What do you mean by meaning? What is 'abstraction'?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.)
Taking what you have described literally:
I have some axioms:
- Wibble
- Flurgle
- Quangwilly
- Shootangle
- Foxnaggle
- Woowoohunks
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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.
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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.
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.
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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?