons here acting like full_time_job bodyguards.
It is very important to check any new idea before we air our view about it, but it cannot be done if we are full_time_job bodyguards, which means, we do nothing to really see something by our own eyes, and instead we want that the mountain will come to us, instead of us to come and at least first see the mountain by using our own eyes.
And when we see the mountain by using our own eyes, then and only then we are in a position to decide if we want to clime on it, or not.
You have to understand that if you do not let yourself to see things by using your own eyes, we cannot move further in our dialog.
In short, I need your active participation on order to develop the dialog between us.
I hope that I explained my point of view about the dialog between you and me.
1) is patently false since you reject the proper definition of cardinality for example, so at best you occasionally use the proper definition of words, however, none of the words you've used in this item have a formal standard mathematical meaning that we can apply here with any certainty that doesn't require you to accept standard mathematics first. And even then "scale" is not well-understood.
2) Lowest and Highest imply an ordering, you've not proven any such exists in your system: what is the highest and lowest bound of the circle? See, you need to actually have the Real numbers as we know them already extant, and that is a contradictory position for you to adopt.
3) makes no sense (how do you define the ratio of an element of that set to the set? and that is a set yo'uve not even bothered to define)
and since you now say that you've not removed the requirement to be defined by tautology you muist still have some proposition in mind, so for the, what, 10th time what is that proposition?
#428
Lama
466
0
what is the highest and lowest bound of the circle.
x1 and y1 are R members.
x1 and x2 are its lowest and highest bounds where x1 < x2, and its length is |x2-x1|*xs, where xs is any R member.
Matt why do you ignore my request for help that can be seen in my previous post?
you could also clarify what you mean by "indivisible" since I can "divide" the interval [0,2] into [0,1] and (1,2].
As it is your grasp of set theory is looking even shakier, and I bet I can make all your statements apply to Q or C, or P(3) if I felt like it, since they are so ill-defined. Of course, you are still presuming the existence in your model of something that you've not shown to exist. Your definitions are becoming increasingly circular: your set of reals is the set of points and intervals somehow, presumably, yet you can't define a point or interval without using the properties of your reals (order and equality, it is for instance not part of your axioms that this set even has an ordering, yet you use it).
Where did you set of reals obtain its ordering? Why is length of an interval not well defined? What is R in your system? What is the tautology that defines x1 and y1? were did the abs value operator come from?
#431
Lama
466
0
Matt Grime said:
you could also clarify what you mean by "indivisible" since I can "divide" the interval [0,2] into [0,1] and (1,2].
I clearly and simply clarify what I mean by "indivisible".
In my system [0,1] and [1,2] are two {.}_AND_{._.} indivisible elements where [0,2] is also {.}_AND_{._.} indivisible element.
Also I have a different interpretation to (r1,r2].
Again you cannot see (understand) my framework in terms of your framework,
and if you can't grasp this, then we cannot communicate.
It is not that I cannot understand your terms, but that you haven't defined them. incidentally the correct mathematical term is connected, not indivisible.
You have still not given this tautology that you say exists.
A note for everyone: shall we refuse to comment or in anyway communicate in this thread until that simple request has been met?
Let us absolutely make clear what is required:
something is alleged to be defined (only) by a tautology. That thing is a set. A tautology is a proposition. Please give the (an) example of this (or any other) tautology that defines this (or any other) set.
Until you can do that you have no need to post anything else, Doron.
#433
Lama
466
0
Matt, please read this again and pay attention to * and * propositions.
Tautology:
x implies x (An example: suppose Paul is not lying. Whoever is not lying, is telling the truth Therefore, Paul is telling the truth) http://en.wikipedia.org/wiki/Tautology.
(tautology is also known as the opposite of a contradiction).
(EDIT: instead of the above definition, I change Tautology to: The identity of a thing to itself.
It means that in my framework we do not need 'if, then' to define a Tautology)
Set:
A set is a finite or infinite collection of objects in which order has no significance, and multiplicity is also ignored.
Multiset:
A set-like object in which order is ignored, but multiplicity is explicitly significant.
Singleton set:
A set having exactly one element a. A singleton set is denoted by {a} and is the simplest example of a nonempty set.
A definition for a point:
A singleton set p that can be defined only by tautology* ('='), where p has no internal parts.
A definition for an interval (segment):
A singleton set s that can be defined by tautology* ('=') and ('<' or '>'), where s has no internal parts.
(Sign '<' means that we look at the segment from left to the right.
Sign '>' means that we look at the segment from right to the left.
When both '<' , '>' are used then we have a directionless segment.)
By the definition of a segment we get {._.}, which is the indivisible singleton set that exists between any two {.}.
Now we have the minimal building-blocks that allow us to define the standard R members.
(edit:
*A statement for a point:
A point is an indivisible finite content of a non-empty set that has no directions.
*A statement for a segment:
A segment is an indivisible finite content of a non-empty set that also has directions.)
The axiom of independency: p and s cannot be defined by each other.
By the above axiom {.} and {._.} are independed building blocks.
Ok Matt since you say that you understand my first two axioms, then please help me and write their logical propositions (statmants), because I do not know what do you mean when you say 'propositions' in this case.
A proposition is a statement of the form If X then Y where X and Y are some mathematical statements to which we can assign truth values.
Example:
If an integer, n, is divisible by 4, then n is even.
We start from the antecedent (n is divisible by 4) and come to the conclusion (n is even).
This proposition is true.
A proposition here would be:
If P is a point then p is defined only by a tautology.
Note, that that in itself is NOT the tautology that defines p, if it were you'd have a circular argument, which is not the tautology you require since it does not tell us what p is (ie offer any indication p even exists). You cannot rely on the definition of p to define p like that, since it does not prove that any such "p" exists. You must therefore offer another model of some such p, and by doing so you will contradict the minimality that it is only defined by (this alleged) tautology as you will be defining it in some other way. Sorry, but as we've been telling you your definition is not consistent.
In short, you cannot define things by tautologies like this, and you cannot prove that any such p exists since it contradicts its own definition. That is not to say it cannot exist, but that we cannot know it exists.
It makes someone wonder what you thought you were doing? An ungenerous person might think you'd been displaying ignorance of things you claim to understand. After all, it's you who's been posting the wikipedia definitions, did you not understand them?
And I didn't say I understood your two axioms, I said I was happy to accept them, and asked you to provide examples and models of things satisfying those axioms. You didn't, and, it appears, can't.
An example of a tautology (in mathematics)
If A then (if A then A)
which is equivalen to (not(A)) or ((not(A)) or A), which is equivalent to (not(A))orA, which is always true irrespective of the truth of A.
note that this of course requires you to have boolean logic and excluded middle and so, some things you reject, so actually you are once again possibly misusing a mathematical term, since your tautologies, may not be our tautologies, even after you've made a propositional form out of it. More contradictions again.
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#437
Lama
466
0
The best I can do is:
If a content of a set is a singleton and a urelement and has no directions, then it is a point.
If a content of a set is a singleton and a urelement and also has directions, then it is a segment.
I understans tautology in a different way than you.
For me tautology is not if,then proposition but a self avident existence of a thing to itself for example:
{}={}, {.}={.}, {._.}={._.}, {__}={__}
This is the reason why I call '=' 'only by tautology'.
And you have removed the requirement of some tautology then. Which is good, because that was wrong.
Now, you just need to explain what a direction is.
It is still not clear of you've "defined R" and now define points and intervals in terms of R, or if you are defining points and intervals and then will define what R is.
Which of those two orders is it?
It appears, since you are talking about direction, and equality and ordering (<,> etc) that R must exist in your alleged system, so what is R?
So, your next requirement is to tell us all what YOUR reals are. Until you do that we can do nothing.
#440
Lama
466
0
Ok, let us write is again:
If a content of a set is a singleton and a urelement and has no directions, then it is a point.
If a content of a set is a singleton and a urelement and also has directions, then it is a segment.
(more detailed explanation of the first two definitions:
Let us examine these first two definitions by using the symmetry concept:
1) {.} content is the most symmetrical (the most "tight" on itself) content of a non-empty set.
It means that the direction concept does not exist yet and '.' can be defined only by '=' (tautology), which is the identity of '.' to itself.
2) {._.} content is the first content that "breaks" the most "tight" symmetry of {.} content, and now in addition to '=' by tautology (which is the identity of '._.' to itself) we have for the first time an existing direction '<' left-right, '>' right-left and also '<>' no-direction, which is different from the most "tight" non-empty element '.'
In short, by these two first definitions we get the different non-empty and indivisible contents '.'(a point) or '_'(a segment) .
For me tautology is not if,then proposition but a self avident existence of a thing to itself for example:
a tautology is not required to be a proposition, merely a statement that is true irrespective of the truth values of the elements in it.
A=>A is a tautology
A=>B is not
A<=>A is not
(A=>(B=>C) => (A=>C) is i think a tautology.
it is not a self evident truth.
so stop saying tautology and pointing to the mathematical definition as if it is the one you are using. everything is in some sense equivalent to itself (tautologous in your new fangled interpretation) so it is impossible to say what is only defined by a tautology, since we cannot define it in any other way. that attempt at a definition is illogical.
I almost give up: what you've shown is that the definition of something implies itself. that is always true and not that useful directly. of course, your use of the word definition is different from everyone elses.
Now, what is R? in you system of course.
#443
Lama
466
0
Since you missed it then here it is again:
If a content of a set is a singleton and a urelement and has no directions, then it is a point.
If a content of a set is a singleton and a urelement and also has directions, then it is a segment.
(more detailed explanation of the first two definitions:
Let us examine these first two definitions by using the symmetry concept:
1) {.} content is the most symmetrical (the most "tight" on itself) content of a non-empty set.
It means that the direction concept does not exist yet and '.' can be defined only by '=' (tautology), which is the identity of '.' to itself.
2) {._.} content is the first content that "breaks" the most "tight" symmetry of {.} content, and now in addition to '=' by tautology (which is the identity of '._.' to itself) we have for the first time an existing direction '<' left-right, '>' right-left and also '<>' no-direction, which is different from the most "tight" non-empty element '.'
In short, by these two first definitions we get the different non-empty and indivisible contents '.'(a point) or '_'(a segment) .
(I have changed Tautology to: The identity of a thing to itself)
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#444
Lama
466
0
Matt, there are some points that we have to check before we examine R by my framework:
1) In an included-middle reasoning contradiction is not used because any two opposites are simultaneously preventing/defining their middle domain.
2) A thing is defined by its identity to itself, and we do not need the 'if, then' proposition (as we do in an excluded-middle reasoning) in order to define the existence of something.
3) By this reasoning we distinguish between a true statement and a tautology, which by included-middle reasoning is simpler and stronger then any existence that must an 'if, then' to exist.
4) The included-middle interpretation of a Tautology is circular only if we look at it from an excluded-middle reasoning. But then we must realize that we see and understand things which are not from an included-middle point of view.
If you stay in standard point of view, then you cannot understand my new framework (and in this sentence I used the standard 'if, then' reasoning)
Included-middle reasoning is the logic of mutual communication between opposites (and in this sentence I used the self identity of a thing to itself)
In short, I gave just now a simple demonstration that I understand very well the standard point of view.
Can you do something which demonstrates that you are able to write something from an included-middle logical reasoning?
If you cannot do it at this stage, then your mathematical skills will not help you to understand my system (and here I used again the 'if, then' reasoning).
An included-middle reasoning can be understood only by an included-middle reasoning (and here I used an included-middle reasoning).
Some examples:
By my system 4 not= 2+2.
By my system 4=4 and 2+2=2+2.
Now can say: But you can do nothing with these two trivial and circular equations, right?
My answer is: In included-middle reasoning we can do very interesting things because 4 not= 2+2, for example:
For example, let us represent the variations of cardinals(*) 2,3,4:
Let Redundancy be more then one copy of the same value can be found.
Let Uncertainty be more than one unique value can be found.
Let XOR be #
Let a=0,b=1,c=2,d=3 then we get:
Code:
b b
# #
{a, a, {a, b}
. . . .
| | | |
|__|_ |__|
| |
{x,x} {{x},x}
c c c
# # #
b b b b b
# # # # #
{a, a, a,} {a, a, c} {a, b, b}
. . . . . . . . .
| | | | | | | | |
| | | |__|_ | |__|_ |
| | | | | | |
|__|__|_ |_____| |_____|
| | |
| | |
{{x,x,x} {{x,x},x} {{x},x},x}
[B]Uncertainty[/B]
<-[B]Redundancy[/B]->^
d d d d |
# # # # |
c c c c |
# # # # |
b b b b |
# # # # |
{a, a, a, a} V {a, b, c, d}
. . . . . . . .
| | | | | | | |
| | | | |__| | |
| | | | | | | <--(Standard Math language uses only
| | | | |_____| | this no-redundancy_
| | | | | | no-uncertainty_symmetry)
|__|__|__|_ |________|
| |
={x,x,x,x} ={{{{x},x},x},x}
============>>>
[B]Uncertainty[/B]
<-[B]Redundancy[/B]->^
d d d d | d d d d
# # # # | # # # #
c c c c | c c c c
# # # # | # # # #
b b b b | b b b b b b b b b b
# # # # | # # # # # # # # # #
{a, a, a, a} V {a, a, a, a} {a, b, a, a} {a, a, a, a}
. . . . . . . . . . . . . . . .
| | | | | | | | | | | | | | | |
| | | | |__|_ | | |__| | | |__|_ |__|_
| | | | | | | | | | | |
| | | | | | | | | | | |
| | | | | | | | | | | |
|__|__|__|_ |_____|__|_ |_____|__|_ |_____|____
| | | |
{x,x,x,x} {{x,x},x,x} {{{x},x},x,x} {{x,x},{x,x}}
c c c
# # #
b b b b b b b
# # # # # # #
{a, b, a, a} {a, b, a, b} {a, a, a, d} {a, a, c, d}
. . . . . . . . . . . . . . . .
| | | | | | | | | | | | | | | |
|__| |__|_ |__| |__| | | | | |__|_ | |
| | | | | | | | | | |
| | | | |__|__|_ | |_____| |
| | | | | | | |
|_____|____ |_____|____ |________| |________|
| | | |
{{{x},x},{x,x}} {{{x},x},{{x},x}} {{x,x,x},x} {{{x,x},x},x}
a, b, c, d}
. . . .
| | | |
|__| | |
| | | <--(Standard Math language uses only this
|_____| | no-redundancy_no-uncertainty_symmetry)
| |
|________|
|
{{{{x},x},x},x}
Also please pay attantion that the last form is the standard R members 0,1,2,3:
Code:
0 = .
1 = 0______1
2 = 0____________2
3 = 0___________________3
And the standrard [B]R[/B] is nothing but the above 2-D representation
in a 1-D representation:
0______1______2______3
And because no R member is both Multiset_AND_Set, I call it: The "shadow" of my new number system.
(*) Please pay attention that we are not talking about the natural numbers 2,3,4 but the cardinals 2,3,4.
It means that our Organic Natural Numbers are actually a general representation of information-trees, where any finite quantity of names of R members can be described by them, for example:
Instead of a=0,b=1,c=2,d=3 we can use a=0,b=.5,c=3,d=pi and then we use the same information-trees above.
I called these general information-trees 'Organic Natural Numbers' because:
1) These information-trees of cardinals are always having a structure, which is based on N members.
2) They can be used as natural (not forced) and general representation for any interaction between complementary states, which simultaneously preventing/defining their middle domain.
3) Because no R member is divisible by my system, it has its own organic (complete) unique and independent self existence.
Matt, please reply your comments to this post before we continue, thank you.
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#445
sparkster
153
0
Since you're trying to redefine another word, in this case "tautology," it's better to just take it out of your definition.
1) I do not start every single concept from scratch, so words (for example) like, finite, and scale have their standard meaning.
If "finite" has it's standard mathematical meaning, then you are assuming the existence of the natural numbers. Here's Rudin's definition of a finite set
For any positive integer (natural number) n, let J_n be the set whose elements are the integers 1, 2, ..., n; let J be the set consisting of all positive integers (or natural numbers)...A is finite if A has the same cardinality of J_n for some n
Again, if you're using finite as a concept in your definitions, you must accept the standard definitions for the natural numbers (and also cardinality). If you do not concede this, then you cannot use the concept of finite, or else define it yourself.
By accepting the natural numbers, you must either construct them yourself, or accept the standard construction. This in turn forces you to accept standard ZF set theory. Here are the axioms:
1. The empty set is a set as is every member of a set.
2. If X is a set and, for each x in X, P(x) is a proposition, then
{x in X: P(x)} is a set.
3. If X is a set and Y is a set, so is {X, Y}.
4. If X is a set, {z: z in x for some x in X} is a set.
This set the “union of X”.
5. If X is a nonempty set, {z: zin x for each x in X} is a set.
This set is the “intersection of X”
6. If X is a set, {z: z is a subset of X} is a set.
This set is called the power set of X.
7. The set N of all natural numbers is a set.
8. No set is a member of itself.
9. If X is a set and Y is a set, so is X cross Y.
I left off the axiom of choice, which is often included.
Are you prepared to concede all this to have your real numbers? If not, you must provide your own construction of the naturals.
My simple comment is that you evidently do not understand the distinction between equivalent and equal, but heck, you're not very well educated mathematically, so whose fault is that?
99% of your last post is nonsensical unless you actually get round to defining all of the terms you use so that the rest of the world might know what you're doing. You should also put a huge disclaimer saying:
contrary to what I said earlier, when I use words I do not in fact use them with their proper well understood meanings.
You're talking about something you've labelled R butnot actually saying what it is. AGAIN.
You also don't seem to understand that, whilst you say your interpretation of "tautology" is different from that in formal mathematics, you cannot then cite its definition! And there is a difference between statement being tautologous, and it being true, since everything satisfies its own definition, then including this "tautology" is unnecessary.
It reminds me very much of your confusion over the Collatz conjecture, that you thought you'd figured out.
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#447
Lama
466
0
Matt,
Matt Grime said:
My simple comment is that you evidently do not understand the distinction between equivalent and equal, but heck, you're not very well educated mathematically, so whose fault is that?
I know exactly the difference between:
Code:
_____
_____ AND ______ NOTATIONS
_____ ______
The first one is stronger then the other.
By standard Math, if internal structural properties are omitted and only quantity remains, then and only then (and because of this trivial, again not abstract but trivial approach) there is a difference between 'identical' and 'equal'.
But you see, in my framework a '=' notation has one and only one meaning, which is the identity of a thing to itself, and the reason is:
Any given element in my system is unique and cannot be represented by any other element but itself.
And by this fundamental approach, my system is sensitive to any information change between elements, and can use these differences to make richer and deeper Math.
You are still standing in your spot and observe my system from your standard point of view, and I am talling you again, it will not work.
Matt Grime said:
You're talking about something you've labelled R butnot actually saying what it is. AGAIN.
I alreadry crearly and simply defined my number system, and it is very easy to see how R members are the "shadows" of my number system.
Here it is again, and if you still standing in your spot, you will not understand it:
Tautology:
The identity of a thing to itself.
Set:
A set is a finite or infinite collection of objects in which order has no significance, and multiplicity is also ignored.
Multiset:
A set-like object in which order is ignored, but multiplicity is explicitly significant.
Singleton set:
A set having exactly one element a. A singleton set is denoted by {a} and is the simplest example of a nonempty set.
A definition for an interval (segment):
A singleton set s that can be defined by tautology* ('=') and ('<' or '>'), where s has no internal parts.
(Sign '<' means that we look at the segment from left to the right.
Sign '>' means that we look at the segment from right to the left.
When both '<' , '>' are used then we have a directionless segment.)
By the definition of a segment we get {._.}, which is the indivisible singleton set that exists between any two {.}.
Now we have the minimal building-blocks that allows us to define the standard R members.
(more detailed explanation of the first two definitions:
Let us examine these first two definitions by using the symmetry concept:
1) {.} content is the most symmetrical (the most "tight" on itself) content of a non-empty set.
It means that the direction concept does not exist yet and '.' can be defined only by '=' (tautology), which is the identity of '.' to itself.
2) {._.} content is the first content that "breaks" the most "tight" symmetry of {.} content, and now in addition to '=' by tautology (which is the identity of '._.' to itself) we have for the first time an existing direction '<' left-right, '>' right-left and also '<>' no-direction, which is different from the most "tight" non-empty element '.'
In short, by these two first definitions we get the different non-empty and indivisible contents '.'(a point) or '_'(a segment) .
*[/color]A statement for a point:
A point is an indivisible finite content of a non-empty set that has no directions.
*[/color]A statement for a segment:
A segment is an indivisible finite content of a non-empty set that also has directions.
(In standard Math we had to write:
A point: If a content of a set is a singleton and a urelement and has no directions, then it is a point.
A segment: If a content of a set is a singleton and a urelement and also has directions, then it is a segment.
But in this framework A=A is a tautology, and we do not need an ‘if, then’ proposition in order to define it)[/color] )
The axiom of independency: p and s cannot be defined by each other.
By the above axiom {.} and {._.} are independed building blocks.
By the above axiom we define the basic property of the middle domain between {.} and {._.}
The axiom of minimal structure:
Any number which is not based on |{}|, is at least p_AND_s, where p_AND_s is at least Multiset_AND_Set.
The above axiom allows us to:
1) To define the internal structure of standard R members.
2) To define the internal structures of my new number system.
The axiom of duality(*):
Any number is both some unique element of the collection of minimal structures, and a scale factor (which is determined by |{}| or s) of the entire collection.
The above axiom allows us to construct a collection of R members and also a collection of my new number system.
First, let us see how we use my method to construct a collection of R members.
R members are constructed like this:
1) First let us examine how we represent a number by my system:
=>> is ‘represented by’
a) |{}|=>>0
b) There is 1-1 and onto between ‘0’ and the left point of {._.} and we get {‘0’_.}
c) |{{}}|=>>|{0}|=>>1
e) There is 1-1 and onto between ‘1’ and the right point of {._.} and we get {‘0’_’1’}
In short, {.} is the initial place of R collection, which is represented by ‘0’, where {‘0’_.} is the initial place of the second place of R collection, which is represented by ‘1’, and we get our first two must-have building-blocks of R collection.
2) When we get {‘0’_’1’} we have our two must-have numbers, which are ‘0’ and _’1’.
Be aware that ‘0’ is the representation of {.} where ‘1’ is the representation of {._.}.
3) If we get {.}_AND_{._.}, then and only then we have the minimal must-have information to construct the entire R collection because:
a) We have ‘0’ AND _’1’ that give us the to basic scale factors 0 and _1.
b) We also have our initial domain _1, which standing in the basis of any arbitrary scale factor that is determined by the ratio between the initial domain _1 and another segment that is smaller or bigger than the initial domain _1 , for example:
The negative numbers are the left mirror image of the above numbers.
There is no division in my number system because both {.} and {._.} are indivisible by definition.
In short, any segment is an independent element, that clearly can be shown in the above 2-D representation.
If we use a 1-D representation, we get the standard Real-line representation, but then we can understand that division is only an illusion of an overlap of independent elements when they are put on top of each other in a 1-D representation, for example:
Code:
0__.5 __1_____2_____3__pi
(*) The Axiom of Duality is the deep basis of +,-,*,/ arithmetical operations.
Since in my system nothing is divisible, then '/' stands for a ratio between at least any given two (indivisible) numbers.
If "finite" has it's standard mathematical meaning, then you are assuming the existence of the natural numbers. Here's Rudin's definition of a finite set
Let us stop here (before we continue to my new number system) to get your remarks.
I already gave you my answer to 'finite', which is:
The axiom of completeness:
A collection is complete if an only if both lowest and highest bounds are included in it and it has a finite quantity of scale levels.
Then you asked me what is lowest and highest bounds, and my answer is:
The ends of some given element, where beyond them it cannot be found.
This last definition (and its opposite) is for {},{.},{._.} and {__} sets.
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#449
CrankFan
137
0
No matter how many times you repost your incomprehensible crap it's still incomprehensible crap.
