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My boss claims that distance in $\Bbb{R}^n$ between two vectors, $a=[a_1, a_2, \dots a_n]$ and $b=[b_1,b_2,\dots b_n]$ is defined as:

$$d = \sqrt{\sum (b_i-a_i)^2}$$

I claim that no, under a geometric treatment one only needs to define length in one dimension. From there, one can apply Pythagoras theorem to get derive the Euclidean distance in $\Bbb{R}^2$ and inductively from there to $\Bbb{R}^n$. My question is two-fold:

  1. Is it true that Euclidean distance in $\Bbb{R}^n$ must be defined and can't be derived once we define it in $\Bbb{R}^1$ and just apply Pythagoras theorem repeatedly (and the definition of $\Bbb{R}^n$ as the space spanned by orthogonal vectors)?
  2. In modern geometry, how is distance/ length in $\Bbb{R}^1$ defined? I understand everything is built on Euclid's five postulates. The first postulate talks about being able to draw a line segment between two points. But I see nothing defining length in $\Bbb{R}^1$.

Also, is there a textbook on Geometry that covers these concepts?

Rohit Pandey
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    Where do you think that the definition at the top comes from, if not from iterative application of Pythagoras? – Xander Henderson Oct 17 '23 at 23:31
  • Exactly my point. I think it comes from iterative application of Pythagoras. My boss says "why don't you agree that the definition of length in Rn is the square root of the dot product of the vector with itself and instead you try to "prove it" ?" How can I respond to this? – Rohit Pandey Oct 17 '23 at 23:34
  • I mean, it depends on where you start. In my world, I tend to start with some set, then define whatever metric I like on that set. If my underlying set is $\mathbb{R}^n$, then the "usual" Euclidean distance is one of many metrics I might use. But I don't really think of it (or even care about it) with respect to Euclidean geometry or its postulates. – Xander Henderson Oct 17 '23 at 23:37
  • Articles like this: https://math.libretexts.org/Courses/Lake_Tahoe_Community_College/A_First_Course_in_Linear_Algebra_(Kuttler)/04%3A_R/4.04%3A_Length_of_a_Vector "define" length in $\Bbb{R}^n$ and this reinforces the notion that it must be defined. – Rohit Pandey Oct 17 '23 at 23:38
  • I see. My take though is treating Euclidean distance in $\Bbb{R}^n$ as a definition isn't appropriate given it can be derived from the concept of distance in $\Bbb{R}^1$ and Occam's razor. – Rohit Pandey Oct 17 '23 at 23:40
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    In "modern" mathematics, we have a multitude of methods by which we can define distance so long as they hold with the basic axioms of the distance metric. Then we define $u\cdot u = |u|^2$ as a distance metric and then show that this is indeed a metric and that it corresponds with the Euclidean concept of distance and name it the Euclidean metric. So, your boss is right. – user317176 Oct 17 '23 at 23:41
  • But why would we define it that way if we can derive it from more fundamental ideas? – Rohit Pandey Oct 17 '23 at 23:59
  • Let me put it this way - my boss says deriving Euclidean distance by repeated applications of Pythagoras in invalid. – Rohit Pandey Oct 18 '23 at 00:00
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    You are both wrong, and you are talking past each other. From one point of view, we can start with any ol' set, and define a metric on that set in any way we like, so long as it satisfies the axioms of a metric. Given a set which already has an inner product structure, there is a "natural" choice of metric coming from that inner product. You don't need to know anything about Euclidean geometry to define that metric, though it happens that this metric coincides with the Euclidean metric when the underlying space is $\mathbb{R}^n$, thought of as a vector space over $\mathbb{R}$. – Xander Henderson Oct 18 '23 at 01:46
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    You are wrong to assert that this is not appropriate, or that this requires reference to the Euclidean postulates. – Xander Henderson Oct 18 '23 at 01:46
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    On the other hand, your boss is wrong to assert (if, indeed, your reporting of what your boss has said accurately reflects their views) that it is invalid define a metric on $\mathbb{E}^n$ (where I am defining $\mathbb{E}^n$ to be $n$-dimensional Euclidean space, constructed per Euclid's axioms) via iterated application of the Pythagorean identity. However, in that context, "length" is, essentially, a fundamental concept---a unit of length can be defined by constructing a segment, and declaring that this segment represents one unit. – Xander Henderson Oct 18 '23 at 01:49
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    I also don't see that Occam has to do with this. Mathematics is not empirical, and is not fundamentally about the "real" world or physical quantities. Mathematics is a game we play. We establish rules, then investigate the consequences of those rules. One rule set is "Euclidean geometry". Another is "inner product spaces". These rule sets are similar (and have some common ancestry), but are distinct. – Xander Henderson Oct 18 '23 at 02:01
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    In hindsight, your definition and your boss's definition turn out to be equivalent. So it's more a matter of pedagogical choice, or philosophy, than of actual mathematics. There are many mathematical objects for which we can "choose" a definition. Is the exponential function defined as the solution to $f' = f, f(0) = 1$? Or as the sum of the series $\sum \frac{x^n}{n!}$? Or as the converse function of logarithm? Or a solution to $f(x+y) = f(x)f(y)$? Likewise how is the "metre", the SI unit of length, defined? – Stef Oct 18 '23 at 11:11
  • “I understand everything is built on Euclid's five postulates.” This was the case in Ancient Greece. AFAIK, in modern engineering education, geometry is built on $\mathbb{R}^n$ (concrete linear algebra). Pythagoras' theorem talks about lengths, so it requires the definition of length. Thus, you can't use Pythagoras' theorem to define length. In concrete linear algebra, length is defined as your boss says. You can read what is length as Euclid understood it in the answer to the following question. https://math.stackexchange.com/questions/3856142/what-is-length-in-euclidean-geometry – beroal Jul 31 '24 at 15:47

2 Answers2

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I assume that when you and your boss are talking about distances in $\mathbb{R}^n$ you mean the ordinary distance we use most of the time, not the fact that there are other metrics on that space - that is, other consistent ways to define "distance".

I think this is (roughly) the answer to (2):

If you have a good set of axioms for synthetic geometry (that is, geometry from geometric postulates like Euclid's, modernized to fill gaps, as Hilbert did) then if you choose a segment and decree that it has length $1$ you can prove that allows you to introduce coordinates on a line such that the absolute value of the difference defines the distance you want. The Pythagorean theorem (which is in fact equivalent to the parallel postulate) allows you to introduce the usual coordinate system in the plane (once you choose axes). The "distance formula" follows, in all finite dimensions.

So if you are starting from geometry, you introduce coordinates and the Pythagorean theorem tells you how to calculate distances as real numbers. (Euclid didn't do that. For the Greeks the square of the hypotenuse was a literal square with the hypotenuse as one side.) If you start from a coordinate system then with the usual algebraic definition of lines you can prove Euclid's postulates, including the parallel postulate and the Pythagorean theorem.

Ethan Bolker
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Mathematically we can identify lengths with the norms and distances with metrics. What you've noticed is that given some norm $\|\cdot \|$ then we can define a metric as $d(x,y):=\|y-x||$, called the induced metric.

The Kuratowski embedding theorem ensures that ever metric space can be embedded into some Banach space so we can always think of a metric as being induced by some norm. So that means if we're giving a a means by which to measure length we can define distances and every means of defining distances can be related to some means of measuring length.

I first encountered these concepts in linear algebra and analysis. Norms are often related to inner products and so show up naturally in linear algebra. Metric spaces are fundamental to all considerations in analysis and so norms are too.

CyclotomicField
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