Note: This was another short paper I wrote for my Foundation of Mathematics class back in March of 2015 (can’t believe it’s been 3 years!). I will try to publish some papers I wrote in the past on this blog so I can (1) have a more complete record of my thoughts, (2) better organize my thoughts, iA.
Update on life: passed CPA exam, thinking about marriage and looking for a job; miss math
11 March 2015
My Reaction to the Continuum Hypothesis Story
The continuum hypothesis was first proposed by Georg Cantor in 1878 and states that there does not exist an infinite set whose cardinality is between that of the integers and that of the reals. Ever since then, mathematicians had been trying to prove the hypothesis to be either true or false, but none succeeded. Due to the difficulty of the problem, Hilbert listed the continuum hypothesis problem as the first of the 23 problems he presented in 1900. In 1940, Godel proved that in any axiom system, there are statements that are not provable, and these statements are termed “undecidable”; he also found that the the continuum hypothesis cannot be disproved in the axiom system of set theory. In 1963, Paul Cohen proved that it cannot be proved, either. And with this proof, we finally have an answer to Hilbert’s first problem: continuum hypothesis is an undecidable mathematical statement in set theory.
My immediate emotional reaction to the story was fascination and slight unease. I was fascinated at the possibility that a mathematical statement could be undecidable, but somehow uneasy because I felt that the perfect world of mathematical objectivity in my mind was being harmed in some way. Fortunately, my slower intellectual reaction has provided some relief to my unease (but only elevated my fascination). This intellectual reaction will be the topic of this paper.
In order for my reaction to make sense, though, I will first have to describe my deeply held view of mathematics. I have always thought of mathematics as having its own existence independent of our minds. I cannot imagine, for example, that the statement 1+1=2 is false just because human beings don’t exist. To me, mathematical entities seem to exist in an eternal and unchangeable state (in a platonic mathematical world, to use Roger Penrose’s phrase), and only some mathematical entities and some statements about their relationships to each other have been discovered by our mental efforts so far. As a consequence of this belief, I think of the mathematical world as having a more constant reality than the mental world, meaning that a human thought may or may not have a truth value, but a mathematical statement definitely has to be either true or false because they are about entities that are constant, eternal, and unchangeable. To be clear, what I mean by a mathematical truth is what corresponds to the mathematical reality.
When I learned that there are undecidable mathematical statements, my first intellectual reaction was to try to reconcile this fact with my existing view of mathematics as described above. In my view, it is necessary for every mathematical statement to be either true or false. Does this contradict with Godel’s incompleteness theorems? The answer seems to be no, for even if a statement is not provable, meaning that there is no way to logically show it to be true or false, it does not necessarily follow that the statement does not have a truth value. There seems to be no contradiction here. If, however, in addition to the aforementioned platonic view of mathematics, one also holds the belief that every mathematical statement can be accessed or discovered (to use the term I used before) by our mental effort, then one will perhaps find a contradiction here.
Personally, I am still not sure whether I hold this latter belief, for I do not have a very deep understanding of how human intelligence is able to access mathematical knowledge (this seems to a question that lies in the intersection of epistemology and philosophy of mathematics). But suppose that I do hold this belief—how then can I reconcile my platonic view of mathematics (every mathematical statement has to be either true or false) with the fact that there are undecidable statements such as the continuum hypothesis?
One possibility is to adopt classical modal logic and classify the statements in the mathematical world as either necessarily, possibly, or contingently true or false. Furthermore, we should also apply these notions to the physical world to make things even more clear. In this way, we will have a system in which even undecidable statements can be classified as true or false.
To see how this works, consider the parallel postulate. If we accept the parallel postulate as an axiom, then we have Euclidean geometry. If we don’t accept the parallel postulate as an axiom, then we have a non-Euclidean geometry like hyperbolic geometry. So the key question here is: is the parallel postulate itself true or false? According to the modal logical framework I presented above, the statements “If parallel postulate, then Euclidean geometry.” and “If not parallel postulate, then non-Euclidean geometry.” would both be necessarily true in the mathematical world, but the parallel postulate itself wouldn’t be necessarily true, but only contingently true, since its truth value depends on the axiom system you choose to adopt. In the physical world, however, it would appear that the parallel postulate has to be either necessarily true or necessarily false, depending on which geometry (Euclidean or one of the various non-Euclidean geometries) is the one that actually describes the universe. So, if it turns out that it is hyperbolic geometry that is actually the geometry of the universe, and not Euclidean geometry, then the parallel postulate will be necessarily false in the physical world.
Similarly, the statements “If we accept the continuum hypothesis, then we have set theory A.” and “If we don’t accept the continuum hypothesis, then we have set theory B.” are both necessarily true in the mathematical world. However, the continuum hypothesis itself would be only contingently true in the mathematical world, since its truth value again depends on which axiom system (set theory A or B) you choose to adopt. As for the truth value of the continuum hypothesis in the physical world, the answer would depend, for one thing, on whether one believes that physical (e.g. temporal, spatial) infinities are possible. Then there is also the question of what would be the physical equivalent of the set of integers? The set of reals? Perhaps string theorists who study the multiverse can better answer this question.
So, in this framework, the fact that the continuum hypothesis is undecidable poses no threat to my view that every mathematical statement has to be either true or false even if I believe that all mathematical knowledge is accessible by the human mind.
Another possibility is that the continuum hypothesis and other undecidable statements do have definitive truth values, we just haven’t developed the mathematical technology (e.g. new rules of proof; a new kind of logic etc.) to be able to show that. Of course, this is merely speculation, but I can see the possibility that the only thing that prevents us from determining whether the continuum hypothesis is true or not is just that we aren’t seeing the big picture. So, just as in the story of the blind men and an elephant, we are currently “blind” to the totality of truth concerning set theory. Another analogy is how we can only see 3D representations of a spinning tesseract, but never see the tesseract directly in its four-dimensional shape. In this analogy, the tesseract is “truth about set theory” and the 3D representations of different aspects of the tesseract are all the different possible axiomizations we could have of set theory. Hopefully, if we continue to examine all the different 3D representations of the tesseractset (different set theories) and how one representation changes to another (what makes one set theory different from the other; the overall logic etc.), then perhaps we will have the chance to really “see” the tesseract (the totality of truth concerning set theory) directly. When that happens, we may be able to see the exact status of the continuum hypothesis in the mathematical world. But, needless to say, all this is wildly speculative.
Yet another possibility is that the continuum hypothesis is a badly posed problem based on insufficient knowledge or the same issue discussed in the last paragrah. But I don’t know nearly enough about what qualifies a problem as “well-posed” or “badly-posed” to elaborate on this possibility.
To conclude, although the continuum hypothesis story has made me feel slightly uneasy, it has challenged my view of mathematics and forced me to find ways to modify my view to allow the possibility of undecidable statements. In this process, the continuum hypothesis story has enriched my intellectual life and launched me on a quest to develop an ever more accurate picture of the mathematical world.
According to the Wikipedia Article “Modal Logic”: In classical modal logic, a proposition is said to be possible if and only if it is not necessarily false (regardless of whether it is actually true or actually false); necessary if and only if it is not possibly false; and contingent if and only if it is not necessarily false and not necessarily true (i.e. possible but not necessarily true).