The Foundations of a Non-Naive Mathematics

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The discussion centers on a new mathematical framework proposed by Lama, which aims to redefine fundamental concepts such as tautology, sets, and the real line. Key axioms include the independence of points and segments, the duality of elements, and the completeness of collections. The framework emphasizes the relationship between absolute and relative properties, suggesting that the real line is a fractal system with invariant cardinality across various scales. Critics question the validity of the proposed definitions and their equivalence to established mathematical constructs like Dedekind cuts and Cauchy sequences. The conversation highlights a clash between traditional mathematics and Lama's innovative approach, which seeks to address complexities overlooked by conventional methods.
  • #391
arildno said:
Lama's "tautologies":
1. = (Here, it seems to be some sort of equivalence relation.)
2. x implies x (don't we have any longer: only if x then x?)
3. The opposite of contradiction
4. Self-evidently true
(So combining 3. and 4., we might possibly infer that a contradiction is a non-self-evidently true statement?)
Please see for yourself in:

http://en.wikipedia.org/wiki/Tautology.
 
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  • #392
Lama said:
Please see for yourself in:

http://en.wikipedia.org/wiki/Tautology.
The point everyone is making is that a tautology is a proposition--even the link you cite says this. When you claim that a point is defined by tautology, what is the proposition that you're referring to?
 
  • #393
ex-xien said:
The point everyone is making is that a tautology is a proposition--even the link you cite says this. When you claim that a point is defined by tautology, what is the proposition that you're referring to?

Dear ex-xien,

Please read all of:

https://www.physicsforums.com/showpost.php?p=278285&postcount=390

And then please give your remarks, thank you.
 
  • #394
Lama said:
Dear ex-xien,

Please read all of:

https://www.physicsforums.com/showpost.php?p=278285&postcount=390

And then please give your remarks, thank you.
Ok...That still doesn't answer my question.

You state,
A definition for a point:
A singleton set p that can be defined only(*) by tautology ('='), where p has no internal parts.

(*) only by tautology means: the minimal possible existence of a non-empty set.
All I'm asking is what proposition is tautologous. Your own link makes it clear that a tautology is a statement.

Are you going to redefine this word too?
 
  • #395
Do you understand that in both definitions (of {.} or {._.}) the conclusion cannot be different from the premise ?(http://mathworld.wolfram.com/Tautology.html)

To understand this you have to read https://www.physicsforums.com/showpost.php?p=278285&postcount=390
until the end of it, thank you.

I erased this part ( (*) only by tautology means: the minimal possible existence of a non-empty set) to avoid understanding problems.

A statement for a point:
A point is an indivisible finite content of a non-empty set that has no directions.

A statement for a segment:
A segment is an indivisible finite content of a non-empty set that also has directions.
 
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  • #396
Lama said:
Do you understand that in both definitions (of {.} or {._.}) the conclusion cannot be different from the premise ?(http://mathworld.wolfram.com/Tautology.html)
No, I don't. What is the conclusion and what is the premise? That's what I've been getting at. I don't see a proposition in {.} or {_}.

To understand this you have to read https://www.physicsforums.com/showpost.php?p=278285&postcount=390
until the end of it, thank you.
I've read it already, thanks.

A statement for a point:
A point is an indivisible finite content of a non-empty set that has no directions.

A statement for a segment:
A segment is an indivisible finite content of a non-empty set that also has directions.
If you're going to adjust your definitions to these, it's a bit clearer. Why don't you just define a point as a set with exactly one element. This seems to be the concept you're dancing around.
 
  • #397
ex-xian said:
Why don't you just define a point as a set with exactly one element.
Because only by this definition we cannot distinguish between a point and a segment, which are two different elements that are not defined by each other.

Please read this again:

A statement for a point:
A point is an indivisible finite content of a non-empty set that has no directions.

A statement for a segment:
A segment is an indivisible finite content of a non-empty set that also has directions.

This is exactly the basic idea that makes the whole change between the standard point of view and my point of view.
 
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  • #398
If you're going to adjust your definitions to these, it's a bit clearer. Why don't you just define a point as a set with exactly one element. This seems to be the concept you're dancing around.

I don't think that would be satisfactory anyways... Organic is focused on telling us what points and segments are, but he's not telling what we know about / can do with them.

In the usual context, defining something in terms of sets would be fine and dandy, but past experience, and his own admission, has shown that what Organic means by "set" does not coincide with what mathematicians mean by "set".



And back to Organic:

A statement for a point:
A point is an indivisible finite content of a non-empty set that has no directions.

A statement for a segment:
A segment is an indivisible finite content of a non-empty set that also has directions.

The same problem I mentioned last time applies here. The axioms you give still have the defect that you can't combine them to prove things.


Let me give you an example of an accepted selection of axioms; these are the usual axioms of incidence from Euclidean geometry:

If A and B are points, then there is a line L such that A and B both lie on L.
If L and M are distinct lines, then there is at most one point P that lies on both L and M.
There are points, A, B, and C, that do not all lie on the same line.

Notice that there are 3 statements above, yet they only define three terms: "point", "line", and "lies on".

Furthermore, two of these are constructive: the last one "makes" some points and tells you something about them, and if you have some points, the first one let's you "make" some lines.

Finally, the middle axiom is a restriction.



Look at your axioms:

A statement for a point:
A point is an indivisible finite content of a non-empty set that has no directions.

A statement for a segment:
A segment is an indivisible finite content of a non-empty set that also has directions.

You have only two axioms, yet you are defining at least four terms: "point", "segment", "indivisible finite content of a non-empty set", and "directions".


Furthermore, they tell you very little. They don't tell you how "make" points and "make" segments, and they don't many any useful restrictions about what you can make.




addendum:

by "make", I don't mean the act of physically making things. I merely mean that you can prove they exist, and have certain properties.
 
  • #399
Lama said:
Because only by this definition we cannot distinguish between a point and a segment, which are two different elements that are not defined by each other.

Please read this again:

A statement for a point:
A point is an indivisible finite content of a non-empty set that has no directions.

A statement for a segment:
A segment is an indivisible finite content of a non-empty set that also has directions.

This is exactly the basic idea that makes the whole change between the standard point of view and my point of view.
I can understand what you mean by point except for the "has no direction" part. So the only difference in a point and a segment is that a segment has direction?

What does that even mean? What does "direction" mean in this sense?
 
  • #400
To understand this please first read all of:

https://www.physicsforums.com/showpost.php?p=276964&postcount=383

and only then all of:

As I showed in post #383, each number which is not 0 is at least a representation of {.}_AND_{._.}.

Also each {._.} has 3 basic states which are: '<' for left-right direction, '>' for right-left direction, '<>' for no-direction.

Let us write again The axiom of minimal structure:
Any number which is not based on |{}|, is at least p_AND_s, where p_AND_s is at least Multiset_AND_Set.

Let us examie this part: "...where p_AND_s is at least Multiset_AND_Set."

We know that the elements of a non-empty "normal" set, which its cardinality > 1,
cannot be identical.

But the elements of a multiset, which its cardinality > 1, can be identical.

If p_AND_s is at least Multiset_AND_Set, then any given element which its cardinality > 1 has several variations that can be found between '<>' to '<' or '>'.

For example, let us represent the variations of cardinals(*) 2,3,4:

Let Redundancy be more then one copy of the same value can be found.

Let Uncertainty be more than one unique value can be found.

Let XOR be #

Let a=0,b=1,c=2,d=3 then we get:
Code: 
    b  b                                        
    #  #                                        
   {a, a,  {a, b}                               
    .  .    .  .                                
    |  |    |  |                                
    |__|_   |__|                                
    |       |                                   
                                                
    {x,x}  {{x},x}                              
                                                
                                                
                                 
                                                
                                                
     c  c  c                                    
     #  #  #                                    
     b  b  b          b  b                      
     #  #  #          #  #                      
    {a, a, a,}       {a, a, c}       {a, b, b}  
     .  .  .          .  .  .         .  .  .   
     |  |  |          |  |  |         |  |  |   
     |  |  |          |__|_ |         |__|_ |   
     |  |  |          |     |         |     |   
     |__|__|_         |_____|         |_____|   
     |                |               |         
     |                |               |         
    {{x,x,x}         {{x,x},x}       {{x},x},x}

              
                [B]Uncertainty[/B]
  <-[B]Redundancy[/B]->^
    d  d  d  d  |
    #  #  #  #  |
    c  c  c  c  |
    #  #  #  #  |
    b  b  b  b  |
    #  #  #  #  |
   {a, a, a, a} V   {a, b, c, d}
    .  .  .  .       .  .  .  .
    |  |  |  |       |  |  |  |
    |  |  |  |       |__|  |  |
    |  |  |  |       |     |  | <--(Standard Math language uses only 
    |  |  |  |       |_____|  |     this no-redundancy_
    |  |  |  |       |        |     no-uncertainty_symmetry)
    |__|__|__|_      |________|
    |                |
    ={x,x,x,x}       ={{{{x},x},x},x}



============>>>

                [B]Uncertainty[/B]
  <-[B]Redundancy[/B]->^
    d  d  d  d  |          d  d             d  d
    #  #  #  #  |          #  #             #  #        
    c  c  c  c  |          c  c             c  c
    #  #  #  #  |          #  #             #  #   
    b  b  b  b  |    b  b  b  b             b  b       b  b  b  b
    #  #  #  #  |    #  #  #  #             #  #       #  #  #  #   
   {a, a, a, a} V   {a, a, a, a}     {a, b, a, a}     {a, a, a, a}
    .  .  .  .       .  .  .  .       .  .  .  .       .  .  .  .
    |  |  |  |       |  |  |  |       |  |  |  |       |  |  |  |
    |  |  |  |       |__|_ |  |       |__|  |  |       |__|_ |__|_
    |  |  |  |       |     |  |       |     |  |       |     |
    |  |  |  |       |     |  |       |     |  |       |     |
    |  |  |  |       |     |  |       |     |  |       |     |
    |__|__|__|_      |_____|__|_      |_____|__|_      |_____|____
    |                |                |                |
    {x,x,x,x}        {{x,x},x,x}      {{{x},x},x,x}    {{x,x},{x,x}}     
 
                                      c  c  c
                                      #  #  #      
          b  b                        b  b  b          b  b
          #  #                        #  #  #          #  #         
   {a, b, a, a}     {a, b, a, b}     {a, a, a, d}     {a, a, c, d}
    .  .  .  .       .  .  .  .       .  .  .  .       .  .  .  .
    |  |  |  |       |  |  |  |       |  |  |  |       |  |  |  |
    |__|  |__|_      |__|  |__|       |  |  |  |       |__|_ |  |
    |     |          |     |          |  |  |  |       |     |  |
    |     |          |     |          |__|__|_ |       |_____|  |
    |     |          |     |          |        |       |        |
    |_____|____      |_____|____      |________|       |________|
    |                |                |                |
    {{{x},x},{x,x}} {{{x},x},{{x},x}} {{x,x,x},x}      {{{x,x},x},x}
 
    a, b, c, d}
    .  .  .  .
    |  |  |  |
    |__|  |  |
    |     |  | <--(Standard Math language uses only this
    |_____|  |     no-redundancy_no-uncertainty_symmetry)
    |        |
    |________|
    |    
    {{{{x},x},x},x}
Also please pay attantion that the last form is the standard R members 0,1,2,3:

Code: 
 0 = .

 1 = 0______1

 2 = 0____________2  

 3 = 0___________________3

And the standrard [B]R[/B] is nothing but the above 2-D representation 
in a 1-D representation:

0______1______2______3
And because no R member is both Multiset_AND_Set, I call it: The "shadow" of my new number system.

--------------------------------------------------------------------------------

(*) Please pay attention that we are not talking about the natural numbers 2,3,4 but the cardinals 2,3,4.

It means that our Organic Natural Numbers are actually a general representation of information-trees, where any finite quantity of names of R members can be described by them, for example:

Instead of a=0,b=1,c=2,d=3 we can use a=0,b=.5,c=3,d=pi and then we use the same information-trees above.


I called these general information-trees 'Organic Natural Numbers' because:

1) These information-trees of cardinals are always having a structure, which is based on N members.

2) They can be used as natural (not forced) and general representation for any interaction between complementary states, which simultaneously preventing/defining their middle domain.

3) Because no R member is divisible by my system, it has its own organic (complete) unique and independent self existence.
 
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  • #401
Hurkyl said:
I merely mean that you can prove they exist, and have certain properties.
Thank you dear Hurkyl for your reply, but I am afraid that you did not see the rest of my axiomatic system and what I can do with it, so for a better picture of it please read all of it until the end of it, at lease twice before you reply about my axiomatic system, that can be found here:


The axioms of Complementary Set Theory:

http://www.geocities.com/complementarytheory/My-first-axioms.pdf



Thank you,


Lama
 
  • #402
oops, sorry: I have found some mistakes in the above pdf file and I fixed them.
 
  • #403
Lama said:
Please see for yourself in:

http://en.wikipedia.org/wiki/Tautology.

Nowhere have you shown any understanding of the actual meaning of tautology in the sense used by logicians.
I know perfectly well what a tautology is; you don't seem to understand it.
 
  • #404
arildno said:
Nowhere have you shown any understanding of the actual meaning of tautology in the sense used by logicians.
I know perfectly well what a tautology is; you don't seem to understand it.
Please first read at least tiwce all of http://www.geocities.com/complementarytheory/My-first-axioms.pdf including all of its links, and only then you can raply meaningful remarks on my work.
 
  • #405
Lama said:
Please first read at least tiwce all of http://www.geocities.com/complementarytheory/My-first-axioms.pdf including all of its links, and only then you can raply meaningful remarks on my work.
Please read at least 3 times my questions about where the premise and conclusion is on your "tautology", read at least 5 times what wikipedia actually says about tautology, post a reponse explaining what the premise and conclusion is in your "tautology", proofread it 2 times, then present it for us to read.

Only then can anyone make meaningful remarks on your "work." THat is, you have to actually respond to what people ask, rather than give reading assignments or reposting the same stuff over and over.
 
  • #406
Lama:
As yet, you have not convinced anyone that your first axiom is not total gibberish.
Instead of wasting virtual space with tons of cooked-up definitions from your own fancy, please focus only on elucidation of your first axiom.
If you think it cannot be "understood" out of context with the rest, then it is not an axiom at all, but just gibberish.
 
  • #407
How do complex numbers fit into your new theory? I see only talk of real numbers here. I have read through all 28 pages of this thread and I have had time to look at all of your files on your website. Please explain i=sqrt(-1) in terms of your new notation. Please do not refer me to one of your pdf files, as I know that there is no reference to the complex numbers.
 
  • #408
Dear oreopoj,

When we have {},{.}_AND_{._.},{__} then we can construct any information form that we like.

Complex numbers are based on R part + Z part, so all they are is no more then a technical extension that help us to solve equations where numbers like √-1 are involved.

My research about the language of Mathematics is exactly in the opposite direction, which means:

Instead of searching unsolved problems within the standard framework (based on 'How' questions) I went back to the most fundamental concepts of this beautiful language and used most of the time 'Why' questions, that helped me to develop my non-standard framework.
 
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  • #409
ex-xian said:
Please read at least 3 times my questions about where the premise and conclusion is on your "tautology", read at least 5 times what wikipedia actually says about tautology, post a reponse explaining what the premise and conclusion is in your "tautology", proofread it 2 times, then present it for us to read.

Only then can anyone make meaningful remarks on your "work." THat is, you have to actually respond to what people ask, rather than give reading assignments or reposting the same stuff over and over.
Dear ex-xian, all the answers to your questions are in http://www.geocities.com/complementarytheory/My-first-axioms.pdf

All you need is to read and (I hope) understand my work.

Good luck.
 
  • #410
arildno said:
Lama:
As yet, you have not convinced anyone that your first axiom is not total gibberish.
Instead of wasting virtual space with tons of cooked-up definitions from your own fancy, please focus only on elucidation of your first axiom.
If you think it cannot be "understood" out of context with the rest, then it is not an axiom at all, but just gibberish.
Why do you need more people around you to be sure that my work is (by you) just gibberish?

At least respect yourself and say it clear: "I, arildno, think that your work is nothing but a gibberish!".

Believe me it will look much better then using the "not convinced anyone" style.

Because my framework is a paradigm-shift in the Langauge of Mathematics,
you have no choice but to understand it by the most fine intuition/reasoning
interactions abilities that you have.

Naturally well educated mathematicians are the first persons that have the biggest problems to understand a paradigm-shift because of more then a one reason, for example:

1) The foundation of their own work can be changed for better but also for worse, but in both cases they have to reexamine their work according the paradigm-shift, which is a very unpleasant situation, for persons who afraid from deep changes.

2) They are already trained by another 'school of thought', therefore it is hard for them to see fundamental things from a different point of view.
 
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  • #411
Lama said:
Naturally well educated mathematicians are the first persons that have the biggest problems to understand a paradigm-shift because of more then a one reason, for example:

1) The foundation of their own work can be changed for better but also for worse, but in both cases they have to reexamine their work according the paradigm-shift, which is a very unpleasant situation, for persons who afraid from deep changes.

2) They are already trained by another 'school of thought', therefore it is hard for them to see fundamental things from a different point of view.

Mathematicians are the first people that would be happy for some deep changes. You just don't explain your definitions well. Every new term you introduce in your axioms should be explained using the most fundamental concepts (ex. < ,> = ...)
 
  • #413
Lama, it isn't fair for you to criticize anyone for not understanding what you're doing, nor is it fair to say that the fact that virtually everyone has panned what you've done.

You've failed to explain what the premise(s) and the conclusion(s) are in your tautology. You've invoked something called "number," which you've not defined, in order to explain your first two axioms. This isn't the way math, any math, is done. You can't just pull concepts out of the air.

If you want to be taken seriously, you need to present your definitions, then your axioms. Your axioms need to be understandable and explainable only by appealing to the definitions or previous axioms. When you have to pull out mathematical words that you haven't defined in order to explain your axioms, you're not being at all productive.

I'm trying my best to understand you and to give you the benifit of the doubt. Here's what we have so far:

Set: A set is a finite or infinite collection of objects in which order has no significance, and multiplicity is also ignored.
1) Since your building up the numbers from scratch you need to give definitions for "finite" and "infinite."

2) Also, do you have commonplace definitions in mind for "order" and "multiplicity?" That is, is {x, x, y} the same set as {y, x}?

Tautology: x implies x.
1) Since you refernced wikipedia, then any mention of a tautology necessarily means you have a proposition in mind.

2) That is, "x implies" is the same as "if x, then x."

A definition for a point:
A singleton set p that can be defined only by tautology ('='), where p has no internal parts.
1) What is a singleton set? Since you're building the numbers up from scratch, you either have to admit to the standard way of constructing the counting numbers or explain "singleton set" w/o referencing numbers.

2) What are internal parts? Remember, you have to define this only with previously defined terms.

3) You say that a point can be defined only by tautology, which you equate with the operator "=". When one writes "x = x" is this the same as saying "if x, then x"?

See if you can take care of these questions, please?
 
  • #414
Dear ex-xian,

There comes a time where you have no choice but to use your own finest internal intuition/reasoning interactions abilities to understand something.

From this moment you are in your own and nobody, included me, can help you to understand this thing.

In short, I did my best to represent my ideas in the simplest and clearest ways that I can.

All what I have to say about my work can be found in http://www.geocities.com/complementarytheory/My-first-axioms.pdf and its links.

From this point I can only answer to questions that shows that the person who asks them already made his paradigm-shift in his mind and leaped by using his own abilities to my new framework.

for example, you wrote:
ex-xian said:
1) Since your building up the numbers from scratch you need to give definitions for "finite" and "infinite."
All you need is to understand how the axiom of completeness

(The axiom of completeness:
A collection is complete if an only if both lowest and highest bounds are included in it and it has a finite quantity of scale levels.)

is related to the rest of my work, because one of the important things in my system is, that you have to understand the whole of it in order to understand a part of it, where what I call 'the whole of it' is the simplest level of understanding, which stands in the basis of my work, and cannot fully shared with others (by using written axioms) that are not already share within them this common source of the simplest state, which is beyond any definition.

And this is exactly the deep meaning of the words ‘paradigm-shift’.

There is always an unclosed gap that has to be closed by an intimate-private journey that a person does in the deep and fine silence of his both heart and mind.


Good Luck.


Lama
 
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  • #415
oreopoj said:
How do complex numbers fit into your new theory? I see only talk of real numbers here. I have read through all 28 pages of this thread and I have had time to look at all of your files on your website. Please explain i=sqrt(-1) in terms of your new notation. Please do not refer me to one of your pdf files, as I know that there is no reference to the complex numbers.


The very well known mathematision Michael Atiya, gave a very interesting lecture about a year ago at the conference about "The unity of mathematics". He said that we are waiting now to a new-Newton that will break the enigma of mathematics and Physics.

He gave the example of I ( root of -1) to how new concept bring high resistant when they are appearing.

Well, I hope that you can see the analogy to this complitly
new theory about mathematics as one whole - M(i).

Moshek :approve:
 
  • #416
What in the world has this forum become?! Lama, what is the point of repeatedly posting links to your pdf files and saying the same thing over and over again to show the community your brilliant new way of doing math if you say:

"nobody, includ[ing] me, can help you to understand this thing".

Asking people to abandon their current way of thinking about math is not reasonable. If you want to share the wonderfulness of what you've found, you must prove that it works to others in a way that is comprehensible to people. Why else would you have tried to share these ideas, producing 28 pages of a single thread. This is what forums and discussion are for. That is why you are here. If you just want to advertise your new ideas without accepting criticism, then please just use your website.
Lama, please try to understand that the reason some people have demanded that you demonstrate some kind of knowledge of the current system of math is because there hundreds of years of theoretical and applied research to back up its use. Math is taught in schools and university the way it is because it works and has been put to use in so many real world applications; ie, computers, physics, economics, etc. That's not to say that math is a static thing, certainly not. It, as you have said, evolves through research and new ideas by bright minds, but the great thing is, each time a new breakthrough in mathematics is made, it ENCOMPASSES and is CONSISTENT WITH all the old ideas and methodologies that came before.
I have read through all of the previous messages in this thread and have gone through every single pdf file on your website. I am not convinced that you have discovered a new branch of mathematics in which the old mathematics is merely the "shadow" of yours.
Why not attempt to re-express your ideas in a new way that is compatible with the way it is done normally. I think you should start by defining every single unique term in your axioms in your own words and not simply use the definitions found on Wolfram's website. If your definitions and following axioms are inconsistent in some way as some people have pointed out so far, go back rework them further until they are consistent.
And don't be so defensive when someone points out a mistake in your reasoning! It's annoying. Please do not say, "I have clearly shown [such and such] on this file at my website." You sound like a broken record when do this. Everyone has probably looked at the link you're referring to already. We need a new and clearer explanation of what you're trying to say in order to understand you.
Lama, you can brand me as a "bodyguard of math". Yes, there are bodyguards of math, and for a good reason. Research in science and math is a vigorous process involving the constant judgment of one's submitted ideas by the other peers of the research community. A new idea cannot be accepted as truth without first being questioned and scrutinized. This is the power of the research process. Imagine how crazy and disorganized science and math would be if every single idea was immediately incorporated into the current body of knowledge. Your ideas should be able to predict and explain the behavior of a certain model or system, and, if this is successful, then those same ideas should be able to do the same on a different system or model. If the new idea fails in the respect just once, then it must be rejected in favor of the old ideas that can already correctly explain some phenomenon. At that point, it's time to go back to the drawing board. I can tell you from experience that less than 5% of any research that is done in anyone research group or laboratory in an academic setting ever goes somewhere significant. Don't be surprised if someone shows that you are wrong. A good mathematician or scientist is able to admit when they are wrong.
And back to my original beef with what has been going on here in this thread. I’ll quote you again:

"nobody, includ[ing] me, can help you to understand this thing".

Let us suppose for the moment that you actually have come up with some new way of doing math that will lead to tremendous advancements in the field and in science. What a shame it would be if that idea were to die with you some day then. You must be able to convince people who study math that you are correct in order for that advancement to take place. If you cannot make us understand, then it is all for nothing and you should not be wasting our and your time on this forum. If you cannot do this, go and learn how to; go and share your ideas with a professor in mathematics that is near you. Maybe he/she can help you express your ideas in a way that most people can understand. Believe me, I think it’s good that people like you want to explore and discover new things in mathematics. Mathematics is a beautiful thing as you have said before and is a very noble pursuit; but it is not one to be taken alone. If you want us to embrace your ideas, you have to be willing to embrace our thoughts and ideas as well.
 
  • #417
Well, I guess that settles it. You're not seriously trying to do math after all. Your trying to do mysticism and smear math words over it.

If you want to be a mystic, that's fine and good. But at least be honest about it.

All you need is to understand how the axiom of completeness

(The axiom of completeness:
A collection is complete if an only if both lowest and highest bounds are included in it and it has a finite quantity of scale levels.)
Then you need to give definitions for lowest bound, highest bound, finite, and scale. And you have to define them all w/o using numbers, since you're building everything from scratch. Either that or concede the standard way of defining finite sets--which would concede the standard way of building the natural numbers.
 
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  • #418
I agree, sometimes it sounds more like mysticism to me.
 
  • #419
oreopoj said:
I agree, sometimes it sounds more like mysticism to me.
And I actually don't have a problem with mysticism per se, but I wish more of them subscribed to Wittgenstein's philosophy of mysticism.

"Of that which we cannot speak, we must pass over in silence."
 
  • #420
I with you on that one. I would prefer mysticism to religion any day of the week. But look at how many times people in history have looked silly for trying to mix mysticism/religion with science; for example, Kepler tried to explain the orbits of the planets in terms of known 3D polyhedrons of the time because they were "perfect" or "divine". That's garbage. It was only after he let go of the idea of "perfect" shapes that considered using elliptical orbits and sucessfully formulated his laws.
 

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