Help
blackshoe, on 2014-May-28, 23:28, said:
I have no idea what this means. Can you explain?
Let's take English for example. I will also assume here that there is only one English dictionary out there and it is authoritative (this is already untrue but we don't even need conflicting definitions to get in trouble), and also that every word has only one meaning (this is not per se necessary but it makes my point easier to follow).
An axiomatic basis for English would be some words that are defined by something other than words. Maybe "tree" would be defined by a picture of a tree. I have no idea how many such (visual) aids you would need, I guess a lot of them. Because any word in English is defined by only other words in English, it has no basic, indisputable starting point.
OK let me show you what I mean: let's say I want to define length and I accept everything I see on www.dictionary.com. OK, off we go (taking definition number one):
Quote
length: the longest extent of anything as measured from end to end: the length of a river.
I already see the word "long" there which hints at circularity but let's say I look up extent.
Quote
extentthe space or degree to which a thing extends; length, area, volume, or scope: the extent of his lands; to be right to a certain extent.
Oops! Length is defined as an extent, extent is defined as a length (I've seen longer circles too of course but you can find circles no matter where you start form)! The point, of course, is that the dictionaries can do something for people who know the meanings of some words, but not all words, but they cannot provide anything for people who (pretend to) know no words at all. Of course you can have English-to-other language dictionaries in which you can take knowledge of that language for granted but in principle we could take all words written in all alphabets (with some sort of a system disambiguating homographs) and make a big dictionary linking them and you will always need to have some words that you already know if you are going to use it for anything.
Of course the whole point of any dictionary is that you are assumed to know at least some words. It is also a very good idea not to have any prescribed "required words" in the vocabulary. Let's say the word "cat" is assumed to be known by everyone. What if someone by some freak accident never heard it or for some strange reason forgot it? Why can't he look up the word? It's impossible to be prevent all such cases, so it makes sense to just define all words, even the simple ones, with the help of other words, even if that leads to circularity.
What about something that does have axiomatic basis? The Peano axioms are perhaps the most famous:
Peano said:
1. 0 is a natural number.
2. For every natural number x, x = x. That is, equality is reflexive.
3. For all natural numbers x and y, if x = y, then y = x. That is, equality is symmetric.
4. For all natural numbers x, y and z, if x = y and y = z, then x = z. That is, equality is transitive.
5. For all a and b, if a is a natural number and a = b, then b is also a natural number. That is, the natural numbers are closed under equality.
6. For every natural number n, S(n) is a natural number. {S(n) is the successor function, i.e. S(n)=n+1}
7. For every natural number n, S(n) = 0 is false. That is, there is no natural number whose successor is 0.
8. For all natural numbers m and n, if S(m) = S(n), then m = n. That is, S is an injection.
9. If K is a set such that:
* 0 is in K, and
* for every natural number n, if n is in K, then S(n) is in K,
then K contains every natural number.
2. For every natural number x, x = x. That is, equality is reflexive.
3. For all natural numbers x and y, if x = y, then y = x. That is, equality is symmetric.
4. For all natural numbers x, y and z, if x = y and y = z, then x = z. That is, equality is transitive.
5. For all a and b, if a is a natural number and a = b, then b is also a natural number. That is, the natural numbers are closed under equality.
6. For every natural number n, S(n) is a natural number. {S(n) is the successor function, i.e. S(n)=n+1}
7. For every natural number n, S(n) = 0 is false. That is, there is no natural number whose successor is 0.
8. For all natural numbers m and n, if S(m) = S(n), then m = n. That is, S is an injection.
9. If K is a set such that:
* 0 is in K, and
* for every natural number n, if n is in K, then S(n) is in K,
then K contains every natural number.
If I'm trying to understand a discussion between two very patient mathematicians on natural numbers, and I ask them repeatedly questions like "Yes but why?" "How do you know that?" "What exactly do you mean?" If they are very, very patient, then they can lead me to these 9 axioms through a lot of different steps and I can get back to their high-level theory if I care to. However, they will never answer questions like "yes but why is y=x if x=y?" They will just shrug and will probably ignore my question from that point on. That is assumed to be true. Luckily for us, in this case, the axioms are quite easy to understand and are intuitively acceptable. Certainly if you say "Johnny is the president of the club" and "the president of the club is Johnny," it takes little suspension of disbelief to accept both as equivalent statements. So in this case you cannot say that natural number as a concept is undefined, or it is circular, well, in a way it is, everything is circular, but the mathematicians did the best they could by isolating and naming the axioms necessary to make the rest of it non-circular. There is no such mechanism for language. Not that there's anything wrong with that. Most sciences have likewise no axiomatic basis, including physics. In fact, mathematicians asked for clean, noncontradictory axioms in physics for more than 100 years:
http://aleph0.clarku...t/problems.html check out number 6.