Breaking Euclid's Back

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Treatid
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Re: Breaking Euclid's Back

Post by Treatid »

A_Seagull wrote:Why do you want 'meaning'? What do you mean by 'meaning'?

A consistent and simple way of viewing a system such as mathematics is to consider it as being entirely abstract. In other words it has no direct correspondence with the 'real world' - ( as derived from dense-data.)
What do you mean by meaning? What is 'abstraction'?

Taking what you have described literally:

I have some axioms:
  • Wibble
  • Flurgle
  • Quangwilly
From these axioms I have derived a number of results:
  • Shootangle
  • Foxnaggle
  • Woowoohunks
Since the meaning of axioms and their results don't arise until we find a correspondence with real world observations; then these axioms and results are as meaningful as any other axiom set.

...

The whole point and purpose of axioms is that a statement by itself has no meaning (or has every meaning). A statement only has a definite meaning with respect to a particular set of axioms (according to mathematics).

This is the basis of axiomatic mathematics. It is the reason for axiomatic mathematics. Mathematics tends to be pedantic for a reason...

You have shifted the meaning of 'meaning' in an entirely arbitrary way. It is true that we find mathematics useful when we find correlation with real world experience... But 1+1 has to be more than just a set of symbols. We need a way to treat those symbols consistently.

It is certainly reasonable to view mathematics as nothing more than symbol manipulation. But in order to manipulate symbols in a consistent way you have to define the particular manipulations that are allowed. What those manipulation apply to.

Axioms are supposed to be the rules that describe what manipulation of what symbols. Without those rules then any manipulation is possible. 1+1=567,877,935.743fhjhgfdjmktr9084557 is just as valid as 1+1=2 if you haven't defined the rules of the system.

...

Part of what you are arguing is that axioms (the rules of the system) don't need to be justified. Axioms can be as arbitrary as we like.

Which misses the whole point.

Axioms cannot be defined.

Axioms cannot be stated.

There are no axioms.

It isn't a question of whether a particular set of axioms is useful, or consistent, or meaningful.

Axioms don't exist.

Don't be fooled by what you were taught.

Look at what axioms are.
  • A statement only has a defined meaning with respect to a set of axioms.
  • Axioms, themselves, are statements.
This is a closed system. There is no starting point. There is no way to construct axioms.

It doesn't matter that you (and lots and lots of other people) think that they have written down or read many sets of axioms. It doesn't matter that we do perceive meaning in symbols and can act in a consistent way based on that perceived meaning.

Within mathematics, axioms are not an ambiguous concept. What an axiom is supposed to be is quite clear.

And based solely on what axioms are supposed to be - we can see that they cannot exist.

Our world hasn't suddenly vanished in a puff of logic. To the extent that mathematics is useful - there must be a mechanism that permits that usefulness. Computers still work and your pay-check still has taxes taken out of it.

It has never been possible to do the impossible. Just because large numbers of people thought that axioms existed and thought they were building logical systems on those premises doesn't make it true.

...

Addendum:

Seriously; what do you think 'abstract' or 'abstraction' means? Your use of the word suggests to me that you think it is some sort of phase shift that does... well... magical stuff...

My understanding of abstraction is that irrelevant or surplus material is removed to leave you with the essential elements.

Taking an unknown system and removing bits doesn't turn the unknown system into a known system. It may make it easier to see the essential components or mechanisms of the system.

However, the big, huge, whopping, massive issue is that naive abstraction assumes that a system has a fixed reference frame.

That is - naive abstraction believes that it is possible to remove an element of a system without changing the other elements of that system.

Take Distance, Velocity and Time as a system. d=vT. Now remove any one of these concepts (abstract the system to just e.g. distance and time).

A naive approach to abstraction would assume that distance and time remain the same in the absence of velocity. Or that velocity and distance remain as they were in the absence of time. Can velocity really exist without time? Or without distance?
Londoner
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Re: Breaking Euclid's Back

Post by Londoner »

Mathematics is an example of an abstract system that does provide theorems which can be effectively and usefully mapped onto the real world.
Can you give an example? I do not see how you can ever make a bridge between 'abstractions' and 'reality'. One or the other.

And how do we know that maths is effective and useful? If it is because it provides results that coincide with sensory evidence, then the validity of maths does not rest on maths but upon sensory experience. We do not know maths delivers good answers or confirms correctness, rather we are saying we assume a mathematical method is good if it produces those results we had already decided are correct.

To put it another way, we humans need to order our sensory experiences in a particular way in order to operate. This makes sense of the world thus we find it satisfying. Maths orders ideas in the same way, so of course we find that satisfying too. So maths describe us, our mental architecture, perhaps our limitations. We have no idea if it represents anything 'out there', any more than we can access some 'real world' that might exist beyond our sensory impressions.
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A_Seagull
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Re: Breaking Euclid's Back

Post by A_Seagull »

Treatid wrote:
A_Seagull wrote:Why do you want 'meaning'? What do you mean by 'meaning'?

A consistent and simple way of viewing a system such as mathematics is to consider it as being entirely abstract. In other words it has no direct correspondence with the 'real world' - ( as derived from dense-data.)
What do you mean by meaning?

I generally don't use the word except in reference to a dictionary.

What is 'abstraction'?

As I stated: it has no direct correspondence with the 'real world' - ( as derived from dense-data.)

Taking what you have described literally:

I have some axioms:
  • Wibble
  • Flurgle
  • Quangwilly
From these axioms I have derived a number of results:
  • Shootangle
  • Foxnaggle
  • Woowoohunks
Your 'results' do not follow from your axioms. Perhaps there are some axioms missing?

Since the meaning of axioms and their results don't arise until we find a correspondence with real world observations; then these axioms and results are as meaningful as any other axiom set.
But they do NOT have any correspondence with the real world!

...

The whole point and purpose of axioms is that a statement by itself has no meaning (or has every meaning). A statement only has a definite meaning with respect to a particular set of axioms (according to mathematics).


You've lost me.

This is the basis of axiomatic mathematics. It is the reason for axiomatic mathematics. Mathematics tends to be pedantic for a reason...

You have shifted the meaning of 'meaning' in an entirely arbitrary way.

As stated above I generally don't use the word 'meaning' , if I used it, it was only to humour you.
It is true that we find mathematics useful when we find correlation with real world experience... But 1+1 has to be more than just a set of symbols. "has to be".. why?
We need a way to treat those symbols consistently.

It is certainly reasonable to view mathematics as nothing more than symbol manipulation. But in order to manipulate symbols in a consistent way you have to define the particular manipulations that are allowed. What those manipulation apply to.

Exactly, that is what axioms are for. The difficulty with mathematics is that most of its axioms are implicit rather than explicit. So no one really knows what all the axioms of mathematics are.

Axioms are supposed to be the rules that describe what manipulation of what symbols. Without those rules then any manipulation is possible. 1+1=567,877,935.743fhjhgfdjmktr9084557 is just as valid as 1+1=2 if you haven't defined the rules of the system.
Quite so.


Part of what you are arguing is that axioms (the rules of the system) don't need to be justified. Axioms can be as arbitrary as we like.

Which misses the whole point.

Axioms cannot be defined.

Axioms cannot be stated.

There are no axioms.

How about the axioms for Conway's 'Game of Life'? Aren't those well defined axioms which enable interesting 'theorems' to be deduced?

It isn't a question of whether a particular set of axioms is useful, or consistent, or meaningful.

Axioms don't exist.

Don't be fooled by what you were taught.

What was I taught?

Look at what axioms are.
  • A statement only has a defined meaning with respect to a set of axioms.
  • Axioms, themselves, are statements.
This is a closed system. There is no starting point. There is no way to construct axioms.

It doesn't matter that you (and lots and lots of other people) think that they have written down or read many sets of axioms. It doesn't matter that we do perceive meaning in symbols and can act in a consistent way based on that perceived meaning.

Within mathematics, axioms are not an ambiguous concept. What an axiom is supposed to be is quite clear.

You seem to be contradicting yourself. Didn't you just say a few lines back that "axioms don't exist"?

And based solely on what axioms are supposed to be - we can see that they cannot exist.

Our world hasn't suddenly vanished in a puff of logic. To the extent that mathematics is useful - there must be a mechanism that permits that usefulness.

What do you mean "must"?

Computers still work and your pay-check still has taxes taken out of it.

It has never been possible to do the impossible. Just because large numbers of people thought that axioms existed and thought they were building logical systems on those premises doesn't make it true.

...

Addendum:

Seriously; what do you think 'abstract' or 'abstraction' means? Your use of the word suggests to me that you think it is some sort of phase shift that does... well... magical stuff...
I have already answered that above.

My understanding of abstraction is that irrelevant or surplus material is removed to leave you with the essential elements.
I think you misunderstand the meaning (as defined in a dictionary) of the word.

-- Updated January 22nd, 2015, 3:03 pm to add the following --
Londoner wrote:
Mathematics is an example of an abstract system that does provide theorems which can be effectively and usefully mapped onto the real world.
Can you give an example? I do not see how you can ever make a bridge between 'abstractions' and 'reality'. One or the other.

Pure mathematics is an abstract system. "1+2=3" is a string of symbols that would be considered to be 'true' in the system of mathematics. I.e that string can be generated as a theorem of mathematics from the axioms of mathematics.

I can apply a 'mapping' to the real world: "The counting of apples can be mapped onto the arithmetic of positive whole numbers."

Suppose I have 1 apple and someone gives me 2 apples. I can then apply the relevant mathematical theorem: "1+2=3", to deduce that I then must have 3 apples.


And how do we know that maths is effective and useful? If it is because it provides results that coincide with sensory evidence, then the validity of maths does not rest on maths but upon sensory experience.
No. The validity of maths depends upon the consistency of its axioms and its derived theorems.

It is its relevance to the real world that depends upon sensory experience.


We do not know maths delivers good answers or confirms correctness, rather we are saying we assume a mathematical method is good if it produces those results we had already decided are correct.

To put it another way, we humans need to order our sensory experiences in a particular way in order to operate. This makes sense of the world thus we find it satisfying. Maths orders ideas in the same way, so of course we find that satisfying too. So maths describe us, our mental architecture, perhaps our limitations. We have no idea if it represents anything 'out there', any more than we can access some 'real world' that might exist beyond our sensory impressions.
Yes I think I agree. The difference is that when we 'order our sensory experience', it is based upon sense data from the real world. In contrast, pure and abstract mathematics emanates from axioms. The two can then be linked using a mapping as described above.
The Pattern Paradigm - yer can't beat it!
Londoner
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Re: Breaking Euclid's Back

Post by Londoner »

Suppose I have 1 apple and someone gives me 2 apples. I can then apply the relevant mathematical theorem: "1+2=3", to deduce that I then must have 3 apples.
I do not think there is any such thing as a 'three apples', (as indicated by the awkward grammar).

There to the left is this red apple, there is that green apple, there is another small apple on the right. I can only say there are 'three apples' if the term 'apple' no longer refers to those actual apples, which are all different objects separated in space, but to something abstract; a disembodied and uniform concept 'apple' that is distinct from any actual apples.

To put it another way, is the 'three' meant to be descriptive of the apples? In that case, what attribute of those apples does it describe? If I had said 'the apples are red' then (if it was true) each apple separately has the property 'red'. But if I say 'the apples are three', which apple has the property 'three'? If it is none of them; if it is only all together they have that property, then we are saying 'three' describes not the apples but 'togetherness'. So again, I would say that number does not describe objects, only itself. It only applies to apples if we strip away all the 'appleness'; so 'three apples' just means 'three'.
The difference is that when we 'order our sensory experience', it is based upon sense data from the real world.
I do not see how we can validate the way we choose order our sensory experiences, based on those sensory experiences.

The photons impacting on my retina are just that. They carry no meaning. That I then process that event in terms of shape or colour or infer the existence of classes of objects is something I do. I could do so another way; if I had a different brain then I would interpret those photons differently, or if I was blind then that whole class of sensory experience would not exist.

I cannot show that my ordering of sensory experience is more valid than your ordering of sensory experience, or that of a blind man or an insect. I think we can only work backwards; I first decide (or am obliged to assume for practical purposes) that the world is a particular sort of place, therefore I order my sense experiences accordingly.
In contrast, pure and abstract mathematics emanates from axioms. The two can then be linked using a mapping as described above.
I agree pure maths emanates from axioms. In my opinion, these axioms concern things that are not - cannot possibly be - objects. For example Euclidean geometry applies its logic to things that are only two dimensional. (I switch to geometry for the same reason the ancients did; to avoid the extra problem that maths is not logical in its own terms). It is logical, but not logical about the world.

But I would say applied maths is different. A mathmatical model is validated by whether or not it corresponds to what it describes. If I made a mathematical model of the weather system but it did not describe the weather, then it would be incorrect, even if it were logical i.e there was no error of calculation.

So I am not clear what 'mapping' describes. An actual map reduces objects to symbols, but those symbols do not take on an existence of their own; drawing a contour line does not shape a hill. Nor can things only be validly mapped in one way; to symbolise is to select, to choose to ignore certain sorts of differences. I think this is also true of mathematical descriptions.
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TimBandTech
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Re: Breaking Euclid's Back

Post by TimBandTech »

Well Treatid, Seagull has found a grammatical conflict in your own rendition. I accept that wee humans start out as a blank slate roughly, versus the orb state. A natural consequence of this position is that we must take the freedom to construct, whether what we construct is accurate or not. We should beware that we are engaged in a progression, and that we are somewhere along in that progression. Unfortunately the accumulation of information is getting a bit dense, so I can appreciate your attempts at starting from scratch. It's a good attempt and you are being very careful, but whether you can actually pull it off is another thing. Thus far you've laid out a system and yet have no practical examples. I often criticize parts of mathematics for failing this same way. What good is a long series of definitions(or relations) which carry no instantiables? That would be pedantic, and we all have to tread carefully here.

I think a fine instance is countability. It does abstract well, by which I mean that it is generally applicable. There would certainly be plenty to discuss among primitives with regard to what is countable, but it would be fairly easy to agree that sheep in a pen are countable, versus the weight of an individual sheep being not precisely countable even while a counting method to say a pound unit will be a reasonable approximation. Thus the quantity of pounds of sheep in a pen is likewise countable. Shades of gray do enter into the discussion, and the realization of continuous versus discrete properties quickly ensues, and lo, mathematics can be borne from a primitive people. Even those who have not come to symbolic definitions for numbers may use rocks in a leather bag to account for transactions.

I do think that this could form an instance of distinction within Treatid's construction, and particularly a device that can measure pounds to the pound unit will be applicable across many objects, while obviously we needn't confuse the quality of ten apples with ten sheep, or even ten pounds of mutton versus ten pounds of apples. Their commonality is that they are quantifiable, even while their other qualities may be distinct. The concept of units is here and soon we would flush out a fair amount of standard mathematics if we carried on.

Within the ultraprimitive societies of countenance it's pretty obvious that every time you count a sheep in the pen you should both remove that sheep from the pen and add a rock to the bag. Whether such a rule is an axiom at this primitive stage may not be important. Still, this idea of being careful with the numbers to the point of having provable results is a matter of refinement, and we could regard the axiomatic approach as simply an instance of such care.

We are merely somewhere in a progression from such a simple starting point and no matter how many preceding philosophers and mathematicians have pronounced dead ends the fact is that the next generation are free to play with their own constructions and variations. This freedom to construct seems to me an axiom of humans which goes unstated most of the time. I think the cogito is a fine instance, and an unlikely truth at that. The rock which does not think still is as far as I can tell. I'm still waiting on the supercomputing rocks to speak my language, at which point they cannot tell me everything all at once, and so quantum physics will be falsified.

Another context to consider is one of parallel theories. This certainly can exist and while two parallel theories are not identical there will be subtle differences which are instructive. Perhaps Treatid's attempt can function as that. You may not actually need to falsify the axiomatic approach to find another route. Clearly you want to avoid it and that is a pursuit worthy of your attempt; no matter how many pronounce it futile.
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A_Seagull
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Re: Breaking Euclid's Back

Post by A_Seagull »

Londoner wrote:
Suppose I have 1 apple and someone gives me 2 apples. I can then apply the relevant mathematical theorem: "1+2=3", to deduce that I then must have 3 apples.
I do not think there is any such thing as a 'three apples', (as indicated by the awkward grammar).

What 'awkward grammar'? I have no idea what you are talking about. Is English your first language?

There to the left is this red apple, there is that green apple, there is another small apple on the right. I can only say there are 'three apples' if the term 'apple' no longer refers to those actual apples, which are all different objects separated in space, but to something abstract; a disembodied and uniform concept 'apple' that is distinct from any actual apples.
That is where the 'mapping' comes in.

The difference is that when we 'order our sensory experience', it is based upon sense data from the real world.
I do not see how we can validate the way we choose order our sensory experiences, based on those sensory experiences.

That is probably because you have not read 'The pattern Paradigm' : http://bookstore.xlibris.com/Products/S ... adigm.aspx

The photons impacting on my retina are just that. They carry no meaning. That I then process that event in terms of shape or colour or infer the existence of classes of objects is something I do. I could do so another way; if I had a different brain then I would interpret those photons differently, or if I was blind then that whole class of sensory experience would not exist.

I cannot show that my ordering of sensory experience is more valid than your ordering of sensory experience, or that of a blind man or an insect. I think we can only work backwards; I first decide (or am obliged to assume for practical purposes) that the world is a particular sort of place, therefore I order my sense experiences accordingly.

Not so.
In contrast, pure and abstract mathematics emanates from axioms. The two can then be linked using a mapping as described above.
I agree pure maths emanates from axioms. In my opinion, these axioms concern things that are not - cannot possibly be - objects. For example Euclidean geometry applies its logic to things that are only two dimensional. (I switch to geometry for the same reason the ancients did; to avoid the extra problem that maths is not logical in its own terms). It is logical, but not logical about the world.

But I would say applied maths is different. A mathmatical model is validated by whether or not it corresponds to what it describes. If I made a mathematical model of the weather system but it did not describe the weather, then it would be incorrect, even if it were logical i.e there was no error of calculation.

So I am not clear what 'mapping' describes. An actual map reduces objects to symbols,

No it does not. It relates a correspondence between 'objects' and 'symbols'.

.
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Londoner
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Re: Breaking Euclid's Back

Post by Londoner »

What 'awkward grammar'? I have no idea what you are talking about. Is English your first language?
I like to think I am reasonably fluent. So don't you find the phrase 'a three apples' awkward? I think we will have to differ on that one.
Me: I do not see how we can validate the way we choose order our sensory experiences, based on those sensory experiences.

That is probably because you have not read 'The pattern Paradigm'
I haven't, but I have read lots of other philosophy books and they all seem to be of the opinion that nothing is self-validating. If this book has proven that we can know our sensory experiences as 'valid' (meaning what exactly?) it would clear up a lot of philosophical problems so I am surprised it isn't better known.
Not so
Unless the book enables its readers to present a more rounded argument I'm reluctant to part with $3.31.

Are there any reviews? I can't find one.
Me: So I am not clear what 'mapping' describes. An actual map reduces objects to symbols,

No it does not. It relates a correspondence between 'objects' and 'symbols'.
If a map has reduced an object to a symbol, then how does that contradict saying that the symbol corresponds to the object?

So I still don't understand what you mean by 'mapping', and since you won't explain and I won't buy the book I think this philosophical breakthrough will have to remain in obscurity.
Treatid
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Re: Breaking Euclid's Back

Post by Treatid »

This is not soft science.

We can count the angels on the head of a pin all we like....

This is not a matter of opinion or taste.

I'm not saying this to exert rightness through authority.

This is the context. Almost everyone believes that axioms exist or can be justified.

The Earth is not flat. The sun does not revolve around the Earth.

There are no axioms.

This isn't a question of philosophical belief. This is mathematics.

Without axioms we do not have logic. We have no proof. Of anything. Ever.

This isn't an obscure result of esoteric ideas. There are no axioms.

Everything constructed around the idea of axioms is wrong.

Not slightly right. Not approaching some understanding. Simply, straight-up, wrong.

Imagine something that has been demonstrated for over a hundred years.... but mathematicians still pretend that there are axiomatic mathematics.

There cannot be a fixed reference frame.

It isn't that we inconveniently don't have a handy fixed reference frame... the concept of a fixed reference frame is fundamentally unworkable. There cannot be a fixed reference frame.

I'm not demolishing what has gone before. There has never been any axiomatic logic.

Constructional philosophies are wrong.

There is no grey area here.

There never has been, never was, and never could be a fixed reference point or reference frame.

The notion that there could be a fixed reference system is wrong.

Black and white wrong.

This isn't a difficult topic. It just goes against everything we are taught.

Euclidean Geometry is so deeply a part of our culture that it is quite difficult to get your head around what there being no axioms means.

There is nothing wishy washy, wooly, hand wavey, vaguey concepty, about this.

There are no axioms.

Right after our own existence, this is the only thing we can know with zero ambiguity.

Absolute is a null term.

I'm not trying to force my own personal philosophy down your necks. There is only one correct way of thinking. You don't have to like it.

Axiomatic thinking is wrong.

It was never possible to construct something in an axiomatic way.

And there is no middle ground. There is no slightly-axiomatic approach.

There is nothing to be preserved. Axiomatic thinking is definitionally tautological. (a closed loop).

Even if axioms were possible they would demonstrate their own non-existence.

Axiomatic thinking is the single greatest mistake of modern thought.

...

A_Seagull:

You claim that my conclusions don't follow from my axioms.

What makes you think that? You have said that axioms and their conclusions don't acquire meaning until mapped onto real world experience. If there is no meaning - there is nothing to judge with. You have nothing to evaluate whether the results follow from the axioms.

Even if you just view axioms and their results as symbols - you still have to interpret those symbols in a consistent way. This interpretation IS meaning.
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TimBandTech
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Re: Breaking Euclid's Back

Post by TimBandTech »

Well when pressed we see that you don't have much of an investigation. You do not falsify fairly clear and fundamental concepts. For instance, are sheep in a pen countable? Are rocks in a leather bag countable? I agree that we all likely suffer from Euclidean thinking within our early work on pencil and paper. I agree that language is a critical limit of interest and that words become self referential. We are humans caught here as prisoners of spacetime. As its prisoners whether we can actually access its basis is dubious. This leaves theoretical construction as our sole means. The experimental physicists are way out ahead. Their idea of a theory is a matter of fitting an equation to an observed curve. That's quite primitive compared to the sort of universal solution we are after. The relative reference frame is of interest, but I don't see a blanket denial of the usage of reference frames as a productive means of analysis, particularly when no replacement has been posited. Likewise with regard to the axiom. I do wish you luck with you own theory, but I don't feel compelled to buy into your repulsion from the axiomatic technique. To work carefully is to work axiomatically as far as I am concerned. It is that simple.
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Re: Breaking Euclid's Back

Post by Mgrinder »

TimBandTech wrote: To start over, arithmetic products mixed with sums such as 3 + (5)(4) = 23 are taken by the mathematician as sensible, but if these numbers are of the same type then to the physicist there is a conflict in units, for the second term is in square units and so cannot mix with a term in the base units. Even if these values are real values in this thinking that second term is is square real units according to this construction. Allowing such values to take their place within geometry (i.e. allowing an entity such as (5)(4) to be positioned at (5,4) is akin to physical field theory, and though this simple construction is unlikely, the freedom which builds it is established.
If such an equation came up in science it would read like: 3m^2 +(5m)(4m)=23m^2, where meters are just example units. There is no problem here between math and science. Math just uses unitless numbers for simplicity, but when you apply math to science you just tag on the units. there is no problem.
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A_Seagull
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Re: Breaking Euclid's Back

Post by A_Seagull »

Londoner wrote:
What 'awkward grammar'? I have no idea what you are talking about. Is English your first language?
I like to think I am reasonably fluent. So don't you find the phrase 'a three apples' awkward? I think we will have to differ on that one.

I made no reference a "a three apples", there is no 'a' in my statement , which is why I questioned your familiarity with the English language.
Me: I do not see how we can validate the way we choose order our sensory experiences, based on those sensory experiences.

That is probably because you have not read 'The pattern Paradigm'
I haven't, but I have read lots of other philosophy books and they all seem to be of the opinion that nothing is self-validating. If this book has proven that we can know our sensory experiences as 'valid' (meaning what exactly?) it would clear up a lot of philosophical problems so I am surprised it isn't better known.

So am I!
Me: So I am not clear what 'mapping' describes. An actual map reduces objects to symbols,

No it does not. It relates a correspondence between 'objects' and 'symbols'.
If a map has reduced an object to a symbol, then how does that contradict saying that the symbol corresponds to the object?

So I still don't understand what you mean by 'mapping', and since you won't explain and I won't buy the book I think this philosophical breakthrough will have to remain in obscurity.
"Mapping" is a mathematical term that relates the elements of one set with those of another. So, for example, the elements of the set of positive even numbers can be 'mapped' onto the elements of the set of positive even numbers under the mapping: E1(i) = E2(i) + 1. In other words they can be correlated under the operation of the addition of 1.

In the instance of mapping the elements of an abstract system, such as mathematics, with the data relating to the real world, as for example in measurements in physics, then the mapping will be one between abstract symbols and real-world 'objects'.

-- Updated January 31st, 2015, 11:24 am to add the following --
Treatid wrote:A_Seagull: What is it that you think a definition defines? A definition specifies the meaning. Definition and meaning aren't two different things.

Definitions give an indication of the meaning of a word. They do not specify it, (except in the instance of axioms and abstract systems.

You are being inconsistent.
I am not! Perhaps I have not explained myself clearly or perhaps you have not understood my intention, but I do not do inconsistency.


If axioms are just symbols - they don't tell you anything. If they tell you something (like how to manipulate other symbols) then they have meaning.
As I see it, axioms are the definers of an abstract system. They specify both the elements and the operations for that system. These axioms act as a blueprint to allow a 'machine' to be constructed which will allow 'theorems' to be generated from the axioms.


Given a set of symbols, there are infinitely many sequences of symbols we can place after them. Without 'meaning' or 'definition', there is nothing to choose between those possibilities.
Some of these sequences of symbols can be generated by the 'machine' cited above and some will not. Those that can be generated are then 'true' within that system. Those that cannot be generated, cannot be so labelled.


Alternatively, having arrived at a set of (abstract) symbols that you have arbitrarily chosen with no justification... there are then infinitely many theories/models/lumps-of-reality that you can map that set of symbols to. Without a definition or meaning already existing within those symbols - there is no reason to prefer one mapping over another.

There are only a limited number of mappings between an abstract system , such as mathematics and the facts of the real world that can be deemed 'useful'. As mentioned above the arithmetic of whole numbers provides a useful mapping when counting apples. And the use of complex numbers is beneficial to the electrodynamics of alternating currents and inductance.

You are simply regurgitating what you have been told about axioms.

You do talk rubbish sometimes!


You aren't actually thinking about a consistent mechanism. You are assuming that the mechanism works and then slapping words around it and assuming that they make sense.
You have no idea!


It is a trait I have come across unfortunately frequently - and I'm being more abrupt with you as a consequence than your comment in isolation, perhaps, deserves.

Your past history does not interest me.

What happens is that people are told that axioms exist and that they work. They are shown pieces of the puzzle that support the assertion that is supposed to be illustrated. People are then under the impression that they are describing a consistent system. If you are genuinely describing a consistent system, then your description will be consistent. So - you KNOW the system is consistent, you KNOW that your description is consistent - you forget to actually check that what you are saying makes sense.

I can't recall anyone telling me that.

But axioms aren't consistent. Mathematics does not provide theorems.

If axioms are not consistent then no working theorem-generating 'machine' can be constructed from them and you have a null system.

Take it step by step from the beginning.

You say we start with an abstract system. Axioms are just symbols. What do you do next? Symbols don't tell you how to behave - they are just symbols. Anything you do with 'just symbols' is entirely arbitrary.

You have an arbitrary set of symbols and map them to reality. Which mapping? Why that mapping? How does that mapping create meaning?

Abstract is not "undefined" or "without meaning" - or if it is - abstract systems tell you nothing.

Abstract systems can tell you the relationship between elements under a logical sequence of operations.

-- Updated 30 Jan 2015, 20:32 to add the following --

Please ignore this post (last post?). Got confuddled over post order - so this is out of context.
Too late! lol
The Pattern Paradigm - yer can't beat it!
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