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Define 0^0 to be the smallest natural number.

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Where I post from

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Ah, that definitely cleared up the index thing - I thought "Well a vector v in R3 consists of three scalars v1, v2, v3, so thats three indices", but under this viewpoint thats obviously only one index.

By "why do this", I didnt mean "why use tensors", I mean "why introduce vectors and tensors in this weird way instead of just defining vectors as elements of vector spaces (or, if thats too abstract, as "a tuple of three or four numbers such that scalar multiplication and vector addition are associative, commutative, distributive etc.)" In my opinion, defining vectors in this mathematical way and then proving the symmetry and translation invariance properties by showing they correspond to linear maps is way easier and way more clear than doing this whole dance that feynman does here, and while I'm not super familiar with tensors, I would assume theres an analogous way of defining a tensor of order n as just being an n-dimensional array of numbers such that the relevant operations follow axioms x,y, and z (or, even more intuitively, as a multilinear map)

I'm currently reading the first volume of the feynman lectures on physics and just got to the chapter on symmetries and vector algebra and for some reason he insists on this incredibly odd definition of "vector" thats like. "a vector is a set of three numbers such that rotational and translational symmetry are respected". And he'll say like "For two vectors a = (a1,a2,a3) and b = (b1,b2,b3), we define an object (a1+b1,a2+b2,a3+b3). Of course, we havent proven yet that this is a vector." and then he "proves that this sum is a vector", by which he means "prove that rotating the sum is the same as rotating a and b first and then adding them" (i.e. by using the fact that rotation is linear)

And this all seems so backwards to me? Why would you define vectors like this? This definition doesnt even make any sense mathematically? Do physicists actually think about vectors like this????

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What benefit does this provide that we don't get from just defining tensors in a more "normal" mathematical way (which to me would seem both more rigorous and also more clear and intuitive) and then proving that symmetry actions like translation, rotation etc. are linear maps? What's meant by a rank one tensor (which from what I know is a vector) requiring "one index to define" given that we don't know the dimensionality?

I've already heard "a tensor is something that transforms like a tensor" brought up many times in the past as an unhelpful and confusing definition so I'm curious why it would suddenly be a good thing in the physics context (this is a genuine question I'm not trying to be cheeky)

I'm currently reading the first volume of the feynman lectures on physics and just got to the chapter on symmetries and vector algebra and for some reason he insists on this incredibly odd definition of "vector" thats like. "a vector is a set of three numbers such that rotational and translational symmetry are respected". And he'll say like "For two vectors a = (a1,a2,a3) and b = (b1,b2,b3), we define an object (a1+b1,a2+b2,a3+b3). Of course, we havent proven yet that this is a vector." and then he "proves that this sum is a vector", by which he means "prove that rotating the sum is the same as rotating a and b first and then adding them" (i.e. by using the fact that rotation is linear)

And this all seems so backwards to me? Why would you define vectors like this? This definition doesnt even make any sense mathematically? Do physicists actually think about vectors like this????

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This is honestly making me wonder how many famously confusing and difficult physics things are only seen as confusing because people have only ever heard some incredibly strange physicist definition instead of the actual well thought-out mathematician definition (cough cough entropy cough cough)

I'm currently reading the first volume of the feynman lectures on physics and just got to the chapter on symmetries and vector algebra and for some reason he insists on this incredibly odd definition of "vector" thats like. "a vector is a set of three numbers such that rotational and translational symmetry are respected". And he'll say like "For two vectors a = (a1,a2,a3) and b = (b1,b2,b3), we define an object (a1+b1,a2+b2,a3+b3). Of course, we havent proven yet that this is a vector." and then he "proves that this sum is a vector", by which he means "prove that rotating the sum is the same as rotating a and b first and then adding them" (i.e. by using the fact that rotation is linear)

And this all seems so backwards to me? Why would you define vectors like this? This definition doesnt even make any sense mathematically? Do physicists actually think about vectors like this????

[emmatization]

I'm currently reading the first volume of the feynman lectures on physics and just got to the chapter on symmetries and vector algebra and for some reason he insists on this incredibly odd definition of "vector" thats like. "a vector is a set of three numbers such that rotational and translational symmetry are respected". And he'll say like "For two vectors a = (a1,a2,a3) and b = (b1,b2,b3), we define an object (a1+b1,a2+b2,a3+b3). Of course, we havent proven yet that this is a vector." and then he "proves that this sum is a vector", by which he means "prove that rotating the sum is the same as rotating a and b first and then adding them" (i.e. by using the fact that rotation is linear)

And this all seems so backwards to me? Why would you define vectors like this? This definition doesnt even make any sense mathematically? Do physicists actually think about vectors like this????

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there are more symmetries in heaven and on earth, euclid, than are dreamt of in your geometry

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big things happening in differential geometry based metaphysics!!

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as expected there is in fact a strong correlation between "professor who does the most buzzword heavy AI stuff possible and offers classes on entrepeneurship" and "professor who is extremely bad at explaining things and contradicts his own slides while presenting them"

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Bastelstunde!

Yes, this is real, actual math research.

No, I'm not procrastinating, no, I'm not...

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combinatorics questions are so conductive to making you waste hours messing around with incredibly complicated equations only for you to suddenly realize the actual way to do it is incredibly trivial

^ the culprit

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now I just need to know peoples favorite math classes, am I getting the right impression that mathblr adores algebraic topology?

also very curious about peoples opinions on functional analysis and on differential geometry -w-

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Update to this: I'm starting a bsc in math next semester, I'll get about 66 ECTS worth of courses marked as already completed during my compsci degree (out of the full 180 ECTS), and I even found out I don't have to do my compsci thesis with a compsci professor, I can also do it at one of the engineering chairs if I want to, which means that a lot of really cool topics like molecular dynamics / fluid dynamics simulations or numerical control algorithms just entered the picture :D

once again there is a deep-seated desire growing within me to do an extra bonus math bachelor, for enrichment...

would cost 1-2k extra and I could do it in two years with the knowledge I already have

I did the maximum amount of math my bsc in compsci let me do and it feels like it wasnt nearly enough to be satisfying :(

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I just think it’s neat

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Kindergarten-level math research papers:

Field: Combinatorial Number Theory

Journal: Communications in Addition and Subtraction

Title: A Lower Bound on the Largest Natural number

Abbreviated Abstract: We prove a lower bound on the largest number. The proof proceeds in two steps: we begin with 1, and proceed by induction as until we lose count. We then add that number to itself. The main advance in the first step is to get a big number, and the second step notably avoids using multiplication (they don't teach that until 3rd grade).

Field: Topological Geometry

Journal: Advances in Nonlines

Title: The Four-Color Scribbles

Abbreviated Abstract: In this work we show a zoo of examples of nonlines (curves) with the unique property that they are either red, blue, orange, or purple, or some combination therein. The key idea is to use less colors rather than more, creating a clear and easy to follow proof. This provides a clear basis for simplifications to further work, such as scribbling with 5 colors.

Field: Playground Analysis

Journal: Slide Dynamics

Title: Sufficient Conditions to Yell Weeeeeee on Spiral Slides

Abbreviated Abstract: We identify sufficient conditions for a slide to cause joy. We identify a notion of a "fun slide," and prove that fun slides are a sufficient condition to make someone go Weeeee on a slide. We then verify a spiral slide is a fun slide, and provide numerous examples and non-examples (notably, a ramp is really not fun to slide down).

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anybody wanna help me debug this

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