Sunday, August 22, 2004

Is Mathematics a Science?

Today's article is a guest article written by my good friend Arturo Magidin who was kind enough to write the following essay at my request.



Is mathematics a science? Well, the answer depends on what one means by "mathematics", and what one means by "science". There are well known and respected philosophers of science that will tell you the answer is 'no', while others are just as emphatic that the answer is 'yes'.

So before we can provide an answer to the question, we must agree on what we are talking about.


WHAT IS SCIENCE?

This is in itself a rather tricky question. People bandy about the words "science" (with or without first or last names) and "scientific" quite easily, but most would be hard pressed to provide a coherent answer to the question of what qualifies something as science and what does not.

Since I am writing at the behest of Andrew, I asked him that question when he asked if I was interested in writing this essay. So I will quote his answer and use that as a general guide.
Science is a form of epistemology that, like any good epistemology, attempts to distinguish true statements from false statements thereby leading to an accumulation of knowledge. One of the primary things that distinguishes science from other epistemologies is that it is a) systematic and b) nondogmatic. A proper science must have a means of validating its claims as well as a means of identifying and rejecting false claims. This, too, ought to be systematic.
Let us agree to this definition, and also agree that here "true" and "false" refers to an objective agreement with the 'outside' world.

Mathematics seems to have the deck stacked against her: the axioms are very close to dogma, and it seems not to care about truth or falseness, or even the outside world for that matter. There are proofs, but there seems to be no experimentation; mathematics does not follow the scientific method. Is that really the case?

SOME HISTORY

Most people are never exposed to modern research mathematics. They either know mathematics as a collection of recipes, algorithms, and rules (e.g. the formula for solving a quadratic equation, the rules of differentiation, how to multiply two numbers together, etc), or perhaps know the axiomatic model of Euclid (lemma, proof, theorem, proof, corollary, proof, all based on abstract notions and 'self-evident truths' known as axioms). Or both. The truth is that it is in reality neither, though at one time it (sort of) was close to those impressions.

Up until the 19th century, people who did "abstract mathematics" were called Geometers; they were usually either philosophers, amateurs, or physicist/mathematicians in their spare time. Major mathematical works dealt either with the sort of development similar to Euclid, or were collections of recipes and algorithms to solve problems (e.g., the works of Diophantus and Fibonacci). Almost everyone else who did mathematics was also doing physics. In fact, the most famous mathematical names up to the early 1800's were connected with physics in one way or another: Newton, the Bernoullis, Fourier, even Fermat; Gauss, the Prince of Mathematicians, officially held a post as Astronomer, and did much work in physics. That kind of mathematics was intermingled with physics.

About the only kind of mathematics that did not fit in either one of the above molds was number theory, which was, until Gauss's landmark Disquisitiones Arithmeticae, considered recreational mathematics, and in particular, not a subject of serious study but of play.

Then, during the 19th century, something happened. Mathematics began to develop as an independent field. In part, this had to do with a number of problems that had been accumulating: the foundational problems in calculus, the constructions of monsters that highlighted the problems with naive and intuitive notions being used in proofs, and the discovery of non-Euclidean geometry. Another part was the explosion of new ideas and methods: the beginnings of group theory, complex analysis, and algebra; in the final years of the century, the development of naive set theory (later replaced with an axiomatic variant) and the appearance of non-constructive existence proofs (most notably, Hilbert's finite basis proof). [Please follow this link for a discussion of some of the relevant issues of this proof]

During this crisis, a break developed between physics and mathematics. While most mathematicians still engaged in solving problems derived from physics, and most physicists still solved mathematical problems, the emphasis became different. Physicists were, by and large, not very concerned with the foundational problems, since calculus and its derivations obviously worked. The monsters, paradoxes, and antinomies might be interesting to philosophers, but they were not things that one was likely to encounter in "real" problems. On the other hand, Mathematicians were very concerned with the problems, and struggled to try to place their edifices on solid grounding.

Out of this crisis, there arose two main schools of mathematical thought. Both agreed that mathematics needed to be placed on a more solid foundational footing. On the one hand there were those, led by Kronecker, who believed in an emphasis on algorithms and recipes that followed from clearly defined concepts which were based on some empirical reality; "empirical" here must be taken loosely: Kronecker's famous dictum was "God gave us the integers; the rest is the work of Man", meaning that he considered the (potentially infinite) set of integers to be an 'empirical reality', for example. They were called 'constructivists' or 'intuitionists'. On the other hand there was the school led by David Hilbert, sometimes known as the 'formalists'. To the Hilbert school, the ultimate arbiter was self-consistency and interest. A mathematical theory should be based on clearly stated axioms and rules, but it makes no sense to ask whether the axioms are "true" or "false". The only questions that one must ask are: (i) Is it possible to use the axioms and rules to prove both a proposition and its negation?; and (ii) Is the resulting theory interesting? Upon an answer of "no" and "yes" (respectively), the theory would be deemed acceptable. (The reason for asking the first question is that, under the rules of classical logic, if it is possible to prove both a proposition and its negation, then it is possible to prove anything. Such a theory, needless to say, is both uninteresting and useless.)

However, the axioms need not have any relation to "reality". Here again we have a famous dictum, this time attributed to Hilbert: "It must always be possible to substitute 'table', chair' and 'beer mug' for 'point', line' and 'plane' in a system of geometrical axioms." That is, the actual meaning of the axioms is immaterial; their semantic content plays no role in mathematics.

In the end, it was the Hilbert school that triumphed among most mathematicians (with come caveats). As a result, most mathematicians will describe mathematics as following the classic Euclidean axiomatic model. Perhaps the quintessential example is found in the works of Bourbaki. It has also strongly influenced the way in which mathematics papers are written and advanced mathematics is presented. More on this below.

These days, mathematics tends to be divided into two: applied mathematics and pure mathematics. Applied mathematics are the parts of mathematics that deal with problems that arise out of empirical concerns: statistics, differential equations, and the like. Pure mathematics deal with problems that arise out of theoretical frameworks, often purely mathematical. The distinction, however, is in large part artificial. For example, at one time Number Theory was considered the most pure of pure mathematics, a branch of mathematics that had absolutely no possible practical application. In recent years, however, it has become the cornerstone of modern cryptography and developed a very robust applied side.


WHY THE HISTORY?

What is the point of all of the above? Well, the point is that the Hilbert School exercised a very strong influence in mathematics in the 20th century and through today. This influence in turn helped to produce and enshrine a particular style of writing when one deals with mathematical research. This is the dry definition-lemma-theorem-corollary style that many are aware of.

The "problem" with this style is that it obscures how mathematics is done. A professional mathematician engaging in research does not produce a definition, then write a theorem and its proof, perhaps separating some crucial step as a lemma. The way in which mathematics is reported in research articles and books is very different from the way in which mathematics is done. (I should point out that it is my impression that there has been a slow but steady shift towards a more engaging style in mathematics articles in the past two or three decades; a style that allows the reader a look at some of the process the writer went through to produce the proof; there have always been gifted mathematical writers who did this anyway, such as Gauss, Dedekind, or Paul Halmos, for example, but it seems to me to be becoming more widespread.)

The effect of this prevalent style is that everyone but the professional mathematical researcher tends to get a skewed and inaccurate view of how mathematical research is done. This in turn has led some philosophers to conclude that mathematics is not a science, because its methods are (apparently) so different from those of other empirical sciences. I will argue below that, once we go past the facade provided by the writing style and into the way mathematics is done, that this conclusion is in fact quite unwarranted.

Another effect is that the Hilbert School, from its beginnings and especially in the wake of the work of Goedel, Turing, Church, and others, abandoned the ideas of "true" and "false" in favor of the ideas of "provable", "disprovable", and "undecidable". Our working definition of science places a strong emphasis on truth, and so it would also seem to follow that, insofar as mathematics seems unconcerned with truth and falsity anyway, it could not be considered a science or a scientific pursuit.


BUT IS IT SCIENCE?

Gauss called mathematics "the Queen and Servant to Science". Pretty much everyone agrees at least to the "servant" clause: it is undeniable that mathematics plays a major support role in science. And no longer just in physics, or chemistry, but increasingly in other sciences as well. Statistics was the cornerstone of the change in medicine from art to science. And most social sciences seem to feel that the more math they can put in, the more robust and scientific they will be. It is the claim that mathematics is a science (let alone the "Queen of Science") that seems to lack support.

For starters, does mathematics even follow the scientific method? Observation, hypothesis, experimentation, testing, verification?

In what may come as a surprise to some, yes, it does. This is where the prevalent style does a disservice to an accurate perception of research mathematics. A mathematician engaging in research does not produce a statement for a theorem and proceed to prove it. She is usually feeling her way in the unknown as much as any scientist. She will consider some specific examples (observations), and try to see if they have a property or not. She will formulate some questions, both general and specific, and try to see how she can answer them for specific cases. She may then attempt a general statement (hypothesis), and proceed to attempt a proof (experimentation); sometimes, if that fails, she will attempt to construct a counterexample (falsification and testing). This process continues until the mathematician finally obtains an argument establishing her hypothesis, or she manages to disprove it (or, finding herself unable to do either, sets it aside and tries something else...)

For particularly troublesome problems, working out specific examples is considered a worthwhile pursuit, akin to experimental confirmation of details of difficult theories. Checking all odd numbers up to a large bound to see if any of them are perfect may not be considered mathematical evidence that no perfect odd number exists, but it still helps. Proving that the ABC Conjecture implies certain results which are known to be true does not establish the conjectures, but it lends them some "street cred" (and makes people more interested in trying to establish them as true or false). And so on.

On the other hand, there are some notable differences with sciences like physics: while there is observation, there seems to be no observation of the "real world". And mathematicians always demand "proof", a far more stringent standard than is met by any other science! Take, for example, Newton's Law of Universal Gravitation. The claim that this law applies everywhere (the "universality" of it) would never satisfy a mathematician. The fact that it has never been contradicted is not enough for such a claim. Only proof, meeting mathematical standards, would be. Compare this, for example, with Fermat's Last Theorem: 350 years of being unable to find a counterexample or proof was not considered (mathematical) evidence of truth or falsity. It was merely annoying. Only when a proof was produced (and checked and verified) was this accepted. Or the Goldbach Conjecture, which has been verified to very large numbers; such verification, while indicative, is not enough. Likewise with the existence of an odd perfect number. (As an illuminating aside, a joke has a mathematician, a physicist, and an engineer traveling on a train through Scotland, when they see a black sheep in the distance. The engineer promptly asserts "In Scotland, sheeps are black." To which the physicist replies "No, in Scotland, some sheeps are black." The mathematician then gently corrects him: "In Scotland, there is at least one sheep which is black... on at least one side.")

So: how do we deal with the apparent lack of observation, and the demand for "proof"?

Taking the last point first, I would argue that the demand for mathematical proof does not seriously distinguish mathematics from other sciences. It is merely that the standards which a mathematical result must meet are more stringent than those of, say, physics. But other sciences also set up their own thresholds for acceptance: no more than a certain amount of error, statistical significance to a certain degree of confidence, sufficiently varied observations, predictions, etc. The standards of math are merely quantitatively different (in that they appear to be stronger and with higher thresholds), not qualitatively different.

On to the first point, then. Aren't axioms arbitrary statements? Aren't they dogmatically followed, never questioned or modified? Is there any concern for "truth" and "falsity"? For the outside world?

The idealized model of mathematics presented by the Hilbert School would argue that the answers are, respectively, "Yes, they are", "Yes, but they are arbitrary and we can change to some other system at will", "no", and "no". But like all idealized models, this is not an accurate presentation.

Axioms can be arbitrary statements, but they almost never are. There is usually some reason for presenting a particular set of axioms over some other. They represent not so much "arbitrary statements" as they represent the "ground rules" for a particular development, the minimum agreed-upon assertions from which we will proceed. Often, these are distillation of actual observations, or attempts at abstracting real world situations in a way that makes it amenable for mathematical treatment. The ideas of differential calculus (an eminently empirical development, made for the express purpose of providing the tools to study motion) have been distilled into a series of "axioms" for the real numbers from which we proceed, based on centuries of work and observation. They represent the compromise between avoiding the paradoxes, contradictions, and antinomies which faces mathematicians in the 18th and 19th century, and keeping the properties that allows calculus to be practical and useful.

The axioms of group theory were developed from the study of equations; they represented the minimum conditions on which the arguments would follow. And while one works with groups, these axioms are not changed. On the other hand, people are free to drop, discard, add, or remove axioms at will to produce other systems with which they can also work: this is the case of semigroups (obtained by removing one axiom), or ring theory (obtained by adding a number of axioms).

The axioms are, to some extent, "unquestioned", because from the mathematical side there really is little concern on whether they are "true" in an empirical sense. On the other hand, when a mathematical theory derives from a real world situation it is attempting to abstract and study, the axioms seldom go unquestioned or unmodified, as people attempt to make sure the abstract theory stays as close as possible to the situation it is attempting to model. There is continual feedback and fine tuning between the mathematical theory and the real world.

And while it is true that mathematics usually refrains from saying "true" and "false", and instead talks about "provable" and "disprovable", this does not mean there is no contact or application with the outside world. Mathematical theorems are never simple declarative statements; rather, they are always implications. All mathematical theorems are of the form "If (some conditions are met), then (this conclusion will follow)." Moreover, this will hold true whenver we interpret the theory in a specific model.

As Hilbert noted with his comment on geometry, an axiomatic system should not depend on any specific meaning given to the undefined terms or the axioms. What this means, however, is that any mathematically correct conclusions we draw from those axioms will be true in any interpretation. If we take a theorem of geometry, and interpret "point" to mean "table", "line" to mean "chair", and "plane" to mean "beer mug", then the theorem will provide us with a true statement about tables, chairs, and beer mugs (assuming that the Axioms, when thus interpreted, are also true). In this respect, there most certainly is a connection to the real world and an ability to test and check the validity of the interpretation. In addition, even though we recognize that the semantic meaning we might give to undefined terms like "points", "lines", and "planes" should play no role in a proof, nonetheless these semantic meanings are often used to inspire proofs and theorems. We will make a drawing of a circle and a line to help fix ideas or inspire a proof, even though "circle" and "line" are terms that should carry no semantic content in the proof itself.

The standards of proof that mathematics requires in fact ensure that the conclusion will be true whenever the premises (including the axioms) are also true; and that at least one premise will be false if the conclusion can be seen to be false. What the reliance on provability instead of truth gives us is flexibility and certainty. By relying on abstract rather than concrete considerations, we ensure (or at least, attempt to ensure) that our deductions will indeed follow for any particular interpretation.

Most mathematicians will usually have some specific interpretation in mind when doing mathematics. The danger is that we may use specific properties of that interpretation inadvertedly in a proof, and thereby obtain a result which will not be valid in other interpretations. This was the pitfall into which Euclid himself fell, for example. Proposition 1 of Book 1 relies on the obvious fact that two particular circle segments have a point in common. However, this "obvious" fact does not in fact follow from the axioms. It takes an enriched set of axioms before the theorems of Euclid become actual theorems that will be true whenver all the axioms (both classical and new) are true.

It is because this danger is present that mathematics has developed its standards of proof; just as other sciences have developed their own based on their own experiences. In this it is also that mathematics presents the characteristics of a science.


CONCLUSIONS

Is mathematics a science? I believe so. It follows the scientific method (though this fact is sadly obscured by the prevalent writing style). And while it seems (and sometimes claims) to live in its own little world, cast adrift from concerns about reality, the truth is that, even in its "purest" guise, it keeps an eye out to reality for both applications and inspirations. In its "applied" guise there can be no doubt that it sticks close to reality, and that its assumptions, problems, and conclusions are continually tested and refined against that backdrop. Nonetheless, it is also different from other sciences in that its standards are a bit higher and a bit more certain. But this part of its strength as a science, not a disqualifying property.

So, coming back to our working definition of science, does mathematics meet the requirements? It attempts to distinguish true statements from false statements. However, we must understand here that a "true statement" does not refer to a theorem or lemma in isolation, but rather to the implicit statement given by a theorem, that whenever all axioms and hypothesis are interpreted in such a way as to make them true, then the corresponding interpretation of the conclusion from the theorem will also be true. Likewise, "false statement" would mean that there is at least one interpretation of the axioms and hypothesis which makes them true, while at the same time making the conclusion false.

It is certainly true that mathematics accomplishes this in a systematic manner, through the use of proof. The proofs are open to scrutiny by any and all investigators, who are encouraged to "repeat the experiments", as it were, by going over the proofs line by line and agreeing to their validity (or requesting clarification, or even pointing out mistakes). It is not unheard of that results that had been considered correct are suddenly cast into doubt by a mathematician pointing out a flaw in the argument; sometimes this damns the entire original enterprise, sometimes it merely requires a "fix".

And mathematics has a very systematic way of validating its claims, again through the use of proofs; false claims may be identified by presenting counterexamples or pointing out gaps in proofs. This is done in a systematic and commonly accepted way.

It should then be clear that mathematics does meet the requirements; its particular interepretation of the scientific method, its particular thresholds, may be quantitatively different from other sciences, but they are qualitatively the same. It plays, moreover, a singular role among sciences, being an indispensable tool to so many other facets of science.



Dr. Magidin is a professor of mathematics who teaches at the University of Montana.

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Daniel R. Page said...

Very good read. I definitely agree. To get around a point in the difficulty of dealing with natural sciences in comparison. Usually science is split between two main groups, empirical science and formal science. Empirical sciences are your Biology, chemistry, physics, and so on that also could be called the natural sciences. Formal sciences on the other hand follow the same procedure but, are not in particular aiming the natural world for example, Mathematics, Computer Science, and Statistics. Formal sciences tend to give the models and applications which empirical sciences apply to attack the natural world. Formal sciences are very important especially in these days in such technology.

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