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I teach a Theory of Knowledge course in the I.B. curriculum when I'm not teaching Algebra. I presented this thought experiment to a class: If God, or some supernatural entity, could tell you the exact weight of your school mathematics textbook, would the resulting value be rational or irrational?

Sifting through the written responses, one kid had a pretty sharp response:

"The weight of a book is most definitely rational because it is composed of a specific and finite number of atoms so its mass is non-moving. Those atoms are not leaving the textbook so it is staying a specific weight no matter what."

While I agree with the specific/finite language and can set aside concerns about the image of atoms not "moving," I directly question whether the weight of individual atoms can be treated as uniform, rational units. Yes, there are integer combinations of subatomic particles defining the weights but does not each atom (or subatomic particle for that matter) inspire its own rational/irrational debate?

Anyway, this is where the discussion moves out of mathematics and into quantum physics where I tread with caution. So I throw it out to the crowd...Are atomic weights best viewed as rational values determined by integer combinations of mass/energy or unique irrational values which we statistically approximate for utility sake?

[Question inspired by Tobias Dantzig's outstanding "Number: the Language of Science"]

Samuel Williams
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  • What units are we measuring in? – florence Jun 14 '17 at 16:13
  • That kinda depends on the units used, no? You could just be in units where a quark has weight $1$. So your question is really meaningful. – Bobson Dugnutt Jun 14 '17 at 16:13
  • @Lovsovs That's not accurate. First of all, there are multiple kinds of quarks with different masses, and those masses need not obviously be rational multiples of each other; and this also leaves out electrons, which do have mass, and again that mass need not obviously be a rational multiple of any of the other masses. The most direct way to trivialize the question would be to measure weight in [thatbook]s. – Noah Schweber Jun 14 '17 at 18:34
  • One thing I was trying to get at with the thought experiment was the Kronecker quote "God created the Integers, everything else is the work of man." Kids intuitively grasp that no two snowflakes (or textbooks) are truly alike and that we approximate at some point whenever we use rational numbers to measure. The student comeback stymies me, though, because I know the danger of applying human scale intuition to quantum scale phenomena. Assuming quarks as a basic unit and factoring in @Noah's caveat about different quark types, is it safe to assume that no two quark masses can ever be the same? – Samuel Williams Jun 15 '17 at 20:27
  • Sorry if this seemed off-topic. I'm looking at the responses a year later and they seem quite thoughtful. Thanks for taking the query seriously! – Samuel Williams Aug 02 '18 at 14:59

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