Topics Mathematics

Question To Whom it May Concern: Mathematical results are assumed to be precise. But how can mathematics be precise if results are rounded up or down? Don't such small incremental "roundings" add up to imprecision? So, in general, don't "roundings", in some way, betray the advertised precision of mathematics? Sincerely, Alexander

Accepted:
February 1, 2006

Comments

'Mathematics' covers a lot of ground: from pure geometry, for example, which is not quantitative at all -- that is to say, is not concerned with numbers and calculations -- to statistics which is not only quantitative but must do some version of the 'rounding' you speak of.

Within statistics, however, part of the science (and in fact perhaps the most important part of the science!) is its ability to describe the imprecision of its results. For example, with the notion of a 'confidence interval'. This has two implications, it seems to me, one practical and one theoretical.

First, the way in which statistical results are distributed through non-scientific forms of publication (reported in Newspapers, for example) often leaves out this analysis of precision, and every result is reported with equal confidence. This is not precisely a philosophical problem, to be sure, but certainly has implications for the ethics of the media, and for how scientific research is translated into political decision-making.

Second, it might be argued that statistics (and other quantitative forms of mathematical inquiry) are indeed precise within their own sphere. But, that our ability to model the world mathematically, and in return apply the results of mathematical analysis to the world, is intrinsically imprecise.

You're right, mathematical results will not be precise if they are rounded off -- which is why mathematicians usually don't round off their results. I think such rounding is much more common among people who are applying mathematics to real world problems than among mathematicians doing theoretical work. For example, consider the question: What is the height of an equilateral triangle whose sides have length 1? A mathematician would most likely say that the answer is sqrt(3)/2, which is exactly correct. But someone who needed the answer to this question in order to apply it to a real world situation might prefer to have a decimal value, and so they might round it off to 0.87.