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Apples

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Title text: The experimental math department's budget is under scrutiny for how much they've been spending on trains leaving Chicago at 9:00pm traveling at 45 mph.

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Explanation

In the comic, a group of three "experimental mathematicians" has experimentally confirmed the answer to a math story problem that might normally appear in elementary school: "If Cueball has seven apples and Hairbun has five, how many apples are there?" Cueball counts the two groups of apples and states that the total is twelve. Blondie agrees that this is noteworthy.

Most people with a basic level of math would represent this as 7 + 5 = 12 and be confident of the answer without needing to count groups of physical objects. However, the title text states that there is an entire experimental math department dedicated to testing out common story problems in the real world, as if there was some doubt that the theories were sound.

It may also be an allusion to the most basic step of human mathematics, that of realising that seven of any conceived item plus five more of it will be twelve such items in total, and that numbers alone can therefore represent items without there being actual items to prove their own totals. Early accounting methods initially used proxy representations of the items, in a form of hybrid literal/symbolic manner, which meant that the combining of numbers of apples and combining numbers of livestock could be considered almost as different concepts, even though they had the same total sum applied only to different products.

This particular kind of abstraction sometimes fails in the real world when combining different things. For example, when measured volumes of two different substances are combined to make a solution, the volume of that solution is usually not exactly equal to the sum of the volumes of the original substances. Even in the case of combining equal volumes of nearly-freezing and nearly-boiling water, the result will not exactly equal the sum of the two volumes, since the density-vs.-temperature curve of water isn't a straight line (and the specific heat capacity of water varies with temperature).

It is possible that this Experimental Mathematics department has been working on this particular level of problem, as part of a mostly pre-mathematical culture. They are just now checking that 7 apples plus 5 apples equals 12 apples, after perhaps extrapolating from the recently confirmed fact that (e.g.) 7 sheep plus 5 sheep equals 12 sheep. Their theory that this extends to apples (and any other items they have tested before this point) has so far not managed to support the null hypothesis in which it might not.

Many branches of science have a known division between the empirical approach (gathering direct evidence or practically demonstrating that something works) and the theoretical (developing abstract models that fit the available information without fully testing them). High-quality experiments tend to be difficult and expensive, so rigorous testing is normally reserved for problems that someone considers sufficiently important or interesting. Math often deals with numbers and situations that cannot be reliably reproduced. The department's focus on confirming what most people already know may face difficulties when applying for grant funding. In reality, experimental mathematics is the branch of mathematics which uses computation as opposed to "pure" deductive proof methods. This does not involve "verifying" simple arithmetic, but could encompass e.g. calculating long runs of the digits of pi in search of patterns that may not be 'obvious' from known principles but which could be proven once identified as a candidate for proof.

On top of the simple problem that requires simple addition (and possibly subtraction) to fully understand the answer of, the title text goes on to cover a slightly more complicated schoolroom mathematical problem, one which generally requires at least some understanding of multiplication and division (though more advanced problems of this type might require moving into the realms of algebra, and the nature of simultaneous equations in particular). These may take the analogous form of a train (or other vehicle) setting off at a given time and constant speed along a given hypothetical route, and comparing that against other trips made to/from the same location. As with the hyper-practical experimentations with the number of apples, these more advanced queries are being investigated by directly examining the real-world incarnations of the terms of the problem. It seems that enough identical repetitions have been attempted, at least of a particular Chicago-departing rail service, to have worried those who oversee the financial accounts. (Presumably the accountants at least know enough about numbers to know that the acceptable number of purchased train tickets plus yet more purchased train tickets is adding up to more train tickets purchased than the accountants can consider to be justified.)

As every regular train calling at Chicago Union Station either originates or terminates there, a train has to accelerate first before reaching 45 mph. To leave the station at this speed at 9:00pm, the department has to rent a train using one of only two through tracks, and resolve possible conflicts with other scheduled trains.

A flaw in the system is that with irrational numbers and infinitesimals. Those cannot be represented with physical objects easily and will probably need very precise things or are just impossible.

Transcript

[Hairbun and Cueball stand at the left of the panel. Blondie stands at the right. Between them are two piles of apples, one of seven apples (stacked four on the bottom, two in the middle row, and one on top) and the other of five apples (stacked three on the bottom, and two on top).]

Cueball: Okay, with my seven apples added to your five, we have ... let's see ... twelve apples!

Blondie: Incredible!

Blondie: Perfect agreement with the theory!

[Caption below the panel:]

Experimental mathematicians

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