All mathematical* statements are tautologies.
This statement, along with its’ many variations such as mathematics only produces tautologies, is categorically false.
I know that many of the readers are sophisticated so let me apologize up front for stating the obvious. (In fact the arguments that I am about to give are pretty much old hat, with the possible exception of some personal perspective)
Definition: B is a tautology of A if and only if (A implies B) AND (B implies A)
The reader should notice that if B is a tautology of A then A is a Tautology of B.
From a philosophic point of view, tautologies are generally viewed pejoratively because the statements are considered redundant and therefore no meaning will pass with the reference to B. (In the world of Mathematics tautologies can be great for linking together two separate theorems that may not appear to have anything in common. If you have read the book “Fermat’s Enigma” by Simon Singh, Simon Singit, and John Lynch you will recognize the importance of this comment).
Finding a counter example to the statement “all mathematical statements are tautologies” is easy. For example:
If x = 5 then x is greater than 3. Note that if x is greater than 3 then x = 5 is not a true statement.
Similarly, if x = 5 then x is a prime number. Note that if x is a prime number then x = 5 is not a true sentence.
Unfortunately, virtually every person that seriously addresses this subject has to undo the wrong that philosopher Ludwig Wittgenstein (a biography is interesting and morbid) has done.
Wittgenstein invented his own private definition** (private language is another subject) where he changed the meaning of the word tautology.
Originally he said that mathematical statements are tautologies because the conclusions are implied by logic. Latter (after realizing that removing the parallel postulate yielded different viable geometries) that statement was modified to claim that, in a given axiomatic system, all mathematical statements are tautologies.
In normal usage Wittgenstein’s sentences, similar to “all mathematical statements are tautologies”, should be replaced by “all mathematical statements are implied by their associated axioms”. The meanings are vastly different. I believe he kept the misleading pejorative terminology because he felt that mathematical theorems had no more content than the underlying axioms.
So, if I say all mathematical statements are implied by their associated axioms, what do we gain? The answer is almost, but not quite***, nothing.
What we loose is understanding and insight into how mathematics works, and the creativity and ingenuity required to structure proofs.
***Kurt Godel came along and constructed a mathematical sentence that could not be implied by the axioms. So now Wittgenstein’s statement is not only misleading, pejorative, lacking understanding and insight, it is actually rigorously wrong.
In a classic bit of irony, an alternative rigorous refutation can be constructed by a Turing Machine analysis. In theory, if the axioms imply the conclusion, there should be a sequence of statements starting with an axiom A and ending with a theorem T. The construction should go something like A > B1 > B2 >…> Bn > T (read A implies B1 implies B2 implies and so forth implies Bn implies T). These implications can be restated as an algorithm and a theoretical computer should be able to draw this conclusion. At first you might think so what? (At least I did). However it turns out that there is something called the Halting Theorem which ultimately allows one to test whether or not these algorithms, with given properties, do actually yield a result.
We know that this Turing Machine analysis provides an alternative proof for the Godel Theorems. In addition, from my readings anyway, it also shows that entire classes of mathematical statements can not be implied from given axioms.
The reason for the irony is that Turing was a student of Wittgenstein. Turing’s biography is also interesting but tragic.
Implications from a Kantian perspective would be that some Mathematics is analytic and some is synthetic.
- Here I am using standard Boolean mathematics. (The same as Wittgenstein, Russell, or Hilbert used)
**I can not stress strongly enough that one should be extremely skeptical of anyone that misuses the standard meanings of a given word or phrase. (I understand the plus side, but my caution still stands)