Theorems and TheoriesTwo words which are often confused by people not familiar with mathematics and the other sciences are "theorem" and "theory". Despite their similar sound, these two words refer to quite different kinds of things in mathematics and the other sciences. Adding to the confusion is the fact that, unlike "theorem", which shows up only in mathematics, the word "theory" has an everyday usage which is also quite different from the word's meaning in the sciences. In this article, we will provide some clarification of the different meanings of these words. First, we'll look at the mathematical term "theorem" and see some examples; then, we'll go on to the scientific meaning of the term "theory", looking at how this technical usage differs from the common everyday use of the term. Finally, we'll see how the word "theory" is used in mathematics. Theorems are what mathematics is all about. A theorem is a statement which has been proved true by a special kind of logical argument called a rigorous proof. A rigorous proof is simply a sound deductive argument, meaning that it starts with statements which we know to be true and then makes small steps, each step following from the previous steps, until we reach our conclusion. One statement follows from another if it's impossible for the first statement to be false while the second is true. For example, the statement "Socrates is mortal" follows from the statement "All Greeks are mortal, and Socrates is a Greek" -- it is impossible for "All Greeks are mortal, and Socrates is a Greek" to be true without "Socrates is mortal" also being true. Hence, if we start with statements which we know to be true, then any statement which follows must just as certainly be true. Once a theorem has been proved, we know with 100% certainty that it is true. To disbelieve a theorem is simply to misunderstand what the theorem says. Here are some simple examples. Theorem: There is no largest counting number. Proof: Suppose there is a largest counting number, and call it n. Then n+1 is a counting number, and it's larger than n, so our choice of n was wrong. Maybe n+1 is the largest counting number then? No, because again (n+1)+1 is a greater counting number. Indeed, we can apply this same procedure to any candidate for "largest number", always with the same result. This shows that every choice of largest counting number is wrong, which is to say there is no largest counting number.
Here, the argument shows that you can only disagree with the conclusion by disagreeing with our meanings of the terms "largest" or "counting number". If you understand what these terms mean, you cannot help but agree that there is no counting number that's larger than all of the others.
Another example, this one due to Gauss:
Proof: Set x = 1+2+3+...+(n-1)+n. Then adding x+x yields a sum of n copies of n+1:
You can check for yourself that this statement is true for any number n that you like. For example, if n=5, then 1+2+3+4+5 =15 while 5 times 6 divided by 2 is 30/2 = 15. There is a famous story which says that Gauss thought of this when he was in grade school and a teacher assigned the students to add up the numbers from 1 to 100 as busy work. Little Gauss amazed the teacher by finishing in only a few minutes.
Another famous theorem with an nice easy proof: Theorem: (Pythagoras) If a right triangle has short sides with lengths a and b, and long side (hypotenuse) with length c, then 
Proof: Consider the square below. We can compute its area in two ways, either as the square of the length of its side (a+b) or as the sum of the area of the inner square and four triangles. We can pair up the triangles to make two rectangles with one side of length a and one side of length b, so the total area of the square is c squared plus twice a times b. This must then be equal to (a+b) squared, since these two quantities are both describing the area of the square. Some simple algebra then gives us the result.
Here, we might object "But what about non-Euclidean geometry?" It is true that by changing what we mean by "distance", we can get relationships between the sides other than the one in the theorem. But again, this requires changing the meaning of the words in the theorem -- with the usual meanings of "distance" and "right triangle", the conclusion of the theorem is inescapable.
Mathematics is about proving theorems. In pure mathematics, we study problems which we find interesting in their own right, answering questions by starting with what we know and proving theorems. Theorems are sometimes called different things depending on their importance or their relationship with other theorems. A theorem which is proved primarily as a step toward proving another theorem is called a lemma, while a theorem which follows as an easy consequence of another theorem is called a corollary. Theorems are often called propositions when they're first introduced. The foundation of modern science is applied mathematics. In the sciences, observations lead us to discover both facts and laws. A law is an observation about a regularity in the observed facts, such as "objects fall when dropped". An idea that might explain a law or other set of observations is called a hypothesis, and hypotheses often take the form of mathematical models, that is, mathematical systems whose structure is intended to mimic as closely as possible the behavior of the phenomena being studied. Applied mathematics is about developing mathematical models of real-world phenomena and applying theorems we can prove about the mathematical models to draw conclusions about the corresponding real-world phenomenon. Scientists start with careful observations about the world, then devise mathematical models which replicate the observations. One can test a scientific model by using the model to predict the results of an observation, then making the actual observation and comparing the results. If the model fails to accurately predict the result of the observation, then the model is in need of refinement; if the model's prediction matches the observation, then the model is consistent with the observation. Consistency with an observation does not guarantee that the model will always match the phenomena for every future observation, but consistency with enough repeated observations does say that the model is reasonable candidate for an explanation of the phenomena. Over time, a model gets tested and refined, with details filled in by repeated observations and tests. As mathematical models get tested and refined over time, we have increasing confidence in the effectiveness of our overall collection of mathematical models. That is, over time the current version of a set of models becomes a better and better description of reality. There is never any stage at which the process ends and we declare that we have a 100% accurate description of reality, since that would require having made every possible observation; however, we can and do have mathematical models in which we have a very high level of confidence, higher even than the level of confidence that we usually require to call a statement a "fact." 99.9% certainty may not be 100%, but it's a lot closer to 100% than it is to 0%. Note that, by contrast, common claims are often granted "fact" status after only a single observation or inference! Because of the difference in the levels of testing, the "facts" of everyday life are actually much more likely to be incorrect than are our scientific models. As our hypotheses are being tested and refined until our level of confidence in them is very high, we seek a set of principles which provide a coherent explanation for the various laws and facts which we've assembled. This kind of detailed explanation of some aspect of reality, incorporating all of the various well-tested hypotheses and mathematical models and explaining the various facts and laws that we've observed, is what we call a scientific theory. This is quite a different kind of thing entirely from what one might call a "theory" in day-to-day life. Our usual non-technical meaning of "theory" is much closer in meaning to the scientific term "hypothesis", that is, a simple idea which can be tested. For example, a detective might have a "theory" about who committed the murder, or a student might have a "theory" about the best way to get a good grade. These are not "theories" in the scientific sense! A single individual never creates an entire scientific theory alone, for scientific theories are much too large and complex. Even theories which have an individual's name associated, such as "Einstein's theory of General Relativity" or "Darwin's theory of Evolution" are not the work of a single individual but are the cumulative results of the collaboration of many individuals over time. A scientific theory is an extensive body of knowledge which brings together a great number of well-tested hypotheses and mathematical models, weaving them into a coherent explanation for the facts and laws we can observe. An everyday hypothesis is no more a scientific theory than a single bolt is an automobile. A common related confusion is the idea that scientific theories are waiting to be tested and proved before becoming accepted as a fact or a law. This is a serious misunderstanding -- the ideas that make up a scientific theory are already as well-tested as the current technology permits before they achieve the status of a Scientific Theory. Theorems are proved, not theories. In mathematics, before a theorem is proved, it is called a conjecture. In the sciences, only well-tested hypotheses can become part of a theory. In this way, the term "scientific theory" is very poorly chosen, because for most of the population, the word "theory" suggests weakness and doubt when in a scientific theory there is only well-earned confidence. In other words, to say that something is a "scientific theory" is to say that it is backed by all available evidence and that our confidence in its correctness as a description of reality is as strong as it can get with our current ability to test it. Theories do not get proved and become facts or even theorems; if a model or hypothesis is part of a scientific theory, then it already is as "proved" as it can ever get. It is true that scientific theories are not static and absolute; as technology matures, we constantly find new ways to refine our previous ideas. Occasionally, there are so-called "paradigm shifts" in which a new theory replaces and older one, but in every such case the new theory must be able to explain why the old theory matched up with observations as well as it did. For example, relativity was only able to replace Newtonian mechanics as the accepted scientific theory of gravitation because at low speeds and in low gravitational fields, relativity looks a lot like Newtonian mechanics. This confusion between the ordinary and scientific meanings of the word "theory", between "simple guess to be tested" and "extensive collection of thoroughly tested and confirmed hypotheses which together provide an explanation of observed facts and laws", forms the basis for a Monty Python sketch -- a character has "a new theory about the Brontosaurus", namely "that all Brontosauruses are thin at one end, thick in the middle, and thin at the other end." Sadly, like much of MP's humor, this joke is lost on the uneducated. A more serious consequence of the confusion of the two meanings is the ability of ideologically motived groups to exploit this confusion, misrepresenting very well-tested and robust scientific theories as "only a theory". This creates the false impression that there are serious doubts about the theory, when in fact the models and hypotheses which comprise the theory are known to be accurate descriptions of reality with a very high level of confidence. Examples are numerous, but climate research and biology come to mind. In particular, the attempt to criticize of any aspect of science by saying "X is only a theory" is a dishonest rhetorical trick that uses the fact that the word "theory" has two different meanings to mislead people. It's akin to telling someone they can feel free to ignore a Supreme Court ruling since it's only the justices' Opinion, or like the roadrunner trying to break the Law of Gravity. Finally, the word "theory" is used in mathematics in a way that is similar to the "scientific theory" sense, meaning something like "the study of". For example, "knot theory" does noes not mean "a hypothesis about knots", but rather "the mathematical study of knots" -- it's really just a short way of referring collectively to the entire body of articles, theorems, books and works relating to the subject. Where a scientific theory is composed of well-tested hypotheses and mathematical models, a mathematical theory is composed of lemmas, theorems and corollaries. Both are forever incomplete, in the sense that there are always new theorems to be proved and new hypotheses to be tested; both refer to fields of study in which a great deal is known but there is still more to be discovered.
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