### Is Mathematics an Exact Science?

This is a hard question to answer because there are many views on what mathematics is and whether it is a science at all, let alone an exact one. And, of course, there is a question of what constitutes an exact science. To compound it all, not only it is possible to question the meaning of the word "is", but it is also a fact that the word "is" has many meanings.

It's all of course a philosophical matter which means that any one who struggles to answer that question may expect disagreements and objections. The entire business of airing the question may simply prove a fruitless philosophical exercise. Steven Weinberg, a Nobel laureate in physics, dedicated two chapters in his book *Dreams of a Final Theory*, to subjects beyond physics: mathematics and philosophy. He writes that time and again he is surprised to discover how useful mathematics is and how futile is philosophy. (Quoted from *Arithmetic for Parents*.)

So why to even try answering that question? The reason is that it is being often asked by students so that the teachers (and parents) need to be prepared to to handle the inquiries. Cecil Read tells of her personal experience:

Probably a teacher learns as much from his students as the student learns from his teacher. Some years ago when I was attempting to introduce the concept of negative and fractional exponents to a class in college algebra I had a student bring the text which he had studied in high school. He pointed out the statement: "The student will see that it is impossible for an exponent to be anything but a positive integer." I could find no qualifying statement. The student asked, "In the face of this contradiction how do you justify your statement that mathematics is an exact science?"

In mathematics this is not unusual to have terminological discrepancies, especially regarding the terms borrowed from the common language. After several similar examples, Cecil Read concludes her article:

Yes, mathematics is an exact science, but mathematicians use inexact terminology.

This is tricky. Yes, mathematician often use inexact terminology but this does not make mathematics either exact or inexact science. But what does? This may be a harder question than the one in the caption. Ms Read found a way to avoid the philosophical trap by listing several instances where the exactitude of mathematics had been put in question but which she was able to dispel by a reasonably sensible argument. She managed to refute several arguments that were supposed to chisel away the notion that mathematics is exact science. I'll try to follow in her footsteps while staying away from the prevarications of philosophy.

A recent post at the Mathematics and Multimedia blog raised exactly this question, Is mathematics an exact science? The author showed several reasons why any one could doubt the affirmative answer. Let's check them one by one.

- Mathematics is often made to deal with inexact quantities. 3.14 is only approximation to π. There are better approximations than that, but none of them is exactly π. (However, note that π is an exact quantity.) In the same vein, any cutoff of a repeating decimal is only an approximation to the exact value of a rational number. (However, note that the rational number itself is an exact quantity.) What is important here is that mathematics makes a distinction between a number and its approximation and deals with either depending on the circumstances - and this without confusion. 22/7 is close to π but not quite it. In mathematics, we know exactly what is what.
- In Proposition I.1 of his
*Elements*, the great Euclid showed more than two millennia ago how to construct an equilateral triangle based on a short list of axioms. Euclid's method is still in use. However, some 150 years ago, it was realized that Euclid's axioms were lacking and that some of his proofs relied on informal intuition as much as on the list of axioms. Mathematicians patched the axioms up. Now we know exactly which intuitive notion has been overlooked in Euclid's*Elements*. - The most basic notion of Calculus is the limit. 1/n→0 as n→∞. 1/n tends to 0 without ever reaching 0. We know exactly that this is how it is. No worry here. In the nonstandard analysis, we even can plug into 1/n an infinitely large number and get an infinitely small one. We know exactly how to do that.
- Mathematicians often recourse to the proofs of pure existence. Georg Cantor proved that the transcendental numbers are prevalent among the reals, albeit, even today, only a very small amount of numbers are known to be transcendental. Isn't it rather wonderful? But I find it even more wonderful that the result had surprised Cantor himself. He did not expect that. However, what's proven proven, and we now know without any doubt that the transcendental numbers are prevalent among the reals and we know exactly what that means.
- Mathematicians use axioms not only to organize mathematics but also to avoid contradictions. True enough. Now we can finally feel safe.

This is hard, if not impossible, to exactly define an exact science and determine without a shadow of doubt whether mathematics is exactly one such science. Nonetheless, it may be even harder to come up with an argument that would shatter the common belief that mathematics is exact.

### References

- Cecil B. Read,
__Is Mathematics an Exact Science?__,*National Mathematics Magazine*, Vol. 17, No. 4 (Jan., 1943), pp. 174-176 - Ron Aharoni,
*Arithmetic for Parents. A Book for Grownups about Children's Mathematics* - Steven Weinberg,
*Dreams of a Final Theory*

Numbers are the Supreme Court of science. However Godel proved that we may not prove everything. There are Physics Foibles!!

October 23rd, 2011 at 1:18 pmNo, it's not what Gödel proved. As a matter of fact, depending on the selection of axioms practically anything can be proved. Gödel showed that for in any axiomatic system there are true but unprovable statements.

October 23rd, 2011 at 8:21 pm