This metric definition is overbuilt... but does it matter? (exercise at the end) | Quick Math

The commonly given metric definitions can have some ...fat trimmed, but should the fat be trimmed?

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[ARBB]

EMT slowly becoming an 80s sci-fi show with these aesthetics

[mranonymous]

I agree, the definition should most definitely reflect the underlying intuition. That being said, I would personally enjoy reading about such a redundancy as a remark just after the definition, but that is just my personal opinion.

[alexblokhuis]

There's something to say for both. I think rather than deciding on "the" definition it would be useful to distinguish equivalent definitions for the same object by adjectives. i.e. we have a "parsimonious" definition that shows us what is strictly necessary, and an "intuitive" definition that summarizes the properties we intuitively seek to capture (and that may a posteriori be shown to have redundancies if it doesn't coincide with a parsimonious definition).

[AndrewDotsonvideos]

I must be misunderstanding something. For a metric that has a time component, there can certainly be negative metrics in the sense that ds^2 = g_{\mu\nu} dx^\mu dx^\nu <0 is allowed. It's how we may distinguish space-like from time-like separated events. I assume pseudo-Riemannian spaces are excluded from this definition, or am I missing something else?

[saschabaer]

For the question at the end: For arbitrary a, b we have (setting x=y=a and z=b in property 2)

d(a, b) ≤ d(a, a) + d(b, a) = d(b, a)

Analogously, we also get d(b, a) ≤ d(a, b) and conclude d(a, b)=d(b, a). This shows the missing axiom for being a metric.

[NikolajKuntner]

Another good example for this situation would be the Zermelo-Freankel set theory axiom: Here, in 1908, Zermelo wrote down the bulk of the axioms of his set theory Z, including the Separation axiom, which states that given any set X and any class P (a property), the intersection of the two gives a subset S of X. For example, X={1, 2, 3, 4, 5} and P denoting all even naturals {2, 4, 6, ...} gives the subclass S={2, 4} of X that's a set by the axiom. Then in 1922, Freankel added the stronger Axiom of Replacement to Z set theory, stating that the image of any X under a functional predicate is again a set. You can now take a fixed element c in X and any P and and define a function which maps any x in X to itself if it has the property P and otherwise map it to the default value c. The empty set case must be treated separately. This way, the axiom of Replacement now also allows subsets to be given just via properties P. For the example, the function f mapping all evens to themselves and all unevens to c=2 maps the elements 1, 2, 3, 4, 5 to 2, 2, 2, 4, 2, respectively. So the function maps, in effect, {1, 2, 3, 4, 5} down to the subclass {2, 4}, now also deemed a set by the Replacement axiom. So in conclusion, 16 years later, the theory was strengthened (more claims provable but less models permissible) and Separation became derivable. However! People adopted the ZF set theory as Z plus Replacement, and now ZF has a redundant axiom.
Now back to your question: While I don't have a strong opinion on this, I tend to favor the approach of removing the redundancies. For example in the above tale, arguing why Separation is outshined in power by Replacement is rather simple and convincing in plain text, but people who glimpse at set theory tend to be scared off by the logical formalism and usually never bother to learn interpreting the axioms for what they say. So from just looking at the raw axioms, this connection will not stare at them right away. If we don't adopt a standard to expectation that axioms are given in an independent fashion, people will more often than not be left unsure about the relation of the axioms to each other.
Great production quality of the video btw.

[WrathofMath]

I agree, just like with the group axioms typically including closure, even though that's usually a defining property of binary operations anyways. Though with the metric definition, I think the redundancy of a point in the definition makes for a great first proof using the properties just listed! On another note, sometimes I think "man I'd love to have a lightboard set up" for my math videos. Then I think "but I wouldn't be able to decorate that!", then I see an EMT video and I am once again unsure of my desires.

[hopelessdove]

If I were the one teaching, I would leave it out from the definition, but it would be the first thing I'd show as a consequence of the definition.

[holofech]

I think it's good to include it, since it's necessary in some generalizations, for example quasimetrics. It helps to group them all together

[tomkerruish]

I'm fine with some redundancy in definitions as long as it provides clarity. I just have this image of rewriting Principia Mathematica using only the Sheffer stroke as the outcome of eliminating redundancy. Scary.

[JM-usfr]

Kinda like how you don't need the axiom of the empty set in ZFC (since it follows from the axiom schema of restricted comprehension), but it's often included for funsies.

[-sideddice]

I think the best solution is to give the definition as we are used to (with the redundancy) and then give the exercise to the students : is positivity necessary ? It is the best of both worlds.

I did not see your trich immedatly but you just re-introduced some symmetry there ! Nice trick !
I will do the proof (since you ask so nicely).

You take x=y giving d(z, y)>=d(y, z) for all y and z. By inter changing the roles of y and z you conclude that d is symmetric.
Since d is symmetric, you find back the triangle inequality : d is indeed a distance !

Did you find this exercise in a book (and in that case which one ?) or did you find this alone ?

[lucanalon]

Let X be the set of clock hands' configuration on standard clock. If x and y are two configurations in X, we define d(x, y) as the time you need to wait to get configuration y starting from x. The two properties holds but d is not symmetric.

[korayacar]

Since the definition is often used for identifying metrics as such, I would suggest that it doesn't belong in the definition, but must be immediately stated as a _corollary_ from the definition.

A lot of our intuitions about an object overlap in many ways, and these intuitions should be presented along with the object itself. This means that associating a metric with positive distances is useful, but it will only get in the way once there is a seeming need to prove it for metrics that fit the more concise definition.

This is why corollaries are very useful: they allow us to derive our intuitions from the definition, and allow the definition to streamline the identification process.

[leekeewei]

Well, I do believe it is essential to leave the intuition on positivity. Like positivity is somewhat essential because we still want to understand some abstract notion of 'distance'.
The metric we are used to is an abstraction of the notion of 'distance', but there can be other forms, like quasimetrics or metametrics. In spaces with these metrics, the idea of positivity becomes non-redundant becomes of the weakening of some of the axioms.
So, I guess by placing the positivity as an 'axiom' just emphasizes the fact that this property is something essential in our discussion of what an abstract 'distance' should have.

[Felixkeeg]

Being clear is always more important than keeping things 'clean', especially in the context of teaching.
Thinking something is "obvious" and leaving it out because it is trivial is the best way to alienate your students.

[petargameplay]

The axiom of commutativity of "vector addition" in Vector space also follows from the rest of the axioms. But its not quite obvious at first glance. Let alone when students learn about these things. :)

[peterosudar]

Try using the triple quad formula. If you replace distance with quadrance.... and angle with spread... and instead of just using the "blue" or euclidean notion of geometry you will find a whole world opens up to you...

And if you were looking for motivation to pick up the rubiks cube on your desk and begin to get into speed cubing... well that is it... you got it.

[tomkerruish]

I wanted to find a counterexample, but I wound up showing it is a metric space. (Abbreviated for space.)
d(x, y)+d(x, y)>=d(x, x) so d always nonnegative
d(x, x)+d(y, x)>=d(x, y) and d(y, y)+d(x, y)>=d(y, x) so d(x, y)=d(y, x)
Triangle inequality immediately follows.

[_okedata]

i think we should have axioms that are minimal but also include definig corolaries and make the distinction. that way the full intuition is shown and people writing proofs have less things to consider