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

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.

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

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.

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).

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.

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 u} dx^\mu dx^ u

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.

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.

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.

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.

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.

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

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 ?

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

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. :)

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.

Let d:X*X to R be a function such that it satisfies the two following properties 1. d(x,y)=0 iff x=y 2. d(x,y) + d(z,y) >= d(x,z) for all x,y,z in X We will show it satisfies the four properties of a metric. To make things clear we will use a,b,c for the metric properties 1m. d(a,b)=0 iff a=b 2m. d(a,b)>=0 for all a,b in X 3m. d(a,b)=d(b,a) for all a,b in X 4m. d(a,b) + d(b,c) >= d(a,c) for all a,b,c in X First note that property 1 and 1m are equivalent under the variable change x to a and y to b. Next consider the variable change x to a, y to b and z to a on property 2. this gives us d(a,b) + d(a,b) >= d(a,a) for all a,b in X 2d(a,b) >= d(a,a) for all a,b in X recall property 1 of the function d says d(x,y)=0 iff x=y and a=a so d(a,a)=0 2d(a,b) >= 0 for all a,b in X d(a,b) >= 0 for all a,b in X This shows the function d satisfies property 2m. Now consider the variable change x to a, y to a, and z to b on property 2. this gives us d(a,a) + d(b,a) >= d(a,b) for all a,b in X recall property 1 of the function d says d(x,y)=0 iff x=y and a=a so d(a,a)=0 d(b,a) >= d(a,b) for all a,b in X Furthermore consider the variable change x to b, y to b, and z to a on property 2. this gives us d(b,b) + d(a,b) >= d(b,a) for all a,b in X recall property 1 of the function d says d(x,y)=0 iff x=y and b=b so d(b,b)=0 d(a,b) >= d(b,a) for all a,b in X By the anti-symmetric property of the inequality we see d(a,b) = d(b,a) for all a,b in X which shows d sadisfies property 3m. Finally consider the variable change x to a, y to b and z to c on property 2. This gives us d(a,b) + d(c,b) >= d(a,c) for all a,b,c in X recall we now know d satisfies property 3m d(a,b)=d(b,a) for all a,b in X so d(c,b) = d(b,c) for all b,c in X. This gives us d(a,b) + d(b,c) >= d(a,c) for all a,b,c in X which is property 4m and finally shows d satisfies all four of the properties in the definition of a metric so d must be a metric. Your videos are always awesome. If i don't get into grad school this next fall for my second attempt these will be my exercises to keep a mathematical mind.

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.

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.

I have always seen the definitions specify R_>=0 as the range of the function. That makes the most sense to me. It makes the notion of distances being non-negative explicit without needing to be stated as an axiom or take up any additional space at all. I don't see the point in specifying a range larger than what any of these functions can ever map on to if it can be specified in a simple, explicit manner. I mean, why not specify the complex numbers as the range instead? Addition is still defined there and the order is too, if you extend the order of the reals by taking its reflexive closure with regard to the complex numbers (it turns into a partial from a total order, but still). Of course, that is nonsensical, just like specifying all of R.

I'm all for giving more details than rigorously necessary anyway (to be more precise, I'm all for that if a) it helps understanding a definition/theorem etc. better or b) you're dealing with "beginners" in a subfield of maths). Since both cases apply here (usually one sees the definition of a metric space in a Semester 1 or 2 course, at least at my uni, and it clearly helps to understand the definition better). In conclusion, I agree with your opinion.

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.

Very nice

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.

The editing on these videos is insane

The usual definition of a group says any element has a right and a left inverse and they are equal. But it would be enough to say every element has one of the two inverses.

I disagree. When defining things, I find a minimalist approach to be the most efficient, I.e. I don’t want to have em check that a function always outputs positive values every time I’m trying to check said function is a metric. Furthermore, while I do agree that intuitively distances should be positive, that’s something that, at the end of the day, is grounded in the physical world. I find that relying too much on everyday intuition makes abstraction that much harder.

I like it being part of the definition. Here's a thought - include it in the definition for "introductory" texts, but if you're writing a more advanced texts, feel free to leave it out, if you prefer. I feel we should do a better job of giving intuition and insight into things (results _and_ definitions) than most texts/classes currently do. I view this as a similar thing with vector spaces. The commutativity of addition is redundant as an axiom. However, I feel it is beneficial to include as an axiom in introductory texts. I also like how Dummit and Foote handle this same issue in rings. They actually show that if you have a unital ring (which all rings should be, right?), then commutativity of addition is redundant.

Let X=R+ and d(x,y)=ln(x/y) for x, y in X. Clearly d(x,y)=0 iff x=y. Also, d(x,z)+d(z,y)=d(x,y), so the triangle inequality is satisfied. Finally, it is clear that d(x,y) is not always equal to d(y,x). Hence, a function satisfying (1) and (2) is not necessarily a metric.

i think axioms should follow an order of importance, i think the most relevant axiom here is #4 that the distance between a point and itself is zero, from this we pretty much derive all the others.

From a teaching point of view I like having it in the definition (it was in there for my point set topology class too) but from a more abstract point of view I don't, so I guess conflicted.

I agree positive property should be in definition because it is something we know in our hearts.

I think it should be part of the definition that distances are positive

I do not know if anybody posted this already but I have found the following: From d(x,x) = 0 and d(x,y) + d(z,y) >= d(x,z) we cannot derive the symmetric property, because: Let x=y and z arbitrary d(x,y) + d(z,y) >= d(x,z) => d(x,x) + d(z,x) >= d(x,z) => d(z,x) >= d(x,z) But now if we let z=y und x arbitrary d(x,y) + d(z,y) >= d(x,z) => d(x,z) + d(z,z) >= d(x,z) => d(x,z) >= d(x,z) we get no new information, since d(x,z) is a real number and all real numbers are greater or equal to themselves I am all for eliminating redundancy in a definition, because it seems just a bit more elegant to me, if you define only the absolute necessary and prove everything else. But I am also a strong advocate for making definitions as intuitive as possible.

not a proof, but the second axiom is the triangle inequality with d(x,y) = d(y,x) d(x,y) + d(y,z) = d(x,z) d(y,z) = d(z,y) so d(x,y) + d(z, y) = d(x,z) the normal axioms imply these two

Metric space 🎖👏🏻

I help build understanding early so I’m fine with it

Do you think the definition of a group homomorphism should include f(e_1) = e_2 and f(x^-1) = f(x)^-1 ? Because technically they are required for it to be a homomorphism we just exclude it from the definition because they follow for groups. abut they for example don‘t follow for monoids which is why the definition of a monoid homomorphism include that neutral elements are mapped to neutral elements.

How do you know the first line thaugh, because although not explicit, triangle inequality assumes x, y, z to be different..

Don't think the symmetric property can be implied from the other properties so it's not a metric?

You could also just embed it into the type of the function if you want to specify it without making it a rule

Can axion 4 be derived from the others?

No, corollaries should follow axioms. I don't think it's a question of preference either.

I think the smallest counter example is a directed graph with one arc from node A to node B. d(A,B)=1 but d(B,A)=infinity and therefore not symmetric

I think d(a,b) = sqrt(a^2+b^2)*(1-(cos P)/k) works for R^2, where d(a,b) =/= d(b,a), k > 1, P is the angle formed with respect to the x axis when going from a to b. Still working out if it respects the triangle inequality.

Jesus Christ he has the omnitrix now

WHY NOT MAKING ANY NEW LONG VIDEOS?

Anyhow, it is confusing to leave the positivity out and then never mention it (Happened to me)

You assumed d(x,x)=0

I don’t care either way tbh