By ZF set theory we know that {a,a,a,b,b,b,c,c,c} = {a,b,c}

It means that concepts like redundancy and uncertainy are out of the scpoe of set's concet in its basic form.

When we allow these concepts to be inherent properties of set's concept, then we enrich our abilities to use set's concept, for example:
Code:
<-Redundancy->
c   c   c  ^<----Uncertainty
b   b   b  |    b   b
a   a   a  |    a   a   c       a   b   c
.   .   .  v    .   .   .       .   .   .
|   |   |       |   |   |       |   |   |
|   |   |       |___|_  |       |___|   |
|   |   |       |       |       |       |
|___|___|_      |_______|       |_______|
|               |               |

Where:

c   c   c
b   b   b
a   a   a
.   .   .
|   |   |
|   |   |  = {a XOR b XOR c, a XOR b XOR c, a XOR b XOR c}
|   |   |
|___|___|_
|

b   b
a   a   c
.   .   .
|   |   |
|___|_  |  = {a XOR b, a XOR b, c}
|       |
|_______|
|

a   b   c
.   .   .
|   |   |
|___|   |  = {a, b, c}
|       |
|_______|
|
I think that any iprovment in set's concept has to include redundancy and uncertainty as inherent proprties of set's concept.

The above point of view leading me to what I call Complementary logic, which is a fading transition between Boolean logic (0 Xor 1) and non-boolean logic (0 And 1), for example:

Number 4 is fading transition between multiplication 1*4 and

These fading can be represented as:
Code:
(1*4)              ={1,1,1,1} <------------- Maximum symmetry-degree,
((1*2)+1*2)        ={{1,1},1,1}              Minimum information's clarity-degree
(((+1)+1)+1*2)     ={{{1},1},1,1}            (no uniqueness)
((1*2)+(1*2))      ={{1,1},{1,1}}
(((+1)+1)+(1*2))   ={{{1},1},{1,1}}
(((+1)+1)+((+1)+1))={{{1},1},{{1},1}}
((1*3)+1)          ={{1,1,1},1}
(((1*2)+1)+1)      ={{{1,1},1},1}
((((+1)+1)+1)+1)   ={{{{1},1},1},1} <------ Minimum symmetry-degree,
Maximum information's clarity-degree
(uniqueness)
============>>>

Uncertainty
<-Redundancy->^
3  3  3  3  |          3  3             3  3
2  2  2  2  |          2  2             2  2
1  1  1  1  |    1  1  1  1             1  1       1  1  1  1
{0, 0, 0, 0} V   {0, 0, 0, 0}     {0, 1, 0, 0}     {0, 0, 0, 0}
.  .  .  .       .  .  .  .       .  .  .  .       .  .  .  .
|  |  |  |       |  |  |  |       |  |  |  |       |  |  |  |
|  |  |  |       |__|_ |  |       |__|  |  |       |__|_ |__|_
|  |  |  |       |     |  |       |     |  |       |     |
|  |  |  |       |     |  |       |     |  |       |     |
|  |  |  |       |     |  |       |     |  |       |     |
|__|__|__|_      |_____|__|_      |_____|__|_      |_____|____
|                |                |                |
(1*4)            ((1*2)+1*2)      (((+1)+1)+1*2)   ((1*2)+(1*2))

4 =                                  2  2  2
1  1                        1  1  1          1  1
{0, 1, 0, 0}     {0, 1, 0, 1}     {0, 0, 0, 3}     {0, 0, 2, 3}
.  .  .  .       .  .  .  .       .  .  .  .       .  .  .  .
|  |  |  |       |  |  |  |       |  |  |  |       |  |  |  |
|__|  |__|_      |__|  |__|       |  |  |  |       |__|_ |  |
|     |          |     |          |  |  |  |       |     |  |
|     |          |     |          |__|__|_ |       |_____|  |
|     |          |     |          |        |       |        |
|_____|____      |_____|____      |________|       |________|
|                |                |                |
(((+1)+1)+(1*2)) (((+1)+1)+((+1)+1))  ((1*3)+1)        (((1*2)+1)+1)

{0, 1, 2, 3}
.  .  .  .
|  |  |  |
|__|  |  |
|     |  |
|_____|  |
|        |
|________|
|
((((+1)+1)+1)+1)
Multiplication can be operated only among objects with structural identity .

Also multiplication is noncommutative, for example:

2*3 = ( (1,1),(1,1),(1,1) ) or ( ((1),1),((1),1),((1),1) )

3*2 = ( (1,1,1),(1,1,1) ) or ( ((1,1),1),((1,1),1) ) or ( (((1),1),1),(((1),1),1) )

More about the above you can find here (the first 9 lines defined by Hurkyl):

http://www.geocities.com/complementarytheory/ET.pdf

More about Complementary logic, you can find here:

http://www.geocities.com/complementarytheory/CompLogic.pdf

http://www.geocities.com/complementarytheory/4BPM.pdf

Organic

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Adding redundancy into the basic definition of a set doesn't give us any new abilities though. Using sets without redundancy, we can construct sets with redundancy.

Whether redundancy is an intrinsic property, or one that we add afterwards doesn't have any relevance.

I think it is relevant because when we use concepts like redundancy, uncertainy, symmetry-degree, information's clarity-degree and complementarity, as fundamental set's properties, it immediately effcts on all things that using set concept as their building-block.

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Hurkyl
Staff Emeritus
Gold Member
it immediately effcts on all things that using set concept as their building-block.

Only in the sense that you have to rebuild all of them from scratch.

And, incidentally, the things you seem to be focused on studying are things that don't use sets as their building block. In particular, the theories of natural numbers and real numbers.

But using your "enhanced" sets we can also construct the more "traditional" sets. So your enhancments don't affect objects that are usually constructed with the more traditional concept of a set.

If "traditional" sets are private case of "enhanced" sets , then we can still use use "traditional" sets if we want to, but our abilites of using sets are now beyond "traditional" abilities.

Hurkyl
Staff Emeritus
Gold Member
And we can build enhanced sets using just traditional sets, so we have gained no ability we didn't have before.

Hi Hurkyl,

And, incidentally, the things you seem to be focused on studying are things that don't use sets as their building block. In particular, the theories of natural numbers and real numbers.

This is one of the new abilities of the new set theory, to conncet between set and number concepts in a very simple way, which its result can be very complex (but not complicated).

1) http://www.geocities.com/complementarytheory/GIF.pdf

2) http://www.geocities.com/complementarytheory/LIM.pdf

3) http://www.geocities.com/complementarytheory/RiemannsBall.pdf

4) http://www.geocities.com/complementarytheory/RiemannsLimits.pdf

5) http://www.geocities.com/complementarytheory/CL-CH.pdf

6) http://www.geocities.com/complementarytheory/CompLogic.pdf

7)http://www.geocities.com/complementarytheory/4BPM.pdf

Organic

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I should point out, that traditional constructions such as constructing $\mathbb{N}$ using $0=\varnothing$ and $a+1=a\cup\lbrace a\rbrace$ do not tell us anything about the intrinsic properties of numbers.

For example, real numbers are not in fact Cauchy sequences of rational numbers. And zero is not in fact the empty set.

These proofs are only relative consistency proofs.

So studying set theory doesn't tell us anything about natural numbers. Replacing ZFC set theory with another theory won't change the properties of other number systems.

Dear master_coda,

Thank you.

Organic

I recognize that you've made some effort to include the more rigorous concepts that've been discussed here. But you have to stop explaining your ideas by throwing more and more terminology at them, and making more and more pdfs.

Look at the posts from pheonixtooth. His ideas are somewhat grandiose and vague, when he states them in words. But he then when he tries to state his ideas mathematically, he does it using standard mathematical terminology.

what is zero, then? are you saying there is a definitive definition of 0? i know for a fact that the empty set isn't the only definition...

in other words, you should define what your terms mean.

for example, 0 can be defined as Ø vs 0 can be defined as the qualia of emptiness. the second definition is more like how you define things like uncertainty and information clarity degree, iow, not mathematically rigorous though intuitively clear. the goal is to obtain a rigorous framework for such ideas and then dispense with the foundation and actually put it to use. to satisfy the mathematician, the foundation needs to exist.

using words i can find in www.mathworld.com,[/URL] define things such as redundancy (which i see as being some class function whose domain is the class of all sets and range is the natural numbers), uncertainty, etc. i'd recommend defining one word at a time and posting it to be revised and critiqued by us.

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Dear master_coda,

Please let me put it this way:

1) I am not a professional mathematician, therefore I don't know how to express my ideas in the standard mathematical terminology.

2) Moreover, maybe this is the reason why I can see things differently form well educated professional mathematicians, which, in general, is not bad (in my opinion).

3) Because I see things not in their common way, I may develop tools, which are not standard, to explore and research my personal point of view on what is called accepted mathematical definitions.

So, there is some paradox, because sometimes to see things in a non-traditional way, is easier to a non well-educated person, but then it is very hard to put his new point of view in an understandable format.

Maybe this original point of view is so fundamental, that we have no choice but to change our paradigm about what we call well-defined terms.

I'll appreciate any cooperation that will result in rigorous mathematical definitions.

Yours,

Organic

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Dear phoenixthoth,

what is zero?
I think this is a beautiful question, because we can learn a lot of thinks out of nothing.

I think to give deferent definitions to the same object is a legal mathematical attitude.

If you don't agree with me, then please tell me why it can't be a legal mathematical attitude.

Some examples:

1) |{}| = 0 is a quantitative point of view on the emptiness concept.

2) x - x = 0 is the "balanced" point of view of some system.

3) 0^0 = 1 is a structural point of view of the continuum concept

4) x=0,y=0,z=0,w=0,... is the absolute coordinate of n-dimensional system.

5) 0 is the south pole of Reimann's ball (the opposite of oo XOR -oo and the middle of -oo AND oo).

6) (0,oo) is a quantitative point of view of any one of our positive real numbers.

7) Also (0,oo) can be the open interval of any segment existing between 0 and oo .

8) 0 is an even number.

9) 0=false 1=true

10) '0' = jump on this place in '0.0010...' but remember its power.

11) By Complementary Logic ( http://www.geocities.com/complementarytheory/CompLogic.pdf )
0 = |{oo ... nor <--> Emptiness <--> nor <--> Emptiness <--> nor ... oo}|

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Organic, if you want to talk math you have to use at least some standard terminology. That doesn't mean you have to use terms from higher mathematics. Indeed, I wish you would stop using terms from higher math, because you aren't using them correctly and it only makes your ideas more confusing.

For example, when non-Euclidean geometry was first being explored, it wasn't a popular idea. It went against the philosophy of math that was accepted at the time.

But that didn't mean that the proponents of non-Euclidean geometry did things in a non-mathematical way. It was still a mathematical way, just a way that no one had ever thought of before.

What I'm trying to say is that if your ideas are so different from math that you can't use anything from contemporary math to describe them at all, then your ideas probably aren't mathematical. That isn't necessarily a bad thing...there are a lot of good things that aren't math. But there isn't much that mathematicians can do to help you develop your ideas.

Hurkyl
Staff Emeritus
Gold Member
Some examples:

Which of those examples is supposed to be a definition of zero?

Dear master_coda,

Please be more specific, take for example: http://www.geocities.com/complementarytheory/GIF.pdf

Please read all of it and then return and find any thing that you can't understand mathematically, and please write your remarks on it.

Thank you.

Organic

Dear Hukyl,

This is the all point, in my opinion there cannot be one definition for Zero, and any definition of it depends on the way it is being used through some system.

Hurkyl
Staff Emeritus
Gold Member
As far as I can tell, none of those statements could be construed to be a definition of zero.

I'm not even sure what point you're trying to make, though; of course the same entity can have multiple definitions. The point is, though, that the definitions are equivalent; so that no matter which definition you use, you can prove all of the others are still true.

If you have inequivalent definitions, then you're talking about different concepts (such as zero the natural number and the zero vector in R^2)

Ok Hurkyl,

Originally posted by Organic

What zero do you want? Zero the natural number? Zero the vector in R^3? The zero matrix?

In general, the zero element of a set is the additive identity. But that isn't really a rigorous definition...just more of a standard guideline. To get a rigorous definition, I would need to know what zero you want me to define.

Originally posted by Organic
Please be more specific, take for example:

http://www.geocities.com/complementarytheory/GIF.pdf

Please read all of it and then return and find any thing that you can't understand mathematically, and please write your remarks on it.

One example of a problem:

You talk about the open interval ({},{__}). But you aren't using the standard definiton of an open interval, and you don't explain what your definition of an open interval is.

Another example:

You talk about the statements "for all x" and "for any x" as if they are two different statements. These are quanifiers from first-order logic. But they're both the same quantifiers. But not according to you.

That tells me that you aren't even willing to use first-order logic, and that you want to replace it with your own form of logic. Fine. But you have to explain what logic you are using. And you can't explain it using "sets" and other more complex objects, since we need logic before we start defining other objects.

So before you can define anything, you have to provide us with the rules of inference that we are allowed to use in your system of logic. And we need to know what connectives and quantifiers we can use. And you can't explain them by making up more terminology, and throwing around big words.

Hi master_coda,

One example of a problem:

You talk about the open interval ({},{__}). But you aren't using the standard definition of an open interval, and you don't explain what your definition of an open interval is.
I use open interval in its standard meaning, which is: {} and (__} are out of the scope of anything that exist between them.
You talk about the statements "for all x" and "for any x" as if they are two different statements. These are quantifiers from first-order logic. But they're both the same quantifiers. But not according to you.
You are right, because from my point of view, which is "General Information Framework" (GIF) point of view, infinitely many objects cannot be connected to words like "all" or "complete" because they exist in the open interval ({},{__}) where {} AND {__} cannot be reached simultaneously.

In other words: [{},{__}] = no information.

Another point of view:

If 'all x' means 'all not_x is left out' then it means that x must be complete, but infinitely many x cannot be completed otherwise they are finitely many objects.

The word 'any' is good for both finitely or infinitely many objects.

What zero do you want? Zero the natural number? Zero the vector in R^3? The zero matrix?

Organic

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a very nice

Truth

A speech given by L. Sue Esch,
in conjunction with receiving the Beachley Distinguished Teaching Award
at Juniata College

"Veritas Liberat". Truth sets free. It's more than just a motto; it's what we're all about. Education could easily be defined as "the pursuit of truth". The problem is "what is truth?". A sticky question, to say the least, but you would think that if anyone could get a handle on it, it might be a mathematician. After all, the basic building block of mathematics is logic; the basic component of logic is the proposition; and the very definition of a proposition is "a statement which is either true or false". We also have the Law of the Excluded Middle to guarantee that a statement must be true or false and the Law of Contradiction so that it can't be both. Surely, we must know the difference -- or at least have ways of determining it. In applied mathematics we seek to determine new truths (in the form of equations) about the relationships among physical objects. In theoretical math, the "objects" become more abstract, but truths about relationships are still the goal. In both, we must not only determine the relationships but be able to prove that they are in fact truths. In any such proof, we assume axioms to be true and deduce conclusions or theorems which are then said to be true. It certainly seems that truth is an essential ingredient in mathematics -- but you have to watch out for that word "true" in mathematics. The truth is that there are probably few disciplines more ambivalent about its meaning. As Bertrand Russell said, "Mathematics is the subject in which we do not know what we are talking about nor whether what we say is true." Despite the fact that most of you agree wholeheartedly with Russell, and have probably been searching ever since kindergarten for such a quote to confirm your suspicions, let me backtrack a bit and tell you how mathematics arrived at such a state and what it all might mean to those of us committed to education and the pursuit of truth.

In the beginning, truth for mathematics was the same as truth for science; it was empirical in nature. A statement was true if it accurately described the physical world. If one needed to know the area of a rectangular plot of land, one just measured the lengths of the sides and multiplied. Therefore the equation, "the area of a rectangle equals the length of the base times the length of the height" was deemed a truth for the simple reason that it worked. Mathematics at this point was totally applied; it was a science, concerned with relationships among physical objects. Then along came the Greeks, who insisted that mathematical truth be more than empirical truth. It was no longer sufficient for an equation to merely work; one must be able to prove that it works. One must be able to logically deduce it from first principles, called axioms. With this new emphasis, mathematics entered a new era, an era in which its job was not only to unearth empirical relationships about the physical world but also to prove them to be truths.

This view of mathematics continued until the eighteen hundreds with the discovery of non-Euclidean geometry. As opposed to Euclidean geometry, where parallel lines are everywhere equidistant, in non-Euclidean geometry, parallel lines either do not exist at all or if they do, they are "curved" so that the distance between them varies. This may seem strange to us, but then there are no doubt many things in mathematics that seem strange. The real problem arose in that although one could demonstrate that non-Euclidean geometry was logically deducible, it wasn't empirically consistent. Mathematicians could prove the logical deducibility of parallel lines that "curve" and even come arbitrarily close to one another, but everyone knew that empirically, in our physical world, parallel lines are "straight" and remain equidistant. After all, the physical world is Euclidean, or at least in the eighteen hundreds it was. There wasn't any doubt about it. The scientists assumed it, and the best mathematicians had spent 2000 years devoted to proving it. Later, with Einstein and relativity, would come "curved space" and the discovery that the Euclidean model of the world wasn't quite so obvious after all. But, for the nineteenth century mathematician, the discovery of non-Euclidean geometry was extremely problematic. It was "true" because you could logically deduce it, but could not be "true" because it did not accurately describe the physical world. Clearly, mathematics had to redefine its notion of truth. It had to choose between empirical consistency and logical deducibility. It chose logical deducibility. A new era. Mathematics ceased to be a science in the physical sense of the word. Its game became logical deduction and proof. A statement would be true if and only if it was a theorem, i.e., could be proved.

However, nothing ever turns out to be quite as simple as we hope. The first problem that arose should not surprise anyone. It had been there since the Greeks, lurking just beneath the surface. A statement is true if and only if it can be proved, i.e. logically deduced. But, logically deduced from what? First principles, axioms. But where do they come from? For the Greeks, that was not a problem. Their mathematics had been fundamentally empirical. Their first principles were merely self-evident statements about the physical world, statements so basic that everyone accepted their truth. Who could deny that "two points determine a line" or even more basic, that "equals added to equals are equal"? But when non-Euclidean geometry forced the nineteenth century mathematicians to discard empirical consistency in favor of logical deducibility, mathematics had a problem. Assuming axioms or first principles and deducing theorems, and then claiming that the truth of the theorems follows from the truth of the axioms, in no way deals with the truth of the axioms. The classic approach is to assume it. But, on what basis? Well, the primary criterion for a statement to be an axiom in the first place is that it be self-evident. But, self evident to whom? You can see that this quickly becomes a philosophical quagmire. In the shift from empirical to logical, "truth" had lost most, if not all, of its meaning. In fact, if truth be known, mathematics rarely uses the word "true". Rather it uses "deducible" or "provable". It leaves truth to the philosophers. "Fools rush in where angels fear to tread."

But, can mathematics totally avoid any mention of truth. What about those propositions and the Law of the Excluded Middle, which demanded that "statements must either be true or false"? O.K. We can deal with that. If "truth" now means "provability", we can just translate and demand that "statements must either be provable or disprovable." But, again, it never turns out to be that simple. Enter Godel. 1931. What he did was prove that there exist statements which are neither provable nor disprovable. How he did this is rather interesting. I hope you like mind games...

Somewhere, you have probably encountered The Liar's Paradox, in one version or another. At its center lies the statement S which says about itself "this statement is false." There's a Catch-22 here though. Is the statement S itself true or false? If S is true, i.e., "this statement is false" is a true statement, then since "this statement" refers to S itself, what we have is "S is false" is a true statement, or just "S is false". In other words, if S is true, it follows that S is false. Oops, that can't be! So, since assuming that S is true leads to a contradiction, the only alternative is that S must be false. But if S is false, i.e., "this statement is false" is a false statement, then "this statement" S must be true. Oops, again. In short, S can't be either true or false, but, by the Law of the Excluded Middle, it must be. Catch-22. We could refuse to consider S, claim that it isn't a legitimate proposition, but we have no justification for doing that other than avoiding the consequent paradox, which hardly seems sporting. So, what do we do? What else! Let the philosophers worry about it! After all, it doesn't have anything to do with numbers or equations. For most mathematicians, it was and is just a cute little oddity -- probably not a paradox at all, merely an anomaly. For others, however, it raised some serious questions -- first about the foundations of logic and then, given the close tie between the two, the foundations of mathematics. And, even though the paradox can be translated into set theory, still for most working mathematicians, the source of the problem seems to lie more in the realm of logic and philosophy than mathematics.

continue

What Godel did was to bring it home! Essentially he translated the paradox into mathematics, replacing "truth" with "provability". First, he translated each mathematical symbol into an integer, then each axiom (or string of symbols) into a string of integers which was nothing but a bigger integer. Next, he translated the rules of logical deduction into arithmetic rules for deriving new integers (representing theorems) from old ones (representing axioms). The net effect was that, unlike truth, "provability" became as mechanical as adding and multiplying integers. If you began with only the integers representing axioms and used only the arithmetic rules representing logical rules of deduction, then whatever integers could be produced must represent provable statements, i.e., theorems. The final coup was the arrival at a statement G which said about itself "G is not provable." Now, as in the Liar's Paradox, ask yourself whether G itself is provable or disprovable. If G is provable, i.e., you can prove the statement "G is not provable", then G must not be provable. Oops, here we go again. If, on the other hand, G is disprovable, i.e. you can disprove the statement "G is not provable", then G must be provable. Oops, again. Our old Catch-22. But what does all this mean? Well, "truth" was supposed to be "provability". However, in G we found a statement which must be true or false by the Law of the Excluded Middle, but, following our stellar reasoning, can be neither provable nor disprovable. Therefore, "truth" can't be the same thing as "provability". And if that weren't enough, another even more troubling consequence arises if we give this whole thing a slightly different twist. Let's start again with Godel's statement G that "G is not provable." This time, appealing directly to the Law of the Excluded Middle, we know that G must either be true or false. If G is false, i.e., "G is not provable" is a false statement, then G must be provable, and since if you can prove something, it follows logically that it must be true, we have again shown that if G is false, then G is true -- which can't be. And as before, since it can't be false, it must be true. But now we're really in trouble -- because what we have is a statement which is true but not provable! In other words, what Godel established was that in mathematics, there will always be true statements which are not provable. To put it mildly, this set the mathematical world on its ear. To begin with, it once again demonstrated conclusively that truth cannot be the same thing as provability. But, more fundamentally, in one ingenious stroke, it established that mathematics not only never will but never can prove all true statements, even arithmetical ones. There will always be truths which are not theorems. In short, it put truth out of the reach of mathematics. In the words of Hofstadter in Godel, Escher, Bach, "truth transcends theoremness." Therefore, if anyone can be expected to answer the question "what is truth?", it won't be a mathematician.

O.K., now that I have shattered whatever faith you may have had in mathematics, what does all this have to do with education and the pursuit of truth? On a trivial level, it certainly guarantees that I will never advocate that Juniata students take 120 credits of mathematics to graduate. On a not so trivial level, it should raise some questions about the basic tenets of the western intellectual tradition, upon which our whole educational system is founded. If "truth transcends theoremness", doesn't it seem rather futile to conduct a search for it within the confines of such a rationally based tradition? The answer is an unequivocal "yes and no". Remember, many truths do not transcend theoremness. Reason and rationality have in fact brought us a long way in the pursuit of truth. Granted, Godel showed that they are not sufficient to go the distance, but it took some pretty fancy footwork. In mathematics the vast majority of truths are theorems; they can be proven. It was only when we made a statement refer to itself ("This statement S is false." "This statement G is not provable.") that we ran into trouble. So, the lesson is certainly NOT to abandon rationality, only to realize its limitations. Number one, we must recognize the basic tenet that some truths are and will remain beyond the reach of rationality and theoremness. Number two, even those "truths" that appear to be within their grasp are at best conditional. They are deducible from axioms or assumptions, and therefore our acceptance of their truth depends on our acceptance of the truth of the axioms. And since each discipline, if not each individual, advances its own axioms and often its own rules for deduction, any kind of universal acceptance seems almost unthinkable, even in the most harmonious of communities. This, at the very least, suggests the need for careful communication. We will undoubtedly never share each other's truths. We cannot even agree where and how to search for them. But if we can, through thoughtful communication, somehow come to recognize and respect alternate avenues of exploration, perhaps acceptance becomes a bit more thinkable. But, as even we at Juniata have discovered, communication alone is not enough. Plans and reports do not necessarily lead to understanding and community. We also need TRUST. Trust that there exist truths beyond theoremness, truths that can neither be determined by nor irrefutably proven by rigorous rational deduction. Trust that others' truths follow from their axioms, even though the passage may seem strange to us. And especially trust that different and seemingly contradictory axioms systems can be equally valid. If this last one seems particularly unattainable, we need only return to mathematics for inspiration. Euclidean and non-Euclidean geometry would seem to provide prime examples of systems which are not only different but contradictory. Far from it. Not only are they not contradictory and mutually exclusive, but mathematics has shown that logically, they stand or fall together.

So perhaps mathematics is not as much of a lost cause as we originally thought. It may not be able to totally capture truth, but it can certainly teach us a thing or two, not only about theoremness, but also about the importance of communicating our assumptions and modes of thought, trusting alternative realities, and even recognizing the limits of our own self-imposed way of looking at the world. Since for the Greeks, mathematics was geometry, perhaps it is more appropriate than traditionally recognized that above the entrance to Plato's Academy was inscribed the maxim, "Let no one ignorant of geometry enter this door."

4/30/91

Hi Moshek,

I think it will be a better idea if you give only the address of this article, and let people to read the original text.

Hrere you can write your personal opinion on it, or evan better, you can open your own thread on this inteserting subject.

Yours,

Organic