0, 1, infinite model, question

In summary, the conversation revolves around the use of the numbers 0, 1, and infinite in a dialectic and their relationship to mathematics. The original poster was seeking help in understanding this concept and had a discussion with a former math professor from Princeton. The professor mentioned that the numbers 0, 1, and infinite are used in math as basic distinctions of idea, but the other participants in the conversation do not see the connection to mathematics. The original poster believes that the concept is a principle in mathematics and wishes to understand it in order to incorporate it into their dialectic. However, the other participants argue that it is not mathematics but rather a personal philosophy. The conversation ends with the original poster seeking clarification and further discussion on the
  • #1
Moonrat
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Hello. I was hoping someone here can give me a pointer or a link..

I am not a mathematician, or even in a math course, but I
am doing research regarding a dialectic that refrences the numbers 0, 1, and 2 in it's conceptual environment.

I was having a discourse in my research with a former math professor from Princeton about how it works, and he informed me that the numbers 0 (mystery), 1(true) and 2 (false) how they are used in the dialectic (as basic distinction of idea) is called in mathematics 0, 1 and 'infinite'. refrencing the three basic distinctions of number... Is that a higher math, a specific branch? What is that called? I am wishing to write about this and can't find any source or even where to look. Any help would be great in this regard.

MR
 
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  • #2
Sorry, but it looks to me like this has nothing to do with mathematics. As far as I can see you are only using "0", "1", and "2" as labels and could as easily use "a", "b", "c".

"0, 1 and 'infinite'. refrencing the three basic distinctions of number... " is just nonsense.
 
  • #3
HallsofIvy said:
Sorry, but it looks to me like this has nothing to do with mathematics. As far as I can see you are only using "0", "1", and "2" as labels and could as easily use "a", "b", "c".

"0, 1 and 'infinite'. refrencing the three basic distinctions of number... " is just nonsense.

mmm, thanks, I guess..however, it's not nonsense, and it's not like quanitfying things down into a, b, c at all, that was a bit of an assumption on your part. This was a former math professor at Princeton whom informed me of this, so I think he would have an idea about what he was talking about. perhaps it is a refrence in a branch of math you do not yet understand...

using common language, it can be explained simply that there are three basic distinctions to number.

0 is the value for number with an unknown or non value. 1 is the value of a finite basic unit , and 2 , or 3, or 4, ad infinitum, etc etc is merely an arrangment of '1's.

3, count them, three distinctions.

I don't have the mathematical languages to explain this, but I do know it is a princaple, and a mathematical one...


MR
 
  • #4
Just because an ex=professor explained something to you doesn't mean he understood what you were asking, and more likely that you didn't understand what he was getting at.

One interpretation might be this:

take 0, the algebraically we obtain no more elements.

so we add 1, which is the multiplicative identity, then algebraically we can get more things, but we never obtain anything *algebraically* that we think of as infinite.
Perhaps. But only that professor knows what he meant. We do not, as this isn't mathematics ion the sense of a universally agreed convention but his personal philosophy about something.
 
  • #5
matt, thank you for your time and response,

hear me out a bit, please...I have been developing a dialectic for the past two years, and am just now getting to working through some of the complex math aspects, which is not my forte, I admit...


matt grime said:
Just because an ex=professor explained something to you doesn't mean he understood what you were asking, and more likely that you didn't understand what he was getting at.

yes, I agree, that may be true, however, what may also be true is that he understood me, I understood him, but you do not understand us. And that certainly is not your fault, but mine. So, I have to explain this feature as best I can, rationally, using common language. However, this is a distinction that is found in number, and you just pointed it out.


take 0, the algebraically we obtain no more elements.

yes, and this I would explain, in my common language, that this is a 'non-value'.

so we add 1, which is the multiplicative identity,

yes, we add 1.Or, from 0 we obtain 1. One, in the dialectical sense, is simply 'distinction'. Now, I don't know how 'distinction' is refrenced in mathematics, but each thing that can be distinguished is always '1' thing. Now, there is one, and then there is what happens to 1 when multiplied, divided, and sumed.

It becomes a multiplicable identity because of the 'infinte' number of possible combinations of multiplying 1.

then algebraically we can get more things, but we never obtain anything *algebraically* that we think of as infinite.

yes, agreed. but we do have an infinite number of possible combinations, and that is the point of the infinite environment. Each one of those combinations is assigned a number, any number, but 1.



So, to put it poetically, we can say that mathematics is then the 'play', of 0, 1, and infinite. 3 distinct mathematical environments that all numbers decontruct into...

Perhaps. But only that professor knows what he meant. We do not, as this isn't mathematics ion the sense of a universally agreed convention but his personal philosophy about something.


hmm, ok, I can see that perhaps, but to me, he did say 'Ahhh, I see what you mean now, and in mathematics, we call that 0, 1, infinite...( i don't know if it would be parsed that way, written, but those were his exact 'words' to me)


Are you suggesting that this is merely 'philosophy of mathematics' or something?

the dialectic distinquishes this relationship, found in math, found also in perception, and in logic creates a ternary system of true, false, and mystery as a gaming dynamic inside of a dialectic.

Now, I just want to understand that relationship in math so I can understand it clearer inside of the dialectic, so I would love to keep this discussion moving forward.

Thank you once again.

MR
 
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  • #6
What you are talking about is NOT mathematics.
 
  • #7
HallsofIvy said:
What you are talking about is NOT mathematics.

hmm, puzzling. I ask one mathematician, one whom was rather highly valued and with respectful published work, and he says, yes, this is mathematical, and you say no...interesting...

So to understand you clearly, there is NO mention, parsing, noting, or perception in all the annals of mathematics that simply distinguishes the basic ternary interplay of number?

Surely mathematicians have noted this perception in number before...are you suggesting this is philosophical and not mathematical?


How so? If it can be quantified, how so is it not mathematical, can you explain? Do not all 'normal' numbers fall into either of those three categories? And if it is not mathematical, then how is it the nice fellow who posted after you noted the algebraic relationships immediatly?

MR
 
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  • #8
You certainly sound like a philosopher.

There is, as far asI am aware nothing that states "in mathematics there are 3 kinds of ... 0, 1, and infinite". Though some mathematicians may draw an anaolgy.

Hurkyl, is right, though, this isn't mathematics, as in "something done by a mathematician", more precisely what you're writing about is n't mathematical:


You say something about an infinite number of combinations making 1 the multiplicalbe idetnitty. this isn't meaningful to a mathematician. Also, we cannot speculate as to what "non-value" means to you. I do not see how *you* have obtained 1 from 0.


And I do not know what you mean by numbers deconstructing into 0,1, infinite.


however, relatign them to true false and mystery is speculative to say the leas. Assiginin 0 and 1 to false and true is boolean logic, nothing more. mystery is a mystery.
 
  • #9
Matt,

matt grime said:
You certainly sound like a philosopher.

hey, be careful, those are fighting words ;-)

There is, as far asI am aware nothing that states "in mathematics there are 3 kinds of ... 0, 1, and infinite". Though some mathematicians may draw an anaolgy.

ok. but the analogy is mathematical. i.e it is a rational observation or perception of number, yes?

Hurkyl, is right, though, this isn't mathematics, as in "something done by a mathematician", more precisely what you're writing about is n't mathematical:

Well, yes, that I agree with and see. However, what I am refrencing is also found in mathematical 'perception', or what I am calling 'the perceptional environment of number'. It's not like i am talking metaphysics here or anything of the sort. Just simple observation.

You say something about an infinite number of combinations making 1 the multiplicalbe idetnitty. this isn't meaningful to a mathematician.

Sorry, that was me 'attempting' to make it meaningful to a mathematician. Actually, that is not really meaningful to me either. I was trying to parse what i am signifying into a lanuage you can understand...and am failing miserably thus far, but perhaps you nice intelligent fellows will allow me a few more rounds...


Also, we cannot speculate as to what "non-value" means to you. I do not see how *you* have obtained 1 from 0.

Here it is practicaly. I have no money. 0 Dollars. I get a job. Now I have 1 dollar that the job pays me. Before, my monetary value was 0, now it is 1. the 'philosophy' would be that having 0, or nothing, or non-value, inspired me to 'get up off my ass and get a job' so I could obtain the 1.

Or, mathematically, I am looking at a number line with 0 in the middle, and I can count, in an infinite number of directions, 1's coming from the 0 and spanning out, both positivly and negativly. 0,1,2, 3..etc

And I do not know what you mean by numbers deconstructing into 0,1, infinite.

Not being a mathematician, I can only speak for 'linear numbers and number lines'. I am sure there are qualities of number that I am simply unaware of that can express marvelous vectors and all sorts of things. The math professor I met with was fascinated a bit with the dialectic, and he said I may be tapping into mathematics used in quantum computing. Which is real neato to say, but hopeless for me to understand in relationship to anything.

So, let me know if this makes sense to you. Numbers are symbols, yes? They are symbols that represent, like a language, quantative units. Are we in agreement in my poor man's mathematical language thus far? In this sense, there are three qualities of what numbers express, summed up simply as 0, 1, and 2. Let me explain the 0 here so you can come to see what I mean by 'non-value'.

0 in math, historically as I am sure you are aware of, has presented troubling notions for mathematicians in the past. In my common speak, I am simply observing that 0 is a number that is a numerical value for that which has either no known value or an unknown value. I mean value in the sense of that which can be quantified into a unit. Thus, on a number line, what I mean is simply expressed as 0, 1, 2, 3 etc etc. We 'begin' with 'nothing' and that 'nothing' we signify with only 1 number, 0.

There is no number that holds this value in linear mathematics other than the number 0. In the dialectic, we place this same value onto perception and idea as 'mystery'. Mystery, as defined in the dialectic, is simply an idea, proposition, or perception where you cannot distinguish yet what is true, from what is false. You could say that to me, much of mathematics is 'mystery', and as you can see especially in this instance, i have one mathematician telling me one thing, and another mathematician another thing. I cannot yet deduce the 'true' value of this yet, it is suspended in 'mystery'.


however, relatign them to true false and mystery is speculative to say the leas. .


0 is to number what mystery is to perception of true and false.

Ok, so how then would 1 be true in perception or idea? Simple. All numbers, in the linear sense, are simply combinations of 1, like we have already mentioned. 1 is the number of 'unit'. A unit is always refrenced as 1. 1, or a combination of 1's, either postivly or negativly, is what we are left with when we decontruct all numbers into their basic 'units'. 1,000, 000 is just a 'million' 1's. A 'million' units. So in this sense, 1 is the only 'true' number, and all numbers are simply 'expressing combinations' of 1.

Thus 2 is the first pure 'false' number. It is false in the sense that it is merely in service to expressing a number of combining 'truths' or 1's. The 'ding an sich', what we call the 'thing in and of itself', of 2, is two 1's. The ding an sich of 1 is 1. We could say 3, 4, etc etc is false in the same sense, however, 2 is the first pure expression of this quality in linear numbers, and philosophically it elucidates with more subtle princaples that are irrelevant to discuss now.

This is not speculative, it is observational. In the dialectic, 'false' is a quality of absense of truth in decontructed form, yet it points to truth in combination or expressive form.

Thus, we have three distinctions or values of 'truth' for number, perception, and idea.

0=Mystery or Mysterious Idea
1= True or Objective, rational Idea
2=False or Subjective and artistic idea.


I am not qaulified to mathematically advance this concept, that will be someone else's job. But I am qaulified enought to hammer out the basics. I have been developing this for over two years, and am pretty clear about these functions and how the apply, and I also can make them work when I apply them. The trick for me now is understanding how they read to mathematicians.

Assiginin 0 and 1 to false and true is boolean logic, nothing more. mystery is a mystery.

yes, that I am aware of. However, although many have drawn similarities to Boolian concepts, this is not boolian. It's another language, and indeed, if correct, and it appears thus far that this is correct, it is a 'natural' mathematical or rational perception language basic to human being.


matt, thank you for your time.I would appreciate if we kept this going a bit. I still have plenty of work to do, but your really helping me thus far.

Glad to see that the mathematicians are horrible at spelling just like the filosophers;-) :yuck:

MR
 
  • #10
Moonrat said:
but the analogy is mathematical. i.e it is a rational observation or perception of number, yes?

if that's what you think "mathematical means" that is.

Here it is practicaly. I have no money. 0 Dollars. I get a job. Now I have 1 dollar that the job pays me. Before, my monetary value was 0, now it is 1. the 'philosophy' would be that having 0, or nothing, or non-value, inspired me to 'get up off my ass and get a job' so I could obtain the 1.

But there is nothing mathematical about that; 0 has not created 1; you have creted symbols to express something; in mathematics 1 predates 0. Perhaps I should explain that in mathematics generally things are mathematically constructed from other things. For instance one constructs the integers from the naturals, then one makes the rationals, reals, complexes and so on.


Or, mathematically, I am looking at a number line with 0 in the middle, and I can count, in an infinite number of directions, 1's coming from the 0 and spanning out, both positivly and negativly. 0,1,2, 3..etc

but nothing here "makes" you construct 1 from 0.

So, let me know if this makes sense to you. Numbers are symbols, yes? They are symbols that represent, like a language, quantative units. Are we in agreement in my poor man's mathematical language thus far?

presuming by this you mean the natural numbers. no, this is backwards to this mathematician. mathematical objects are used to describe things in the real world. they can be used to accurately describe (some) quantities, and this is quantitative thing is the origin of some part of our study. one model for the natural numbers is by, say, finite set cardinals. but it is all a matter of opinion: some will call me a formalist for this opinion in a derogatory manner.

In this sense, there are three qualities of what numbers express, summed up simply as 0, 1, and 2. Let me explain the 0 here so you can come to see what I mean by 'non-value'.

no. define it.

0 in math, historically as I am sure you are aware of, has presented troubling notions for mathematicians in the past. In my common speak, I am simply observing that 0 is a number that is a numerical value for that which has either no known value or an unknown value.

no, this isnt' true: if the temperature of the water is 0 degrees C I know exactly what that temperature is.



I mean value in the sense of that which can be quantified into a unit. Thus, on a number line, what I mean is simply expressed as 0, 1, 2, 3 etc etc. We 'begin' with 'nothing' and that 'nothing' we signify with only 1 number, 0.

There is no number that holds this value in linear mathematics other than the number 0.

there is no number that holds the value 2 apart from 2 as well. so?

mathematically, 0 is the additive identity, 1 is the multiplicative identity.


In the dialectic, we place this same value onto perception and idea as 'mystery'. Mystery, as defined in the dialectic, is simply an idea, proposition, or perception where you cannot distinguish yet what is true, from what is false. You could say that to me, much of mathematics is 'mystery', and as you can see especially in this instance, i have one mathematician telling me one thing, and another mathematician another thing. I cannot yet deduce the 'true' value of this yet, it is suspended in 'mystery'.

it's not suspended in mystery, just undefined unmathematical terms.


Ok, so how then would 1 be true in perception or idea? Simple. All numbers, in the linear sense, are simply combinations of 1, like we have already mentioned.


not until you define what a combination is, because you've yet to do so.


1 is the number of 'unit'. A unit is always refrenced as 1. 1, or a combination of 1's, either postivly or negativly, is what we are left with when we decontruct all numbers into their basic 'units'. 1,000, 000 is just a 'million' 1's. A 'million' units. So in this sense, 1 is the only 'true' number, and all numbers are simply 'expressing combinations' of 1.


the same observation still holds. and all you're saying is the N is generated by 1 under addition. this is not a mystery.

Thus 2 is the first pure 'false' number. It is false in the sense that it is merely in service to expressing a number of combining 'truths' or 1's.

this is an odd use of the word false. usually two truths make a truth.

i think i'd prefer the words irreducible and reducible

The 'ding an sich', what we call the 'thing in and of itself', of 2, is two 1's. The ding an sich of 1 is 1. We could say 3, 4, etc etc is false in the same sense, however, 2 is the first pure expression of this quality in linear numbers, and philosophically it elucidates with more subtle princaples that are irrelevant to discuss now.

This is not speculative, it is observational. In the dialectic, 'false' is a quality of absense of truth in decontructed form, yet it points to truth in combination or expressive form.

Thus, we have three distinctions or values of 'truth' for number, perception, and idea.

0=Mystery or Mysterious Idea
1= True or Objective, rational Idea
2=False or Subjective and artistic idea.


I am not qaulified to mathematically advance this concept, that will be someone else's job. But I am qaulified enought to hammer out the basics. I have been developing this for over two years, and am pretty clear about these functions and how the apply, and I also can make them work when I apply them. The trick for me now is understanding how they read to mathematicians.



yes, that I am aware of. However, although many have drawn similarities to Boolian concepts, this is not boolian. It's another language, and indeed, if correct, and it appears thus far that this is correct, it is a 'natural' mathematical or rational perception language basic to human being.


matt, thank you for your time.I would appreciate if we kept this going a bit. I still have plenty of work to do, but your really helping me thus far.

Glad to see that the mathematicians are horrible at spelling just like the filosophers;-) :yuck:

MR

this contains a lot of non-mathematics that mathematicians won't care about.


my spelling is almost flawless; i continually obtain 100% in such tests. however my typing is abysmal.
 
  • #11
Well, I'll give you one concession, moonrat:
The conceptions of "no thing", "the unit/basic amount of finiteness" and "boundlessness/infinite" are certainly ideas about quantity which ordinary people have vacillated back and forth between through history.
What you seem to be missing, however, is that these extremely imprecise ideas (which, hence, enable their users to embark on flights of fancy) have practically no relevance to actual mathematics.
Hence, you should rather try to see if you can utilize your distinction in a sort of history of mentalities; to think these ideas have any deep relation to mathematics as such, is simply to bury yourself in a dead-end.

While, conceivably, your tertiary distinction might be an interesting classifying tool for the shifting, internal dynamics in numerology as evidenced in its history, dragging them onto mathematics proper is simply a misapplication of the tool.

For example, the distinguishing tool is not really able to clarify the close resemblance of 1 and 0, in that they are both identity elements for distinct binary operations.
Furthermore, the "no thing" concept is not really a good way in understanding how "0" appears as a reference level, in some uses of maths.
 
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  • #12
Moonrat said:
So, let me know if this makes sense to you. Numbers are symbols, yes? They are symbols that represent, like a language, quantative units. Are we in agreement in my poor man's mathematical language thus far?
As they are usually defined, numbers are not symbols; Numbers are what the symbols represent. Numbers, as all mathematical objects, are usually also considered to be abstract objects.
In my common speak, I am simply observing that 0 is a number that is a numerical value for that which has either no known value or an unknown value. I mean value in the sense of that which can be quantified into a unit.
You just "quantified" it when you said you had 0 dollars.
 
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  • #13
Matt

matt grime said:
if that's what you think "mathematical means" that is.

well, like I mentioned in my previous post, much of mathematics is 'mystery' to me. However, I do assume that when I can quantify something, that the process of 'quantifying' itself is mathematical. Any quality of 'adding, dividing, or multiplying' I would assume would be mathematical, or would be able to be parsed into a mathematical language. Are you suggesting that I am mistaken in this?


But there is nothing mathematical about that; 0 has not created 1;.

hmm, it appears to me that here we are having perceptional snafus. I of course do not mean that '0 creates 1' in the same sense that I can create a work of art, or, for example, in a few months I am expecting a son, so I don't say that I and the mother created a child in the sense that 0 creates 1, that would be absurd and irrational.

I simply mean that, from a specific POV, we can say that it appears that from 0 comes 1. We can distinguish nothing from one thing, and when we distinguish 'nothing' mathematically, we give it a 0. When we approach this with a linear P.O.V., then we count..0, 1, 2...right? I am only refrencing this simple component, nothing more or nothing more complex than that.

you have creted symbols to express something; in mathematics 1 predates 0

Here I disagree. I have not created any symbol. I have merely noted that perceptional values (true, false, or mystery) to indeed 'true to numbers', and indeed true to those numbers assigned specifically. It is a basic perceptional unit that is found in all language, including mathematical language, or rather that is my arguement.

and yes, in terms of history, i do agree and see that 1 predates 0, however, that is not my point, historical record is not how they relate to each other.


Perhaps I should explain that in mathematics generally things are mathematically constructed from other things. For instance one constructs the integers from the naturals, then one makes the rationals, reals, complexes and so on.

Ok, I can follow that.


but nothing here "makes" you construct 1 from 0.

agreed.


and ok, now i see a bit more...and this is interesting to me what you write below...


mathematical objects are used to describe things in the real world. they can be used to accurately describe (some) quantities, and this is quantitative thing is the origin of some part of our study. one model for the natural numbers is by, say, finite set cardinals. but it is all a matter of opinion: some will call me a formalist for this opinion in a derogatory manner.

Okay, now this is where it gets interesting for me. This opinion you proscribe is a 'quality of perception', not a 'mathematical 'p.o.v.'. but a 'p.o.v. of mathematics'.

this quality of perception is defined as 2, false. Notice how you mentioned there is more than one way to view this issue in mathematics. I would imagine that although each mathematician would appear to have a sound argument regarding how he arrives at his 'opinion about mathematics', there is no 'sound way to determine' which mathimatical perception is empirically true. Yet all mathematicians would agree about the numbers being viewed, just not the relationships that they have.

This is what the dialectic of 0, 1, and 2 defines, the perceptional p.o.v.




no. define it (0,mystery)

Please don't expect me to parse things into a language I don't understand, that would not be fair. However, I can define mystery as it is defined in the dialectic.

0 = mystery, an idea that is both true and false at once. It cannot be determined, defined, or perceived any value in relationship to it's linear or extending environment.

For example, you mention that

no, this isnt' true: if the temperature of the water is 0 degrees C I know exactly what that temperature is.

. It has 'no celcius value', as opposed to 1 degree C, 2 degree C, etc etc..

0 degrees suggests a centering value in relationship to 1 degree C, etc etc, it's a POV. Without the POV, there is no distinction between the levels of temperature at all, and we could not have -1 degree celcius either without the 0. It's value is relational. Like I said, 0 apears is either 'non, unestablishing, or unknown' value.

there is no number that holds the value 2 apart from 2 as well. so?

the value of 2 is 1 and 1. the value of 1 is 1. the value of 3 is 1 and 1 and 1. All numbers relate to 1.

2 IS 1, 1. again, this is a relationship of perception. 2 is a refrence for two 1's.


mathematically, 0 is the additive identity, 1 is the multiplicative identity.

what do you mean by 'additive identity'? I understand multiplicative identity, but what does it mean that 0 is additive? does it mean that it is the number that we can only add to, and not subtract from? In this sense, yes, I can see that.

And if so, then what I am refrencing would agree that 0 is the additive indentity, 1 is the multiplicative identity, and 2 through infinity are how many combinations identity is expressed or parsed.


it's not suspended in mystery, just undefined unmathematical terms.

that is the same thing in relationship, just expressed with two distinct qualities of perception.

not until you define what a combination is, because you've yet to do so.


ahhh! now this is where I run into trouble...allot. here I run into trouble because I understand that there is a 'method' to parsing, so to speak, that I may be completely and perceptionally unaware of. The best I can do is describe what I mean by 'combining' in common speak, and then request you try to see what I mean intuitivly at first, and then rationally parse afterwards..

So, to me, mathematical combination would be adding, dividing, and multiplying. I mean combination in the same sense as this.

There may be much more to combination than what I describe here in mathematics that I am totally unaware of.



all you're saying is the N is generated by 1 under addition. this is not a mystery.

yes, I agree. Nor was I suggesting it was mystery. So I don't understand your rebuttal here. Here we are not even in conflict.

this is an odd use of the word false. usually two truths make a truth.

yes, I agree and can see that it is 'odd'. however, in relationship, it is accurate. This is a rather tricky aspect of the dialectic to percieve at first. It is how it 'uses' false as another function of truth.

I am sure I am going to ruffle many a feather when I suggest this, but herin lay the three perceptional distincions of truth. For now, we can simply agree that 'truth' is merely that which holds observable function.

1=pure truth. a=a. ( it's function as true is objective.)
2= false truth. a=b. metaphor. opinion. (it's function as true is purely subjective)
0=mysterious truth. it is undetermined, unknown what value of truth can be obtained from it, other than it is true that we cannot determine finite value.

i think i'd prefer the words irreducible and reducible

that is fine, as long as we agree that we are refrencing the same function. I can simply say that all (whole?) numbers are reducible to 1. 1 is reducible to an finite number of .1's. 1 defines the relationship of the 'finite'. finite is reducible to 1. any finite number is reducible to 1. there are an infinite number of 1's, thus, an infinite number of combinations of finites.

I agree that I am describing this descriptivly, yes, using a distinguishable language, but we are on the same page here.

this contains a lot of non-mathematics that mathematicians won't care about.

Now, if you don't mind, let me show you what I mean by 'parsing' perception into these relationships. What you wrote above was a false idea.

However, I can see what you mean from your POV, and from your POV, I can translate it away from an opinion, subjective, into an objective statement. "So far, what it appears you have presented to me is not something that is applied in mathematics"

And yes, I agree. however, I am not wishing to introduce a new mathematical proof, I am wishing to show where this perception exists in mathematics in relationship to mathematics, and then I am requesting you help me find how this can be expressed into mathematical language.

If something has a pattern, it can have a mathematical expression, no?



my spelling is almost flawless; i continually obtain 100% in such tests. however my typing is abysmal.

yeah, that's what us filosophers say too;-)

Thanks Matt, I really appreciate your time.


MR
 
  • #14
honestrosewater said:
As they are usually defined, numbers are not symbols; Numbers are what the symbols represent. Numbers, as all mathematical objects, are usually also considered to be abstract objects.
.

yes, agreed. we are refrencing the same thing, just explaining it differently

You just "quantified" it when you said you had 0 dollars.

yes, I know, however, I quantified it as having non value. 0 is value for no value.it is only a unit from the POV of numbers, it is a 'unit' which contains no value, as opposed to 1, which is a unit that expressed the value for unit.


Thanks for keeping your eye out.

MR
 
  • #15
The choice of 0 as the freezinv point of water is completely arbitrary, as the other scales of temperature indicate.

0 does not necesarily indicate an absence of something. This is where you need to separate mathematical objects, such as the ste of natural numbers, from their uses.

You've still not explained what you mean by "combinations".

To give you some idea of the oddities in mathematical philosophy how about

www.dpmms.cam.ac.uk/~wtg10/reals.html

Your decision to assign 0,1, 2 or infinity to some other context is not something I encourage. It has led to some awful things in the social sciences. One needs only recall the Sokal episode. Also your assignment is completely aribtrary and says more about you than either mathematics or philosophy, but then that is possibly true of any stance one takes about a philosophical argument.

Incidentally 0 is an (the) additive identity since 0+x=x for all x, just as 1*y=y for all y.

Yes, all natural numbers are *by definitoin* just those things that can be obtained from adding 1 to itself repeatedly.

However it is then that we define 0 from 1, if we are of the desire that 0 is not in our natural numbers. Some people declare 0 to be the smallest element of N.

From these we can by adding in all the "combinations" in your sense, only obtain the rational numbers. There are numbers that cannot be described from these things, such as root 2, e, pi. See that link.

But this is different from applying it to some quantifiable thing.

This is just a reply "for now" not a complete reply. I will try and write more later
 
  • #16
matt grime said:
The choice of 0 as the freezinv point of water is completely arbitrary, as the other scales of temperature indicate.


yes, that too is my point.

0 does not necesarily indicate an absence of something.

i would say that 0 necessitates the absence of 1 by default, no? That is all I am suggesting. Again, this is not a metaphysical discussion, although I can see how it may be perceived as such

This is where you need to separate mathematical objects, such as the ste of natural numbers, from their uses.

the function of numbers is true in all environments, I would imagine. I would imagine both conceptually and imaginitavly that 0 or nothing always and permanently suggests the absense of 1 or a thing.


You've still not explained what you mean by "combinations".

I have the best I know how. perhaps this is something you can help me with. Can you intuitivly see what I mean first? Combining, to me, is simply 'adding, multiplying, or dividing' of objects (1's) with other objects (1's). I am not sure what I am being unclear here on, as I am refrencing simply that and nothing more.

To give you some idea of the oddities in mathematical philosophy how about

ahh, now that's a link. thanks so much for that. Will study it soon.

Your decision to assign 0,1, 2 or infinity to some other context is not something I encourage. It has led to some awful things in the social sciences. One needs only recall the Sokal episode.

hmm, let me look into that.

Also your assignment is completely aribtrary and says more about you than either mathematics or philosophy, but then that is possibly true of any stance one takes about a philosophical argument.

it depends if the ideas are true or not in the objective sense, right? Believe it or not, I am not wishing to develop a philosophy, nor is this about philosophy in my eyes at least either. This is about perception, and how perception relates to how we view or value information, be it mathematical or otherwise.

Incidentally 0 is an (the) additive identity since 0+x=x for all x, just as 1*y=y for all y.

Yes, all natural numbers are *by definitoin* just those things that can be obtained from adding 1 to itself repeatedly.

yes, we are on the same page.

However it is then that we define 0 from 1, if we are of the desire that 0 is not in our natural numbers. Some people declare 0 to be the smallest element of N.

hmm, i am not sure if I follow, are you saying that some suggest 0 is a natural number, and some suggest that it is not a natural number?

If is not a natural number, then what qualitive of number is it?

From these we can by adding in all the "combinations" in your sense, only obtain the rational numbers. There are numbers that cannot be described from these things, such as root 2, e, pi. See that link.

yes. So my model is refrencing natural and rational numbers, and is true in that sense?

Would this proposition be true then? 0 is to mystery what 1 is to true what 2 is to false in perception and in rational and natural numbers.


But this is different from applying it to some quantifiable thing.

how so? this is a quantifiable thing in perception. It is basic to human perception.

This is just a reply "for now" not a complete reply. I will try and write more later

thanks, looking forward


MR
 
  • #17
Sokal

matt,

By the way, I just freshened up on Alan Sokal's 'hoax', and I must say, it is similar, but not in the way you suspect. What he did was an example of what I refrence as the 'political approach' and showed how perception can percieve a false truth and confuse it's function as a pure truth. I don't know Sokals reasons for doing this, but what he exposed is a perceptional snafu that this dialectic of perception addresses and decontructs. But that is beyond the scope of what this discussion is refrencing, but thanks for the great link..

MR
 
  • #18
Moonrat said:
yes, I know, however, I quantified it as having non value.
What does it mean to quantify something and for something to have a value and for something to have a non-value? Value, amount, quantity, magnitude etc. are all closely related if not synonymous, so you'd have to clarify how something can be "quantified as having non-value".
0 is value for no value.
I don't understand this. It seems to mean "0 is the value of x if and only if x has no value", but that's a blatant contradiction.
it is only a unit from the POV of numbers, it is a 'unit' which contains no value, as opposed to 1, which is a unit that expressed the value for unit.
What is the distinction between a unit and a number? From "the value for unit", I gather all units have the same value. In that case, neither 0 nor 1 is "a unit from the POV of numbers", because from the POV of numbers, 0 and 1 have different values.
 
  • #19
honestrosewater said:
What does it mean to quantify something and for something to have a value and for something to have a non-value?

Hmm, I like those questions. I am going to get a bit dangerous here and open up some territory I am still understanding myself, but In common language I would respond that when we quantify something, it means we identify either number or measurement into mutual language or symbol referencing a distinguishable unit. For example, I can distinguish 1 glass of water, and then the amount of water in it, and then add a value for the total volume, etc etc.

From the POV of perception, we can then say that all things that can be distinguished can be assigned a value appropriate.

Value in the objective sense that I reference would mean it's equivalent in relationship to other objects. the value of 1 being a finite can only be in reference to an infinite or at least a continuous set. thus, we assign that which has no finite value, 0, to help define the finite value, 1, and vice versa.

Value, amount, quantity, magnitude etc. are all closely related if not synonymous, so you'd have to clarify how something can be "quantified as having non-value".

are you saying that I would have to have an objective criteria for what quantifies something not having a finite value?

0 is the absence of a finite value. Where ever there is a signifier for that which is the absence of a finite value would be synonymous with 0. Having that defined, we can then distinguish the finite value (1) with ease..

I don't understand this. It seems to mean "0 is the value of x if and only if x has no value", but that's a blatant contradiction.

hmm, no more so than 0 in general, I would imagine. You act as if I am introducing the concept of 0 into mathematics for the first time! 0 is a paradox a bit after all. I even think Aristotle, or perhaps his cult, feared the number and even had murdered the Greek mathematician who suggested it's existence and what it did to their system of mathematics.

remember, 0 is non value in relationship to the natural numbers that are counted from it.

What is the distinction between a unit and a number?

hmm, I have written a response for this, but I just deleted it. this is a wonderful question. I may need a bit of time to think about this answer, but for now, let me say that to me it appears as if a unit is a finite, functioning in relationship to other finites, and any whole rational number would be simply a collection of finites or units or that which can be distinguished or quantified into units. A unit is the deconstructed finite that appears when we make basic distinction.

Or, you could say that the basic unit of number is 1.

and then to add even a bigger mess, we could then say that all finites are abstractions or conceptual objects or 1's, but that is probably getting too philosophical here.

From "the value for unit", I gather all units have the same value. In that case, neither 0 nor 1 is "a unit from the POV of numbers", because from the POV of numbers, 0 and 1 have different values.

YES! Now here is where it gets rather delicate since we are overlapping two environments confusing them as one, and I can accept that my explanations are a bit faulty or needing some help, but 0 and 1 have different values in relationship only to each other. 0 is no sum, and 1 is finite sum. However, from the pov of number, 0 is a unit, 1 is a unit, 2 is a unit in the sense that they are are distinguishable from each other, each one a 'unit of or measurement of number', 0 being the 'number' for non value, the 'unit' for 'no unit'.

perhaps with all of this confusion about 0, you can see that 0 being an idea that is 'both true and false at once' may seem a bit apropos here.

Good questions, and you got me really thinking...it's a treat to receive such rational questioning, thank you so much

MR
 
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  • #20
Moonrat said:
i would say that 0 necessitates the absence of 1 by default, no? That is all I am suggesting. Again, this is not a metaphysical discussion, although I can see how it may be perceived as such

I can't make that make sense mathematically.


Is not 2 the absence of 3? Are you talking of the zero element in Z, Q, R C, or some other field or ring? Do you think that the 0 elements of all rings are equal?


I think you're talking about an application of some numbers, to be honest, and not about any of the inherent mathematical properties of the numbers.



the function of numbers is true in all environments, I would imagine. I would imagine both conceptually and imaginitavly that 0 or nothing always and permanently suggests the absense of 1 or a thing.

what does it mean fo a function to be true?

hmm, i am not sure if I follow, are you saying that some suggest 0 is a natural number, and some suggest that it is not a natural number?

If is not a natural number, then what qualitive of number is it?


i am suggeting that. you find that odd because you think that the name we give to something is somehow intrinsic. there are lots of mthematical objects/concepts that are different and have the same name.



yes. So my model is refrencing natural and rational numbers, and is true in that sense?

what sense of true is this?



Would this proposition be true then? 0 is to mystery what 1 is to true what 2 is to false in perception and in rational and natural numbers.

not in mathematics.
 
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  • #21
Matt

Again, thank you for your time and keeping this discussion going with me. I hope you realize what an assistance your being.

I think I may have a false idea about what mathematics ‘is’, this may be true. I assume that it is a rational language that assigns mutually agreeable symbols to observable functions, and measurements. I want to come to understand what mathematics is then in the pure sense. I want to see what you see.

I said: I would say that 0 necessitates the absence of 1 by default, no?


I can't make that make sense mathematically.

Hmm..puzzling that you can't see this, and your ‘reason’ is…

Is not 2 the absence of 3?

Yes, 2 is the absence of 3, but it is NOT the absence of 1, it is the inclusion of 1. All numbers, at least the real and the rational, are the inclusions of 1 except for 0. I am merely making an observation about this number.

In relationship to all other numbers, be they real, rational, or natural, is not 0 then the only number that holds this value? It permanently suggests that ‘1’ is as ‘nada’ as the boogie man and Bigfoot. Where there is 0, no 1 can exist…

Are you talking of the zero element in Z, Q, R C, or some other field or ring?

I really wish I knew the answer to that question, however…when you ask….

Do you think that the 0 elements of all rings are equal?

I think I see your point. In relationship to what they may signify individually on each ring, no, but in relationship to 1, yes.

I think what I am having difficulty explaining here is the fact that I am only referencing 0 in relationship to 1 and 1 in relationship to 2, and that is it. I am going no further than that, I cant, I am not a mathematician!

I can assure you that I am only referencing this relationship, and I understand what your telling me, your saying that although this relationship between numbers may be perceived to exist, it is not utilized or acknowledged, even needed, in all of mathematics that you are aware of.

Is that what you are telling me?


I think you're talking about an application of some numbers, to be honest, and not about any of the inherent mathematical properties of the numbers.

Well, yes and no. I am talking about, for sure, inherent properties in the numbers 0, 1, and 2. Your telling me, and I can accept this if you can continue to argue this as true with me rationally, that what I am observing is not a mathematical function.

The question I am asking now is, if what your saying is true, how is it that these inherent properties of number that I am able to observe and define not mathematical, and how so?



what does it mean for a function to be true?

a function itself is true by default how I define function and truth. If a function is not ‘true’, then it is an irrational or imaginary conceptual object that is still has function, just not the function it is signifying. Once something has an observable ‘function’, (let me define function as ‘relationship’ to other objects or 1’s to other 1’s), it’s ‘true’ in the objective sense. A simple common way would be to describe function as an object’s ‘cause and effect’, it is not just conceptual, it is observably physical, and can be broken down and dissected and communicated intoa conceptual formula…this is ’physicalism’, no?….(and now we are getting off topic and I am most likely confusing the issue here, so if non of this makes sense, let's move on!)


I am suggesting that. you find that odd because you think that the name we give to something is somehow intrinsic. there are lots of mathematical objects/concepts that are different and have the same name.

so to be clear, your saying 0 is not a natural number, and therefore by default has no ‘intrinsic quality’ that can be quantified? Or it can be a natural number or even reference a natural, depending upon how it is defined in a formula?


And if so, that 0 is permanently not a natural number, then I am saying that yes, if this is true, then 0 is the integer that represents the absence of natural number.

I am not trying to find what I am defining in math, but rather the math in what I am defining. I assume that if something has an observable pattern, it can be quantified, am I mistaken in this?

Thank you once again

MR
 
  • #22
a function itself is true by default how I define function and truth.


good, cos that is how mathematics works - simply by deducing things from the rules.

If a function is not ‘true’, then it is an irrational or imaginary conceptual object that is still has function, just not the function it is signifying
.

to be honest, as soon as i read that i switch off, as will almost any matheamtician. in what sense are you using irrational or imaginary? the mathematical one?


so to be clear, your saying 0 is not a natural number, and therefore by default has no ‘intrinsic quality’ that can be quantified? Or it can be a natural number or even reference a natural, depending upon how it is defined in a formula?


And if so, that 0 is permanently not a natural number, then I am saying that yes, if this is true, then 0 is the integer that represents the absence of natural number.

some people declare that the set of naturals contains zero, some do not. I am abivalent and use which ever is more convenient.

you need to define "absence"

as it is all you appear to be doing is stating

"I want 1 to be the smallest natural, i will declare that the other naturals are combinations of 1, and i will assign the phrase "indicates an absence of" to say "is less than""


the other 'personifications' you are talking about appear to be suggestions and analogies. They are not true or false, merely useful or not useful. And I suspect that what that mathematician at princeton was doing was trying to explain to you some of the shorthand that mathematicians often use to explain the underlying gist of their work. For instance, I often say things like: using derived categories is the morally correct way to define cohomology.
I am drawing an analogy about the importance of the mathematics, but you would be foolish to draw the conclusion there is actually anything ethically correct in this choice.
 
  • #23
arildno, sorry I missed your post earlier,

The conceptions of "no thing", "the unit/basic amount of finiteness" and "boundlessness/infinite" are certainly ideas about quantity which ordinary people have vacillated back and forth between through history.

Yes, I can see that..

What you seem to be missing, however, is that these extremely imprecise ideas (which, hence, enable their users to embark on flights of fancy) have practically no relevance to actual mathematics.

I also see that and I certainly hope it doesn’t read that I am suggesting otherwise, however, we are still left with numbers none the less. And again, I am only identifying the relationships of these numbers to each other. It doesn’t matter, at least from this POV that I reference, how these numbers are used.

Hence, you should rather try to see if you can utilize your distinction in a sort of history of mentalities; to think these ideas have any deep relation to mathematics as such, is simply to bury yourself in a dead-end.

Hmm, that is an interesting thought indeed! The history of perception in relationship to number…Have you read the book ‘Zero, the History of a Dangerous Idea’? It goes into some of this…

But what I am suggesting is of course functioning in math at least as perception, there is a built in perception quality that is ternary in nature, and if this did not exist, there would be no 'math' by default..without these qualities that are found in 0, 1, and 2, we would not be able to 'distinguish' conceptual objects from external objects..

While, conceivably, your tertiary distinction might be an interesting classifying tool for the shifting, internal dynamics in numerology as evidenced in its history, dragging them onto mathematics proper is simply a misapplication of the tool.

Lol, well, I can sure accept that, I certainly hope I am not giving the impression of assuming that this ‘should’ be applicable to mathematics, I am just asking how is this basic distinction referenced as 0, 1, and 2 can be understood in mathematical principles…which your saying that it cannot be understood it mathematical principles, but it does have a tertiary relationship in perception.. (is not ‘tertiary’ also a mathematical principle, or no?)

For example, the distinguishing tool is not really able to clarify the close resemblance of 1 and 0, in that they are both identity elements for distinct binary operations.

Hmmm, I am not sure if I follow, so if I don’t, PLEASE let’s elaborate on this a bit.

First off, from what I understand (very primitively in this regard, yes I admit) 0 and 1 is base two counting just like 0, 1, and 2 is base three counting. So binary systems create ‘simple choice’. Either this, or that.Distinguishing ONE thing from the other ONE thing. Now, this, or that, may both have value, however, one is always off and the other always on. Off is to 0 what on is to 1, no? 1 turns on it’s function, and 0 turns it off. Two distinct states.

If this sounds jumbled, then let me put it this way, we can apply binary counting to a ternary system, I can explain this philosophically, but certainly not mathematically, however, I do assume that since I can find a pattern, one must therefore be said to exist and thus must be able to be quantified into a mathematical expression.

Furthermore, the "no thing" concept is not really a good way in understanding how "0" appears as a reference level, in some uses of maths.

Yes, I can see that. However, the relationship of 0 to 1 is always the same when we ‘divide, add, and multiply’, no?

MR
 
  • #24
All right:
To clarify the term "binary operation":
This does NOT refer to writing numbers in a binary base rather than a decimal base.

A binary operation means that you pick out two elements from a set and "produce" from them according to a given rule another element in the same set.

For example, addition is basically a binary operation in that you pick out two numbers (i.e, elements) and, by the rule of "adding" gain another number.

0 is a number called the additive identity (or neutral element), in that whenever 0 is one of the two elements you add, the other being, say, "a", your output is "a".
1 is analogously the multiplicative identity for the binary operation "multiplication".

That is what I meant with 0 and 1 being similar, in that they fulfill similar roles in distinct binary operations.
 
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  • #25
Philosophy, to be sure!

I can see a particular argument for 0, 1, and infinity as the building blocks for numbers (taken to be the set [tex]\{0,1,2,\ldots\}[/tex] adjoin infinity): 0 is the base element, and is the successor of no other; 1 is the 'typical' number, the first successor of 0, and the additive building block of the counting numbers (1, 1+1, 1+1+1, ...). Infinity is the only 'number' that is its own successor.

However, outside of some unusual philosophy, I can't see the relationship between 0 and mystery, 1 and true, or 2 and false. That's arbitrary at best. This is better done as abstract algebra, perhaps...

T, F, U (representing true, false, unknown). Would you like to define for us how you think these should combine? True and true is true, mystery and mystery is mystery, but what of the other 7 combinations? Maybe then we can better understand how these relate to numbers and give you input regarding how well 0, 1, and 2 work for these.
 
  • #26
In infinite ordinals w has a sucessor, and it isn't w.
 
  • #27
matt grime said:
In infinite ordinals w has a sucessor, and it isn't w.

While I of course agree, I was looking at this more as one possible corner interpretation where Moonrat's observations could be given some mathematical meaning. (That is, the first part... the second part, about mystery and all, just isn't math as far as I can tell.)
 
  • #28
matt grime said:
good, cos that is how mathematics works - simply by deducing things from the rules.

yes, I agree, I would assume, and that is also how I come to the conclusion that what I am referencing must have some mathematical language to be parsed into. Since I can define the 'rules' in common speak, or in the language of the dialectic, and even make a 'game' out of the rules, then I assume there must be a way to quantify that mathematically.


Can a pattern that is arrived or distinguished by logical or rational means then be said to have a mathematical component, at least in principle?





to be honest, as soon as I read that (referring to 'imaginary conceptual object' I switch off, as will almost any mathematician.


thank you for your honesty. I can clarify that no problem. what makes discussing this particularly challenging for me is the fact that I am used to communicating creatively, which has a distinct set of rules than mathematical forms of communication, as you could well imagine. (in the dialectic, you could say that I use the realm of the 'false' to communicate the 'true')


in what sense are you using irrational or imaginary? the mathematical one?

in that particular phrase, no, but i am also aware that I can use a 'false' term to get you to intuitivly percieve a very rational truth. You being a mathematicain, when you see what I see, will use the 'true' language to define it as opposed to my 'false' language. Does that make sense?

I know I am communicating 'falsly' or 'creativly', yet I know what I am signifying is true and functioning in the rational and mathematical sense.


some people declare that the set of naturals contains zero, some do not. I am ambivalent and use which ever is more convenient.

So then for certainty we can say that 0 is both a natural number and not a natural number, depending upon the POV. (in the dialectic this is referenced to 0 as being mystery, in terms of perception, 'true and false at once'. As your note above regarding 0 containing both possible elements of natural or not. By the way, when it is not a natural number, what is it? And in the set of numbers 0, 1, and 2, is it the only number that holds that value of vacillating pov's of natural or not?)

you need to define "absence"

In reference to 0, absence would the negation of the finite, or 1.

as it is all you appear to be doing is stating

"I want 1 to be the smallest natural, I will declare that the other naturals are combinations of 1, and I will assign the phrase "indicates an absence of" to say "is less than""

hmm, thanks for that feedback. that's kind of close, but not..but it was very valuable for me to read how my words read to you...

I usually explain this as the play of finites in an infinite environment. 1 is the finite, and the infinite environment is 2 through infinity and 0.

Let's say 1 is the smallest natural finite, (1 = true finite)

All other naturals are combinations (let's also define combinations as 'expressions') of 1 (2=false or 'expressed' finite)

So here, the first distinction I make using the dialectical principles of 0, 1, and 2 is that there are two distinct qualities of finite numbers. There is only one true finite in terms of perception, and that is 1. Then there is the false finites, expressions of 1 + 1 + 1 is always expressed as 3, for simplicity sake, but 3 is conceptual, and 1 + 1 + 1 is actual.

thus, in perception 1= true, 2 =false (2 is only false in relationship to 1, not to itself or in and of itself)

next, we have 0. I cannot concur that your interpretation of what I am suggesting is correct because 0 is not 'less than' finite. .0000001 is less than 1 numerically, yes, but .000001 still a finite. Absence is not 'less than'. absence in this sense is the negation of finite, which makes it 'infinite' by default. Any concept that defines that 'infinity' other than 0 is 'false' by default. Thus we can say perceptionally that 0=mystery (true infinite), 2 = false (false infinite), 1 = true finite.


the other 'personifications' you are talking about appear to be suggestions and analogies. They are not true or false, merely useful or not useful.

well, I am not sure about what exactly your talking about, what personifications? And in relationship to the perceptional qualities of 0, 1, and 2, then of course useful (function) is true, and not useful (non function) is false.


And I suspect that what that mathematician at Princeton was doing was trying to explain to you some of the shorthand that mathematicians often use to explain the underlying gist of their work. For instance, I often say things like: using derived categories is the morally correct way to define cohomology.

hmm, I don’t think so, I think he was hitting at what the other poster,CRGreathouse suggested.

At the end of the discussion with this former professor, I wanted to be VERY CLEAR about his perception regarding what I was talking about.

We were in logical agreement about what I mention here, and he said that what I am discussing is indeed very rational, i.e. not a belief system.. He was very specific when he said 'oh...this is something, you do have something here, in math, we would call this 0, 1, and infinite'

I am drawing an analogy about the importance of the mathematics, but you would be foolish to draw the conclusion there is actually anything ethically correct in this choice.

yes, I agree, however, we would both agree that mathematics would hold 'importance' because of your analogy, and we could say that your analogy, which in the perceptional dialectic would classify as a '2' to explain or define the '1', which would be 'importance'.

Please keep in mind that everything I mention in regards to 0, 1, and 2 is about perception and factors the 'perciever' into the equation!

matt, this is great! your a gentleman and a scholar, I only wish more 'philosophers' would be able to communicate as rationally and honestly as you...


MR
 
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  • #29
arildno said:
All right:
To clarify the term "binary operation":
This does NOT refer to writing numbers in a binary base rather than a decimal base.

A binary operation means that you pick out two elements from a set and "produce" from them according to a given rule another element in the same set.

For example, addition is basically a binary operation in that you pick out two numbers (i.e, elements) and, by the rule of "adding" gain another number.

0 is a number called the additive identity (or neutral element), in that whenever 0 is one of the two elements you add, the other being, say, "a", your output is "a".
1 is analogously the multiplicative identity for the binary operation "multiplication".

That is what I meant with 0 and 1 being similar, in that they fulfill similar roles in distinct binary operations.


One, that read beautifully to me. Thank you for that thoughtful explanation.

If 0 is a neutral element, that clearly delineates it from 1, no? the fact that you can distinguish one binary operation from another binary operation is the same distinction I am referencing regarding the distinction between 0 and 1.

perhaps, and maybe I am being a bit bold here, but let me say that we can define 'relationships' as 'distinctions'. Things that have 'distinctions' have clearly defined borders, boundaries. Indeed, a distinction 'draws' the finite conceptual object that we model it to be.

So when I say the relationship to 0 to 1 to 2, I simply mean the observable basic distinctions to how we perceive those conceptual mathematical environments that exist with their own natural set of rules, and then I am saying that now that since I CAN make the distinctions, and the distinctions are observable to all p.o.v. in the objective sense, I want to define those distinctions mathematically and rationally.

I see what you mean above about 0 and 1. Do you understand what I mean here now, and do you see how your analogy does not apply to this particular environment?

MR
 
  • #30
CRGreathouse said:
While I of course agree, I was looking at this more as one possible corner interpretation where Moonrat's observations could be given some mathematical meaning.




:rofl:

well, if I have to settle for a possible 'corner interpretation' for mathematical meaning I will happily accept that!

(That is, the first part... the second part, about mystery and all, just isn't math as far as I can tell.)

thank you for your participation in this discussion. I read your earlier posts, and yes, you see the same distinction I see, however, it's application lay when we apply that same distinction to perception and how we arrive at true, false, or mystery for any 'certainty'.

I will respond to those posts tommorow, it's late and I have been working all day and if I continue, I won't be able to be as fresh on it as tommorow will asuredly bring me.

thanks again,

MR
 
  • #31
If you are now going to declare the "infinite environment is 2 through infinity and 0" then I don'et know what you're remotely getting at. Why is 1 not in this set? Because it is "the finite"? This is a departure from the usual ideas, and almost certainly wouldn't agree with whatever notion of numbers the bloke from princeton had.

I don't see whty you need to make a distinction between different symbols for the same object: 3=1+1+1.

You are reading more into it than I do (what I termed "personifiying", such as assigning terms like mysterious to 0).

If however, all this princetonian was doing was explaining to you that there is a initial ordinal, successor finite ordinals, and that the set of finite ordinals is infinite, then I can see what he was getting at.


Oh, and 1 is also a "neutral element" for the operation of mutliplication. Though I don't like those labels for objects - naming them the real and imaginary numbers is one of mathematics' worse choices.

A reason why some people do and some do not take 0 to be natural is that some wish for the naturals to be the"counting numbers" as, say, going back to the romans. Some wish to use 0. The point is that we can view the set of natural numbers, in either version, as a model of an "inductive set". Loosely a set with an initial term, and for every other term there is a "next" element.

Some start coutnign with 0 soem with 1. 0 comes about becuase of the use of the empty set to create the model in peano's axioms, I would suggest.

We could equally start with -1,0,1,...

All that matters is the ordering. It is only if we also wish to define addition to these that we must make more specifications. But the choice of how we add things is what makes 0 the identity.

There is a perfectly valid form of addition on N that says 1+1=3, and 2+2=5.

To see this, hold up two fingers and then two more fingers and you have "1+1=3" by coutnign the spaces between them. So 0 is only special with respect to your common notion of addition.
 
  • #32
Moonrat said:
One, that read beautifully to me. Thank you for that thoughtful explanation.

If 0 is a neutral element, that clearly delineates it from 1, no? the fact that you can distinguish one binary operation from another binary operation is the same distinction I am referencing regarding the distinction between 0 and 1.

perhaps, and maybe I am being a bit bold here, but let me say that we can define 'relationships' as 'distinctions'. Things that have 'distinctions' have clearly defined borders, boundaries. Indeed, a distinction 'draws' the finite conceptual object that we model it to be.

So when I say the relationship to 0 to 1 to 2, I simply mean the observable basic distinctions to how we perceive those conceptual mathematical environments that exist with their own natural set of rules, and then I am saying that now that since I CAN make the distinctions, and the distinctions are observable to all p.o.v. in the objective sense, I want to define those distinctions mathematically and rationally.

I see what you mean above about 0 and 1. Do you understand what I mean here now, and do you see how your analogy does not apply to this particular environment?

MR
I think I'll need to clarify a few points here (hopefully, I won't be obfuscating..:wink:)

1. Concerning the distinction between "addition" and "multiplication"
Now, in order to get to grips with this, we need to focus on those rules mathematicians CHOOSE to regard as the defining characteristics of those operations.
That is, the axioms for these operations:
a) Commutative&Associative rules:
a11) The commutative law for addition
Pick any two numbers "a" and "b" ("a" and "b" need not be different numbers):
Then we say that the number called the SUM of a and b, that is a+b, is equal to the number called the SUM of b and a.
That is, for any choices of a,b the equality a+b=b+a is assigned the truth-value "true".
a12) The commutative law for multiplication
Pick any two numbers "a" and "b" ("a" and "b" need not be different numbers):
Then we say that the number called the PRODUCT of a and b, that is a+b, is equal to the number called the PRODUCT of b and a.
That is, for any choices of a,b the equality a*b=b*a is assigned the truth-value "true".
a21) The associative law for addition:
Given any numbers a,b,c, we state that the number called the sum of (the Sum of a and b) and c, is the same number as the sum of a and (the sum of b and c).
That is:
(a+b)+c=a+(b+c) is fundamentally "true"
a22) The associative law for multiplication:
Given any numbers a,b,c, we state that the number called the product of (the product of a and b) and c, is the same number as the product of a and (the product of b and c).
That is:
(a*b)*c=a*(b*c) is fundamentally "true"

b) The distinguishing rule: The distributive law.
Note that apart from different fancy words (addition, multiplication, sum, product) and fancier symbols (+,*) the laws as specified under a) does not at all make it clear that the "+"-operation is any different than the "*"-operation.
Thus, if we are to be justified in regarding these as DISTINCT operations (rather than being just different names for the same "thing") we need to specify a rule which breaks the operational symmetry under a)
We do so by specifying how addition and multiplication act together:
The distibutive law says that given any numbers a,b,c, the product of a with (the sum of b and c) equals the sum of (the product of a and b) and (the product of a and c).
That is:
a*(b+c)=a*b+a*c

Note that now we are in possession of a distinguishing tool, because, at the outset, it does not follow from the other rules that we also must have:
a+b*c=(a+b)*(a+c) for all numbers a,b,c


That is, we may distinguish "addition" and "multiplication" by choosing a particular set of axioms.

2. The distinction between 0 and 1
Again, this is not something we can deduce unaided by specified axioms, in fact, we CHOOSE to regard the statement [tex]0\neq1[/tex] AS AN AXIOM IN ITSELF.
This emphasis on choice is in fact a rather subtle, but crucial, point, in that it shows that from a logical point of view, there is nothing inherently contradictory in regarding the multuplicative and additive identities as the same number; effectively, what you'd get then is a perfectly consistent mathematics which only deals (in a consistent manner..:wink:) with a single element..(all numbers can be shown to be equal)
It is not a particularly fascinating mathematics, but it is not internally contradictory.


EDIT:
I have tried to emphasize the "CHOOSE"-word throughout this post, because it indicates that there may well exist other internally consistent systems&operations which bear very little resemblance to our "ordinary" way of thinking about numbers, and the operations we may perform upon them.
 
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  • #33
I think I'll post a few comments on what ought to be crucial if we are to perform a philosophical analysis on mathematics.
Then, I hope you agree with me, we need to represent "how mathematicians think about maths" in a correct manner!
In particular, we should pay special attention to where a mathematician's perspective differs from the layman's perspective, i.e, where we can pinpoint a conceptual shift, if you like.

Now, I assert, the ordinary way of thinking about numbers, is first and foremost to think about what numbers ARE.
This is NOT the way a mathematician thinks about it; instead of being interested in what numbers ARE, he is interested in what we may DO with them!

That is, the question of what a particular number is, unrelated to whatever numerical relations we might choose to put it in, is of little interest to the mathematician.
Rather, he focuses his attention precisely onto these relations, and couldn't care less if numbers have independent existence apart from these relations.
Indeed, that tends to be a rather meaningless issue, since we basically "bring into existence" numbers by specifying the defining relation of that number.
For example, we may introduce the symbol "2" to represent that number we gain by adding 1 to itself; i.e, the equality 2=1+1 might be regarded as the definition of the number 2
(Note: There are valid quibbles to this way of thinking, in particular, it may be convenient to introduce definitions more intimately linked to the fundamental level maths, i.e, set theory&logic).

What "2" inherently is, is an issue faded into insignificance, i.e, we may say with Hegelian terminology, "2" is sublated/"aufgehoben".

What remains therefore as mathematics proper, is the study of self-stated axiomatic systems, chosen rules of inference and operations, rather than the study of inherent properties of quantities.


This subtle shift in focus, that is to shift from speculating what numbers are, to study what numbers will "do" under a specified set of axioms is therefore, in my opinion, of the utmost importance to grasp if we are to perform a good philosophical analysis of maths.

The ROLES of "no thing", "finiteness ("one")/infinity) might be assigned to some of the players (numbers) in the logical games mathematicians devise&study, or these roles might not appear, if the mathematician chooses differently.
Thus, they are, essentially "aufgehobene Grössen", rather than the main object of study, i.e, the game itself.
 
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  • #34
Moonrat,
The main point of my questions was just to understand what you were saying. You're using familiar terms in unfamiliar ways and introducing new terms without defining them, so it's very hard to figure out what you mean. I still don't understand most of what you're saying, but here's a relationship to consider (bearing in mind that I don't really understand your terms or what you're trying to do...):
In classical logic, a proposition is tautological if its truth-value is always "Truth" (T); a proposition is contradictory if its truth-value is always "Falsehood" (F); a proposition is contingent if its truth-value can be T or F (but not both). In this way, contingent propositions seem to agree with your "mystery" concept, leaving tautologies and contradictions to be "true" or "false". However, all atomic (the smallest "unit") propositions are contingent, and all tautologies and contradictions are "combinations" of contingent propositions and connectives (i.e. truth-functions). In this way, contingent propositions seem to agree with your "true" concept, while tautologies and contradictions would both seem to agree with "false". This seems to show that "mystery", "true", and "false" can apply to the same object. I know you want to apply these concepts to numbers, but you seem also to want an object to have only one of these properties for each "POV", and this doesn't seem to hold in general.
I don't understand why you dismissed other's points about 0 and 1 having the same property with respect to addition and multiplication, respectively. If 0 has a "non-value" because x + 0 = x, then 1 has a "non-value" because x*1 = x. If "non-value" only applies to addition, why? You want a logical interpretation, so you need a logical reason for the restriction (a definition would work but still seems ad hoc). This is important for your identification of 0 with mystery and 1 with true.
Also, why do you include addition, multiplication, and divison but not subtraction? Because 0 = 1 - 1, making 0 seem to be "false"?
Moonrat said:
It seems to mean "0 is the value of x if and only if x has no value", but that's a blatant contradiction.
hmm, no more so than 0 in general, I would imagine.
The definition is self-contradictory, period. The problem has nothing to do with 0 but with x both having and not having a value. If that is your definition, then there is no object whose value is 0.
Things that have 'distinctions' have clearly defined borders, boundaries. Indeed, a distinction 'draws' the finite conceptual object that we model it to be.
Have you never watched the sun rise or set? :tongue2: I can distinguish between night and day, but the transitions between them are not clearly defined, but perhaps this isn't what you meant.
Also, if you want to be liberal in your interpretation, you get 1 and 2 simultaneously. The very concept of distinction implies 2 objects: x and not x, each of which is 1 object.
Anyway, you want to consider a certain relationship between 0, 1, and 2. As distinct numbers, they are indistinguishable (in that they are all distinct numbers). The properties you want them to have seem to rest on order and addition, does that seem right?
 
  • #35
Hehe, me and my big mouth! okay, it is probably going to take me a bit for me to be clear about where I follow what the three of your are saying, and where I get lost in confusion. However, I do like the switch around, usually, I am used to understanding people, but them not understanding me..now I get my dream come true, you can understand now me, but I am not so sure I understand you!

Really, it makes me wish the world was run by mathematicians, you all are so neat and orderly and elegant.

So, give me a bit to digest this, and let me respond to this one little point to keep this moving a bit forward...

To be honest, this is a bit of a dream come true, you all are providing me with extraordinary feedback...


honestrosewater said:
The properties you want them to have seem to rest on order and addition, does that seem right?

Yes, that does 'seem' right. And I of course am referencing the 'common' way numbers are referenced. But what I am saying is that there is a pattern to the ‘common way’ and this must be able to be parsed.

I am really only dealing with two themes in ONE set, Finite, and infinite. Naturally, this is an infinite set. This is merely describing the players on an infinite number line. Now, keep in mind that how I define finite and infinite may NOT be how it is defined in mathematical principles, however, I do want to define or parse my definitions into mathematical elements. I define 'infinity' not necessarily as an 'eternal' or anything metaphysical, but rather as something that may have a beginning point, just no ending. It is merely 'continuous'. Thus, 0 and 2 represent the 'continuum' in which the only thing continuing is the 1. it is almost like verb to noun. the noun is the finite 1, and the verb is the 'continuing' or 'continuum', expressed as 0 and 2.

I don’t know how to define 'infinite' other than 'continuous'.

At first, finite and infinite appears as a bivalent set, however, what I then define is that there is finite (1), and finite is expressed or perceived via two modes or basic perceptions, both infinite and not finite, 0 and 2. this makes for 'continuous' combination of 1. It is not that the finite 1 does not exist in the set of 2 through infinity and 0, it is that in the environment that is continuous inside of the set 2 through infinite, '1' is the only 'real' thing that is continuing, and '2 through infinity' are merely 'abstractions of the 1 abstraction' that we use for convenience sake. Like you said

Have you never watched the sun rise or set? I can distinguish between night and day, but the transitions between them are not clearly defined, but perhaps this isn't what you meant.

YES! However, the signifiers ‘night’ and ‘day’ clearly distinguish them! We do this for simplicity’s sake. Thus, the bivalency of night and day create a new distinction of ‘sunset and sunrise’. Sunset and sunrise, as you can ‘see’, is both ‘night and day at once’.

The basic patterning of distinction is; this, that, and that which is both this and that (the meta signifier)

Also, if you want to be liberal in your interpretation, you get 1 and 2 simultaneously. The very concept of distinction implies 2 objects: x and not x, each of which is 1 object.

YES! It is implied…hmmm..perhaps that is what I ‘mean’, it is the ‘implication’, built in, to the numbers 0, 1, and 2. 0 implies ‘no 1 or combination of 1 here’. 1 implies only itself, ‘ I am here’. 2 through infinity are always just implying ‘some combination of 1 here’.

You are most certainly correct, The very second we have ‘one’, we get two distinctions, which we can call ‘subject and object’ if you wish. Now, that is the ‘basic’ of distinguishing perception. The next step is to see how when we get 1, we really get 3 distinctions. 0, nothing, or mystery existed before the 1st distinction, however we could not perceive it or distinguish it yet! And not only that, but we don’t perceive it even though it is ‘here’ before we make our next distinction!

The 0 is always ‘implied’ by the 1, whether we recognize that or not.


And I wish to only suggest that what I am observing or distinguishing is essentially, 'common sense'. It's not rocket science, I don’t expect you to go 'wow, that is some pretty advanced formulae!' however, it is profound...and rational, and very very simple.

Essentially, to give the three of you a bit more idea of where this leads to, is I have been developing this dialectic that is viable for internet discussion. It has a crude method of creating a non zero sum 'environment' in conflict to arrive at win win in the ‘conflict of idea’. So it's like a game, a game of perception, and how we classify all ideas we encounter into the discussion (o, 1, and 2). All players are competing for the 1 how 1 is defined in the dialectic.

Although it is logical and rational, it is not logical in the sense that 'tractucus' is logical and only appeals to those with such a philosophical bent, but rather in the same manner that 'checkers' is a logical game with defined moves, kings, etc etc. So it has 'common sense' appeal.

It is still in crude form, but the bottom line that it's 'crude form' at the foundation is very sound and rational, and very novel and unique. the trick of the dialectic is once you have the simple principle, then you apply it to your perception of idea, even the dialectic, and 'game' conflict in a profound new way. It has strategy, moves, all kinds of things. I have a strong feel for applying it, but describing many of it's principles into purely rational language (math or symbolic) eludes me.


this play on perception made my professor friend comment both in the beginning of our chat and at the end of our chat that what I was describing was, in his opinion, touching upon principles used in quantum computing. To me, that sounds, sure, really 'cool' sounding, however, I don’t know what that really means, or if that helps any of you or not.

I can accept if this is what matt was refrencing me 'personalizing', however, what I am saying is that there is a pattern in the 'personalization' and the 'human element', from the p.o.v. of how we ‘naturally’ perceive.

Okay, you three got my head spinning, let me go back and reread those posts of yours about a hundred more times…

MR
 
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