When discussing with non-mathematicians about why I want to become a mathematician, what a mathematician’s work at a university is, and what it is good for, I often answer with the counterquestion what art is good for, because usually, there are two kinds of people asking such questions: the ones that like at least some kind of art, be it just folk music or modern films, and the pragmatic people which believe that investing ressources into research that has no immediate practical application means wasting them, of which most are die-hard capitalists or die-hard communists, which is a kind of people I usually dont try to discuss about such things anyway.
Some people say that art is something good because many people like it, and then mostly admit that mathematics can also be good because many people consider it being beautiful. This is my favourite way of that type of discussions. Some discussions lead into a meta-discussion about the meaning of live at all. And well, some of them argue that art rises creativity and helps you to get a different point of view of some things, and may help you to “find yourself”.
There is no doubt about that, as most artists insist on the fact that they want to tell something with their art. While mathematicians usually dont want to do that in the first place. Mathematicians just want to produce knowledge which they consider interesting and beautiful. Anyway, some theories in Mathematics can be applied to philosophy, especially logic – which is one reason why mathematics are important, even if they dont have a direct practical application. Another thing is that the strictly formal kind of thinking in mathematics taught me a lot of real-life facts which I consider essential for my personality.
So, recently, I was asked to give an example which is the fact that I consider the most essential fact mathematics taught me.
I think, the most essential fact mathematics have taught me is, that most problems can be solved by just properly phrasing them. In fact, most of them will vanish completely. To make it clear, I’ll give some examples.
First, lets look at a mathematical example. It is rather hard to find one which can be understood by non-mathematicians. Something that new math students often seem to be confused about is the set theoretical definition of numbers. In general, the question “what is a number” is a question which is often asked. “How can you be sure that the axioms number theory bases on are really true?”. The answer is very simple, but in fact, not very satisfying: Numbers are defined to be the elements of an arithmetic, and an arithmetic is defined to satisfy my axioms. That is, these axioms are not assumptions about numbers that can be refuted by physics, these axioms are the definition of what numbers are. They may be similar to our intuition of numbers, and in fact, they are motivated by our intuition of numbers. But should our intuition of numbers at any time find something that is not described by these axioms, all the mathematical theorems derived from these axioms will still be correct, they just wont neccessarily describe our intuition on “numbers” anymore. That is, what a mathematician calls “natural number” is something else than what an average person would call a natural number. For most mathematicians it is a finite ordinal (i’ll leave this concept unexplained here), for some mathematicians, it is an object of some arithmetic, but thats no problem, since both concepts are isomorphic, i.e. equal when renaming a few things. In our intuition its “the thingies I can count with”. It may not be a coincidence that both things have the same name, but still, it is something different.
Now, lets go to some real-life-examples. I have heard a lot of stuff about so-called “UFOs”. And in many discussions, people argue whether there are such “UFOs” at all, showing and arguing about evidences like photographies of them, forgetting what an UFO is. UFO is an acronym for Unidentified Flying Object, or sometimes Unknown Flying Object. And actually, I wouldnt be surprized, if there are videos of flying objects which cannot be identified uniquely. So yes, there are probably UFOs, because there is simply a lot of stuff flying around out there, and since there is a lot of place to fly around out there, we cannot know everything of it. Whats the big deal? The real problem is that “UFO” became synonymic with “object from outerspace built by alien life forms”. And of course, that is a question which is more difficult to answer. As far as I know, there is no evidence for the existance of alien life forms, but there is a high probability that some form of live exists out there, which does not imply that it is similar to us in any way, and builds apparatus to fly to visit us. Especially, considering UFOs an evidence for alien life forms of that kind would be like considering the loss of a sock as an evidence for sock-eating gnomes who live in tumble driers.
A similar thing are discussions about the existence of “supernatural” phenomena. It is quite clear that “supernatural” means that something is above the nature. But before we can even talk about that, we must make clear what we mean by “nature”. Actually I mostly use the term “nature” to distinguish between things that the human civilization has produced and things whose existence is mostly independent of the existence of humans. In that sense, even the computer I am typing this in this moment would be supernatural. Another meaning of the word “nature” can be the sum of everything which exists. But in that case, the existence of supernatural phenomena is absurd, because since anything that exists belongs to nature, anything is natural. I have also heard that some people call a thing “natural” when it can be expierienced by humans in some way. Actually, that is my definition of “existence” – something exists when there is some way to expierience it – because if there is something I cannot expierience somehow, it cannot influence my life in any way, and I cannot tell anything about it, so why should I consider its existance? And especially, why should I argue about it? Anyway, since the term “phenomenon” implies that there must be something that can be expierienced, also here, the question is clearly “no” – as soon as it can be expierienced, as soon as there is any evidence of it, it wouldnt be supernatural anymore. Wikipedia gives another definition: The term supernatural pertains to an order of existence beyond the scientifically visible universe. At least this is a little more clear, even though we must define what “scientifically visible” means. There are almost certainly phenomena which cannot be seen with recent scientific methods. Science is not a state, science is a process. So in that sense, there are probably supernatural phenomena – which will become natural as soon as they can be made visible. But I think, talking about the scientifically visible universe in that context means talking about things that can be scientifically detected at all, which means, a supernatural thing is a thing which cannot be detected scientifically at any time in the future. But what does that mean, detecting something scientifically? If humans can expierience it, we should mostly be able to detect it at least through some form of EEG. If humans tell that they have expierienced something, then we can test whether they are lying or believing what they say. Then we can look at their overall healthy, test them for mental diseases, etc., and as soon as we wont find anything, we have detected something which we cannot explain yet – but we have detected it.
Similar to that is the question whether there are miracles. Depends on what you consider being a miracle.
So, back to earth again, well, there are also simpler examples. Lets, for example, talk about a sort of recent topic, gay marriage. What does “marriage” mean? In the past centuries, as far as I know, “marriage” referred to the sacramentum of marriage in the catholic church, and later christian churches in general. It was clearly seen as something god likes and wants us to do. And clearly, the christians in those days considered homosexuality as a sin, and in that sense, there is no point of gay marriage. Today, we have basically two kinds of marriage, the civil ones and the religious ones. Since we are secularized, the latter ones have almost no lawful meaning as far as I know. There is a way to combine both kinds of marriage when the religious one comes from a religion to which a huge part of the people belongs, but this is rather pragmatic – many people want it, so they can have it – but this makes no difference before the law. Religious marriage has a religious meaning – the state cannot force any religion to marry people, nor can it forbid that. So, if somebody wants this kind of gay marriage, he just has to find a religion which marries gay couples. So what about civil marriage. What is civil marriage about at all? Civil marriage gives you some tax privileges, because the state wants to encourage people to get married and found a family. So we must make clear why the state wants to encourage marriage at all – and I think that is something which nobody really did so far. One thing are the children which are usually produced by married couples – same-sex-couples wont produce children, so if this was the only reason for the state to encourage marriage, there is no point in having the same thing for gay couples. Another thing is that spouses can be forced to pay social contributions, even after a divorce, which is – in general – something desirable for the state, and for both mates. But, honestly, if that was what gay couples are concerned about, there would be no point about marriage or living together at all when two people dont trust in each other that far. And civil marriage gives you the right to get information about your husband or wife in case of emergency – but you can as well authorize any other person you trust. The only thing that could really be a difference is, as far as I see at the moment, the question whether gay couples should be allowed to adopt children. But thats a completely different question – there can be no “right” to adopt children, because the welfare of the children is always more important than the happyness of potential parents, so it boils down to the question whether children which are adopted by same-sex-couples have any disadvantages compared to other children, and if so, are they at least less than the disadvantages when growing up in a children’s home – thats a complex topic which unfortunately cannot be solved by phrasing it out, and of course there is a necessity to discuss about that. But this has nothing more to do with marriage than that of course if a couple adopts a child, there should be some formal obstacles ensuring that the couple is really a couple willing to share their life together at least as long as the child grows up. I dont think that such a formal obstacle is what couples want in first place. So in the end, civil marriage is worth nothing more than taxes and social contributions. On the one hand then, I wonder why same sex couples want this so desperately, on the other hand, I wonder what is the problem with it for some politicians.
Yet another example is connected to Intelligent Design. Well, there are people believing that live has completely evolved through evolution, and others, who believe in the christian theology of creation (and of course there are a lot of others, but lets name the most important ones). Of course, in the end, its a question of believe. But somehow, people dont want to accept that, and fundamentalists try to find evidences of their believe, and flaws in the scientific theory of evolution. Of course, there is no evidence for a god so far, which is why – by the occam-principle – science doesnt act on the assumption of its existence. Thats what science does – in the end, its a principle which proved itself, when looking at the amazing technical achievements we have made through it, but in the end its nothing more. So, if somebody wants to make science accept the existence of a god, you will have to give evidences for its existence, because even if it does exist, as long as there is no evidence, science will not accept it. On the other hand, nobody forces you to believe in what science sais. Sometimes people argue that evolution cannot form anything as complex as humans. But in fact, evolutionary systems, i.e. systems having some kind of mutation and some kind of selection, can evolve very complex systems, none of them as complex as a human, but at least there is no reason why it shouldnt be possible to create arbitrary complex systems. That is, maybe live didnt evolve through evolution, but at least in theory it could have evolved that way, and since evolution of life forms can be seen on some islands, where animals have been separated, that is, since evolution of live takes place right now at least in some places, it is plausible to consider that it took place before. Same here: find an evidence that evolutionary systems have an upper bound of complexity, and science will accept it and adapt its theories – but when you dont have evidence, well, there is no reason for believing that evolution cannot evolve arbitrarily complex systems. Still, it boiles down to one simple question: If you are not a scientist, why do you want to change the scientists point of view, rather than considering science as a tool which has brought us a lot of benefits, even though it cannot always be right?
Finally, a nice example is the discussion whether humans have a free will or are deterministic. For some people, its the question of whether a human is just a machine. For me, this is a problem that can be solved by phrasing it out. The first question we have to ask is what we mean by a “free will”. Basically, it means that a human can behave however he wants. If he wants to be good, he can be good. If he wants to be bad, he can be bad. Of course, looking at several mental diseases, this concept has some flaws, but of course, we only talk about “healthy” people in the sense that they act as they are supposed to act. Another thing that is often made wrong is that people dont distinguish between determinism and predictability. Just because something is deterministic, it needs not to be predictable, the calculations for a prediction could be too complex to do them faster than the system itself to give just one example. Some interpretation I have heard was that parts of a human are “outside” the universe we expierience, and thus do not underly the universal laws, thus have a free will. The question then is, do these parts “outside” the universe underly any laws? If not, they are just random, that is, humans have an integrated randomizer, and thus, they act randomly, even though maybe not uniquely distributed, but still randomly. In that case, they wouldnt have anything which I would consider a “free will” – acting randomly has nothing to do with freedom. If these parts outside the universe underly laws, then they are deterministic, and thus, humans would be deterministic – just because something doesnt underly the universal laws we expierience, this doenst mean that it cannot be deterministic. And if humans have no parts outside the universe we expierience, and therefore underlying the universal laws, the question boils down to whether the universe is determined – which is the same as asking if our universe underlies laws and has no random components. If it has random components which influence our decisions, then we would act randomly, like in the above case. If not, we are deterministic. The reader may wonder why I am telling that this problem can be solved by phrasing it out while not giving a real solution. Well, this time, the solution lies in the question itself. The question makes an implicit statement about a coherence between indeterminism and free will. The flaw lies in what we want that “free will” to be: We dont want it to be something that is restricted by laws we can understand, we dont want it to be something that is restricted by laws we dont understand, but we also dont want it to be random. But there is nothing else. Either something is restricted by laws, or it is not restricted by laws, but then it is random. In fact, free will has nothing to do with that. What I think people really mean by free will is that a human as such can make a decision of changing his behaviour according to his inputs, but he is not stateless, that is, he does not only depend on what he receives, but also on what happens inside his mind.
So. I hope you saw that sometimes it is good to phrase some things out in detail. It often boils down to things that are much less problematic to solve. Not always, of course. But the remaining problems are at least real problems, rather than just problems of phrasing.