# How would mathematical demonstratives be used in a court?

I remember in a children´s show when I was little that one episode featured a court case where the villain was accused of having too much land area in contravention of the laws of the town they were in. Part of the point of the story was to tell kids about the concept of surface area and its equivalence regardless of transformation, and because the land area was in a strange shape, to measure the map proving the area of the property, the map would be rearranged into a familiar shape where you can use simple length times width as the definition of area, and a court hearing was held where this was done.

In an actual court case, if you had to rely on mathematical concepts like how much volume a container has, how would you actually bring such evidence into court and prove it before the trier of fact?

• Tangentially related: youtube.com/watch?v=VYQVlVoWoPY Sep 19, 2023 at 4:01
• Here is an example where quite advanced mathematics (Markov chain Monte Carlo methods) has successfully been introduced in court: ams.org/journals/notices/202204/rnoti-p616.pdf Sep 19, 2023 at 6:16
• The Cyberchase scene in question: whyy.pbslearningmedia.org/resource/…
– MJD
Sep 19, 2023 at 13:58
• Also very tangentially related: en.wikipedia.org/wiki/Sally_Clark Sep 19, 2023 at 19:38
• An interesting related case was when Austria's presidential election had to be repeated a few years back because of a (very low) number of faulty mail election forms in a very close race. A prominent Austrian statistician then wrote an article on arXiv reasoning that it was statistically pretty much out of the question that the result was impacted. The court dismissed this concern arguing (basically) that his math was too complicated. Sep 20, 2023 at 11:49

For simple math, it is sufficient to present a demonstrative exhibit that uses data from already admitted evidence to show more or less step by step how something was calculated. And, courts can take judicial notice of pretty much any mathematical formula that could be found in an almanac or a middle school or elementary school textbook. Federal Rule of Evidence 201(b), which many states either copy directly or duplicate in a parallel rule of their own, states:

The court may judicially notice a fact that is not subject to reasonable dispute because it:

(1) is generally known within the trial court’s territorial jurisdiction; or

(2) can be accurately and readily determined from sources whose accuracy cannot reasonably be questioned.

Often this demonstrative would be entered into evidence through the testimony of a client or a client's employee.

For anything more complicated than that, you would typically endorse an expert witness, perhaps a mathematician or a surveyor or engineer or accountant, to present the evidence to the court.

For example, in a personal injury case I once had where someone was crushed by a falling heavy object, we hired an engineer to calculate the force of the object from its known height onto the person killed by it, even though it involved only first year physics and there are formulas in high school and university textbooks for that scenario. For what it is worth, hay bales are a lot heavier and inflict a lot more force when dropped from a height, than your intuition would suspect.

As a practical matter, a court is generally going to allow evidence of a compound interest calculation, or calculating an amount of tax due or an average. But a court generally wouldn't accept without expert testimony, anything that required calculus or anything but the most simple probability and statistics calculation. Calculating the angles in a triangle or trigonometry or other pre-calculus level math would be on a gray area and some judges might allow it, while others might not.

Part of the reason to use expert testimony is that some "obvious" mathematical question don't have the mathematically correct answer in the real world.

For example, the area of the base of a two by four plank of wood isn't actually eight square inches, because a modern "two by four" doesn't actually have dimensions of two inches by four inches.

Similarly, in commercial leasing, the rent per square foot is usually quoted in dollars per usable square foot per year, and not in dollars per exterior dimensions square feet per month, even though rent in a fixed dollar amount for a space is usually stated in dollars per month.

• "a modern "two by four" doesn't actually have dimensions of two inches by four inches": but this doesn't implicate mathematics so much as measurement. The expert witness who could establish that the cross sectional area of a two by four isn't a mathematician but a sawmill operator. (Also, two-by-fours were never 2" by 4" because the nominal dimension is -- or at least traditionally was -- the size of the piece before it is dried and planed.) Sep 18, 2023 at 21:21
• @phoog The point is that is you didn't get someone familiar with the fact involved and testifying a lot of the assumptions that a mathematician or lay person might naively make, and include uncritically in a calculation, are more likely to be mistaken. It is an easy mistake for someone who doesn't deal with it to make to assume that a two by four is two inches by four inches and produce the wrong answer. Because most "simple math" involves lots of simplifying assumptions that are unstated. Sep 18, 2023 at 22:20
• I'll note that when asking the courts to take judicial notice of a particular fact, it is quite convenient if that fact can be cited from one or more previous court cases. For example, in Dorman vs Dorman, a previous federal case was cited to show that Calibri was not available in 2000. Sep 20, 2023 at 18:37
• Could an attorney first ask opposing counsel to stipulate the result of a physics-related calculation before needing to ask the judge? I would think that in many situations, opposing counsel would want to move things along and not demand that everyone waste time demonstrating things which would be perceived as obvious. I can also imagine, however, that some jurors might want to receive information about how to perform some calculations themselves, and that it might be useful for attorneys to be able to stipulate that some particular math textbook is a good reference. Does that happen? Sep 20, 2023 at 21:35
• "Could an attorney first ask opposing counsel to stipulate the result of a physics-related calculation before needing to ask the judge?" This happens frequently in cases where the party who wants the evidence in has a credible way to admit it at trial if the opposing counsel fails to agree. " it might be useful for attorneys to be able to stipulate that some particular math textbook is a good reference. Does that happen?" This almost never happens. Sep 20, 2023 at 22:23

Where the calculation or methodology is complex, it may be properly the subject of expert evidence. Where the calculation is simple and methodology is indisputable, this may be the subject of judicial notice. See Yurchi v. Johnston, 2006 ABQB 25, at para 59 (internal citations removed):

While the Court is reluctant to interpret scientific data, the engineers in this case filed scale drawings. The scale drawings speak for themselves; they have distance bars on them and, in some cases, they have specific distances. Both engineers agreed on the mathematics of the case. Further the Supreme Court of Canada recently, by implication, reconfirmed that courts may take judicial notice of mathematically certain concepts, in Spence they described the taking of judicial notice of things capable of immediate and accurate demonstration by resort to readily accessible sources of indisputable accuracy.

As an example application, see R. v. Trevisan, 2009 ONCJ 34, at para 67:

By making simple mathematical calculations, I am able to take judicial notice of the fact that when a vehicle is moving at a rate of speed of 60 kilometres per hour, it travels a distance of 54.679997 [sic] feet in one second and that when a vehicle is moving at a rate of speed of 70 kilometres per hour, it travels a distance of 63.756878 [sic] feet in one second.1

For a second example, see Corlis v Blue Grass Sod Farms Ltd., 2016 ABPC 55, at para 30:

I take judicial notice of the fact that there is 43,560 square feet in one acre, meaning that there is 130,680 square feet in three acres.

1. Here, I incorporate feedback from the comments. The exact figures are 62500/1143 and 218750/3429 feet per second. Or, to the limits of IEEE 754 double-precision, 54.68066491688539 and 63.79410906969962. One user approximates these as 54.680665 and 63.794109. Another notes that staying within the two-digit precision from the original figures would result in answers of 55 and 64 (although this is a difference in precision not accuracy). Another user notes that the definition of the "foot" changed in 1959, but says this is not likely what led to the discrepancy. Someone else speculates that the discrepancy was due to early rounding.

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– feetwet
Sep 22, 2023 at 2:56

Necessary anecdote

When physicist Dmitri Krioukov received an undeserved fine for not marking a stop, he wrote and published a mathematical paper, The Proof of Innocence, in the scientific humourous newspaper Annals of Improbable Research, in order to demonstrate that the view of the police officer who had given him the fine had been obstructed long enough for him not to see Krioukov marking the stop. He later used the paper in court and got the fine dropped.

Here is how mathematician Jason Brown told the story in an article (pdf) :

There is a delightful story I read about a physicist, Dmitri Krioukov, who was caught recently by a cop for going through a stop sign. Facing a fine he felt he didn’t deserve, the physicist went to court to fight the ticket, not with a lawyer but with a math paper he wrote.

In the paper (entitled The Proof of Innocence) he presented to the judge, he argued that the police officer erred in three ways. First, the officer mistook the car’s speed with its angular speed — that is, how quickly the policeman’s viewing angle is changing. Furthermore, at the stop sign, the officer’s view was obstructed by a longer car. Finally, Krioukov stated that while the view was obstructed, he rapidly decelerated (due to a sneeze) and then rapidly accelerated.

With all the ensuing calculus, Krioukov proved that the officer might have interpolated the car’s apparent speed before and after the short time of obstruction and thought the car didn’t stop. As Krioukov later stated, his paper was awarded a \$400 prize, in that the ticket was thrown out of court.

However, the judge said:

The ruling was not based on his physics explanation ... It was based on the officer's view ... The officer wasn't close enough to the intersection to have a good view.

So, at least in California, you can refer in court to an article published in a scientific magazine and build your argument on its conclusion.

• The paper appears to have been published in the Annals of Improbable Research (AIR). Is that "peer-reviewed" in the usual sense of the term? AIR's description of itself reads, in part, "research that’s maybe good or bad, important or trivial, valuable or worthless". Based on that description I would suggest that papers published there should not be relied upon for any purpose, including evidence in legal proceedings. Sep 19, 2023 at 17:18
• @Nobody : you're right, it is not peer-reviewed... I edit that in. Sep 20, 2023 at 6:21