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The following quote from Malcolm Turnbull is actually not what this question is about, but nevertheless it is a good-fit epigraph:

The laws of mathematics are very commendable, but the only law that applies in Australia is the law of Australia.


Though the scenario I give in this question is exaggerated, hypothetical, edge-case one, it is technically valid and showcases the essence of the question.

A recent question/answer seems to be telling us that it is perfectly fine for a judge to reject an argument even implicitly — i.e. without actually discussing it. Such a judgment might only be vulnerable to appeal, but in no way to judicial discipline / misconduct investigation. If not successfully appealed, it becomes case law — binding on all inferior courts. Where rights of appeal do not exist (e.g. the court is the top court in its jurisdiction), the judgment becomes set-in-stone law — until cancelled either by the same court or via legislation.

So, let's consider the same example as in that question, but more specific one:

Bob contends:

I am right and Rob is wrong because blablabla... 2×2=4 ...blablabla.

Rob defends:

No Bob is wrong, because blablabla... and 2×2 actually equals 5.

Bob replies:

But look, lets see what 2×2 equals [math proof follows]. It is 4, you see.

The judge says:

I accept Rob's contention that Bob is wrong because 2×2=5. Bob has no case.

So, the judge implicitly rejects the math proof that 2×2=4 (as their Honour "does not have to address every argument raised in their judgement") and, effectively, makes it case law that 2×2=5.

Is this scenario technically/legally possible? Would the case law that 2×2=5 stand for some time? Will the judge not face any disciplinary consequences but just public outcry and reputation damages?

I appreciate that one might want to say that no judge would accept such an extremely outrageous merit-lacking argument as "2×2=5" and I totally agree (with hope). But the way this question applies to reality is that there is no clear boundary between "2×2=5" and any mundane argument that Rob might argue — as far as its acceptance by a judge is concerned.

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  • 1
    One could imagine a somewhat realistic example of this as follows: A bridge crashed and the court needs to decide whether this is the fault of the engineer Bob who designed a faulty bridge. Bob presents and explains his design which leads to the result that the bridge is safe. There is a mathematical error in Bobs explanation but to see and understand it requires technical or mathematical expertise that the judge doesn't have. The judge just sees experts that disagree with each other.
    – quarague
    Nov 10, 2023 at 10:06
  • @quarague perhaps, but where is the line between simple common-knowledge mathematical errors such as 2x2=5 and more complex errors such as rounding errors or more advanced errors such as a mistake in differentiation? Does it depend solely on whether the judge has the capacity to understand, or does the threshold lie somewhere else?
    – phoog
    Nov 11, 2023 at 9:41
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    @phoog The threshold for judicial notice is, practically speaking, in the vicinity of Pre-algebra to pre-calculus.
    – ohwilleke
    Nov 11, 2023 at 16:22
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    2.236x2.236=5 - round to 0 digits and you get the math correct: if the formula presented was "squarer root of 5 squared" and the record reads "roughly 2 times itself is 5" that would actually be a proper representation of the piece of evidence
    – Trish
    Nov 11, 2023 at 16:59

3 Answers 3

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Appellate judges make holdings on matters of law, and generally defer to the fact-finder in a given case (the jury, or sometimes the judge) on factual matters relevant to a case. So in a case that involved certain mathematical arguments, they would generally leave it to the jury to decide whether those arguments were reliable. Put simply, Appeals courts don't make binding decisions on issues of fact, only issues of law.

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    Either this answer is wrong, or math calculations are question of law: stuff.co.nz/national/crime/121448383/…
    – Greendrake
    May 8, 2020 at 1:19
  • 4
    @Greendrake Appelate Courts can use/make factual determinations in determining how to rule on a particular case, but those determinations do not become precedent.
    – David Reed
    May 8, 2020 at 1:56
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Will the judge not face any disciplinary consequences but just public outcry and reputation damages?

The judge will be absolutely immune from civil liability for this decision. A dissatisfied litigant can appeal the decision. Disciplinary action or criminal consequences would be possible only if the judge clearly made the decision in bad faith knowing that it was incorrect, which is not the scenario suggested by the question.

Is this scenario technically/legally possible?

Trial judges make decisions about facts that reach impossible conclusions now and then. They are fallible.

Would the case law that 2×2=5 stand for some time?

Trial courts don't make case law. Case law that binds future litigants in cases with different parties is created when someone appeals a decision, and a published decision on the merits of the question presented is resolved by that appellate court and is not reversed by a higher appellate court or overturned by a later statute.

Appellate courts in common law legal systems can reverse the holdings of a trial court for two main reasons (although there are other less common grounds for doing so).

  1. There is no evidence in the record to support a conclusion of fact made by the trial court judge, or a conclusion of fact that a trial court jury implicitly had to make to reach its conclusion; or

  2. The trial court applied the wrong law to the facts of the case.

A trial court ruling that 2x2=5 would probably either be deemed to be unsupported by the trial court evidence and any reasonable inferences from it (unless some properly qualified expert witness at trial testified that 2x2=5 which is vanishingly unlikely, in part, because it probably isn't a valid subject for expert testimony), or, that simple mathematical calculations are questions of law which can be reviewed de novo without deference to evidence before the trial court.

An appellate court would only reach the merits of the issue presented so it could make that ruling in a way that created case law if the issue was preserved by the appealing party in the trial court giving the trial court an opportunity to correct its error (although this mistake might fall within the "plain error" exception to that requirement), and the issue was raised in the appealing party's opening appellate brief, and the error had a material effect on the ultimate verdict in the case rather than being harmless.

Of course, appellate courts can make mistakes too, although the risk of that kind of mistake is reduced because appellate courts hear cases in panels of three or more judges in most cases, reducing this risk.

It is basically unthinkable that an appellate court would make a binding ruling on appeal that 2x2=5 which would constitute binding precedent in a future case without very good reasons that would suggest that there is more to the story than there appears to be on its face (e.g. the holding is limited to calculations in some highly unconventional number system, or related to a motion to set aside a ruling made in arbitration that the courts have no authority to second guess even if they are clearly wrong, in which the rule holding would not be that 2x2=5 but that arbitration rulings can't be disturbed by courts).

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  • "a binding ruling on appeal that 2x2=5 which would constitute binding precedent in a future case": suppose a court did make such a ruling; wouldn't it in any event apply only to cases with similar facts? For example, if such precedent were somehow set in a case where the calculation was to determine the area of a square surface with sides of 2 units, that wouldn't then imply that the relation 2x2=5 would also apply when multiplying kilowatts and hours to determine energy consumption, or speed and time to determine distance, or unit price and quantity to determine cost, etc. Is that correct?
    – phoog
    Nov 11, 2023 at 9:36
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    @phoog As a practical matter it would also certainly be distinguishable in some way, but the scenario is so improbable without more facts that it is probably a better example of why courts don't issue advisory opinions so it doesn't have to consider issue that don't actually come up in reality.
    – ohwilleke
    Nov 11, 2023 at 9:39
  • Improbable indeed. It further seems to me that less improbable scenarios (such as in quarague's comment on the question) are necessarily going to involve questions where the mathematical error is less obvious to people with average mathematical training, whereas that obviousness seems critical to the question being asked. While "it would never happen" is an unsatisfying answer to questions such as this, it seems like it's really the best answer.
    – phoog
    Nov 11, 2023 at 9:46
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    I suggest you either tag this with "united-states" or remove "Trial courts don't make case law" as per this question.
    – Greendrake
    Nov 11, 2023 at 16:36
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Courts do not rule on either physical reality or basic math. If a judge/jury were to find that 2 apples + 2 apples = 5 apples, that would not allow you to put 2 apples on one end of a table, 2 apples on the other, slide them together and magically have 5 apples.

Courts rule on civil and criminal cases, you can’t bring in your math homework and have the court rule that answering 5 to a question where the formula is 2+2=_ is correct or not.

But because they do rule on civil and criminal cases, they can find that in this or that circumstances, 5 is the answer to the question that is before them.

If a defendant in a murder trial testifies (and shows other evidence) that he arrived at 2 and left 2 hours later, and so was not present when the murder happened, the jury could still find that he committed the murder at 5.

In a civil trial, the jury could find that someone invested 2 million, was promised a 2 million return and award them 5 million.

And to give a different example, a court could easily find that a dozen is 13 loaves of bread.

As your quote says, what matters in court is the law, both courts and the law have their own terminology and their own rules, and as with the Bakers dozen examples, so do other areas of society.

Here’s a simple example of where a math based claim might be rejected: An employee is accused of stealing 100 dollars, the defense argues that all agree that at the start of the work day, there was 200 in the till, the business made 200 in cash sales, and that at the end of the employees shift there was 400 in the till. 200+200=400 equals no crime was committed.

Except the employee took out 100 in cash halfway through the shift, went out back and played craps, quitting when up a hundred. Returned half of it to the till, but was caught and arrested for theft. Can the court find the employee guilty of stealing a hundred? You betcha.

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  • The question is not what courts do; it is whether then can do the specified thing. Whether one would be able to magically produce 5 items by combining two pairs as per such a court ruling is irrelevant. What's relevant is that the lower courts would have to obey the ruling and decide bonkers.
    – Greendrake
    Nov 12, 2023 at 5:20
  • As I said, they don’t (and can’t) rule on basic math, what they can rule on applies to specific circumstances. If those circumstances arose again, similar results should be expected.
    – jmoreno
    Nov 12, 2023 at 21:15

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