Anyway, I ordered a gray one for £50.00 (like $65 US). The setlist is totally epic:

1. Intro
2. Fear of a Blank Planet
3. My Ashes
4. Anesthetize
5. Sentimental
6. Way Out of Here
7. Sleep Together
8. What Happens Now?
9. Normal
10. Dark Matter (omgomgomgomgomgomgomg)
11. Drown With Me
12. Cheating the Polygraph
13. Half-Light
14. Sever
15. Wedding Nails
16. Strip the Soul/.3
17. Sleep of No Dreaming
18. Halo
19. Outro

Which is every track from the Fear of a Blank Planet album, as well as three of the four tracks from Nil Recurring.
Which is amazingly epic. The big surprise, “Dark Matter” is probably worth the £50 itself. The rest of the set is pretty much what they played during the FoaBP tour (so “Sever”, “Half-Light” and “Drown With Me” aren’t big surprises). The “Strip the Soul/.3″ is presumably the medley they usually play live and is totally awesome. I’ve never heard “Wedding Nails” live, so that should be tight. “Sleep of No Dreaming” is another favorite, and the only live version I’ve heard is that on Coma Divine.

One might be inclined to ask why “Trains” didn’t make the DVD, but we can probably guess at the answer: there are a kajillion live editions of that song available anyway. It’s nice to have something different.

Finally, the special edition of the album also comes with a Blu-Ray featuring some of the Lasse Hoile videos used in concert, as well as an audio CD of the concert with one bonus track: “Prodigal”!

My only complaint is that there won’t be any video of “Prodigal” :(

Also, they released a teaser of the DVD.

Staedtler Integrity

Some years ago I made the transition from using only wood pencils to using only pens. I favored pens mostly because they don’t need to be sharpened.

However, I realized maybe a year ago that taking notes with pens is possibly the most horrible thing ever; I needed something erasable. Thus I began writing with mechanical pencils.

Mechanical pencils solved the two problems: they were erasable, but they also didn’t need to be sharpened (and refilling lead/erasers is easy). But the thing that drove me crazy was that when the lead gets down to like half an inch, it keeps sliding back into the pencil, and you have to pull it out and advance a new piece in.

This is sort of preposterous, especially given that lead is fragile, and you therefore get even shorter pieces of lead. The Staedtler Integrity is a pencil that solves this problem. It has two grips, which hold the lead very close to the tip, allowing use of almost all of a given piece of lead.

Let me just say this. I lost my Staedtler Integrity in class two days ago, and I bought a new one yesterday because I couldn’t live with out it.

Finally, I should say that this is a well-balanced and solidly-weighted pencil, which can be spun and flipped with ease. If I gave numerical ratings, the Staedtler Integrity would receive 100%

The other day I made a post regarding this guy on the internet who thinks the real numbers are not well-defined. This man, John Gabriel, responded with some unsavory comments about my blag, my math, and indeed my character. After reading his comments, I realized two things:

1. John Gabriel doesn’t believe the things he says
2. He is the most brilliant troll in the history of the internet.

Despite these two findings, I think it’s probably important for posterity if I continue to deconstruct his misguided arguments. However, I regret having called him a “pretentious retard,” and will henceforth refer to him as “my esteemed colleague.”

Part II – The real numbers are uncountable

This part will be directed at my esteemed colleague’s knol, “Are the real numbers uncountable?“

The answer is, of course, yes, but let’s not get ahead of ourselves. In criticizing Cantor’s Countability Theorem, he says,

The word countable is use without any predefinition [1]. If one accepts that one can first define not countable and then determine countable as its compliment [2], then it might appear that Cantor’s first ideas regarding countability had nothing to do with a one-to-one correspondence of a set with its proper subset [3]…

So in statement [1] my esteemed colleague asserts that the word countable is used before having been defined. Now I can’t speak to the historical accuracy of this statement, but certainly in modern times we define countability long before we attempt to prove anything about it. This wouldn’t be worth quibbling about, but statement [2] sort of necessitates it:

[2] asserts that, for some reason, one might define not countable before one defines countable. This is certainly possible and mathematically valid, but it’s sort of backwards. Usually the following is done:

A set is countable if there exists a 1-1 mapping of onto

Now let’s assume that by “compliment” he means “complement.” The notion of a  complement of countability is somewhat misleading. It would be better to say,

A set is uncountable if is neither countable nor finite.

Now we could, as my esteemed colleague hypothesizes, do the exact opposite. That is, define uncountable as being the non-existence of a 1-1 correspondence AND not being finite, then define countability to be the not satisfying of one of the first condition. But that would be sort of strange, so we tend not to.

Statement [3] makes little sense. The traditional definitions of both countability and uncountability require the notion of 1-1 correspondence. I would not draw the conclusion suggested.

The rationals are countable

Cantor’s imaginary list is possible only because the rational numbers are well-defined [1]. This property of the rational numbers acquired a special title: The Countable Rationals. However, the property name was a misnomer from the beginning, because it has nothing to do with the notion of counting or countability [2]. It is a direct result of the
rational numbers being well-defined.

(Blue is his). Statement [1] is pretty much true, much as I cringe to admit it. This particular proof that rational numbers are countable does require them to be well-defined. A different proof would be required for a different set. I’ve pointed out statement [1] because it will be crucial later.

Image blatantly stolen

Statement [2], unlike [1], is just not true. The rational numbers are “The Countable Rationals” because they are countable! The proof (or rather, sketch of a proof) shows that the rational numbers can be arranged in a sequence. A sequence is just a bijective function , so being arranged in a sequence is equivalent to a 1-1 correspondence.

Thus the rational numbers are countable by such a proof. This gives them their name, “The Countable Rationals.” I do have to agree, however, that it would be difficult to call the rational numbers countable if they didn’t exist. I’ll give him that.

So what about real numbers?

My esteemed colleague begins his assault on the real numbers in earnest with the nota bene,

Note that rest of my critique of Cantor’s diagonal argument assumes Cantor believed real numbers are well-defined and that he believed all real numbers can be represented using the decimal radix system.

Not a good sign for your argument. This indicates to me that you are going to claim the real numbers can’t be uncountable because they aren’t well-defined. But I’ve already addressed that. I also readdress this in brief in the closing remarks of this post.

Nevertheless, I’ll give you the benefit of the doubt and assume that real numbers may not be well-defined (!) to the extent I can.

In his diagonal argument, Cantor challenges one to propose an imaginary list that contains all real numbers in the interval . He then uses a flawed proof by contradiction to show that if the digits along the diagonal in any list are changed, the resulting new real number so formed, is not in the list. Cantor’s proof fails at the first step where he begins his proof by assuming that all real numbers are in the list.

The above statement is so remarkable to me that I spent a good half hour trying to rectify it with my preconceived notions of sanity and lucidity. Cantor’s proof is a proof by contradiction, yes, and he does assume that all real numbers are on the list, and yes, bad things happen because of that assumption… but that’s exactly the point!

The point of a proof by contradiction is to assume something incorrect (e.g., or all real numbers can be arranged in a list) and to show that it leads to a contradiction (e.g., or there is a number that is not on the list). Cantor would agree that such a list cannot be written! That’s the entire point of the proof. If such a list could be written then the real numbers would be countable. Cantor shows that such a list cannot be written.

My esteemed colleague goes on to attempt to show that the real numbers can be listed out in a sequence. He draws a tree (which you can see at his article):

…it is possible to stipulate a method for listing every real number as follows:

1) Perform top-down finite traversals that make it possible to list all the finitely (terminating decimals) represented real numbers in the interval

2) Perform left-right infinite traversals that make it possible to list all the infinitely (non-terminating decimals and irrational numbers) represented real numbers in the interval

3) Remove duplicates as required. [For example, 0.2500 is a duplicate of 0.25,  0.00340 is a duplicate of 0.003400000, and so on.]

4) Assign a natural number to each of the real numbers.

Here things get a bit dangerous. The lack of mathematical rigor has thusfar been acceptable because he’s been simply discussing mathematics from a sort of philosophical standpoint. Now, though, he makes a claim and does not back it up with mathematical proof. For example, although 2) may seem at first reasonable, it is not, in fact, correct.

But before I show this, let’s examine another method he proposes of using this tree to list all the real numbers.

Due to the fact that most real numbers cannot be finitely represented in any radix system, it becomes necessary to make a couple of modifications without any loss of generality:

1)  Real numbers in the interval [0,1] are treated as approximations

2)  The representations 0.25 and 0.250000 are treated as the same real number with a different representation and thus mapped to different natural numbers.

These are both acceptable modifications, at least if 1) really means, “we treat any finite decimal expansion of a number to be an approximation.” Now, how do we find out the index of a given real number?

1. Count the number of digits in y, say n.

2. Go to the (n-1)th top-down tree.

3. Locate the penultimate digit in this top-down tree.

4. Now in the penultimate subtree, locate the last digit.

5. Count the position of the last digit in the nth top-down subtree.

6. Add the value in (5) to the sum of the previous top-down trees and this will be x.

Ok, sir. Let’s give it a try. Let’s figure out the index, of, say,

1. has… ??? digits

Woops. Well, that’s okay if this algorithm doesn’t work. He said earlier we need to do left-right traversals in the tree for infinite decimal expansions. Let’s try to see which natural number corresponds to .

Well, it’s not 3, because 3 is the index of .3. Likewise, it’s not 33 which indexes .33. We see a pattern:

f(3)=.3

f(33)=.33

f(333)=.333

f(3333)=.3333

…

and so forth. Thus we can ask the question, which natural number corresponds to That is, for which in is

The answer is, of course, that there is no such Suppose there were. Clearly such a number would be of the form,

For some positive integer But this would imply has a finite decimal expansion, which is clearly false. Likewise, we see that any number that has an infinite decimal expansion (one that doesn’t reduce, like ) cannot correspond to a natural number of finite size using my esteemed colleague’s method.

The above arguments… rephrased

In writing this, I noticed that my esteemed colleague altered the knol in question, and some of my explanations above may appear outdated or perhaps irrelevant. This isn’t the case, but I will go ahead and address his modifications anyway.

Now, can one be assured that 1/3 is in the tree? Yes, and this is how:

From the left-right traversals, there will be infinitely many numbers of the form:

0.3999…

0.33999…

0.333999…

0.3333999…

Hence one knows that every infinitely represented number will be thus be enumerated.

Not so! I don’t know that! Actually, the above argument falls into the same trap as the previous one did. A “left-right traversal,” no matter which way you turn your tree, will require either

1. An infinite path
2. An “inifinityth” layer of the tree

The former case I addressed above. The second case is clearly a reformulation of the same idea, but in a slightly subtler way. Notice,

0.3999… = 0.4

0.33999…. = 0.34

…

Let etc. We then ask what natural number has the property Obviously there is no such number. For any the final digit of is 4. Thus we cannot index using this method. By similar argument, no number with an infinite decimal expansion can be indexed this way.

Now I will anticipate my esteemed colleague’s objection: this isn’t his method of indexing! But, sadly, it is. Here, for you cunning folk, is a more formal argument:

Let

Claim: but there is no such that

Proof: I will not bother proving the first part because it (almost) supports my esteemed colleague’s claims. He can do it himself. Now suppose for some positive integer

Then

This implies

Or,

Which is clearly not the case. Thus no such exists.

Now let’s “interpret” this result! I’ve shown that you can list,

1. 0.3999…
2. 0.33999…
3. 0.333999…
4. …

And yet the number 0.333… cannot be on the list. This means my esteemed colleague’s method of mapping each number on the tree to a natural number is incorrect.

Closing remarks

This is a post of rather impressive length, but I think it’s important that it be that way. It’s difficult to refute my esteemed colleague’s arguments because they are just that—arguments. In mathematics, everything should be made explicit. It is his challenge to do so with his arguments.

Nevertheless, I have clearly taken steps to formalize my refutations of his arguments because a formal refutation is, well irrefutable.

If my esteemed colleague deigns to read this, I propose to him the following. Either,

1. Come up with a rigorous refutation of Cantor’s diagonal argument (saying real numbers aren’t well defined isn’t good enough. We could just hypothesize the existence of a set with the properties mathematicians claim real numbers have, apply Cantor’s proof and thereby show that what we call real numbers are uncountable)
2. Provide a rigorous proof of the existence of a bijection between the real numbers and natural numbers

You might note that the word “rigorous” plays a large—so to speak—role in both options. I ask this of him not because I think he’s incapable, but rather because he is the most magnificent troll in the history of the internet, and if anyone can do it, he can.

As a closing personal remark to my esteemed colleague, I would like to say the following regarding the well-definedness of the real numbers: the real numbers are well-defined whether they jive with your intuition or not. The delta-epsilon definition of limits throws just about every calculus student for a loop; you don’t have to think the word limit means what mathematicians tell you it means. However, if you choose a different definition, then you can’t use that definition to make arguments about the word as mathematicians use it.

Today I will tackle the article “Are the real numbers well-defined?” I will assume in the reader a familiarity with mathematics that is probably enough for the reader to refute this guy on his own. Whatever.

I skipped some of the boring stuff at the top of the knol. Let’s go!

The successor of zero

Mr. Gabriel posits the following:

The successor of zero is an indeterminate quantity (not an infinitesimal because an infinitesimal is an ill-defined concept) that arises in the measure of all irrational numbers because it cannot be represented as a/b. So when one tries to measure any irrational number using sums of averages (a Cauchy sequence for example), there is no such thing as ‘limit’ [1]. It does not matter that a value for the limit can be fixed for any given partial sum as in the epsilon-delta arguments. Why? Provided more terms are summed in a partial sum, the limit continues to change even though it (limit) converges [2](To what? No one knows for certain…the limit cannot be a number because a number is defined as a ratio! [3]). The successor of zero is the reason a limit does not exist for any irrational number.

(I added the numbers myself but the emphasis is his) First let’s consider the successor of zero. What he probably means is the “number that comes after zero”. He suggests this number is ill-defined (which generally means that such a thing does not exist). Well he’s right; think of a small number. Now divide it by two. Now you have a smaller number. This proves that there is no smallest positive real number and therefore no “successor of zero”.

So what about statement [1]? Recall what a limit is:

A sequence in a metric space is said to converge if there is a point such that there is an integer such that

(Above definition adapted from Rudin’s Principles of Mathematical Analysis)

So when we define a limit of a function we require that the values get arbitrarily close to some value. So let’s reconsider assertion [1]—that there’s no such thing as a limit. It’s clear from the above definition that the limit does exist, but only if it happens to be in the metric space we’re talking about. Mr. Gabriel’s argument clearly, then, rests on the assertion that there exist real sequences that do not converge in

Let’s leave that alone for a bit and consider [2]: “Provided more terms are summed in a partial sum, the limit continues to change even though it (limit) converges”

I’m not sure what this means. He seems to be considering a series of the form:

and is suggesting that because a given partial sum

may not be the same as another partial sum with the series thus cannot actually converge to a number. I will neglect his lack of proof and demonstrate that this is not true. Let . Then the sum converges by the integral test. QED.

Okay, that’s a bit silly, but I think I can point out where his misunderstanding lies. Indeed, each sum increases and is larger than the previous term. But this doesn’t matter; the limit doesn’t have to actually be any of the terms . In fact, it often isn’t, as in the above case. All that matters, as in our definition, is that the sum get arbitrarily close to one number (and therefore not to any other number).

Finally, statement [3] (“numbers are defined as a ratio”) is straight up false. Generally, rational numbers are constructed by defining them as ratios of integers. But integers are constructed by defining them as pairs of natural numbers, and natural numbers are constructed by defining them as sets (or using the successor business, which Mr. Gabriel seems to be familiar with). Real numbers have been defined a number of ways. The two most important (for our and indeed most purposes) are as Dedekind cuts and as equivalent classes of Cauchy sequences.

Irrational numbers are pairs of rational numbers!

…one could define an irrational number as: real part (r) + irrational part (i) [4]. However, the values of r and i are indeterminate because i is indeterminate.

Actually, you can’t. I assume he means you could write an real number in the form:

where and is some irrational identity thing. Note that I use different terminology than he does; I do this with the understanding that he thinks irrational numbers are not real numbers (otherwise why have an irrational part?)

Why is [4] wrong? It would contradict a whole slew of other theorems. For example, it would mean the real numbers are countable (which he thinks is true, by the way; maybe this’ll be part II). But we could do a simple proof by counterexample:

Proposition: There are no rational numbers with .

Proof: Since it must be true that the product is also rational. Now and is algebraic, thus is algebraic. is transcendental, so such numbers cannot exist.

Obviously I could rewrite this proof and replace with any algebraic number and the proof would hold. A similar proof would show that the “irrational identity thing” could not be transcendental either.

Now his counterargument would be that pi doesn’t exist or something. Or maybe that transcendental numbers don’t exist because irrational numbers don’t. Fine. The minor quibbles I’ve addressed thusfar stem from his underlying belief that the construction of the real numbers is somehow “wrong.”

Let’s deal with that.

Pi is undefined

According to the theory of real analysis, an equivalence class is understood to represent some real number. However, in the case of numbers such as pi or the square root of two, all one can do is write or

Let’s call this assertion [5]. Recall that an equivalence class is a set of all sequences in with the property that if and are in then

Mr. Gabriel has a good point. We can’t just say that pi is the equivalent Cauchy sequences that converge to pi. That is circular. That’s why we define pi differently! That is,

Of course we don’t like the square root function because it frequently produces irrational numbers (which we don’t like). So let’s make sure there is an irrational number:

Let and

This sequence converges to (this isn’t too hard to show) but consists only of rational numbers (oh, and it’s Cauchy because all convergent sequences are Cauchy). We can therefore define the square root of two to be the set of all equivalent Cauchy sequences to

Conclusion

I can actually understand Mr. Gabriel’s reluctance to accept real numbers. They’re scary; they made Cantor go insane (ish; more on this in part II). But it’s hard to go up against actual, rigorous proof when you don’t even offer your own proofs; all you offer are “arguments.”

Arguments aren’t valid in mathematics. You’re either right or you’re wrong.

You could challenge the philosophy of mathematics, I guess, and say like “Yeah, well there’s no such thing as truth anyway.” But I don’t think Mr. Gabriel makes that claim.

To be continued…

“Seriously, there’s no point to these things.

“All they do is cause noise. It’s not like ‘Oh my God! There must be someone trying to steal that car! Let’s run over and stop the bad guy!’

“What it really means is some asshole has to come outside and shut his car up.”

And this seems like a pretty reasonable point of view, especially for someone who was recently jarred out of deep, meaningful thought by a car alarm.

Such was my first impression of this stranger’s diatribe, but after I myself lapsed back into deep, meaningful thought, I realized we need to be careful when evaluating car alarms.

Certainly he had a good point: people don’t come running, and cops don’t come streaming in every time a car alarm goes off. But this doesn’t mean they’re ineffective. It all stems from what we want our car alarms to accomplish.

We don’t get car alarms so that car thieves can be arrested; we do get car alarms so that our cars won’t be stolen. The subtle difference between these two motivations provides insight into why car alarms are actually effective:

Cars equipped with car alarms are less likely to be stolen.

Actually, this is conjecture. I have no data to support such a claim. But this is probably the theory: thieves don’t avoid stealing car-alarm-equipped cars because people will look at them and call the cops; they avoid doing it because car alarms strike a subconscious fear into their hearts.

Of course, it’d be really hard to test the efficacy of car alarms. For example, if you noticed a negative correlation between car thefts and use of car alarms, you couldn’t actually claim the car alarms are doing it. For example, older, cheaper cars tend to not have alarms. Modern, expensive ones do. Is it easier and more common to steal an old, cheap car? Possibly.

In the end, I think car alarms warrant further investigation for being so obnoxious and potentially so useless.

But what really gets me is when the article (the fucking article) isn’t even very good. Case in point: a story was recently posted on how reducing one amino acid in your diet might actually increase your lifespan. The article the Slashdot post links to is what is called a “review”: a summary of one or more scientific studies or papers written for a less technical audience.

I don’t blame the poster for linking to the review, but it’d be nice if they linked to the actual study the review was about. Reviews, by their very nature, lack the rigor and clarity that is characteristic of most (good) scientific papers. If people at Slashdot would simply spend the time to read the actual paper, then they would realize that the more skeptical comments are completely off base.

Reviews tend to sensationalize stories because they sell more copies that way; scientific studies don’t do that because integrity is important to scientists (but not to journalists for whatever reason). I rarely read Slashdot comments any more because I simply don’t enjoy being bombarded by commenters lambasting a scientific (or sometimes social) study, when they are in fact basing their arguments on the faulty conclusions drawn by a review.

Theorem Suppose f is a continuous real function on a compact metric space X, and

Then there exist points such that and .

And then the proof is one line; it is a corollary of two other theorems:

1. If f is a continuous mapping of a compact metric space X into , then is closed and bounded. Thus, f is bounded.

2. Let E be a nonempty set of real numbers which is bounded above. Let y = sup E. Then . Hence if E is closed.

I was reminded of the same, albeit more specific theorem, in Spivak’s Calculus.

Theorem If f is continuous on [a, b], then there is a number y in [a, b] such that for all x in [a,b].

And then the proof

Proof f is bounded above on [a,b], which means that the set

is bounded. This set is not empty, so it has a least upper bound . Since for x in [a, b] it suffices to show that for some y in [a, b].

Suppose instead that for all y in [a,b]. Then the function g defined by

is continuous on [a, b] since the denominator is never 0. On the other hand, is the least upper bound of A; this means that

for every there is x in [a, b] with .

This, in turn, means that,

for every there is x in [a, b] with

But this means that g is not bounded on [a,b], contradicting continuous functions are bounded.

Phew. The more complicated proof from real analysis is significantly shorter than the one from calculus, even though the one from analysis is actually strictly stronger. I could do a similar writeup for the calculus proof of the fundamental theorem of algebra vs. the complex analytic one using Liouville’s theorem, but the calculus proof would be like 10 hours of typing.

I’ve only owned a couple Staedtler pens in my life, so when I saw these Staedtler pencils in the teeny little university bookstore, I knew I had to try them. There are only a few criteria I look for in a writing utensil:

1. Comfort of grip

The grip must be comfortable! You have to be able to write out the Laplacian in spherical coordinates without your hand cramping up.

2. Ease of use of mechanical features

This includes things like clicking a pen, removing a cap, or in this case, advancing the lead.

3. Weight and shape

This is the most important. It is absolutely essential that my writing utensils be weighted and shaped so that they can be easily spun.

So how does it stack up? Well for (1) it’s pretty average. The grip is just a piece of rubber that wraps around near the head. It’s not particularly awful, but it’s nothing special.

(2) is the pencil’s weakest point. Clicking to advance the lead is an arduous process. It almost feels like there’s something rusty in there.

(3) is the pencil’s strongest point. It’s a natural width, and is shaped a lot like a standard Bic pen. But the weight—oh the weight. The weight is heavenly. These mofos are heavier than a normal mechanical pencil, and the weight is almost perfectly distributed. I daresay they compare to the Bic Softfeel Med.

Did I mention it was less than $2 for three of them?

How many numbers are between ten and twenty?

Of course, this problem is well within the skills of a fourth-grader. Simply subtract ten from twenty and you get ten. In fact, the only appreciable difficulty is in the interpretation of the question. Students, particularly children, struggle with these so-called “word-problems”: problems where the math is relatively simple, but the problem is phrased as a question. You know, using words and stuff.

Being who I am, I completely failed at understanding the question; I answered, “Eleven.”

So I got laughed at… But to this day I maintain that 11 is just as accurate as—indeed more accurate than—10. Why? It all lies in the interpretation of the word “between.” If “between 10 and 20″ means

Then yes, there are 10 such numbers: . But who the hell uses “between” to mean “including the upper limit but excluding the lower limit”? In my opinion, there are only two reasonable ways to interpret “between ten and twenty”:

But then the correct answer is either or . I chose option (1) in fourth grade, but option (2) is perfectly reasonable.

The problem is that this becomes no longer strictly a subtraction problem. What the teacher wants is for the students to compute . But this is wrong, so how do we reinterpret the question?

I propose the following: first consider the question, “what are the numbers between ten and twenty?” This question still has the ambiguity of the word “between,” but no person in his right mind would answer:

or

Then we can ask, after the set of numbers has been identified, “how many of them are there?” I think if the problem were posed this way, no one would ever arrive at an answer of eleven (unless they miscount).

Further Rambling

I’ve decided that word problems like the ones we’re given in grade school don’t really count as “math.” I think they’re physics. Consider the topics covered in college- and higher-level mathematics. They’re abstract and almost completely disconnected from the real world. A mathematician doesn’t care if we live in Euclidean space or a Minkowski space; if they can prove something about both, then that is interesting.

A physicist (or statistician maybe) concerns himself with questions like these word problems—problems dealing with possible real-world scenarios. When we answer such questions, we don’t really learn a lot about math; rather, we learn about the connection between math and the world we live in.

That’s not to say word problems aren’t good or helpful, but I think school teachers should be very careful both in posing questions and in receiving answers. What if my teacher had let me explain my logic? It would have been nice to shut up the kids who laughed at me for a “stupid” answer, and they might learn something at the same time.

Their sound is certainly “progressive” in the sense that it’s complex, powerful, and weird. It’s hard for me to categorize them well, but I think Wikipedia does a fine job:

Genres: Progressive rock, industrial rock, electro rock, post-metal, avant-garde metal

It’s hard to not like at least one of those.