Proof

Since I teach and talk about mathematics I often get a chance to interact with people about the word “proof.” It is a commonly used word which carries a lot of authority. If some proposition is “proved,” we would say, then all debate ceases; the proposition is true, period. Other claims which have not been proved are open to question or criticism. We tend to think it intellectually dishonest to question something which has been proved, especially with regard to mathematics.

Let me give you an example. Suppose I were to ask some random passersby if it is possible that our existence is not what it seems, that we are subject to some grand delusion similar to that portrayed in the movie Matrix or the Harry Potter books. I suspect that many would agree that such a thing is possible. If I were to ask the same passersby if it is possible that a mathematical truth like 1000 + 1000 = 2000 is false, I suspect the vast majority would say no. Why the difference? Why would people be more willing to accept a mathematical truth than a truth about the validity of our everyday experience?

Most of us, if pressed, would probably say something like this: mathematical conclusions have been proved by mathematicians while the reality of our experience has not. Not that we have actually done any mathematical proofs, but that is what we have been told. But what, one may ask, is a proof?

One answer is that a proof is a deduction using valid logic from a set of first principles. This is the most common notion of proof which implies that the conclusion of a proof is undeniable. However, by this definition, the proof is only as good as its first principles. If the first principles upon which the proof is built are questionable, then so is the proof, and first principles have a long history of being called into question. To avoid this difficulty we might try to find proofs for our first principles. But clearly this is impossible, since to do so, we must use other, prior principles upon which to base the proof, and these principles would suffer the same criticism. Thus we must assume the truth of the first principles, leaving this definition of proof with a bit of a gaping hole, at least with regard to undeniability. Thus a proof is only as good as its first principles.

Another answer to the question of what is a proof was suggested to me by a math professor when I was taking an advanced mathematics class as an undergrad: a proof is a convincing argument. At the time, I was taken aback. What if I am convinced of something that is incorrect? Proofs are supposed to be certain, not merely convincing. On the other hand, who but a mathematics professor should know the answer to this question? As I have further explored the issue, I have come to agree with this position: a proof is a convincing argument.

Whichever of these notions of proof is adopted, let us return to our original question and ask why people would be more willing to accept a mathematical truth than a truth about the validity of our everyday experience? Are mathematical arguments more convincing?

Let us approach this question using both notions of proof. We will start with the first notion of proof. Suppose mathematical proofs are more convincing because they are built upon a solid base of true mathematical first principles, whereas the reality of our experience is just something we believe without an argument from first principles. Now, there has been a great deal of debate about which first principles one should adopt when building a mathematical structure such as arithmetic or geometry. Rather than look at this debate, let us simply choose some basic principle that most mathematicians would accept as one of their first principles, the principle of non-contradiction: a statement cannot be both true and not true at the same time. Seems pretty obvious. If it were wrong, we could not really make arguments at all. No wonder mathematics has such a high level of authority; first principles like this seem pretty strong.

With regard to the reality of our experience, are there some foundational principles which could be used to prove we are not deluded? I would suggest, no. Instead, belief in the reality of our experience is a first principle. Why, I would ask, does the truth of the principle of non-contradiction have any greater authority than the reality of our experience? If a person denies either, there is no real recourse to argument. A person may claim to deny either, but the claim goes nowhere. A person can only claim the principle of non-contradiction is false if the principle of non-contradiction is true. Further, if the reality of our experience is a big, undetectable delusion, how do we know if anything we believe is true? Thus for the first notion of proof, it is not clear to me why the first principles of mathematics are to be preferred over the reality of our experience, and thus mathematics has no special standing.

Let us next consider the second notion of proof: a proof is a convincing argument. From this perspective, are mathematical arguments more convincing than the reality of our experience? To address this question, let us first consider some possible reasons for believing that 1000 + 1000 = 2000:

1. Education. We all were taught how to add. We were told over and over again that the procedure that we use for addition is the right one. Everyone does it, and no one ever questions it. Thus the overwhelmingly consistent application of this procedure of adding is agreed upon by all.
2. It works. Our knowledge of addition is applied all the time. We apply it; engineers and scientists apply it. One would expect that if there were a problem with addition, it would have shown up.
3. Intuitive obviousness. We have an immediate and clear perception that when one thing is put together with another we get two things. Further, it is intuitively obvious that groups of things add in the same way individual things add; how could they not? Putting these perceptions together, we do not really need much education to realize that the addition statement is true.
4. The culture. Our culture indicates in many ways that mathematics is very sure. It is one of the only disciplines where conclusions are undeniable; to even question mathematics is seen as odd. Our culture tells us that all of these mathematical conclusions have been proven to be true. There is not any culturally plausible scenario where mathematics is not right.

Now consider our belief in the validity of everyday experience—that is, the rejection of something like the Matrix as possible. Here are some reasons that might be given for that belief:

1. Education. In this case, the education does not start formally in school but informally at birth. We are taught by our experiences, by our parents, by our peers, and by our teachers. Those around us are teaching us through their words and actions that we must take the physical world around us as existing the way we experience it.
2. It works. We all live our lives every day assuming the reality of our experience is no delusion. In a delusion, usually there is some sort of clue that something is not right. But the physical world is consistent in its operation. We may not understand all of the laws and principles, but every expectation of physical behavior is met in practice. We do not drop a ball and watch it go up. You would think someone would have noticed that things are not what they seemed.
3. Intuitive obviousness. Even if we did not have a lifetime of experiences, it is obvious to our senses and minds that the physical world around us is not a delusion. Our minds are set up to believe the truth of our senses. We can’t help it. Try for even a short while to disbelieve your senses, and you will find it impossible.

It is with regard to the fourth reason, culture, that a difference between mathematical arguments and the belief in the reality of our experience appears. On the whole, our culture overwhelmingly supports the validity of experience. However, there are a few culturally ingrained doubts. There have been philosophers who have gained some degree of fame and influence who have questioned the validity of our experience and cast doubt on even the most obvious things. There is a general belief that governments, corporations, aliens, or magical people will go to any end to delude the populace for their dastardly plans. There are lots of science fiction and fantasy stories which make such a possibility plausible.

Thus I would suggest to you that the difference between the character of our belief in mathematical conclusions and the character of our belief in the validity of our experience is not because one has been proved in the first sense of proof. Proof doesn’t have anything to do with it since none of us has ever proved either conclusion. Ultimately the basis for belief in both is the same—namely, education, consistency, intuitive obviousness, and culture. However, with regard to culture, there is no support for disbelieving mathematics while there is support for disbelieving the validity of our experience. For the vast majority, cultural influence is the deciding difference, not the strength of the evidence or proof.

Nevertheless, you may say, there must be some reason why there is no cultural support for disbelieving mathematics. Mathematics really is sure and certain even if we do not exactly know why. Cultural beliefs are rational, are they not?

Undeniably, there is some reason. The question is whether that reason is a good one. To answer such a question would require an in-depth analysis of the nature of mathematics, the nature of knowledge, first principles, the reliability of our senses, and our mental faculties. This question has been hotly debated for thousands of years, and our general cultural beliefs stem directly from this conversation. Such a question is well beyond the scope of this short discussion.

My take, however, is that the issue is not cut-and-dried. The fact that my mathematics professor claimed that a proof is a convincing argument supports this. He recognized that the common notion of proof is not tenable. Further, I believe that the common notion of proof is a result of a mistaken set of philosophical ideas developed in the seventeenth and eighteenth centuries. Whatever the case may be, I think we can safely conclude that for most of us, our belief in the certainty in mathematics is based on general cultural trends and not on the first, more common notion of proof. And if I may be so bold, if you are convinced by my argument, then I have proved it!

While a mistaken common notion of proof is not something of great significance, I think it illustrates the power that culture has on our ideas. Many ideas weave their way into our thoughts due to cultural influence. To recognize them is a very difficult task, and to weigh them judiciously is even harder. Despite the difficulty, it is worthwhile to stop and ask, on occasion, why do I believe “that” after all.