WHAT IS THE HARDEST MATH QUESTION IN THE WHOLE WORLD?

David Zahora

Math 189- To Infinity and Beyond

Sharad Goel

3/3/03

On the purpose of Mathematics

During the span of human existence, math has played an important role. Math has always had a practical aspect to it. From its start as a method for simply counting objects to describing the complicated interactions of particles in the world around us, math has evolved into a complex system for describing the real world. However, in addition to the applications in the real world, math also has grown a side that deals with a philosophical world created only of math. This math, from henceforth referred to as 'Pure Math' seems to have no real practical applications as it doesn't explain the real world. For a long time, however, the emphasis on math had really been the practical applications, henceforth called 'Applied Math'. In recent times there has become a real divergence between the two maths. In the past there was always a bit of linking between the two subjects, but now they are truly becoming two separate studies. So in looking at the current and future situation of math the question arises: Which of the two subjects of math should be studied?

The first step into analyzing this problem is to quantify the value of math. This is probably the most difficult aspect of this whole problem. One must really know the value of math before they can make decisions on what should be done with it. Yet unfortunately, math is probably one of the hardest things to assign value too. How does one put value into something that crosses so many different fields in different ways. Math is an exceptionally important factor in the field of physics where it is used to describe everything. Conversely, Math can be used in philosophical  arguments, such as those , as it is believed to be a standard in all human thought. Either way, the point is that math effects each person in a different way. To some people math is simply a subject. To others it is an art form. So to quantify the value of math would be a nearly impossible task. However, to properly answer this problem, some thought must be given to the situation.

There seems to be two main ways to quantify math: its practical value and its beauty. In A Mathematician's Apology, Hardy believes that the only real value for math is its beauty. He believes that the actually value of math is degraded by its practicality. To him, math is like an art form. Just like art, however, it can only be truly appreciated by those who have studied it enough to understand it. This argument as a whole can be very convincing. It appears that math could be like a fine art, and valued by those who truly understand it. However, this can not truly define the value of math as a whole subject. The main reason why this argument doesn't hold weight is because it subjectifies the value math. Beauty, of course, is individualistic to the beholder. What one person sees as a beautiful math proof, another might see as ugly. This is not to say that beauty is still not a good judge of something's value, because it is that way in art. However, for this particular problem, a discreet definitive value of math is needed. The only other logical measure of math's value is its practicality. This means that math, for the sake of this problem, is only worth as much as it helps the world. This, too, is not the best way to describe math: The real value of math is not a true discreet and definitive concept. But for the remainder of this paper the value of math will be its practical applications.

  So the value of math is its practicality, it would appear that it would make it very easy to deal with the problem of which of the maths is worth studying. Logically it would seem that all efforts should be put into applied mathematics. This is the subject that deals with real life problems. Applied math takes problems that need to be solved, and solves them. If maths true measure is its value, then this is clearly dealing with the valuable part of math. Applied math should be the eventual goal of all mathematical studies. What about pure math though, does it have any value? It would appear that by the definition that it has actually no value at all. Pure math only deals with other math in a non-real world application. It would appear to have no impact on the real world, and hence should have no value. However, this is not true at all. Pure math's value is not in its immediate definite value but rather its future applications. In fact, it will be argued that most applied math can only be done as a consequence of studying pure math.

If applied math is the eventual goal of all math then how does pure math get to have any value. Well first it must be seen how to get a 'valuable' applied math problem. The first step to solving an applied math problem is to actually get the problem. This actually is more difficult than it seems. While some problems which need an applied math solution are obvious, some are hidden. An example of the obvious is the problem of gravity. It was solved by Newton developing the mathematical system of calculus. This was an 'obvious' problem because it was a problem that was seen every day which needed describing. Thus, there was no question of what the problem was, the only thing that needed to be worked out was the applied math of how to solve the situation. But what about situations that aren't so obvious. How does one go about digging up problems which need solutions. One can not simply study applied math forever because all readily  available problems will eventually be solved. To truly keep finding problems, pure math must be studied.

Just how does pure math create applied math problems? Well, this question is hard to answer, but it can be explained by a business metaphor. If a company wants to build the next hot electronic device that's going to change the world, how would they go about doing it. They couldn't just say to their engineers, "go and design this new product" because the engineers would have nowhere to begin. The engineers in this situation are the parallel to applied math mathematicians. If you want something new, one can't just go to the applied mathematicians and say solve this, there needs to be a base problem first. Once the problem gets stated, the applied mathematicians can do their job which will yield the eventual valuable commodity. But until the problem actually gets stated, the applied mathematicians are like engineers working towards an unknown goal. Just like how engineers can't just build towards an unknown contraption, applied mathematicians can't work towards an unknown goal. Without direction, there is no way to get at the true value out of applied mathematicians

So how do mathematicians get this needed direction. Clearly this is where pure math must come in. Going back to the metaphorical example, if the company really wanted to make the next big product they would probably have to form a think tank. This think tank would just sit around and come up with random ideas. Some would be good and some would be bad. But they are ideas nevertheless. If enough time and effort is put in, eventually a good feasible idea would come out. Once it comes out, the engineers can begin to work on it and actually design the product. Without the think tank, however, the product idea would never be brought to fruition and hence no value  would ever be created. Pure math works in the same way. Pure math is like a mathematical think tank. While a lot of ideas are created which have no practical use whatsoever, if enough time and effort is put in eventually a good workable idea will come out. These few good ideas are the key to pure math, because they are the ideas which will eventually have practical applications. These ideas do not have to be direct problems which will clearly have an impact on the world. They might affect other equations or they might show a new way of looking at a situation. Either way, the point is that pure mathematics spurs ideas that eventually effect applied math, and by association the real world. Thus, the value of pure math is to provide problems that can be solved by applied mathematicians so that there can be practical applications.

Pure math, however, is even more valuable than a simple think tank. A think tank simply thinks of ideas, but pure math also creates answers for the future. Pure math sometimes just solves problems which have no use outside of math. But these are just like stepping stones to a more advanced solution. When the eventual problem with value is found, a lot of the work into solving may have already been discovered. In the metaphorical example, it would be as if the think tank was designing component parts as it was planning. So when the final product was decided on, there were already all of these workable devices created that can be used. The engineers would only have to combine the parts into a workable fashion and the product is complete. In pure math, the ideas are created and random 'component' equations are created as well. So when a practical problem eventually comes up, the applied mathematicians have all of these equations to work with as well. So the applied mathematicians are already a step ahead because of the work of the pure mathematicians. Clearly there is a lot of value in the study of pure math.

  However, it still seems that it would be silly to randomly study problem in the attempt that eventually some speck will come out. But, as stated above there really is no other way to go about the situation. Time and effort can not be simply pumped into applied math with the hope that something new will be discovered. In order to keep making practical discoveries, time and effort must be placed into pure math. While a lot of the discoveries for pure math will only effect other math, eventually there will be that little speck that makes it worth while. Pure math is needed to give applied math problems to work on. There will be some problems that are obviously apparent and can be worked on without have to find the problem. But, there are only so much of these obvious problem. In order to truly keep advancing applied math, pure math must be studied to create problems worth solving.

When looking at the two separate types of math initially, it is hard to see which is more valuable. The first problem, of deciding how to measure the value of math, is of course without explanation. However, if you decided to set the value of the math to be the pure practicality in the real world, then you can start to make some valuable comparisons. At first glance, it is obvious that applied math would be the more valuable of the two maths. It deals exactly with problems which are known to be valuable. Applied Mathematicians work on problems that will affect the real world not a philosophical world of math. But as it is shown above, the pure mathematicians are just as important to solving real world problems as the applied mathematicians. Through their work, the pure mathematicians create ideas that will eventually affect the real world. Thus, the pure mathematicians are just as important to the field as the applied ones. So in looking back to the original question, Which of the two subjects of math should be studied, the answer is obvious. They should clearly both be studied. You can't have one really without the other. Without pure  math there would be no way to really continue applied math, and without applied math there would be no need for pure math. Clearly they are both equally as valuable in the whole design of mathematics.