How do hidden assumptions affect our understanding of mathematics?

[WWW]

As much as I know (and please correct me if I am wrong) no fundamental mathematical systems like ZF or Peano axioms include an examination of our cognition's ability to count.

In my opinion, if we ignore our abilities to count, then there is a reasonable chance that our fundamental axioms include hidden assumptions.

For example, let's examine this situation:

On the table there is a finite unknown quantity of identical beads > 1
and we have:

A) To find their sum.

B) To be able to identify each bead.

Limitation: we are not allowed to use our memory after we count a bead.

By trying to find the total quantity of the beads (representing the discreteness concept) without using our memory (representing the continuum concept) we find ourselves stuck in 1, so we need an association between continuum and discreteness if we want to be able to find the bead's sum.

Let's cancel our limitation, so now we know how many beads we have, for example, value 3.

Now we try to identify each bead, but they are identical, so we will identify each of them by its place on the table.

But this is an unstable solution, because if someone takes the beads, put them between his hands, shakes them and put them back on the table, we have lost track of the beads identity.

Each identical bead can be the bead that was identified by us before it was mixed with the other beads.

We shall represent this situation by:

((a XOR b XOR c),(a XOR b XOR c),(a XOR b XOR c))

By notating a bead as 'c' we get:

((a XOR b),(a XOR b),c)

and by notating a bead as 'b' we get:

(a,b,c)

We satisfy condition B but through this process we define a universe, which exists between continuum and discreteness concepts, and can be systematically explored and be used to make Math.

What I have found through this simple cognition's basic ability test is that ZF or Peano axioms "jump" straight to the "end of the story" where cardinal and ordinal properties are well-known, and because of this "jump" Infinitely many information forms that have infinitely many information clarity degrees, are simply ignored and not used as "first-order" information forms of Math language.

In my opinion, any language (including Math) is first of all an information system, which means that fundamental properties like redundancy and uncertainty MUST be taken as first-order properties, but because our cognition's abilities were not examined when ZF or Peano axioms were defined, both redundancy and uncertainty were not included in our logical reasoning or in our fundamental axiomatic systems.

Also in my opinion, through this simple test we get the insight that any mathematical concept is first of all the result of cognition/object (abstract or non-abstract) interactions.

My paper http://us.share.geocities.com/complementarytheory/ONN.pdf is a reexamination of some fundamental mthematical concepts that are based on this insight.

 

[matt grime]

if you aren't allowed to use your memory (why not though?) how on Earth do you remember that you had to count them, or what the word count means, or why you're sat at a table looking at some beads...

 

[Russell E. Rierson]

As much as I know (and please correct me if I am wrong) no fundamental mathematical systems like ZF or Peano axioms include an examination of our cognition's ability to count.

In my opinion, if we ignore our abilities to count, then there is a reasonable chance that our fundamental axioms include hidden assumptions.

For example, let's examine this situation:

On the table there is a finite unknown quantity of identical beads > 1
and we have:

A) To find their sum.

B) To be able to identify each bead.

Limitation: we are not allowed to use our memory after we count a bead.

X

~X

~~X

~~~X

~~~~X

~~~~~X

etc...

2-valued logic

 

[Hurkyl]

there is a reasonable chance that our fundamental axioms include hidden assumptions.

I think this is an unreasonable suggestion.

The main point of the axiomatic method is to make all assumptions explicit; one is not allowed to use anything that cannot be deduced from the axioms.


Now, of course, there is the (non-mathematical) question that asks whether a mathematical theory corresponds to some "real life" thing (like counting beads). This brings up a practical point of the axiomatic method; because the axioms are listed explicitly, you can check each of them to see whether or not they correspond to the "real life" thing in question.




Anyways, now, it almost seems like you're trying to do basic combinatorics

 

[Hurkyl]

On the table there is a finite unknown quantity of identical beads > 1
and we have:

A) To find their sum.

B) To be able to identify each bead.

Limitation: we are not allowed to use our memory after we count a bead.

Ok, I'll go fetch my stack of stickers.

The sticker with "1" on it is on top, I'll place it on one of the beads.
I see a bead without a sticker on it; I'll take the next sticker (which happens to be "2") and place it on the next bead.
I see a bead without a sticker on it; I'll take the next sticker (which happens to be "3") and place it on the next bead.

Ah, all the beads have stickers. "3" was the last sticker I used, so there must be three beads!

And since all of the beads have stickers on them, I can distinguish them too.

 

[geistkiesel]

As much as I know (and please correct me if I am wrong) no fundamental mathematical systems like ZF or Peano axioms include an examination of our cognition's ability to count.

In my opinion, if we ignore our abilities to count, then there is a reasonable chance that our fundamental axioms include hidden assumptions.

For example, let's examine this situation:

On the table there is a finite unknown quantity of identical beads > 1
and we have:

A) To find their sum.

B) To be able to identify each bead.



Limitation: we are not allowed to use our memory after we count a bead.

By trying to find the total quantity of the beads (representing the discreteness concept) without using our memory (representing the continuum concept) we find ourselves stuck in 1, so we need an association between continuum and discreteness if we want to be able to find the bead's sum.

Let's cancel our limitation, so now we know how many beads we have, for example, value 3.

Now we try to identify each bead, but they are identical, so we will identify each of them by its place on the table.

But this is an unstable solution, because if someone takes the beads, put them between his hands, shakes them and put them back on the table, we have lost track of the beads identity.

Each identical bead can be the bead that was identified by us before it was mixed with the other beads.

We shall represent this situation by:

((a XOR b XOR c),(a XOR b XOR c),(a XOR b XOR c))

By notating a bead as 'c' we get:

((a XOR b),(a XOR b),c)

and by notating a bead as 'b' we get:

(a,b,c)

We satisfy condition B but through this process we define a universe, which exists between continuum and discreteness concepts, and can be systematically explored and be used to make Math.

What I have found through this simple cognition's basic ability test is that ZF or Peano axioms "jump" straight to the "end of the story" where cardinal and ordinal properties are well-known, and because of this "jump" Infinitely many information forms that have infinitely many information clarity degrees, are simply ignored and not used as "first-order" information forms of Math language.

In my opinion, any language (including Math) is first of all an information system, which means that fundamental properties like redundancy and uncertainty MUST be taken as first-order properties, but because our cognition's abilities were not examined when ZF or Peano axioms were defined, both redundancy and uncertainty were not included in our logical reasoning or in our fundamental axiomatic systems.

Also in my opinion, through this simple test we get the insight that any mathematical concept is first of all the result of cognition/object (abstract or non-abstract) interactions.

My paper http://us.share.geocities.com/complementarytheory/ONN2.pdf is a reexamination of some fundamental mthematical concepts that are based on this insight.

To my point of view you have adequately described a "hidden Variable" system in thought processes. Think about what thinking is. First memory comes and goes a stimulated. Comupter models fail from memory problems of how does the self-starting computer make the kinds of decisions and accomplish the kinds of thought processes, such as intuitive leaps, as the mind. That thought is restricted to the brain is an unverified assumption. Does the brain have the storage capacity necessary to retain all the thoughts back to childhood when properly stimulated and still be able to think and rationalize? I say no. When you learn to drive, you are aware of the mental and physical coordination when learning, now try to tell yourself how you drive. You haven't enough time left in you life to do this. I mean read your memory, not just make a logically sounding speech on how you drive.

What is not hidden by the thinking mind? I have been trying to capture the secret of how a mind can stack some very complicated words and pharases uttered by a person ad lib, as in a heated discussion, but not know exactly what words will exit the lips.. Are yo aware of the next words or do you have just a hint?

Hidden variables, nonlocal force centers, you name it. Try a book by Rupert Sheldrake "The Presence of the Past| Morphics Resonance |the Habits of Nature." Excellent book on every thing you wanted to know about this and that . . .

Mathemtical models seem helpful, but I assume the modles are relativiely useless when getting to the nitty gritty of thought and consciousness..

What about the body, including the brain, as a mere antenna absorbing and processing information fields? There is data that suggest this.

 

[matt grime]

You (WWW) might also like to consider the thoughts of othere such as Russell about this and the nature of numbers and what they are when we count. But that's philosophy, not maths.

 

[WWW]

if you aren't allowed to use your memory (why not though?) how on Earth do you remember that you had to count them?

I wrote:

Limitation: we are not allowed to use our memory after we count a bead.

It means that in this case we always start to count from the begining, and the result is that without our memory we cannot count beyond 1.

 

[sol2]

I wrote:

Limitation: we are not allowed to use our memory after we count a bead.


It means that in this case we always start to count from the begining, and the result is that without our memory we cannot count beyond 1.

I hope you all do not mind the intrusion here?

What http://www.fortunecity.com/emachines/e11/86/beneath.html is a interesting question. At Planck length things can become very different? What is memory in this instance?

So we have to find this "point" where the math arrives(expands), where it can be reduced to a point and then "spreads out" in terms of probabilistic determinations. But to do so, would produce infinities. So what would determine such pathways in the marble drop? What mathematical discription would reveal the math involved in Pascal's triangle?

George Lakoff and the Cognitive Science of Mathematics What is the Steering Behavior that would calculate all these probabilities and then, move it through that point.

Were the infinities existant?

How woud such things retain the memory as "information" in solid discrete objects?

 

[WWW]

Hi geistkiesel,

First, thank you for your interesting post.

I think that a part of the answer of our cognitive abiliteis to deal with comlexity somehow connected to an inherited abilites to associate between opposite (abstract or real) things.

This (in my opinpon) important property give us the ability to explore in non-trivial ways a lot of Simultaneous input sources which constructed as a very complex probabilistic wave function, which within milliseconds we have to take some useful information out of them.

It cannot be done if we cannot deal with parallel multi-opposite situations and find our non self destructive unique real time way to express ourselves.

Shortly speaking, the memorey of the evolution process is in our genes and it is the companion of every living thing for better or worse.

 

[WWW]

think this is an unreasonable suggestion.

The main point of the axiomatic method is to make all assumptions explicit; one is not allowed to use anything that cannot be deduced from the axioms.


Now, of course, there is the (non-mathematical) question that asks whether a mathematical theory corresponds to some "real life" thing (like counting beads). This brings up a practical point of the axiomatic method; because the axioms are listed explicitly, you can check each of them to see whether or not they correspond to the "real life" thing in question.

I have a very simple claim which is: If a mathematician does not include its own basic abilities when he use fundamental concepts, then there is a reasonable chance that he can make some "automatic" (unconscious) shortcuts that ignore valuable properties of these fundamental concepts.

I gave an example of how Pano ignored properties like redundancy and uncertainty and his own ability to count when he defined his natural number system.

Because Mathematics is first of all a form of language, both redundancy_AND_uncertainty and our cognition's abilities to count must be included right from the beginning of any fundamental definition process.

Ok, I'll go fetch my stack of stickers.

To make it clearer, we do not start mission B, before we finish mission A.

But this was not my goal, my goal was to give you an insight about cognition/object interactions.

If you try to be "smart" by using shortcuts, then you miss the universe of infinitely many information forms that I suggesting, which can be used by us to make Math.

By this attitude you repeat on Peano's shortcut.

 

[WWW]

Hi sol2,

Your post is very intersting.

I'll read and then reply to you.

 

[matt grime]

I wrote:

Limitation: we are not allowed to use our memory after we count a bead.

It means that in this case we always start to count from the begining, and the result is that without our memory we cannot count beyond 1.

and after we've counted one bead, how are we to remember anything at all? what our name is, what our language consists in, what we are doing...

 

[matt grime]

I have a very simple clime which is: If a mathematician does not include its own basic abilities when he use fundamental concepts, then there is a reasonable chance that he can make some "automatic" (unconscious) shortcuts that ignore valuable properties of these fundamental concepts.

I gave an example of how Pano ignored properties like redundancy and uncertainty and his own ability to count when he defined his natural number system.

Because Mathematics is first of all a form of language, both redundancy_AND_uncertainty and our cognition's abilities to count must be included right from the beginning of any fundamental definition process.


To make it clearer, we do not start mission B, before we finish mission A.

But this was not my goal, my goal was to give you an insight about cognition/object interactions.

If you try to be "smart" by using shortcuts, the you miss the universe of infinitely many information forms that I suggesting, which can be used by us to make Math.

By this attitude you repeat on Peano's shortcut.

but you are unable to justify why peano needs to incorporate uncertainty or redundancy into his arithemetic, which is by definition something which apparently does not require these objects. Objects one migh say that you have not defined without referenece to the natural number system that is insufficient. That is a circular argument, by the way.

Nor do you understand that something is exactly what it does. if you are saying they, the naturals as we all know them, are insufficient to count things then perhaps you could offer some evidence to back this up, and more importantly you might explain why it is that we shoudl give a toss? seeing as it is just an axiomatic and self consistent theory where is it wrong? there ought to be some evidence to back up your opinion. so where is it? if you are once more to claim (not clime) it is wrong please justify why, offer some example where it is insufficient. just one example, that isn't unreasonable. and bear in mind the structural nature of mathematics.

 

[matt grime]

because Mathematics is first of all a form of language,

justify that statement. what is a language, what is a form of such and why is mathematics one of these

 

[WWW]

As for your question about language, in my opinion language is first of all interactions among systems that cause internal or external, actual or potential, direct or indirect influences on these systems.

Some systems can be aware to these interactions (and then the word "information" is used) and some are not.

In our case, the language of Mathematics has a tremendous influence on our systems, and therefore any form of information must be examined right from its first-level.

When Peano defined his axiomatic system he used his natural ability to go straight to the forms of information where cardinality, ordinality and the unique identity of each object are well-known.

But in my opinion, when you define the fundamental level of some language, you have to examine your own system's abilities during this process, otherwise there is a chance that you will ignore fundamental properties that can have a significant influence on the "character" of your basic products, and in this case the products are the Natural numbers, and also addition and multiplication operations.

In my system, where redundancy_AND_uncertainty are first order properties, Natural numbers have internal complexity, which is based on cognition/object interactions.

And because of this internal complexity more information forms can be defined and ordered by their symmetrical degrees and information clarity degrees within any given finite quantity.

Also addition and multiplication have two words of operations: internal and external.

By Peano system the Natural numbers world is only the external world, where each operation changing the quantity of n.

Let us look on - and + operations on n from the eyes of the symmetry concept:

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 structure.

Here is a simple example of a theorem and proof which are out of the scope of Peano system:

Theorem: 1*5 not= 1+1+1+1+1

Proof: 1*5 = {1,1,1,1,1} not= {{{{1},1},1},1},1} = 1+1+1+1+1

To understand this example please look at: http://us.share.geocities.com/complementarytheory/ONN.pdf

and after we've counted one bead, how are we to remember anything at all? what our name is, what our language consists in, what we are doing...

You take my analogy as it is instead of use your creative mind and check the "what if...?" Question that I suggesting here.

So (for you) let us say that after we count a bead, all we remember is that we have a mission which is: to count beads.

Also, if you don't mind, I think it will be good for you to read De-Bono's book on parallel thinking: http://members.optusnet.com.au/~charles57/Creative/Books/B20471.htm

 

[ram2048]

hmm my post got deleted for excess { } thingies i think :O

crap

 

[ram2048]

hmmm when i post on this thread it gets deleted.

how very odd

 

[ram2048]

k i'll try once more.

is what you're saying:

5*1 makes the assumption that we want the sum of the items listed, such that 1+1+1+1+1 which creates the union {{{{{1},1},1},1},1)

but without that assumption 5*1 could mean any number of things as:

1,1,1,1,1

1+1, 1, 1, 1

1+1, 1+1, 1

1+1, 1+1+1

1+1+1, 1, 1

1+1+1+1, 1

or

1+1+1+1+1

 

[WWW]

Hi ram2048,

Please look at http://us.share.geocities.com/complementarytheory/ONN.pdf
if you want to understand my point of view

 

[matt grime]

Perhaps you might also care to explain how on Earth your system of the natural numbers (which it cannot beby definition; when will you accept that that is the main criticism of your work? that you do not get to redefine a term) is able to count beads under the limitations of not being able to remember

 

[WWW]

Matt,

Please read post #16.

 

[ram2048]

i read through the PDF twice, but it's really built for people with a higher level of math knowledge.

i just wanted to clarify that I'm deciphering it correctly

5*1 means 5 items of value 1, right? (1,1,1,1,1)

but those values can be held in the number 5 in a varying number of configurations

(1+1, 1+1, 1) (1+1, 1, 1, 1) (1+1+1, 1, 1) (1+1+1, 1+1) (1+1+1+1, 1) (1+1+1+1+1) (1,1,1,1,1)

now where i get confused is. does each addition "phase" define a "level" or "tier" that the numbers get put into? say i have{{{{1},1},1},1} and i want to add 1 to tier 2 (how are they designated anyways?) let's just say i want to add it here {{{{{1},1}1,1}1} is this the same as (1, 1+1, 1, 1) ?

and say you had 12 beads, we know the property of beads is such that you can't "add" them to get a larger bead of greater value, so there is no assumption when you say 12 beads, you're always going to have (1,1,1,1,1,1,1,1,1,1,1,1)

 

[matt grime]

Matt,

Please read post #16.

that post at no point demonstrates any effect of superiority that your system has given the artificial and unjustified and ill-explained rule that we cannot remember. you may not simply say ah look here, if the ''here" has no relation to anything as this does, or does not depending on your point of view.

 

[WWW]

Hi ram2048,

First thank you for reading my work, but I see that it is still not understood.

My main point is that when there is an internal structure of symmetrical levels within any given finite quantity > 1 (and I am talking only on the Natural numbers at this stage) then addition or subtraction operations are also used to change the internal symmetrical structure within any given quantity > 1.

Shortly speaking, the internal changes are series of single steps of addition or subtraction operations, where each step changes the internal symmetrical degree of the given Natural number.

(1,1) + (1) = [ and we get ((1),1) ]

((1),1) - (1) = [ and we get (1,1) ]

((1),1) + (1) = [ the internal symmetry level and the quantitative domain has been changed; therefore we get (1,1,1)]

(1,1) - (1) = [ the quantitative domain have been changed; therefore we get (1)]

The quantitative domain is changed only when we move beyond the most symmetrical or non-symmetrical degrees within some given quantity.

Shortly speaking, when we have finite or infinitely many quantitative domains, no one of their internal symmetrical structures can be ignored when we deal with complexity.

 

[WWW]

that post at no point demonstrates any effect of superiority that your system has given the artificial and unjustified and ill-explained rule...

If the internal complexity (which is the direct result of my cognition/object interactions approach) within any n > 1 can be represented by cardinality and ordinality that are based on Peano axioms, then my Organic Natural numbers can be ignored as first-order elements.

So, show us by your reasoning, that this is the case.

( For example: 4 is not identical to 1+1+1+1 because by my reasoning each Natural number is an interaction between its integral side (memory=connector=4) and its differential side (objects=1,1,1,1), as we can clearly see in: http://www.geocities.com/complementarytheory/ET.pdf )

Shortly speaking, show us why important concepts like symmetry and information clarity-degree do not have be used as first-order properties when we define the Natural numbers.

 

[matt grime]

Shortly speaking, show us why important concepts like symmetry and information clarity-degree do not have be used as first-order properties when we define the Natural numbers.

Er, because they don't have to be. You may wish to define something which has these properties, however they are not necessarily the natural numbers. You've also not been able to produce a non-circular definition of these objects either, ie one that doesn't rely on the natural numbers as we know them - inparticular you require us to distinguish objects remember maximal symmetry degree or something requires us to count n different objects - how can I do this? according to you I can't:- it's something that we aren't allowed to do in your thought experiment. So how exacttly does your definition of your number system allow you to get round the alleged issues your thought experiment throws up? Yep, it's the third time I've asked, and I'm still waiting for an answer. I don't have to justify these things in another system, so stop asking me to, because these alleged problems are a non-issue - it's rather like you asking me why you can't do division in the natural numbers - they weren't intended to have a division operation, that's why we have the rationals.

 

[WWW]

Matt,

I am really amazed by your inability to follow and understand the results of my simple “what if…” test.

Er, because they don't have to be...

A lot of useful things do not have to be but they exist and you can choose if you want to use them or not, so what is your point?

You can look at the Natural numbers only as well-known quantitative-only elements, but tell us why we have to limit owerselves only to these properties?

 

[matt grime]

You can look at the Natural numbers only as well-known quantitative-only elements, but tell us why we have to limit owerselves only to these properties?

We don't, but they are not necessary for their definition. I'm continually amazed by your ability to not understand basic maths, and?

 

[ram2048]

(1,1) + (1) = [ and we get ((1),1) ]

is ((1),1) = 3 then?

would that make:

((((1),1),1),1) = 7 ?

and if so does:

(((1),1,1),1) = 8 ?

 

[hello3719]

( For example: 4 is not identical to 1+1+1+1 because by my reasoning each Natural number is an interaction between is integral side (memory=connector=4) and its differential side (objects=1,1,1,1), as we can clearly see in: http://www.geocities.com/complementarytheory/ET.pdf )
QUOTE]

You are saying that 4 is not identical to 1+1+1+1 but do you still agree that 4 is equal to 1+1+1+1?

if not then define your "equality" concept.

 

[WWW]

I'm continually amazed by your ability to not understand basic maths, and?

Please show us how Peano Axioms can define my system as "second-order".

If you can show this then my system is a "second-order" and N members stays as they are.

 

[WWW]

Hi ram2048,

Thank you for not giving up with my system.

The internal changes are series of single steps of addition or subtraction operations, where each step changes the internal symmetrical degree of the given Natural number.

In the external world of N members each - or + operation is changing the quantity.

In the internal world of each given quantity, the quantity itself remains unchanged during a series of single steps of - or + , where each single step changing the internal symmetry of the given quantity.

But the given quantity can be changed by these single steps when:

a) A single subtraction step is operated on the most symmetrical state of the given quantity for example: (1,1,1,1) - (1) = (((1),1),1).

b) A single addition step is operated on the most non-symmetrical state of the given quantity for example: (((1),1),1) + (1) = (1,1,1,1)

The internal operation steps can move only by a one and only one step for each move, which is not the case in the standard external moves that cares only about quantitative changes.

 

[WWW]

Hi hello3719,

You are saying that 4 is not identical to 1+1+1+1 but do you still agree that 4 is equal to 1+1+1+1?

if not then define your "equality" concept.

Through my point of view, any given n has also several internal symmetrical states that can be defined if and only if any n has an internal structure.

I have found this internal structure by distinguishing between our memory and the objects that it remember, and by combining memory/objects we get the internal world of each given n.

In this internal world of symmetrical changes, our memory is represented by a single notation where the objects are represented by several '1' notations.

From this point of view '4' (the memory notation) is not '1','1','1','1' (the objects notation).

The symmetrical changes are the fading transition between the memory side and the objects side (and vise versa) within any given n > 1.

 

[ram2048]

i think the reason we're all having such a hard time with the system is it has very little in common with the current number system for which we can accurately use to describe common everyday events

that your system would be good for describing events of probability and possibility is indeed useful, it's just difficult to retrain our brains to "think" in that mode.

perhaps if you could outline some example cases it would be easier to get a lock on the system, seeing it in use.