Jennifer Pan

April 5, 2001

Geometry paper

In some academic disciplines, the idea that philosophical logic applies to them seems somewhat absurd. Surely, fields such as physics and geometry are much more concrete in nature than philosophy where some hold to an Aristotelian stance and others Socratic. Strange as that seems, the philosophical principle of using logic to prove or disprove certain "truths" is important and widely used, although the approach varies from discipline to discipline. For the computer science discipline, the existence of truth is more important than, in most instances, the proving of the truth.

In geometry where there are concepts such as "axiom", "proposition", and "theorem", the need to be able to validate them becomes quite important. Without having been proven, any statement could be said to be an axiom, proposition, or theorem and be false, assuming that things are either true or false, but not both or either at the same time. But to be of use, these concepts need to be "true". So, how do we go about showing that they are true? And what is truth?

Truth in geometry appears to be synonymous to consistency. If the whole system holds together, given base conditions (known as axioms in geometry), then the statement (conclusion or otherwise) is considered as true. Computer science takes a somewhat different approach in their view of truth.

The field of computer science is a fairly interesting one and one that is undergoing rapid transition and expansion. As such, there is much interest in it because of its application to many other fields. For instance, people doing cognitive studies use computer science to model artificial intelligence (e.g. Eliza, a well-known, albeit simple program) and geologists use a program known as GIS (graphical information system) to assist them in analysis. Science, according to the Webster’s Revised Unabridged Dictionary is:

Strangely enough, even though the word "science" is part of the title of this particular discipline, unlike geometry which most people don’t even consider as a science, it only loosely follows the first definition of science--that is, knowledge of principles and causes.

By looking at various textbooks in the field of computer science, the existence of axioms, theorems and postulates are somewhat foreign and primarily used in the analysis of computer algorithms which is more closely related to math than a direct application of computer science. By this, I mean that algorithms are used directly in computer science, but the analysis of them is more akin to math because to analyze algorithms, mathematical proof principles and procedures are applied. I have yet to come across a computer science textbook that discusses at any length philosophy of truth, or how truth is defined in computer science. This is not to say that computer scientists never encounter truth issues. In fact, programmers often test for truth (an example in C++, a program used to develop software programs: if (condition= = TRUE) action to be taken;). In this example, the computer scientist does not ask the question "why is this true (or false)", but merely assumes that it is so.
Unlike geometry that builds from the base up (e.g. using neutral geometry to arrive at the parallel postulates), computer science comes up with certain algorithms and, in a sense, "proves" that they work by analyzing them and testing them. For instance, a well-known algorithm used in computer science is that of Dijkstra, an algorithm that has many applications in computer networking or any procedure where there is a need to traverse a graph. There is no real base case it is based on. Dijkastra’s algorithm simply presents people with one of many methods in which to traverse a graph. And computer scientists (as well as people in other disciplines needing to traverse graphs and trees) know that this algorithm works because all one has to do is draw a graph and step through the procedures to see that each node (vertex of the graph) does indeed get visited by using Dijkstra’s algorithm.

That is only one example from computer science where truth is merely verified and not necessarily proved. Or, it is proved, only not in the same way that postulates and axioms are proved in geometry. In computer science, when asked to prove something, proof is done in these cases by the inductive (strong and weak) hypothesis. By having a base case and testing it then logically expanding it to include all cases, that is the method employed by computer scientists to show that an algorithm works. The base case is similar to the way geometry uses previously proven theorems, or given axioms to derive the rest of its theorems, postulates and other rules.

In this regard, it can be said that computer science is like calculus. That is to say, in calculus, there are several different methods for deriving the area under a curve. How does one prove it? Well, in calculus, we showed that these methods worked by taking limits as bounds and observing to see how they behave as they approach a given quantity (sometimes infinity, other times zero or something else entirely). In this way, there are many different "true" methods that all work to solve a specified situation, only these different methods (or algorithms) behave better with different conditions (e.g. lists that are ordered, unordered or ordered in reverse, etc.). That is not to say that there is not a proof for why Newton’s method works, but more that mostly, it is the theoretical mathematicians that are concerned with the proof than the people who use the method and apply it to arrive at solutions in their application.

Truth shouldn’t be confined and defined within only each discipline of study. The different approaches to truth and meaning often times interact with each other, and certainly with the people who are involved in cross-disciplinary fields. What is true in one setting should also be true and valid in another discipline. So, how do I conceive of a somewhat universal concept of truth and meaning when it seems that truths sometimes contradict each other? For instance, how is it that in physics, there are two models for explaining the behavior of light: particle theory and wave theory? Waves are not particles and the concept of particles is not synonymous to the idea of waves, yet with such different properties, light is said to exhibit both the properties of waves as well as particles.

In the sciences, it often appears that truth and meaning are very concrete. It almost sounds funny to have to define what truth means in the sciences. We see it reflected in our interactions with the physical world, that to question the validity of a scientific truth seems absurd. But then again, we should be reminded of the outrage and shock surrounding the "discovery" of cell biology and the heliocentric model of planetary motion. The religious community, at the time, discredited these discoveries because, to them, these ideas contradicted the Truth they felt was taught by the Bible and their religious leaders.

Throughout grade school, we’ve been taught that the scientific approach to truth is as follows: first a hypothesis is arrived at. A hypothesis is a guess as to why something behaves the way it does or an explanation for an observed phenomenon. Once there is a hypothesis, tests are made to see if there are any contradictions to it--these involve empirical studies. From there, if there are inconsistencies, then the hypothesis is either discounted or refined depending on how "major" these contradictions are. If the hypothesis is able to hold up under various test conditions, then the status of the hypothesis becomes a theory. Until something resembling the miraculous happens, a theory stays a theory. Otherwise, as in physics, it becomes a "law" (e.g. Newton’s Laws, Ohm’s Laws, etc.). As long as it remains a theory, it is considered a truth--that things with the given conditions will behave a certain way. Even at the level of "theory", if studies are conducted which yield results that contradict the theory, the theory is invalidated and it is amended or a new hypothesis is needed.

This approach is quite similar to the way in which Lakatos, author of Proofs and Refutations, approaches the idea of truth. You can either try to modify your concept so that it remains consistent to observable reality, or else you can add conditions to the hypothesis in such a way that the basic hypothesis still works by excluding those conditions you know will cause the hypothesis to fail. Otherwise, if the modifications necessary to keep the hypothesis consistent are more than minimal, it could be dismissed as being invalid. While I agree with portions of what Lakatos says, I think I tend towards a more intuitive approach to truth and meaning.

Those things that are true will be true and be in most cases apparently so. There is no need to prove through a bunch of abstraction that this is indeed the case, because to arrive at that, you have to break down each step of the argument to the most fundamental and basic ideas. But, I could argue that that most basic idea is not basic enough because to a certain degree, there is nothing to support it--the basis is usually assumed to be true and used as such (definitions and certain axioms in geometry, for instance). Going in reverse I suppose, if we can assume that something is true and work up from there, cannot we just as soon assume that something bigger is true as long as it is consistent with our body of knowledge without going through the trouble of proving it because our intuition tells us that it works and our observations also show it to be the case that it is true and consistent. Besides, some truths are concrete that proof or no proof, people would agree that it is the case. After all, one only has to look at the definition of the word axiom. According to the American Heritage Dictionary, Third Edition, axiom is defined as:

Using the third definition, something is considered an axiom if it is self-evident OR accepted as true without proof. On those grounds, we could say that anything is an axiom as long as it is accepted as true without proof, and as an example, we can take a look at the following example. We know that cats are felines and dogs are canine: we do not have to prove, but merely correctly assume that in nature, the two cannot interbreed. Basically, if we don’t prove something, but say that it is true, using it as the basis for an argument, then we have made it into an axiom.

Another way of looking at the same situation is this. Take for instance a non-repeating decimal (say, pi). In geometry and trigonometry when pi is used, we typically assign it the value of 3.14 with three significant digits. Is that accurate or true? Well, not in the complete sense, but it IS an approximation. But, at the same time, how far do we need to take it? Are ten decimal places enough? For most of our work, the two places after the decimal are more than enough. How about 20? Even so, it is not the "truth" since pi does not equal whatever we say pi is out to the 20^th decimal position is.

Some people argue that God doesn’t exist. Others insist that there is a God. How do you go about proving it? There is no real base case from to start. There seems no concrete empirical study to test for the existence of God. Does that negate the existence of a Supreme Creator? Does that validate the non-existence of God?

Another way of looking at this same problem where truth is not universally accepted, and is instead debated because existence of proofs on either side seem insufficient to people who oppose the validity of the "truth", is that of examining the question of free will. It is typically agreed that humans have the power to make decisions. In fact, I don’t think anyone would really contradict that because it seems that we do. But can we prove this without a doubt? How about causation? It affects our decisions so that it isn’t entirely always our will. I suppose, in other such cases, truth could be different depending on who you are.

Even in cases where people agree on the truth, I still find that, to a certain extent, the need to prove truth or disprove a statement is unnecessary and essentially stupid, monotonous, and generally uninteresting. After all, take court trials. Sure, judges try to make a judgment based on what they believe to be true from evidence provided (proof). Even then, certain established truths turn out to be false and that fallacy is only discovered many years later. Even in sciences where people have this sense that things are cut-and-dry, die-hard, cast-in-stone truths, this is also the case. We, in science start off with a premise that we assume to be true and come up with theories and models and such from them. All of which are used then to "prove" other ideas and hypothesis. But the body of "truths" upon which they are built could be false, in which case, what good is the proof anyhow? When proving things (concepts, ideas, etc.) we work backwards until a point we think is enough. That "enough" state happens to be a premise or base assumption. The reason I think it’s unnecessary to tirelessly prove things is because we end up showing by cases anyhow that a statement is either true or false. And we arrive at this truth through assumptions of sorts.

Going back to computer science and relating this to computer science as well as other disciplines--it is important in all areas that there are truths. After all, if truth in some form or another was unimportant, then there really isn’t much of a need to study it in detail--any statement would weigh the same and as a consequence have a corresponding contradictory statement that is also assumed to be or taken to be true. As to the universality of these truths, it depends on what field. In computer science, physics, mathematics, etc., I would say that these truths are fairly absolute and concrete, as well as being fairly intuitive. In other cases, truth doesn’t seem to be that universally accepted. And no amount of proof is going to change that because there are just so many conditionals needing to be broken down into smaller and smaller base cases that it is futile to attempt to do so. Where in geometry it is reasonable to prove theorems and postulates used on occasion, it becomes slightly less reasonable to prove that algorithms work, because there are multiple algorithms that do basically the same task. While for both calculus and algorithm analysis, a field in computer science, you can actually prove that certain methods work, it is less important to actually do the proof than to accept that it works. As we don’t prove Newton’s Method in calculus class but merely accept that it works, in computer science, we accept that algorithms work and apply them instead of meticulously proving and reproving that it works. It can be done, but it equates more to proving that a process works rather than that a cause-effect/state-condition is true.

Like the students in Lakatos book, Proof and Refutation, who argued about the validity of V+E=S, I don’t think computer science is much different. As people, we can see possible cases but cannot always account for all cases. What started off as a near theorem wound up being modified and revised as Lakatos’ students saw more and more cases where the "theorem" may or may not work. Computer science is more concerned about the application of what is perceived as true than tirelessly proving it.