The discussion centers on the concept of non-commutative natural numbers, specifically through the construction of a set C that combines natural numbers with rooted finite trees. The participants debate the validity of defining operations on these elements, arguing about the implications of these operations being non-commutative and potentially non-associative. One viewpoint emphasizes that traditional natural numbers represent only a limited case of broader structural and quantitative forms, while another insists that their construction adequately captures the essence of natural numbers. The conversation highlights the need for clarity and rigor in mathematical definitions, particularly when introducing new concepts that challenge established paradigms. Ultimately, the dialogue reflects a deeper philosophical inquiry into the nature of numbers and their representation in mathematics.
In semi-response to Organic's post I thought I'd half take up one of his challenges:
Let S = NxT be the product of the natural numbers, N, with the set of all rooted finite trees (or directed graphs satisfying the obvious conditions), embedded in the plane, with the natural ordering on the branches/leaves.
Let C be the subset of all possible { (n,t) | n in N, t a tree with exactly n edges}
We define an operation + on elements of c:
(n,t)+(m,r) = (n+m,s) where s is the tree obtained by gluing the tree r onto the end of the first leaf.
The subset (n,t) with t the trivial tree with 1 leaf and n edges, forms a copy of N under addition.
We define * to be (n,t)*(m,s) by
(((...(((n,t)+(n,t))+(n,t))+...)+(n,t))
where there are m-1 addition signs.
again (n,t) with t the tree with n edges and 1 leaf, is a subset that forms a copy of N under multiplication.
neither + nor * are in general commutative, and I doubt they are associative either, but I can't be bothered to check, they are both well defined binary operations from CxC to C.
Now shall I claim that C is the new non-commutative natural numbers or not?
but he's already using the naturals inside his definition to define his "new naturals" there is nothing there to suggest these new objects should replace the natural numbers.
So, you don't understand that the standard natural number is a trivial private case of infinitely many structural/quantitative information's forms that ignored by Standard Math paradigm.
Do you get it?
A NUMBER is first of all an information's form, and to understand this we MUST explore our cognition's abilities to define this information's form, as I do here:
Originally posted by Organic So, you don't understand that the standard natural number is a trivial private case of infinitely many structural/quantitative information's forms that ignored by Standard Math paradigm.
Do you get it?
A NUMBER is first of all an information's form, and to understand this we MUST explore our cognition's abilities to define this information's form, as I do here:
When you, in your theory, assign these trees to certain sets, how do you decide which belong in the same set? You do it by counting the things involved in the construction, therefore you are explicitly using the counting numbers. Otherwise how did you decide how to group certain things together?
Nothing to say about my construction here then? This is the answer to your request ot show you something that behaves like your structures do. The difference is I don't make any large and unjustified claims about what I'm doing.
#5
moshek
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Well I am still here.
Organic:
Matt open a great opportunity here
Please see this as a real challenge for you.
I'm not in the slightest against you putting extra structure on the natural numbers, I just would like you to realize that you don't need to, and in fact you are requiring their 'non-existent' quantity only property when you describe the extra structure. The natural numbers come only as the idea of quantity, even their additive and mulitplicative structure arises from this.
It is not your ideas that annoy me; it is your mistaken assertions about them and mathematics.
The philosophical aspects of it are of minimal importance, especially until such time as you even begin to demonstrate you understand what you write.
Here let me demonstrate a rigorous approach using the things I defined above.
Define the complexity of (n,t) to be the number of leaves of t - this is equal to the number of paths from the root to the end of a branch, by the definition of tree.
If all paths from root to branch have the same length, let us define the symmetry thus:
Each set of consecutive leaves, take the smallest subtree containing them, call this a cutting. If the tree can be separated into r isomorphic cuttings that only intersect at the root, we say it has symmetry degree r.
A tree with different length paths we call asymmetric.
Characterization of N inside C: it is the set of elements of symmetry degree 1 with path lengths maximal among (n,t) for each fixed n.
Moreover it is equal to the subset of elements of complexity 1.
That's how you write maths that other people can understand. You use terms everyone knows, or can look up knowing it will agree with your terminology. Anything new you define so that it is unambiguous.
If we denote the degree of (n,t) as n.
Then n is prime iff all trees of degree n are of complexity 1 or asymmetric. How's that for you? Is this the kind of thing you had in mind when you cahlleneged us to find some ordinary mathematics that does what yours does? As you've never even shown us what yours does of course it's a little tricky.
#7
Organic
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0
Matt Grime,
Nothing to say about my construction here then? This is the answer to your request ot show you something that behaves like your structures do. The difference is I don't make any large and unjustified claims about what I'm doing.
Your construction has nothing to do with my natural numbers because,
by your definitions, uncertainty AND redundancy are ignored and also the complementary relation between multiplication and addition.
To understand it better please look at my ET's and find by yourself
that the Equations Trees can be changed by their structure, when quantity remains unchanged: http://www.geocities.com/complementarytheory/ET.pdf
Please show me this property in the standard natural numbers, where each change is a quantity change (because any change in no-redundancy_no-uncertainty information form immediately changes its quantity).
I don't make any large and unjustified claims about what I'm doing.
You right because you don't know what you are doing.
Let S = NxT be the product of the natural numbers, N, with the set of all rooted finite trees (or directed graphs satisfying the obvious conditions), embedded in the plane, with the natural ordering on the branches/leaves.
Why you so afraid to understand that the "quantity-only" natural numbers are only partial information of the structural/quantitative information that I show in my system?
Do you really cannot understand the paradigm shift the QM gave to the scintific world?
Each one of these structural-quantitative products is unique, therefore can be used as a building-block for much more interesting and richer information form, then your “quantitative-only" unique [n] result, which is nothing but a private-case of no-redundancy-no-uncertainty structural-quantitative product of my number system.
This paradigm's shift, does not exist in the basis of Standard Math language, because Boolean Logic or Fuzzy Logic are private cases of what I call Complementary Logic, that an overview of it can be found here: http://www.geocities.com/complementarytheory/BFC.pdf
Through my point of view Natural numbers are complementary elements, based on discreteness(particle-like)-continuum(wave-like) associations.
The information structure of the standard Natural numbers, is only a private case of these associations, for example:
I defined uncertainty and redundancy for my trees, which are not just quantity. Given any quantity, which I called degree there are many elements of that degree. We can label [n] as all the trees of degree n.
Show that my construction has less right to be called the correct interpretation of the new QM paradigm of mathematics? I've got uncertainty, redundancy, quantity, and operations that I call complmentary addition and multiplication. So why am I wrong and you right? They're all based on discrete and continuous constructions such as nodes as branches which encompass all the requirements of the continuum and such.
So I now contend that my number system is the correct one, yours but a pale attempt to obtain this level of complexity and accuracy.
Demonstrate I am wrong.
#11
Organic
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0
Matt,
It is very simple to show that you don't understand my system, for example:
Please find the unique labels of {1,1,1,1}.
Be aware that what you call a function is first of all a reflection of your memory.
It is very simple to show that you don't understand my system, for example:
Please find the unique labels of {1,1,1,1}.
Be aware that what you call a function is first of all a reflection of your memory.
What did I call function?
I'll will call the tree (4,t) where t is the tree with just the root node and 4 leaves from it) the element (1,1,1,1), or perhaps I want the trivial tree of degree 4? I can't remember what you're claiming (1,1,1,1) represents, tell me and I'll tell you which one it is.
I am not claiming my system is your system, but it that it has all the features of your system, with the added bonus of containing a genuine copy of N inside it, as well as a non-commutative deformation of it. And therefore has just as much right to be called the correct system of the 'natural numbers' of mathematics. It contains all possible tree structures of degree n, so it remembers the 'structure' of each number as well as its quantity. I can change the structure and not change its quantity (degree).
Looks like it's a winner.
Oh, and to make it more rigorous, I only defined symmetry degree for symmetric trees, so I will define the symmetry degree of an asymetric tree to be -oo,
#13
Organic
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BY writing (4,t) you did nothing.
Show us how this general (n,t) can define the number of (6,t) unique elements, for example.
the number of elements in [n] ie the set of all (n,t) t a tree with n edges, is exactly the number of (rooted) trees with n edges. You can count them if you like.
There is one element of[1], 2 elements in [2], there are I believe 5 elements in [3]..., call the number of elements in [n] the rank of n:=r(n)
there are then (r(n),t) elements of degree n, where t is the trivial tree with r(n) edges and 1 leaf. Recall the trivial trees form the trivial N structure of quantity only.
Why, how many structures are there for each quantity in your theory?
Perhaps, if my answer didn't satisfy you, you could tell me what the question actually means because I don't understand what you want. None of the things you've asked me about have been defined in this theory using those words. Please exlpain what you want me to define now, and I'll endeavour to add it in.
For every question you ask, as well, could you show what the answer is in your system too, so we know that you're not just asking deliberately meaningless questions, which is how they appear as you don't adequately explain what you want.
Find the rule and give the number of (6,t)unique elements.
That question doesn't make sense in english. I can tell you there are as many elements in my set [6] as there are rooted trees with 6 edges, off the top of my head I don't know how many of those there are. There is probably a simple generating function for them but I can't be bothered to work it out. Is that waht you wanted to know? The rank of 6? Using my definition of rank as above?
Anyway, that diagram refers to your trees, not mine. I am not saying our systems are the same, I'm just saying that mathematically I can define a system properly with all the innate structure of yours but in a way that anyone can understand.
Try rewriting the question so it conforms to the basic rules of English and perhaps I can answer it.
And why are you using my notation for your objects?
#17
Organic
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And why are you using my notation for your objects?
You know what? show how you can find the number of trees of (6,t)
in your system.
It's the number of trees with 6 edges. Get a pen and paper and work it out. It is tedious but doable, hint find the correct recusrive formula. Why, what are the number of diagrams in your theory with quantity 6?
#19
Organic
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67 unique ordered trees (by their symmetry degrees) for quantity 6, without left-right switches.
But you see, what is important here is not just the quantity 67 but the unique structure of each tree.
Any number is at least an association between our memory (the continuum) and some element(s)(discreteness).
Therefore these trees are the basic models of the associations between our memory and a given quantity.
The standard natural numbers are private case of one and only one association tree, which is the maximum broken symmetry tree of any given quantity.
Because any number is first of all memory AND element(s), then no association's structural form can be omitted by us, when we count.
Therefore N members of standard Math are trivial elements, and any other number system that constructed by them is also a trivial number system.
My theory of numbers fixes this triviality by exposing the hidden internal information structures of any given quantity.
Because my organic numbers are an ordered number system by symmetry degrees, we can use them as non-trivial powerful tools that can help us to start a non-trivial research of the complexity itself.
A simple example:
To say 2+3 is not enough because in my natural number system we need to know what internal structure of 2 and what internal structure of 3 we are going to add to each other.
Quite possibly true, you are now into the philosophy of mathematics. There you are free to state what you will. I will not and have not argued against that position. I have always argued against you misusing mathematical terms, and not telling anyone what you mean clearly.
Here I've demonstrated a mathematical object that does all the things you want and that you claimed mathematics wasn't able to do. It possesses attributes labelled by all the terms you gave, has well defined binary operations (for all inputs, unlike yours) and is a genuine non-commutative extension of N, which it contains as a subobject.
Incidentally is 67 the answer in your system or mine?
The formula for the number of trees (rooted, ordered, etc) with n edges is I believe
sum r(i)r(j) where i+j=n-1 and i,j are non-negative, and r(0) is defined to be 1. Which gives, r(1)=1, r(2)=2, r(3) =5, r(4)=5+2+2+5=14 r(5)=14+5+4+5+14=32 r(6)=32+14+10+10+14+32=112 I think.In my system the Natural numbers are the subobject corresponding to the case of zero redundancy and zero uncertainty, just as you wished.
#21
Organic
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Matt,
I do not say that Math cannot do it, I say that Math not use these internal symmety degrees as the fundamental property of The natural number definition.
If we use these symmetry degrees as the basis of Natural number definition, then any Natural number become a much for powerful/sensitive tool for Math language research.
But if you add this structure to the definition of natural numbers, then you are changing the definition, tautologically. This stucture is not needed to define the natural numbers, it can be deduced from their existence, and is therefore not a "first order" property of them. I use that in a non-technical sense. All the objects, trees etc, you use are out there already known and studied. You've given no compelling reason to justify the assertions you've made about why the natural numbers must have links to uncertainty, or quantum objects. Philosophically I believe you have the wrong emphasis, but that difference stems from your need to define mathematics subjectively and use this idea of being aware of its being observed. I don't see any mathematical reason for that. Philosophically that is a different matter; I have no strong opinions either way. But that is my personal philosophy of mathematical research, which is of no interest to anyone but me at the moment.
I just wish you would at least define unambiguously the terms you use so that misunderstandings cannot arise.
#23
Organic
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0
Matt,
Please show me some Mathematical research where trees as symmetry degrees are studied, including detailed structural results.
#24
Organic
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0
This stucture is not needed to define the natural numbers, it can be deduced from their existence, and is therefore not a "first order" property of them.
How do you know that these symmetry degrees are not "first order" property of them?
Show me some research that proves it.
For example: 6 gives, let us say, 67 unique ordered symmetry structures.
In 67 we are talking on very big quantitative result of unique ordered symmetry structures that are ignored by Standard Math point of view, and so on...
If you explain what you mean by symmetry degree perhaps I could.
And I said that I wasn't using "first order" in its proper meaning. Intuitively, you are using the quantity idea to enumerate your 67 trees of quantity 6. If the (structureless) natural numbers weren't available how could you use your "natural numbers" to count your "natural numbers"...? Especially as the 'number' of trees of a given quantity is more than any of the quantities defined so far - you need to "define 67" to count the structures of "quantity 6" ... This isn't rigorous, nor do I intend to make it so; I'm offering a philosophical discussion of the ideas because that way none of us can be wrong, just disagree, possibly.
Let us say that notation '4' is the integral side of ET4,
and '1 1 1 1' notations are the differential side of ET4.
Now, take each ET as a one organic element.
From this point of view the quantity of each ET remains
unchanged , but the structure of each ET is unique and it is not
ordered by quantity change but by the symmetrical degree of each ET.
Again we count the ET's not because of quantitative change, but because of
The inner structural change of each ET.
Now, we can find exactly 9 unique ET's in quantity 4:
By Peano axioms we are using only this ET (((((1),1),1),1),...) Which is a private case of one and only one ET structure.
But what about the rest ET's of ET4?
Any changing in ((((1),1),1),1) form cannot be but a quantitative change, because he has no "inner space" that can be changed.
Any ET with "free space" for structural changes is ignored by standard Math.
By my structural/quantitative approach this information is not ignored.
1) we indeed using peano axioms to find the next basic quantity, then for each basic quantity we define all its inner structures, and then we move to the next basic quantity by using again Peanos axiom.
2) Through this attitude we systematically define any possible information structure That can be expressed by ET's.
3) Then we can research their relations upon infinitely many scales, and create more and more interesting models of complex information structures in very efficient ways.
I suppose a philosophical position moshek might approve of is that your constructions require more ontological commtiment than is necessary to define the natural numbers.
You need to have trees, symmetry (which you've still not explained) and indeterminacy (ditto) and the notions of very complex sets lying around. In fact your (well, stratman's) construction uses the language of computer programming (let n=1...) and so uses explicilty the natural numbers (as understood ordinarily). Of course that last one isn't a fatal problem in itself. So you not only require the existence of the natural numbers as counting objects, but further ideas, that require ontological commitment that is not necessary in classical mathematics.
Can you prove something about quantity that cannot be proven through usual mathematical methods?
Allow me to emphasize that I'm talking about purely quantitative statements, not statements where we've substituted your new concepts for existing concepts (e.g. your use of aleph0 seems vastly different from the "quantitative" use of aleph0)
#29
Organic
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0
Hyrkyl,
Can you prove something about quantity that cannot be proven through usual mathematical methods?
By my point of view -, +, and * have two worlds, the internal world and the external world.
In standard Math the Natural numbers world is only the external world, where each operation chenging the quantity of n.
Let us look on - and + operations on n from ET eyes:
The external result of ((((1),1),1),1) - 1 is (((1),1),1)
The internal result of ((((1),1),1),1) - 1 is (((1,1),1),1)
The external result of (((1),1),1) + 1 is ((((1),1),1),1)
The internal result of (((1,1),1),1) + 1 is ((((1),1),1),1)
So as you see - and + do not changing the quantity but the symmetry degree of each ET.
In this case (1,1,1,1) is the maximum symmety degree and ((((1),1),1),1) is the minimum symmetry degree, for example:
Let us say that we have here a transformation between
multiset {x,x,x,x} to "normal" set {{{{x},x},x},x} and vise versa.
Let XOR be #
Let a,b,c,d stends for uniquness, then we get:
Code:
Uncertainty
<-Redundancy->^
d d d d |
# # # # |
c c c c |
# # # # |
b b b b |
# # # # |
{a, a, a, a} V
. . . .
| | | |
| | | |
| | | |
| | | |
| | | |
|__|__|__|_
|
={x,x,x,x}
{a, b, c, d}
. . . .
| | | |
|__| | |
| | | <--(Standard Math language uses only this
|_____| | no-redundancy_no-uncertainty_symmetry)
| |
|________|
|
={{{{x},x},x},x}
[b]
============>>>
Uncertainty
<-Redundancy->^
d d d d | d d d d
# # # # | # # # #
c c c c | c c c c
# # # # | # # # #
b b b b | b b b b b b b b b b
# # # # | # # # # # # # # # #
{a, a, a, a} V {a, a, a, a} {a, b, a, a} {a, a, a, a}
. . . . . . . . . . . . . . . .
| | | | | | | | | | | | | | | |
| | | | |__|_ | | |__| | | |__|_ |__|_
| | | | | | | | | | | |
| | | | | | | | | | | |
| | | | | | | | | | | |
|__|__|__|_ |_____|__|_ |_____|__|_ |_____|____
| | | |
{x,x,x,x} {x,x},x,x} {{{x},x},x,x} {{x,x},{x,x}}
c c c
# # #
b b b b b b b
# # # # # # #
{a, b, a, a} {a, b, a, b} {a, a, a, d} {a, a, c, d}
. . . . . . . . . . . . . . . .
| | | | | | | | | | | | | | | |
|__| |__|_ |__| |__| | | | | |__|_ | |
| | | | | | | | | | |
| | | | |__|__|_ | |_____| |
| | | | | | | |
|_____|____ |_____|____ |________| |________|
| | | |
{{{x},x},{x,x}} {{{x},x},{{x},x}} {{x,x,x},x} {{{x,x},x},x}
a, b, c, d}
. . . .
| | | |
|__| | |
| | | <--(Standard Math language uses only this
|_____| | no-redundancy_no-uncertainty_symmetry)
| |
|________|
|
{{{{x},x},x},x}
[/b]
The internal structure of any given quantity (finite or infinite) cannot be ignored, therefore the natural numbers are at least structural/quantitative information forms: http://www.geocities.com/complementarytheory/ETtable.pdf
Let us say that in the first stage we get an ordered table of infinitely many symmetry forms, and by this ordered table we can start to explore the relations between highly complex different forms of symmetries.
Shortly speaking, we have in our hand a Mendeleiev-like table of symmetries, where Peano axiom symmetries, are only some private case in it.
Another very interesting thing is that from ET point of view any number is an organic form with internal complexity, that can exist iff we take each ET as a whole, which is a paradim shift in the concept of a NUMBER.
And this paradigm shift is based on this simple test:
Any number is first of all an information form, therefore any aspect of information form MUST be researched by us, where our cognition’s abilities to research information MUST be included too.
Form this point of view, redundancy AND uncertainty cannot be ignored, and through this approach(which is not an extra approach but the MINIMAL approach to understand the natural number concept) we can clearly show that the standard natural numbers are only a one and only one private case of verity of information forms, which are ordered by their vagueness degrees from maximum vagueness to minimum vagueness when a given quantity remains unchanged.
Man is no longer an observer but a participator, which its influence must be included in any explored system.
The above is the QM paradigm shift that is not understood yet by the current community of pure mathematicians.
For example: Be aware that what you call a function is first of all a reflection of your memory.
Hurkyl asked you to use your theory to prove something abou quantity that cannto be proven with the usual concept of the natural numbers.
#31
Organic
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0
My answer was, first you have to understand that we are talking about new dimension of the natural number that its results cannot be reduced to quantity only picture.
My point of view is a comprehensive point of view on the concept of a number that can open a new unexplored yet mathematical dimension.
For example because you don't do this paradigm shift in your cognition, you cannot understand these proofs:
The idea that, given a natural number, n, one can assign many structures to it that are more complex and richer is not a new idea. And until such time as you explain what the terms in your construction mean you won't win any plaudits.
#33
Organic
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0
Please show me a mathematical research about natural numbers that are ordered by their vagueness degrees from maximum vagueness to minimum vagueness when a given quantity remains unchanged.
you have no choice but to agree with that that the minimal existence of the natural number cannot be less then structural/quantitative information form.
Symmetry and probability have to be used right from the most fundamental level of any mathematical system, and I mean from the level of the set concept, or form the natural numbers (what you call "first order").
The most simple element that can combine symmetry and probability, has an information form of a tree.
And when the natural number is a least a tree, all other number systems are also trees.
Let us do some pause for my little song:
Every number is a tree,
climb on it, its all for free,
full of branches, brids and winds,
no more losers no more wins.
Peace on Earth
and in the sky
every heart
can bloom and fly.
But I don't agree with your "test" (there's nothing there that is a test as far as I understand the word) as it is not mathematical, and is rather silly. What does it matter that we may or may not be allowed to "use our memory" whatever that may mean, nor do I see why you've negated whatever problem it is that you think is there. And that doesn't alter the fact that you have not told me what you mean be the sentence
'vagueness is any n>1'
#40
Organic
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0
...it is not mathematical,...
1) There is no such a thing like mathematics where our cognition's abilities
are not deeply involved within it.
2) Therefore any mathematical research MUST include our abilities to define and develop it, and this is exactly the meaning of my test.
3) In this test I show these important things:
a) Without an association between some element AND our memory, we cannot define any quantity beyond 1.
b) Because the minimal conditions to count beyond 1 are at least memory AND element associations, then any minimal information form beyond 1 MUST be the information form of a tree, where the root is our memory and the elements are its branches and/or it leafs.
4) Because when we count, we most of the time using our memory in a sequential way we (without paying attention) jumping straight to the information form of an ordered tree.
5) By checking the ordering process I clearly show that in any given quantity there are several stages (or satiations if you will) that can be defined and be ordered by their clarity degrees, where the first information tree represents the most unclear information and the last tree represents the most clear information.
6) These stages belong to the "first order" level because no information form of memory_AND_elements tree can be ignored.
The numbers exist, you just don't remember they exist, might be a better way of saying it. If you cannot use your memory, how do you even know you were asked to count them in the first place? The question as to whether this has anything in slgihtest to do with mathematics is still open. How did you remember all the squiggly symbols you've just used to write that post? What if...?
#42
moshek
265
0
I was passing yesterday near the bulding of the conference "Cardinals at work" :
1) Resolving the GCH positively ( Woodin, Shelah)
2) Set theory with no axion of coise (Shelah) .
A new paradigem in Set theory in now come even to small talkes.
Moshek
#43
Organic
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0
The numbers exist
What do you need to know that?
What if...?
"What if...?" is one of the most important questions that we ask when we develop something.
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#44
moshek
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0
"Everything is a Number" , Pytagoras
And even the change of the Euclidian Paradigem:
Definition, axioms, theorems , etc
"What if...?" is one of the most important questions that we ask when we develop something.
Seeing as your question in you test is about counting beads it is implicit that you are presuming there is some method to count them. If there are no numbers, how do you count them? Irrespective of the difficulty in counting things and keeping track of which one is which.
Let me offer a different method of counting:
take the set of beads ina pile, pick one up, put it in your pocket and say 'one', then pcik another up and do the smae saying 'two' no difficulty there about people messnig them around or them al looking the same. The last things you say is the number of beads. Or are you asking about the philosophical issues about having a name for a number, and the concept of twoness existing independently of the words two, deux, zwei, dos,... metamathematics at most.
#46
Organic
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0
Matt,
One of the beautiful things that a man can do is a thinking experiment based on a "What if...?" question.
Most of the paradigms shift happened because people did not afraid to use this gifted ability to reexamine the obvious.
When you have eyes it is obvious that you can see, when you have memory it is obvious that you can count and define the concept of a NUMBER.
But "What if...?" questions can go beyond the obvious, and create an evolution in any asspect of life, including the fundamental concepts of Math language.
Numbers do not fully exist without our connition's abilities, and this is for real.
Seeing as your question in you test is about counting beads it is implicit that you are presuming there is some method to count them. If there are no numbers, how do you count them?
Numbers can exist iff there is an association between our memory and some internal(our own thoughts) or external elements.
That is your philosophical position. A platonist would argue differently.
#48
moshek
265
0
Matt:
I think that most of the mathematician in the planet of Earth are Platonic.
They believe that when you discover something in mathematics is a discovery about some true, which is not in the real world. Few of them on not, take me as an example. And what about you?
Most of us are formalists. I think very few are platonicists. We have reasonable consensus on mathetical objects are, in the sense of what class of things are legitmately called mathematical, and as long as we are all honest in the use of logic, we make statements based on these hypotheses that we are in common agreement about, with the proviso that although we never explicitly deal with set theory unless we must, that there is the reassurance of having ZFC behind us that no one has shown to have any problems. And if they did then another person would come along and sort them out. I am perhaps a Wittgenstinian in remove owing to being taught by Gowers. An object is what it does is the paradigm there, for want of a better phrase. The natural numbers are the tools we use for counting things. We never explain it more than that, and few of us would feel the need to. If we had to we might offer some set theory.
Organic's Philosophy is somewhere between the two - although he seems to adopt a pragmatic approach to what the natural numbers are he often veers towards the idea that there is some platonic version that his is, and ours isn't, hence his statements about what properties they must have. Whereas my philosophy would state that we have words, one two three and so on and plus, etc that by common consent we have come to define in the current manner. They are useful for what they were designed for, and using them we can do a whole lot, including formally define all the objects Organic uses (it's a matter of what comes first, the name or the structure). However, when Organic uses the words one two three and so on, his isn't using them in accord with the convention of the rest of us. That ought to mean, in my opinion, that he should change the labels of his 'numbers' and use say, qone, qtwo, qthree, because they don't do the job that we have ascribed the other labels to. He would then need to define objects that are structureless quantities too.
Does it mean that the words are wrong or the usage of them is wrong?
I am happy for Organic to develop his philosophy. he seems to be doing a better job of it now, but I would still like him to properly describe the things he writes down.
Perhaps you should get together with him and talk about ostensive defintion and private languages?
#50
moshek
265
0
Matt:
Thank you for your very kind explanation about your view on mathematics.
If you are formalist I want to ask you does mathematics is only a game with symbols?
Well Organic is Organic ( like me ) !
Yes you are right Organic numbers are defenetly not the conventional number
It will Be better if we write them like this:
1. 2. 3. 4. 5. 6. ...
Do you see the point?
And what is the definition of a point in the Euclidian mathematics ?
