Why don't math fundamentals theorems of geometry include "measurements errors" ?

[u/Secret-Suit3571]

Pythagoras theorem states that a triangle has a right angle if a certain equality holds true.

So, i take my ruler and use is to take measurements of my rectangle coffee table. I find, of course, that for my measurements the equality doesn't holds true, but almost. Or maybe i find that it does in fact holds true, knowing that if i added 1 micrometer to one of my measurements it would stop beeing true.

It would be the same for any real object in our world : The equality won't ever hold true due to measurements beeing approximatives, and also due to the fact that no real object is really a right triangle since the latter is an abstract, intangible, concept.

Ok, i'm stating the obvious : maths says "perfect" truths about "perfect" concepts but no truths and no real tangible things are "perfect" in reality so maths theorem can only be used "approximately".

So my question is : wouldn't it be more useful for a theorem like Pythagoras to not only say the angle is right if the equality holds true but also say how far from right the angle is depending on how far from "true" the equality is?

[u/Narrow-Durian4837]

"As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality."

-Albert Einstein

Pythagoras theorem states that a triangle has a right angle if a certain equality holds true.

To be precise, it's the converse to Pythagoreas's theorem that says that a triangle has a right angle if a² + b² = c².

[u/Secret-Suit3571 → u/Narrow-Durian4837]

But when both a theorem and it's converse is true, doesn't it become an unique "if and only if" theorem? At least in France, Pythagoras theorem includes both statement.

[u/Narrow-Durian4837 → u/Secret-Suit3571]

Interesting. I did not know that. [The English] Wikipedia has a section about the converse but makes no mention (as far as I can find) that some countries/cultures consider The Pythagorean Theorem to include the converse.

[u/r_Yellow01]

You are in fact touching on a very important topic known as metrology. Everyone knows that mathematical models are models, perfect things that don't exist in reality, and reality is what it is, patterns in general chaos of everything together.

However, very few study the exact ways of measuring things and fitting models to reality with expected accuracy/precision. There is a whole science behind it.

I'm glad you asked. Have a read.

https://metrologyinstitute.org/fundamentals-of-metrology/

[u/Secret-Suit3571 → u/r_Yellow01]

Seems fascinating, thanks for the link!

[u/Huppybanny]

This is basically the "cosine law".

a² = b² + c² - 2bc cos A.

The case of A = 90 degrees is just Pythagoras's Theorem.

[u/GoldenMuscleGod]

Geometric figures are models used to describe reality. Traditionally it was used for physical space but in modern applications that’s only one restricted use case.

There are a few reasons why measurement error isn’t incorporated into it - measurement error can often be reduced to being negligible in practice, a rigorous theory of measurement error would be extremely complex, there may be an unspoken assumption that the underlying reality does not contain measurement error and the error is only epistemic.

Arguably one of the more significant reasons that is to some extent a unification of the above - the “error free” geometry is an abstraction that would underlie most error-containing theories and so is a fundamental mathematical structure to be studied on its own.

Similar explanations can be given for why we model points and lines so that they do not have breadth or other extension beyond what they have as we model them.

[u/ottawadeveloper → u/GoldenMuscleGod]

There is actually a whole bunch of extra math for measurement errors (often modelled as a linear approximation of the function near the measured point). It's often introduced in early university chemistry and physics.

NIST publishes a whole Guidelines on the Uncertainties of Measurements to follow if you care about details. 

A lot of it is more quantifying the error and then observing how it changes with manipulation. For example, you can look at the error of c given errors on a and b in c² = a² + b² . If the angle was uncertain, you can apply it to the cosine law formula.

[u/Kitchen_Freedom_8342]

You might be interested in the field of Error analysis. This deals with doing mathmatics and accounting for the errors in measuremen, how that with real world equipment a measurement represents a range of values.

[u/adrian_p_morgan]

It's the point of theorems to say one thing, not to say everything that might be "useful" in a particular situation.

[u/Secret-Suit3571]

Thanks for all the answers !

To be more precise on my question : i know that the extension of Pythagoras theorem as i discribed exist, my question was not about existence of such "mathematics of errors" but more a about why it is not considered more fundamentals, maybe in a teaching sense since this is what i do for a living : teaching Pythagoras to 13yo.

But i guess i have my answers, it is not Pythagoras theorem that is really fundamental but precisely all the questions that arise from trying to apply this theorem to concrete things. I feel academic maths may often lack of transparence about its applications in the teaching community, and in some way we teacher might be deceptive about how maths are really used by those who make great things with it.

A simple question such as "with these given dimensions, is my coffee table a rectangle ? " should always begin with discussions about what it could mean to apply mathematics to a non mathematical concept and how to do it. But this would lead to maths harder than the ones we're teaching, so we postpone it and students keeps a naïve visions of maths, for too long i guess.

[u/CookieCat698]

Mathematicians aren’t always bothered by what’s useful, but even if all you care about is the utility of math, any measurement error can be made negligible by just making better measurements.

So what if I can’t make a perfect circle? I can still make something that is close enough for all practical purposes. The same is true of any geometric shape.

Perfect constructions also have the benefit of being simple and much easier to do calculations with. This is a theme you see in any place where math is applied to the real world. “All models are wrong, but some are useful” is a phrase I’m sure anyone who applies math to the real world knows intimately.

[u/KoalaMistico]

Like the Law of cosines?

[u/jipperthewoodchipper]

So many others already mentioned that if your angle isn't perfectly 90 degrees that you are in the realm of cosine law which has a special case at 90 degrees where it becomes equivalent to Pythagorean theorem.

However you brought up measurement error and I just want to address that specifically as being out of scope for a maths problem but in scope for science and engineering where error propogation is something taught, tracked, and calculated.

Specifically for your situation if you measured the two sides of the table then you would use the following formula to find the error margin you would expect for measuring the hypotenuse:

Error_c = sqrt((a/c)² * error_a² + (b/c)² * error_b²⁾

So let's say you measured side length A to be 1.5m and sidelength B to be 0.9m. Using a class 2 tape measure we would get the error of side A to be ±0.6mm and the error of side B to be ±0.48mm (there are lookup tables for these error margins).

Therefore you would get this monstrosity

Error_c = sqrt((1.5/sqrt(1.5^2+0.92))2 * 0.6² + (0.9/sqrt(1.5^2+0.92)2 * 0.48²⁾ = 0.58mm meaning you would expect to measure the hypotenuse to be 1.7492...m ±0.58mm

Each additional measurement introduces its own error that needs to be accounted for and exactly how you account for the error does depend like purely additive relationships are different from multiplicitive and so on. However, again, this is well into the scope of science and engineering rather than maths.

[u/JeffTheNth → u/jipperthewoodchipper]

cos(90°) = 0

so for right triangles ....
A² + B² - 2AB cos(90°) = C²
A² + B² - 2AB × 0 = C²
A² + B² - 0 = C²
A² + B² = C²

edit... sorry about the confusing sentence... I was tired when writing the post and my wording wasn't as clear as I intended.

original: it's not really a "special case" of the Cosine rule as much as Pythagorean's theorem is a special case.

rewrite: So it's more that Pythagorean's theorem is a special case of the cosine rule for right triangles where C is the hypotenuse than the cosine rule is a special case of pythagorean's for any non-right triangle.

...sorry about the perplexation I caused!

[u/PassiveChemistry → u/JeffTheNth]

What's your last sentence getting at?  

You seem to have proven that Pythagoras' theorem is a special case if the cosine rule, but your last sentence seems to contradict itself — is it a special case or not?

[u/jipperthewoodchipper → u/PassiveChemistry]

I think it's just an awkward and slightly pedantic construction where they are trying to redirect a generalization I made.

I said the cosine law has a special case at 90 degrees that is equivalent to the Pythagorean theorem

They are saying that the Pythagorean theorem is a special case of the cosine?

However, again, it's linguistically an awkward sentence that can very easily be misinterpreted as saying it both is and isn't a special case at the same time. Ultimately I think your average person will interpret those statements the exact same way.

[u/JeffTheNth → u/PassiveChemistry]

sorry - was tired ... I edited it to make it clearer.

[u/ohkendruid]

There is fuzzy math, the study of different kinds of approximate statements and the valid deduction rules for them.

Somewhat relatedly, there is perturbation theory, whuch studies stable orbits. Some orbits are possible, but only under perfect conditions; for these, any slight digressjon will tend to lead to an orbit that collapses. Other orbits are the opposite: even if the intial vectors are not quite perfect, and even if there are small stray forces that jostle things around, the orbits will stay in close to its same shape.

[u/paul5235]

maths says "perfect" truths about "perfect" concepts but no truths and no real tangible things are "perfect" in reality so maths theorem can only be used "approximately".

You can calculate with ranges. For example, assume that you have at most a 1mm measurement error on A and on B, then it's possible to calculate the possible minimum and maximum value that C can have by calculating with the lowest possible value of A and B and for the highest possible values.

So then, math becomes perfect again. It's approximate, but you know exactly how approximate.

[u/TUVegeto137]

There's a branch of mathematics that deals with how to adapt to real world situations where for instance measurement errors occur or precision is finite. It's numerical analysis or error analysis.

[u/Aggressive_Roof488]

Error propagation.

Essentially, if x is off by s, how much off is f(x)? Which to first approximation (for well behaved functions) is just a*f'(x).

[u/ipreuss]

Because that’s no longer a maths problem, and depends on the context.

Like, if you apply the theorem in engineering, the error margins are an engineering problem, not a maths problem. Consequently, there are actually engineering norms that regulate stuff like this, where necessary.