Theorems and Theories
Two 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:
Theorem: (Gauss) The sum of the counting numbers from 1 to n
is equal to ||
Proof: Set x = 1+2+3+...+(n-1)+n. Then adding x+x yields
a sum of n copies of n+1:
Then 2x = n(n+1) says that
| 1 || + || 2 || + ||
... || + || n-1 || + || n
|| + |
|+n || + ||+n-1
|| + || ... || + || +2 || +
|| +1 || =
| (n+1)|| + || (n+1) || +
|| ... || + || (n+1) || +
|| (n+1) || = n(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
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
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
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
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
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
Copyright © 2004-2010 Sam Nelson