Is there any reason to be skeptical about modern mathematics? Is it possible that modern mathematics is culturally biased? Is it possible that there is an element of fraud in modern mathematics? Has mathematics become clever argumentation with no concrete content?
I'm like the reader to consider the possibility that the answer to all those questions is yes.
First of all, it would be absurd to say that the modern academic system based on peer review would preclude fraud and cultural bias. The "Sokal Affair" seems to have proven that when academic writing becomes indistinguishable to the non-expert from buzzword salad, then it's likely that even experts can be fooled. (see http://en.wikipedia.org/wiki/Sokal_Affair) Even physicists can be fooled (see http://math.ucr.edu/home/baez/bogdanoff/ )
But even ignoring the possibility of outright fraud, cultural bias can creep into academia slowly and progressively so that it's hardly recognized by the majority of people involved. The best example might be the takeover of the humanities by the Marxists (or extreme left wing).
The Marxists produce beautiful theories. They produce complex, clever, precise, and apparently logically consistent arguments, which must pass a rigorous peer review process. The Marxists believe themselves to be open minded, unbiased, compassionate, independent thinkers. But, of course, the skeptics don't see it that way at all. According to the skeptics, market forces are simply part of reality, and responding to those forces is a natural and compassionate thing for humans to do, and hence the implementation of Marxism (which tries to replace market forces with governmental planning) requires a brutally oppressive and intrusive government willing to criminalize human nature, so that no matter how beautiful the Marxist theories may be, there is something fundamentally very very wrong with the Marxist world view. Many skeptics in academia believe that they are severely discriminated against by the Marxists.
In other words, the skeptics will claim that despite the cleverness and consistency of the Marxists' arguments, those arguments are built upon a defective (or at least, a culturally biased) model of reality. So the question becomes, is it possible that modern mathematics is built upon a flawed model of reality which gives it a cultural bias? Does this bias have any connection to the left wing bias in the humanities?
For the sake of this argument, let's consider an idealized skeptic. Our skeptic will be intelligent and honest to a fault. She will have a technical background, and will be fully aware of the power of mathematics in technology. She will have no axe to grind, and she will have no philosophical, religious or political biases, other than a propensity for skepticism. That is, she looks for concrete evidence, observable implications, and testable consequences. She is skeptical of mere clever argumentation. And she refuses to be intimidated by appeals to authority or ad hominem attack.
So let's say that our skeptic has been reading popular books about mathematics. For example, books by Hofstadter, Penrose, Rucker, Smullyan, Kline, Singh, Aczel, and maybe others. She is troubled that a lot of the modern mathematics she has been reading about seems to be nothing more than clever argumentation with no concrete content and no testable consequences. She wants to know how the ideas she has been reading about can help us to understand the world in which we live.
Let's look at specific examples of the mathematical ideas she is skeptical about.
1) Set theory and Cantor's Theorem. It seems obvious to our skeptic that the mathematical constructs we actually deal with must be identifiable, and that we can only identify a countable number of such constructs (since our language is countable). So Cantor's Theorem asserting the existence of uncountable sets (and hence the existence of objects which cannot be identified) cannot have any concrete content. The idea that there must exist more unidentifiable objects than identifiable objects appears to be silly word play. Clearly, to our skeptic, set theory includes an element of make-believe. So she concludes that much of what she has been reading is nothing more than clever argumentation with no concrete content; what possible testable consequences are there to the assertion that unidentifiable objects exist? And why don't the books she has been reading address the obvious skeptical objections to such ideas?
2) Godel's Theorem (loosely, no consistent formalism can prove its own consistency) Informally, our skeptic claims, a proof is a compelling argument. It seems clear to our skeptic that if we are to believe that the formal theorems in our formalism should be accepted as compelling arguments, then at the very least it must be the case that we already believe that our formalism is consistent, and hence, no possible formal proof within that formalism could be considered to be the evidence that compels us to believe that the formalism is consistent. And our skeptic asks, is that not already the essential content of Godel's theorem? Even if you argue that Godel's proof is superior because it is actually formal, you still have to deal with the informal notion of proof: does Godel's proof compel us to believe that Godel's theorem is actually true? So, our skeptic asks, what is the concrete content to Godel's theorem? What does it tell us that is not implicit in the definition of "proof"? How can it be tested? Is it anything more than clever argumentation? How can such a theorem be regarded as one of the most important theorems in all of mathematics? Why don't mathematicians raise these kind of questions? Why aren't they at least a little bit skeptical?
3) Self-reference and paradox. (note that some of the popular books our skeptic has been reading do suggest that this is of great importance in mathematics, and essential for understanding Godel's theorem). First of all, our skeptic notes, the assertion that paradox is in some sense "real" (i.e. something more than an illusion or a game or a joke), would appear to be almost equivalent to the assertion that logical reasoning can be used to prove that logic is flawed, which is immediately highly suspicious. But it can be analyzed further: one of the "ground rules" in communication is that we should always intend to tell the truth. That is, when we speak, we are implicitly claiming to be telling the truth, and we need to explicitly comment on the truth value of our assertions (e.g. with modifiers such as 'probably', 'possibly', or 'not') only when we do not feel certain about the truth of what we are saying. Hence, an utterance such as "I am lying" (i.e. the Liar paradox) must be analyzed as if it contained its implicit claim to truth, i.e., it must be deemed logically equivalent to "(implicitly) I am telling the truth; (explicitly) I am lying", which is nothing more than a simple contradiction, with nothing paradoxical about it. So our skeptic wonders, how can the study of paradox can be anything more than a game; how can the contemplation of paradox possibly help us understand the world in which we live; how can it possibly have testable consequences? And yet, whole books have been devoted to its study -- why? Why don't mathematicians ask and address these questions? And as far as self-reference goes, clearly humans can talk about themselves, but to claim that sentences can talk about themselves would seem to be a bizarre anthropomorphization of abstract symbols; natural language gives us no way to create sentences which unambiguously refer to themselves.
4) Fermat's Last Theorem. Our skeptic notes that while FLT itself has clear meaning and a concrete content, there's nevertheless something fishy about the idea that it has been proved. There is something that is immediately clear to anyone who has dared to search for a counterexample to FLT: just due to chance alone, it seems unlikely that there is a counterexample. That is, for an exponent 'p' of modest size or larger, the set of integers which are p'th powers is a very very sparse set of integers, and for an arbitrary set of integers that is that sparse, straightforward probabilistic reasoning tells us that it is very unlikely that the sum of two of its elements will turn out to be another element of the set. In fact, for example, a back-of-the-envelope calculation suggests that for a set as sparse as the set of 50'th powers, the probability that two of its elements will sum to a third element of the set is about 1/10^200, and this can be loosely interpreted as giving a probabilistic proof that FLT is almost certainly true for exponent 50. Going further, given that FLT had been proven for all exponents up to 10^6 before Wiles came along, using the same heuristic argument, the probability (in the Bayesian sense where a probability is a degree of belief) that there could be a counterexample to FLT could be taken to be about 1/10^10^7. So, in other words, Wiles spent seven years locked in his attic (so the story goes) to do nothing more than remove that last little bit (1 part in 10^10^7) of uncertainty that FLT is true, assuming that we generously assign a value of less than 1/10^10^7 to the probability that his proof is flawed! Since the proof tells us nothing that we do not already believe to be true with very very high probability, searching for counterexamples to the theorem in no way can be deemed a test of the proof. So our skeptic has to wonder whether a man who is willing to devote so much energy to such an insignificant task, for no apparent reason other than to seek fame, would he not be willing to pull off a hoax? How could we know? And furthermore, the proof itself is presumably accessible to only the top one tenth of one percent of mathematicians, so our skeptic notes that she has no realistic hope of ever determining for herself whether the proof is consistent. But why should she trust the "experts"? Why should the proof of FLT qualify as headline news? Why don't the books our skeptic has been reading address these kinds of questions?
So, does Cantor's proof compel us to believe that there exist mathematical objects that cannot be identified? Does it compel us to believe that there are more unidentifiable objects than identifiable objects? Of course not! For example, we could simply assert that as part of the definition, mathematics only studies identifiable objects, and then with less magic than was used to prove the existence of unidentifiable objects in the first place, all of the unidentifiable objects would vanish from the mathematical universe! And to be sure, the mathematics that does have testable consequences would hardly be affected at all by such a change of definition.
So what's going on? Our skeptic will note that somehow mathematicians are cheating. When they use words like "proof", "truth", "exists", "logic", and even the word "mathematics" itself, they are not using them in the way the rest of the world uses them. The mathematicians have chosen convenient definitions and convenient axioms which let the mathematicians formally "prove" what they want to prove; they have completely abandoned the idea that mathematics should have testable consequences; they are playing word games; they have insulated themselves from reality.
So now our skeptic asks, given that we see what games the mathematicians play, is it even remotely plausible that the mathematicians could be capable of coming up with important insights into the nature of proof, truth, existence, and logic? For one thing, the mathematicians seem to be totally clueless about what is "important" to anyone but themselves, given that they think testable consequences are not important.
As far as the proof of Fermat's Last Theorem goes, our skeptic admits that she has no special insight. But she has to wonder why mathematicians apparently refuse to even think about such things from a probabilistic point of view. Probabilistic reasoning does produce results with testable consequences, and if we regard mathematics as a science with the purpose of explaining the phenomena we observe within the world of computation (a view completely compatible with the views of those who apply mathematics), then probabilistic reasoning should be accepted as part of mathematics. And furthermore, once we recognize that formal reasoning and probabilistic reasoning complement each other (i.e. probabilistic reasoning works especially well where formal reasoning fails, and vice versa), we have to reexamine the content of Godel's theorem (in this case, the assertion that there exist true statements not provable in a given formalism). Clearly we can come up with an unlimited number of statements which can be shown by probabilistic reasoning to be almost certainly true but for which we have no reason whatsoever to believe that they can be proven true in any particular formalism. So Godel's theorem is true with a vengeance, but it's not Godel's proof which compels us to believe that. And the important question -- are there true mathematical statements having testable consequences which cannot be understood and explained with some combination of formal and probabilistic reasoning -- is most certainly not answered by Godel's theorem.
Our skeptic notes that consistency is the concern of the liar. Those who are devoted to truth get consistency for free. The argument for believing that mathematics is consistent is compelling, but it necessarily comes from outside mathematics itself, and here's the argument: we simply have to believe that we are capable of consistent reasoning (clearly we could not "reason" about the possibility that we lack the ability to reason consistently), and the best model we have of our own minds is that our minds are equivalent to computers. And, the basic laws of mathematics are implicit in our best models of computation. Together, those three assertions compel us to believe that the basic laws of mathematics must be consistent.
To our skeptic, the mathematicians are playing a very twisted game when they try to "prove" that mathematics (e.g. PA) is consistent. They start with the basic principles of mathematics, and then they add on a mythology about a world of the infinite, and then they claim that within this bigger theory they can construct a "model" for the more basic mathematics, and then they claim that that constitutes a proof that the basic mathematics is consistent. Our skeptic notes that the mathematicians' "proof" does not compel us to believe anything that we do not already believe.
Our skeptic notes that "real" mathematics (i.e. the mathematics which has the potential to help us understand the observable world in which we live; the mathematics used by physicists, computer scientists, statisticians, economists, and applied mathematicians) has testable consequences (see appendix). But how can we test statements about a world of the infinite lying beyond what we can observe? What test could we perform to compel us to believe that Godel's proof tells us more than what a simple and immediate informal argument compels us to believe? How can we test the assertion that paradox is something other than pure nonsense? How can we test the proof of a theorem which tells us nothing more than what we should expect from simple probabilistic reasoning? Should we not be skeptical of modern mathematics? Is modern mathematics anything more than clever argumentation with no content? Is it possible that modern mathematics is built on a flawed model of reality? Are mathematicians lost? Given that modern mathematics has so little content, and that it is almost completely inaccessible to the average person, is it not plausible that the mathematicians have created an environment in which outright fraud is possible?
So now, let's pretend that our skeptic ventures into sci.math and sci.logic to explain her reasons for being skeptical of modern mathematics. What will happen? It's not a pretty picture; the mathematicians will go on the offensive. They'll call her a crackpot. They'll claim that she is unqualified to even have skeptical thoughts about mathematics, and that she is unreasonably demanding that the experts come down to her level of understanding. They'll claim that she is trying to impose her religion on others, and that she is trying to take away the mathematicians' freedom. They'll borrow vocabulary from the liberals, and accuse her of being a closed minded, ignorant nut case. They'll try to dismiss all of the skeptical objections as a result of an inability to deal with abstract thought. Ultimately our skeptic will be told that mathematics is rightly defined by the experts, and by definition, mathematics is what expert mathematicians do, and that skepticism is simply not part of what mathematicians do, and hence, by definition, mathematicians cannot be doing anything wrong, and they have no obligation to respond to skepticism. And our skeptic notes that this last argument is an almost perfect example of the mathematicians' clever but vacuous, circular argumentation methods.
>From our skeptic's point of view, those brilliant mathematicians simply cannot respond to skepticism honestly and intelligently, or even civilly and coherently. They just play games. The skeptic is claiming that all statements must have testable consequence, and hence, it's possible for both 'A' and 'not A' to be meaningless if neither has testable consequences. The mathematicians will insist that if 'A' is a grammatically correct sentence, then either 'A' or 'not A' must be true, and they will try to force the skeptic to commit to one or the other. They willfully refuse to even understand the skeptic's position. Our skeptic notes that the problem with modern mathematics lies in the language itself; in order to accommodate Cantor's world of the infinite, the mathematicians had to expunge the notion of testable consequences from their language, and now there is essentially no way to express that idea using their language, and communication between the skeptics and mathematicians has become impossible.
So what's going on? It may not be entirely clear, but one of the books that our skeptic has been reading offers an interesting perspective on the situation. A book by Amir Aczel, "The Mystery of the Aleph : Mathematics, the Kabbalah, and the Search for Infinity" draws parallels between modern mathematics and the Kabbalah. The book itself speaks positively of both modern mathematics and the Kabbalah; it seems to suggest that we should marvel that the Medieval mystics (Kabbalists) were able to anticipate the important results of twentieth century mathematics. But there's another way of looking at what that book is saying: maybe mathematics has been corrupted by Medieval mysticism.
Does the mathematicians' belief in a world of the infinite lying beyond what we can observe have its roots in Kabbalism? There is certainly evidence pointing in that direction. For one thing, Cantor was strongly influenced by his religious beliefs, and those were mystical beliefs. And other mystical influences can be found in the historical record. Some evidence suggests that the mathematicians' belief that important insights are to be gained from the contemplation of paradox has Kabbalistic origins. Where does the mathematicians apparent belief that the knowledge with the very least significant implications for the world we observe is somehow the most important knowledge, come from? Why do mathematicians place so much emphasis on clever argumentation and attach no importance to the testable consequences of their theories? Why are mathematicians unable to respond rationally to skepticism? Such questions suggest that mathematics has been corrupted by mysticism.
If we accept the notion that truth necessarily has observable implications, and some cultures do emphasize that idea, then we must admit that modern mathematics is built on a lie. Modern mathematics is not culturally neutral. Although there is little evidence that anyone is discriminated against on the basis of race or gender or nationality or religious affiliation, there is an extreme bias against those who accept the idea that truth necessarily has observable implications; those people are regarded as untermenschen (crackpots) by the mathematics community.
At the start of this article, I posed the question of whether mathematics has been influenced by the liberalism (humanism, Marxism) which has taken over the humanties and the social sciences. I believe it has. Both Cantorian mathematics and Godel's results, if taken seriously, would appear to validate the liberal view that truth and reality and logic are merely social constructs, and I suggest that that is a big part of the reason why those "theories" are accepted as part of mathematics. As I see it, there is a really big problem here which needs to be addressed; those who question the liberal dogma are severely discriminated against in our universities. I believe that the preferred solution to the humanism (the religion behind liberalism) problem is to recognize humanism as a religion and apply the laws that keep religion separate from government. Likewise, the Cantorian religion (i.e. the belief in the existence of a world of the infinite lying beyond what we can observe) doesn't belong in the publicly funded universities.
Appendix Testable consequences of mathematics
Mathematics that is applied necessarily has testable consequences. For one thing, bridges would fall down, airplanes wouldn't fly, computers wouldn't work, and weather reports would be wrong half the time if mathematics were flawed, and those things could be considered to be tests of mathematics. But even in a more abstract sense, mathematics has testable consequences.
We can think of mathematics as a science which studies the phenomena observed in the world of computation. All of the mathematics that has the potential to be applied can be thought of that way. As a conceptual aid, we can think of the (abstract) computer as both a microscope and a test tube: it helps us peer deeply into the world of computation, and it gives us a way to perform experiments within the world of computation. Mathematics studies what we observe when we look through that microscope. Then, roughly speaking, a statement may be said to have testable consequences if it makes predictions about the results of computational experiments.