Set Theory and the Antimony
I was thinking.....
I found this neat little essay on number theory at a Barnes and Noble in the used pile (why there was a used pile at a new bookstore is mysterious, perhaps it was fortune...) called The Continuum by Hermann Weyl, an early Twentieth Century mathematician [the book was published originally in 1918]. In addition to the standard nearly unintelligible garble, there were a few points worthy of notice. First, Weyl decides to ground number theory (including natural number, rational number and real numbers) solely in the ideas of a few logical principles of inference and two axioms. Instead of using the Euclidean definition of number as a multitude of units, Weyl follows the modern adaptation by using the idea of "immediate successor". Thusly, two axioms: every number has one unique immediate successor, and "one" is not the immediate successor of any number. From this and his rules of inference he proceeds to deduce the rule of number theory.
Unfortunately, number theory is vastly boring to those with no strong interest in it. So I can't go into that great of depth. It just seems important to note that there is a difference since the classical period of mathematics by changing multitude to immediate successor, and also by using rule of inference on strings of purely symbolic logic. In that sense the manipulations he does with these strings do not necessarily have to be about numbers. I digress. Additionally, he uses the same method for deriving other numbers (than counting ones) that I have toyed around with doing, even classifying them as "second degree" "third degree" etc. . . (for the rational numbers). But this is also an aside. Two other things are of note before I mention his antimony.
First, he thinks that a derivation of purely formal number theory is PRIOR to Geometry - which I have not been able to firmly refute thus far (but I just finished the book this morning). But, not to contradict himself, he says that the continuum formed by real numbers is a continuum of individuals different from and not applicable to "real" continua - time and space. Hmm. . . food for thought.
Secondly, and quote:
The failure to recognize that the sense of a concept is logically prior to its extension is widespread today; even the foundations of contemporary set theory are afflicted with this malady. It seems to spring from empiricism's peculiar theory of abstraction; for arguments against which, see the brief but striking remarks in Fichte (1912, 6:133 ff.) and the more careful exposition in Husserl (1913a: 106-224). Of course, whoever wishes to formalize logic, but not to gain insight into it - and formalizing is indeed the disease to which a mathematician is most prey - will profit neither from Husserl nor, certainly, from Fichte.
There is one sense in which it is right to say that mathematics is formalized - since it treats of magnitude as its proper object, but in another very striking way, Weyl has hit upon the problem (even plaguing himself) of modern mathematics. The further abstracted (albeit badly abstracted) mathematics is from reality, the more tenuous its foothold on truth becomes (this post [and this statement] is actually a preface to the next one I'm about to write).
Finally, I'll finish with Weyl's antimony (Bertrand Russell has a similar one, etc. etc. in the realm of mathematics, they all boil down to Epimenides' self-referential statement "this sentence is false").
Some adjectives are what they describe ("brief" when said is brief) - these are autological.
Some adjectives are not what they describe ("long" is only four letters, and one syllable - and therefore not long) - these are heterological.
What about the adjective "heterological" which one is it???
Well, Weyl's antimony isn't a paradox, just like Russell's isn't.
Proof: Let A(x) be a predicate defined on all adjectives such that A(x) iff x describes itself. A(x) for "x" is autological, that is.
Then we'll get that A(heterological) implies -A(heterological), and vice versa. A contradiction.
Conclusion: The predicate A doesn't exist. No contradiction here!
Perhaps A could be "fixed" by removing some strategic set of adjectives from its domain of definition, who knows. But what's interesting is that those antimonies usually arise from defining something that doesn't exist.
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