Department of Mathematics

Trent University

Peterborough, Ontario

Canada K9J 7B8

e-mail:

`sbilaniuk@trentu.ca`

**Abstract:** Mathematics is not a science, but there are grey areas at the fringes.

1991 *Mathematics Subject Classification*. 01

*Key words and phrases*. mathematics, science

Thanks to the members of the Champlain College Senior Common Room for the conversations which provoked this essay.

This essay is also available in LaTeX and PostScript formats.

Mathematics is certainly a science in the broad sense of "systematic and formulated knowledge", but most people use "science" to refer only to the natural sciences. Since mathematics provides the language in which the natural sciences aspire to describe and analyse the universe, there is a natural link between mathematics and the natural sciences. Indeed schools, universities, and government agencies usually lump them together. (1) On the other hand, most mathematicians do not consider themselves to be scientists and *vice versa*. So is mathematics a natural science? (2) The natural sciences investigate the physical universe but mathematics does not, so mathematics is not really a natural science. This leaves open the subtler question of whether mathematics is essentially similar in method to the natural sciences in spite of the difference in subject matter. I do not think it is.

A disclaimer is in order. This essay is a "native informant's" opinion: I am a practicing (if mediocre) mathematician, but not a philosopher or student of the practice of science or mathematics. I have a relevant philosophical bias, in that I am a Platonist where mathematical reality is concerned. (3)

The object of the natural sciences is to devise and refine approximate descriptions or models of aspects of the physical universe. The feature distinguishing science from other means of doing so is its characteristic method. Crudely, this consists of asking a question, formulating a hypothesis, testing it, and then, on the basis of the results, rejecting or provisionally accepting the hypothesis. One usually repeats the process after refining the question, the hypothesis, or one's ability to test it. The ultimate arbiter of correctness is the available empirical evidence: a hypothesis which is falsified -- *i.e.* inconsistent with good data -- is not acceptable. (A hypothesis which could not be falsified by any empirical data is not scientific.) Note that a scientific theory or hypothesis is (at best) only provisionally acceptable at any given time, because a new piece of evidence may force it to be modified or rejected outright.

In mathematics, however, the ultimate arbiter of correctness is proof rather than empirical evidence. This reflects a fundamental diffence in what one is trying to achieve: mathematics is concerned with finding certain kinds of *necessary* truths. For a mathematical statement to be accepted as a theorem, its conclusion must be known to *always* be true whenever its hypotheses are satisfied. We accept it only when we have a proof: a chain of reasoning demonstrating that the conclusion must follow from the hypotheses. (4) Empirical evidence does, to be sure, play an important part in doing mathematics. Conjectures are usually formed by observing a common pattern in a number of examples, and are often tested on other examples before a proof is attempted. However, such evidence is not sufficient by itself: consider the assertion that every even integer greater than 4 is the sum of two (not necessarily different) odd prime numbers. (5) We have lots of empirical evidence supporting this assertion: 6 = 3+3, 8 = 5+3, 10 = 7+3 and 10 = 5+5, 12 = 7+5, and so on. However, we cannot be *sure* it is true unless someone finds a proof. Until then, it is conceivable that someone might find a very big even number which is not the sum of two odd prime numbers. (6)

The essential difference in method between mathematics and science, and the weakness of each, is neatly exploited in the following joke:

Some academics relaxing in a common room are asked whether all odd numbers greater than one are prime.The mathematician is wrong by the standards of her field because a valid proof of the assertion that all odd numbers are prime has not been given. (8) (Leaving the hard -- or impossible! -- parts to the reader is a bad habit regrettably widespread in mathematics. (2)) In view of the available evidence, on the other hand, the physicist is quite correct by the standards of his field to accept the assertion. (9)The physicist proceeds to experiment -- 3 is prime, 5 is prime, 7 is prime, 9 doesn't seem to be prime, but that might be an experimental error, 11 is prime, 13 is prime -- and concludes that the experimental evidence tends to support the hypothesis that all odd numbers are prime.

The engineer, not to be outdone by a physicist, also proceeds by experiment -- 3 is prime, 5 is prime, 7 is prime, 9 is prime, 11 is prime, 13 is prime, 15 is prime -- and concludes that all odd numbers must be prime.

The statistician checks a randomly chosen sample of odd numbers -- 17 is prime, 29 is prime, 41 is prime, 101 is prime, 269 is prime -- and concludes that it is probably true that all odd numbers are prime.

The physicist observes that other experiments have confirmed his conclusion, but the mathematician sneers at "mere examples" and posts the following:

3 is prime. By an easy argument which is left to the reader, it follows that all odd numbers greater than one are prime.(7)

It must be admitted that the difference noted above between science and mathematics is not completely sharp, even aside from the fact that the practice of mathematics does have empirical content. Some of the areas in which mathematics is applied to modelling aspects of the physical universe are very grey indeed. The basic problem is that one can be confident of a fact derived by mathematical methods only to the extent that the mathematical object being considered is an accurate model of the relevant parts of the universe. One can be completely confident this is so in mathematics (where the mathematical object in question *is* the relevant part of the universe) and quite confident in, for example, computer science (where the physical objects being analysed are made to conform to a mathematically precise pattern) and parts of theoretical physics (where some theories have survived very extensive testing). However, one cannot usually be very confident in, say, long-term economic projections. The moral is that in applying mathematics to problems from the "real" world, one must judiciously temper the use of mathematical knowledge and techniques with empirical knowledge and testing.

With increasing interaction between mathematics and the natural sciences, plus the practical problems involved in finding and checking really long proofs, it is arguable that the grey areas are expanding. It has even been argued that proof and certainty in mathematics are nearly obsolete [4], though most of those who agree that "empirical" mathematics has a place still believe that proofs have an important role (*e.g.* [2] and [7]). It is my belief that proofs will remain central for a good while yet.

(1) Which is convenient for mathematicians when grant money is distributed, so

(2) The problem of showing that mathematics is not a social science is left as an exercise for the reader. One could argue that mathematics ought to be classified with the arts and humanities [3], but it doesn't function like one [6]. There is also the argument that mathematics is "not really accessible enough to be an art and not immediately useful enough to be a science" [1], but this assumes that art is accessible and science is useful.

(3) As for non-mathematical reality, who cares?

(4) Of course, this begs the question of just what constitutes such a chain of reasoning. Philosophers really worry about this, but most mathematicians settle for giving arguments acceptable to most other mathematicians. History suggests that it is a mistake to be too rigid about correctness in mathematics: it took over two centuries, for example, to work out rigorous foundations for calculus.

(5) This assertion is called Goldbach's Conjecture. A prime number is an integer greater than one which is not a product of two smaller positive integers.

(6) If you do either, please publish!

(7) What of the others present in the common room?

The chemist [5] observes that the periodic table gives the answer: 3 is lithium, 5 is boron, 7 is nitrogen, 9 is fluorine, 11 is sodium, ... Since elements are indivisible -- nuclear fission being uncommon in chemistry labs --- these are all prime. (The same is true for even numbers!).

The economist notes that 3 is prime, 5 is prime, 7 is prime, but 9 isn't prime, and exclaims, "Look! The prime rate is dropping!"

The computer scientist goes off to write a program to check all the odd numbers. Its output reads: `3 is prime. 3 is prime. 3 is prime. ...`

The sociologist argues that one shouldn't refer to numbers as odd because they might be offended or as prime because the term implies favouritism, and the theologian concurs since all numbers must be equal before God.

(8) If you're still wondering whether it's true, you haven't paid careful attention. (7)

(9) Until subsequent investigation confirms that 9 = 3*3, anyway.

- Robert Ainsley,
*Bluff Your Way In Math*, Centennial Press, Lincoln, Nebraska, 1990. - Keith Devlin,
*The Death of Proof?*, Notices of the American Mathematical Society**40**(1993), p. 1352. - JoAnne S. Growney,
*Are Mathematics and Poetry Fundamentally Similar?*, American Mathematical Monthly**99**(1992), p. 131. - John Horgan,
*The Death of Proof*, Scientific American**269**(1993), pp. 92-103. - Hans H. Limbach, personal communication (1994).
- Kenneth O. May and Poul Anderson,
*An Interesting Isomorphism*, American Mathematical Monthly**70**(1963), pp. 319-322. - Doron Zeilberger,
*Theorems for a Price: Tomorrow's Semi-Rigorous Mathematical Culture*, Notices of the American Mathematical Society**40**(1993), pp. 978-981.

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