Now, hold on. You could define a prime cardinal as a cardinal that has no factors other than one and itself. Then, in particular, [; \aleph_0 ;] is prime, and I'm sure there are others.
EDIT: Well, I'm dumb. See the replies. :(
tfaal ยท 18 points ยท Posted at 18:33:37 on December 5, 2015 ยท (Permalink)
Aleph_0 is not prime since Aleph_0 = Aleph_0*2, and thus 2 is a factor of Aleph_0.
magus145 ยท 8 points ยท Posted at 20:25:54 on December 5, 2015 ยท (Permalink)
If instead you define prime cardinals to be the ones that can't be expressed as the product of a finite number of strictly smaller cardinals (a different, reasonable generalization), then (in ZFC) every infinite cardinal is prime!
[deleted] ยท 6 points ยท Posted at 22:33:26 on December 5, 2015 ยท (Permalink)
So then let's try to think of a formulation of prime for infinite cardinals which doesn't give a trivial result.
Under the presence of the axiom of choice, the cardinals have a very trivial multiplicative structure, since the product of two cardinals is just their maximum unless they are both finite. So a study of multiplication of cardinals boils down to studying chains, and we have ideas like limit cardinals but nothing that is really anything like arithmetic.
As I just mentioned in another comment, there may be some nontrivial things we can formulate about cardinal products without the axiom of choice, but we will never be able to exhibit any examples.
[deleted] ยท 2 points ยท Posted at 22:03:14 on December 5, 2015 ยท (Permalink)
c2 = c for an infinite cardinal c means that, for any infinite set S, there is a bijection between S and SรSโ.
Since a cardinal number is the "size" of a set, this is the usual convention for multiplying cardinal numbers.
Yakone ยท 2 points ยท Posted at 01:32:18 on December 6, 2015 ยท (Permalink)
We define cardinal arithmetic for cardinals. It's the same for the small numbers but extends to cardinals of all sizes.
[deleted] ยท 0 points ยท Posted at 19:41:32 on December 5, 2015 ยท (Permalink)
c2 = c does not necessarily imply that c = 1, c could also be 0. But both of these inferences only make sense in the context of the real numbers, and the set of real numbers doesn't include any infinite elements.
I think that prime ordinals are a more reasonable idea. Then, for any limit ordinal a, a + 1 would be prime and a + p would be prime for any prime p. Not sure if that's all of them.
magus145 ยท 6 points ยท Posted at 22:11:28 on December 5, 2015 ยท (Permalink)
There are others, and a + p isn't prime since (a + 1)(p) is a nontrivial factorization.
faore ยท 37 points ยท Posted at 01:02:30 on December 6, 2015 ยท (Permalink)
In fact this is essentially the only example. The comic is kind of irritating because more than any scientist, a mathematician can actually revolutionise his beliefs very quickly by reading a single proof.
[deleted] ยท 7 points ยท Posted at 13:01:03 on December 6, 2015 ยท (Permalink)
Nowadays, yes. That's mostly thanks to the modern mindset that as long as your definitions/axioms are all logically consistent and your proofs are all sound, there is no such thing as being "incorrect" in mathematics. It's an idea we take for granted, but it only really started taking shape in the 19th century, and it didn't fully solidify until the 20th century.
I can't think of any other examples but are there definitely no more?
faore ยท 6 points ยท Posted at 02:49:26 on December 6, 2015 ยท (Permalink)
I'm sure there are others but none many people have heard of
Vonbo ยท 3 points ยท Posted at 16:59:09 on December 6, 2015 ยท (Permalink)
Greeks (specifically, the Pythagorean school) around 500 BC believed every number could be expressed as a ratio of two integers. According to legend, Hippasus, a member of the pythagorean school, proved the irrationality of โ2 while on a sea trip. His fellow pythagoreans then threw him overboard for betraying their beliefs, where he drowned.
This story is not factual though, and most likely just a tale with a bit of truth to it.
dfqteb ยท 3 points ยท Posted at 20:46:38 on December 6, 2015 ยท (Permalink)
But this is hardly relevant to the current culture within the academic maths community.
I completely agree. I have yet to see anyone give an interesting and important example of this phenomenon other than the creation of set theory. It's a stupid comic.)
(And even the example of Cantor seems rather overblown.)
Tiervexx ยท 29 points ยท Posted at 00:55:39 on December 6, 2015 ยท (Permalink)
Yeah, but set theory was just about the only thing that really shocked the math world that badly. In general, mathematicians are much better than other intellectuals at accepting new proofs.
Didn't someone pitch a fit when Weierstrass demonstrated his nowhere differentiable continuous function? Basically was like "nah man, that's dumb so it doesn't count".
Tiervexx ยท 3 points ยท Posted at 17:09:17 on December 6, 2015 ยท (Permalink)*
I could be mistaken, but I was under the impression that was mostly a delightful shock, rather than a "OMG THE WORLD IS BURNING!" Maybe an individual mathematician lost their mind but as a whole I don't think it shocked them that bad.
Cardinal numbers also include 'transfinite numbers' such as the size of the set containing all the natural numbers and the size of the set containing all the real numbers. When Cantor first came up with the idea and proved for instance, that there are more real numbers than natural numbers (colloquially, that there are different sizes of infinity), he was met with resistance and scorn from much of the mathematical establishment in particular Kronecker. Cantor had a really tough time of it and suffered severe bouts of depression as a result and ended up dying impoverished. Of course, now these ideas are some of the first things you will learn in an elementary set theory class in university.
[deleted] ยท 2 points ยท Posted at 22:22:27 on December 5, 2015 ยท (Permalink)
He's referring to infinite cardinals
[deleted] ยท 28 points ยท Posted at 23:44:44 on December 5, 2015 ยท (Permalink)
Seeing "Your mother's a whore" written in flowy cursive was the funniest thing to happen to me today.
[deleted] ยท 381 points ยท Posted at 16:39:10 on December 5, 2015 ยท (Permalink)
I feel like this would have been funnier, and more accurate, with modern physics replacing math.
redrumsir ยท 124 points ยท Posted at 16:50:22 on December 5, 2015 ยท (Permalink)
Agreed. It's not really true with math. It's nearly antithetical to math and I would argue that stronger if it hadn't taken so long for the notion of limits to be developed (which did settle quite a few disagreements).
le_epic ยท 212 points ยท Posted at 17:38:03 on December 5, 2015 ยท (Permalink)
Didn't a lot of stuff that is now regarded as "basic" like complex numbers, infinitesimals, transcendental numbers, set theory (with that "set of all sets" problem), go pretty much like in the comic for many decades? I guess infinitesimals fall under the "solved with the notion of limits" category. (I'm just a curious layman with only the vaguest notions of both History and Math, I might be completely wrong, it's a genuine question.)
elseifian ยท 137 points ยท Posted at 18:03:29 on December 5, 2015 ยท (Permalink)
Not just the fancy stuff, the definition of a function took centuries to get right, but we expect high school students to just understand the subtleties without even teaching the concept directly.
You guys are making me worried. What subtleties of the definition of a function do I not know? WHAT AREN'T THEY TELLING ME?
magus145 ยท 86 points ยท Posted at 20:21:28 on December 5, 2015 ยท (Permalink)
A function is a set F consisting of a triple of sets: a domain X, a codomain Y, and a subset of X x Y that has exactly one element of the form (x,y) for each x in X.
This definition treats functions as static objects (sets) rather than somehow dynamic processes, and so allows rigorous analysis of sets of functions, or letting the domain elements be other functions.
It also makes rigorous the intuitive concept of what counts as a "rule", which is important if you want things like nowhere continuous functions or weirder things like non-computable functions.
saarl ยท 15 points ยท Posted at 22:48:50 on December 5, 2015 ยท (Permalink)
Why is it a set containing a triple of sets and not just a triple of sets?
It is just a triple of sets. Some people (like me) prefer to just think of as a set of ordered pairs, without the triple part with the domain and codomain.
magus145 ยท 26 points ยท Posted at 00:15:32 on December 6, 2015 ยท (Permalink)
This is a bad way to formalize functions as it becomes impossible in the object language to define surjectivity.
This is the standard way in set theory, where it's much more convenient. In other areas the definition with triples is probably better. However, putting the domain in the definition seems completely redundant to me.
DR6 ยท 3 points ยท Posted at 08:55:19 on December 6, 2015 ยท (Permalink)*
It's not redundant at all. f(x) = x2 with domain R and domain N are drastically different functions: with domain N it's injective and monotonically increasing, with domain R it isn't.
But it is. The definition of a function survives just fine as a collection of ordered pairs (x,y) such that no two have the same x value.
It is true that f:N->N by f(x)=x2 is very different than f:R->R by the same formula... but they are also completely different as sets. In practice, it is useful to know and state a donain and range explicitly, but "behind the scenes" the ordered pairs tell you everything you need to know unambiguously.
DR6 ยท 4 points ยท Posted at 09:44:36 on December 6, 2015 ยท (Permalink)
... well, you can make the domain redundant like that, but you still need the codomain. Taking again f(x) = x2 , f : R -> R+ and f : R -> R are again different functions with different properties(one is surjective, the other isn't) and you can't tell from the ordered pairs alone.
Yeah, I just mentioned elsewhere that I personally don't see surjectivity as a property of the function itself but rather as a property of the pair (f,Y), such that a function is surjective onto its image but not surjective onto any superset. I like the malleability there, no reason to restrict the function to one range.
You're considering a function as a rule here, not a set of pairs. But I see /u/almightySapling already explained this
guilleme ยท 4 points ยท Posted at 01:57:44 on December 6, 2015 ยท (Permalink)
Agreed with this!! Also, the property of the set being ordered is rather important and makes intuitive why something like x = y2 + 3 is not a function (although it is a well-founded relation (actually a pretty standard conic), so something like wolfram alpha or desmos.com should be able to plot it).
Think of it like this: a function is a set of pairs of numbers (x,y) where there is only one (one and only one) pair that has an specific number on the place of the 'x'. This allows us to speak about the order of the pairs of the function.
magus145 ยท 6 points ยท Posted at 00:09:05 on December 6, 2015 ยท (Permalink)
Well, we want a function to be a single object, and in set theory, the only objects are sets. So it IS a triple of sets, but a triple of sets is itself, a set. (And not just a set of the three elements since we really want it to be an ordered triple of sets.)
docmedic ยท 5 points ยท Posted at 22:36:09 on December 5, 2015 ยท (Permalink)
This definition treats functions as static objects (sets) rather than somehow dynamic processes, and so allows rigorous analysis of sets of functions, or letting the domain elements be other functions.
Out of curiosity, is there something inherently illogical/ill-defined about defining functions in the dynamic process manner, if one's goal did not include discussions about nowhere continuous or non-computable functions?
I think mostly the point is that we can define functions in terms of sets, which are define by even lower levels all the way down do ZF(C) and L.PA, logic and such. It makes for a better model. It would be like defining 2 as a separate number rather than as 1+1 (or the number following 1).
docmedic ยท 3 points ยท Posted at 00:14:53 on December 6, 2015 ยท (Permalink)
I see. Rereading the definition of function from wikipedia:
A function f from X to Y is a subset of the Cartesian product X ร Y subject to the following condition: every element of X is the first component of one and only one ordered pair in the subset.
is it really necessary to specify that F is a triplet of sets, rather than just a subset of a Cartesian product satisfying that one condition?
magus145 ยท 3 points ยท Posted at 00:32:02 on December 6, 2015 ยท (Permalink)
Formally, yes, as I explained earlier. If a function is just a set of ordered pairs, how do you write a well formed formula in set theory to identify the codomain? After all, the element (1,1) is in the set R x R as well as R x R+.
So how else could you formally define surjective functions?
But the set R x R+ is in R x R so wouldn't you have the same problem? Couldn't you say the element would be surjective on (1,1) and not (1,1...2) for example?
magus145 ยท 2 points ยท Posted at 03:07:10 on December 6, 2015 ยท (Permalink)
I'm not sure I understand your notation. R x R+ is a subset of R x R, not an element. A function can't be surjective on a single point, and I have no idea what (1, 1...2) is supposed to mean.
Right, sorry about that. You could say The function with element (1,1) would be surjective on {1} x {1} but not {1} x {1,2}. So just as {1} subset of {1,2}, so is R+ with R. Why can't we define surjective to make sense this way, rather than have to assign a couple of extra sets to a functions definition? What am I missing here?
Thanks =)
magus145 ยท 2 points ยท Posted at 14:35:41 on December 6, 2015 ยท (Permalink)
You could, but it depends on what properties of functions you want to be intrinsic.
If you define a function to just be its set theoretic graph, then being a bijection (i.e., being invertible) is no longer an intrinsic property of functions but now depends on what set you embed their ranges into.
This makes cardinality arguments way more laborious as well as formalizing notions in category theory, where the codomain is prior to the Hom Sets.
Ah, now I see what you mean, thanks for taking your time to explain!
magus145 ยท 2 points ยท Posted at 15:17:03 on December 6, 2015 ยท (Permalink)
No problem! Thanks for sticking it out and being polite!
Krestiv ยท 2 points ยท Posted at 13:20:47 on December 6, 2015 ยท (Permalink)
I don't see why one would need to define surjectivity as a function property though. How would one formally define continuous functions? Even if you include the domain and the codomain in the definition of a function, you have no way of telling whether it is continuous for you haven't included the topology yet.
You may have a function which maps certain elements to certain elements. Then you may state "If the codomain has this topology, then this function is continuous." in the same way you may state "If this is the entire codomain, then the function is surjective."
Or, you could try to include the topology, if available, in the function itself. Then you still wouldn't know whether the function is monotonically increasing since you don't have an order on the domain and codomain. Like this, there are many properties of functions that depend on more than just what the codomain set itself is.
So why should surjectivity get special treatment over the other bazillion properties? We could just define f(x) = x2 as a subset of R x R, the simplest way possible. Then f would be surjective from R to R+ and not surjective from R to R.
magus145 ยท 2 points ยท Posted at 15:14:27 on December 6, 2015 ยท (Permalink)*
Of course you could define "function" that way, since what you described is what in standard set theory is called "the graph of the function", and I've been denoting G, the third element of my ordered triple that makes up a function. It's a perfectly well defined set.
Why shouldn't we use the word "function" for that set?
Let me answer from a categorical perspective. For now, I'll just consider concrete categories, so our objects are all "sets with extra structure" and our morphisms are all "set functions with extra properties".
Your question is essentially "Why should the morphisms in the category of Set include the data of domain and codomain?" There is a "top down" answer and a "bottom up" answer.
In category theory, the domain and codomain objects are philosophically prior to the morphisms between them. That is, every morphism lives in a Hom Set, Hom(A,B), so you can't even talk about a function without first taking about which Hom set it lives in. Since this perspective has been so useful in unifying and clarifying the concepts of maps with stricture, it's both convenient and philosophically satisfying to build this data straight into the set theory itself.
So what is a topological space? From this perspective, it's a set with the additional structure of an associated set of subsets called a topology. What are the appropriate morphisms that preserve this structure? Continuous maps. What then become invariants of this category, i.e., what properties should be the same when two topological spaces are homeomorphic but not literally represented by the same set? Things like compactness, connectedness, separability properties, etc.
Similarly, in group theory, the objects are groups, the maps are group homomorphisms, and the invariants are things like order, simplicity, being nilponent, being hyperbolic, etc.
Now, consider set theory, i.e., the category Set. It's the most basic concrete category as all other ones have a forgetful functor into Set. So its objects are just called sets, its morphisms "set functions" or just "functions". Pick a different definition of "function" and you get a different category with different invariants.
So what ARE the invariants of set theory? When should two sets be considered isomorphic (basically the same up to relabeling) when they aren't literally the same set? What invariant should be an invariant for ALL mathematical objects (in concrete categories)?
The answer is cardinality. The notion of a size of a set should be the only invariant of Set. It should be an invariant of every other concrete category (Top, Grp, Met, ...), and set functions should preserve it. In order for this to happen, invertibility (i.e. bijectivity) needs to be a concept intrinsic to the set function.
This is why bijectivity and hence surjectivity is more foundational to functions than continuity or other properties that depend on other structure. The only structure on a set is the containment binary relation, and the only invariant defined just from this structure is cardinality.
So functions with different codomains but identical graphs need to be different functions, or you would lose even this, and have no category at all (composition wouldn't be well-defined).
docmedic ยท 1 points ยท Posted at 03:19:56 on December 6, 2015 ยท (Permalink)
I see, so what you want is for f != g in the set sense if they're not the same function. So rather than associating with f two sets and an element of a cartesian product, you set every element of f to contain that information. That is, let X and Y be sets. Then f = {(X, Y, (x, y)) | x in X and y in Y} with the usual restriction on (x,y) is a function. So if the domain or codomain differ, then no element of either set will be in the other.
magus145 ยท 1 points ยท Posted at 04:57:23 on December 6, 2015 ยท (Permalink)
You've got the conceptual idea right exactly, but that set formalism is wrong. You need an ordered triple of sets (X, Y, G), where G = {(x,y) | x in X} is a subset (not element) of X x Y.
Using the standard way of encoding ordered triples, it would literally look like:
f = {{{{X}, {X,Y}}},{{{X}, {X,Y}},G}}
Of course, that looks so ugly, and so we define the notation (X, Y, G) as soon as possible.
docmedic ยท 1 points ยท Posted at 05:14:04 on December 6, 2015 ยท (Permalink)*
Fair enough, but it should still be possible to write as
f = {(X, Y, (x,y))| (x,y) in G a subset of X x Y with the usual property}
which seems to be more explicit on what elements of f consists of, if I understand correctly.
Or is it in set theory that functions are singletons represented by a triplet of sets (X,Y,G)?
magus145 ยท 1 points ยท Posted at 05:43:57 on December 6, 2015 ยท (Permalink)
You wouldn't want to define it that way because then the points on the graph of the function would be in the same set as the entire domain. Then if you said something like "For each element of f....", you'd need to specifically exclude two points that shouldn't be in there. It's a category error.
As for your second point, I think the answer is "yes". Set theoretically, a function IS the ordered triple (X,Y,G) (which is itself a set).
It's not a singleton though, because an ordered triple is not encoded that way.
docmedic ยท 1 points ยท Posted at 06:28:46 on December 7, 2015 ยท (Permalink)
I guess I don't see it. For instance, f(x) = 3x from R to R has the element (R, R, (0,0)), specifying the domain, codomain, and a point. Which two points aren't on that function?
If we consider f = (R, R, G), then wouldn't an element specify an element of R, R, and G, like (3, 4, (0,0))?
magus145 ยท 2 points ยท Posted at 13:00:09 on December 7, 2015 ยท (Permalink)
Ah, I see the confusion. The function does not consist of ordered triples, rather it is a SINGLE ordered triple (X, Y, G), where G is itself a set consisting of all the ordered pairs.
So f: R -> R, f(x) = 3x gets encoded as f = (R, R, {(0, 0), (1,3), (2,6), (pi, 3pi), ...}).
So a single point on the function still refers to an ordered pair on the graph, i.e., in the set G, like (2,6).
Only the function itself, globally, is encoded as the ordered triple of data. Each point in the function is still a regular old ordered pair of elements.
So how else could you formally define surjective functions?
I guess personally surjectivity is something I don't care about to much. No reason I can't talk about the image of f living inside of any superset, why need to restrict to one of them?
But, I can see in most fields surjectivity is a way more natural property that we would want associated with the function itself, I still don't see why it needs to be a triple... Just give the range and graph, the domain can be ddetermined from the graph.
magus145 ยท 1 points ยท Posted at 14:40:34 on December 6, 2015 ยท (Permalink)*
I mean, definitions are choices, and we could make different choices to formalize different aspects of intuitive ideas.
It's useful in set theory to have functions be distinct when they have distinct codomains, so that we can talk about a function being a bijection rather than it depending on external information (like what set you embed the range into). Universal mappings properties become WAY more difficult to formulate in the object language.
Similarly, yes, you could rederive the domain from the graph, but:
It's way more complicated to write a wff that does that rather than just project onto the first coordinate of an ordered triple, and
From the categorical perspective, the domain and codomain are philosophically prior to the function, so it makes sense to declare them both first in your collection of data that makes up a morphism.
Of course, the informal math won't change if you change the formalism in a consistent way. But for historical and formal reasons, these were the definitions that eventually became standard for convenience.
Cartesian product of what? Saying its a subset of a Cartesian product of X and Y is the same data as just giving X and Y. You still need to give X and Y and the ordered pairs -- three sets.
You might ask 'why not just give the ordered pairs?' but then you couldn't talk about surjective functions anymore.
docmedic ยท 1 points ยท Posted at 08:41:28 on December 6, 2015 ยท (Permalink)
Right, got that like 8 hours ago.
magus145 ยท 4 points ยท Posted at 00:40:05 on December 6, 2015 ยท (Permalink)
Let me give another example of a reasonable question that distinguished the two approaches.
How many functions from the reals to the reals are there?
Obviously infinitely many, but if you want a more precise cardinality answer, you'll need a more rigorous set theoretical definition of function. After all, there are only countably many "rules" you could write in English, but there are uncountable many such functions (in fact, beth2 which is a bigger cardinality than R).
magus145 ยท 3 points ยท Posted at 00:25:29 on December 6, 2015 ยท (Permalink)
Informally, no. And most people, even professional mathematicians writing rigorous proofs, are working informally.
Formally, in the sense of formal languages, mathematical logic, and foundations, we'd like to ground all of mathematics in a single framework so that we can compare their metamathematical relationships and properties. In the case of set theory, everything is a set: numbers, functions, operators, groups, spaces, etc. Everything has be be reducible (or at least interpretable) as a set. So functions need to be sets.
That doesn't mean that if you're, say, a number theorist, you need to worry about how exactly the set theorists are building the natural numbers in ZFC, but it's important that it can be done.
As for functions specifically, a rigorous definition is critical to actually understanding their properties. If you informally think of them as dynamic processes and rules, the second you start to do any functional analysis, you'll run into foundational existence and convergence issues.
[deleted] ยท 6 points ยท Posted at 03:42:08 on December 6, 2015 ยท (Permalink)
Sorry but this is not a subtlety of functions......
This how they are defined in set theory. This matters close to none for practicing mathematicians
magus145 ยท 4 points ยท Posted at 05:14:46 on December 6, 2015 ยท (Permalink)
Do most mathematicians care about the formal definition of an ordered pair and how a function is represented by a set? No.
Do most mathematicians care about the fact that a function has a domain, codomain, and graph, and that if you change any of them, you change the function? And furthermore, that any graph that passes the vertical line test qualifies? Absolutely! And that requires a formal definition of a function more rigorous than "a rule" or "a formula".
[deleted] ยท -2 points ยท Posted at 05:21:40 on December 6, 2015 ยท (Permalink)
why not explain that rather than the unenlightening set theoretic version?
it's important to mention that the function is itself the subset of X x Y, not just that it "consists of" those three things.
magus145 ยท 13 points ยท Posted at 00:14:09 on December 6, 2015 ยท (Permalink)
The standard definition (e.g. in Enderton) is that relations (and thus functions) are a single ordered triple of sets rather than just the subset of X x Y (which is often called the graph of the function). When we informally speak of elements of the relation, we mean elements of the graph, but the relation formally contains the extra data of the domain and codomain separately. Otherwise, the codomain would be formally indistinguishable from the range, and all functions would be surjrctive.
i guess it depends on your definition then. i learned the subset definition, but it makes sense to talk about it that way too.
AgAero ยท 1 points ยท Posted at 06:13:12 on December 7, 2015 ยท (Permalink)
So a function is not simply a mapping of elements of one set onto another one? Never heard it put quite that way. Interesting.
magus145 ยท 2 points ยท Posted at 12:53:48 on December 7, 2015 ยท (Permalink)
Informally, yes, it's a mapping of one set into another set.
Formally, every mathematical object needs to be encoded as a set, so we need a way to represent functions as sets so that we have a single framework to ask about existence, or to rigorously ground functions of functions.
The two big things for me were "set theory is a thing" and "functions are objects in their own right." Before that functions were just sort of these floaty magic things. Being able to make a set of Every Function over a single set... That's A Power I Never Thought Possible
[deleted] ยท 4 points ยท Posted at 22:39:03 on December 5, 2015 ยท (Permalink)
A function's just a specific case of a relation, bro!
So, a set is a small enough collection of things (most mathematicians never consider collections that are too big while doing math, so you probably won't either). A set X is a subset of a set Y if everything in X also belongs to Y. The product of two sets A and B is the set of all order pairs (a,b) with a in A and b in B. A relation R between A and B is a subset of A*B. We say that a is related to b (or aRb) if (a, b) is in R. A function f from A to B is a relation where each in A is related to exactly one thing in B. Instead of writing afb, we write f(a) = b.
[deleted] ยท 29 points ยท Posted at 19:39:42 on December 5, 2015 ยท (Permalink)
Except we don't expect high school students to understand much of any subtlety - quite the opposite in fact. Our standard math curriculum is more or less designed to impart a very rough understanding sufficient for basic physics and engineering courses. Students are expected to have a reasonable intuition for what a real number is, how functions work, etc. but are not at all expected to understand anything deep.
How functions work is deep: the concept of a function as an abstract rule, rather than a formula, took several hundred years for mathematicians to pin down. But we barely teach it: students often see nothing but elementary functions until calculus, when suddenly we expect them to actually understand how functions work.
It is deep, but we aren't teaching that deepness. That's what OP is trying to convey.
Speaking of deepness, have you guys read Mistborn? It's pretty good, but also lacks a lot of subtly, much like how we teach high school math. Sanderson's writing has definitely improved since then.
Yes, I'm saying that we don't teach that depth. The question is whether we expect students to understand that depth without being taught it, and the answer is that we do. The standard pre-calculus and calculus curriculum, formulaic as it is, typically includes the difference between definedness, continuity, and differentiability; expects students to understand domain, codomain, and range; and expects students to be able to use the chain rule and fundamental theorem of calculus on novel or arbitrary functions.
But none of those concepts make sense unless you actually understand what a function is. So while we're not really teaching functions, we are teaching things that depend on functions. (We then turn around and teach those topics half-heartedly and formulaically, because what else are we going to do with students who haven't been given the necessary background to understand it?)
[deleted] ยท 9 points ยท Posted at 23:08:42 on December 5, 2015 ยท (Permalink)*
These are sufficiently robust concepts, though, that one can use them very naively and find it very difficult to get into a logical jam. Arguably, this is why the modern definitions of functions, continuity, etc. were only settled upon in the 19th century: it takes work to find the edge cases where naive intuition breaks down.
A hand-wavy idea of a function as being an assignment of points in one set (pretty much always thought of as a subset of R) to points in another set (pretty much always thought of as a subset of R) is more or less sufficient to build a similarly hand-wavy understanding of continuity, limits, and differentiability which is entirely serviceable for most practical situations. And differentiating arbitrary functions is in practice more or less formal string manipulation; this is why differential algebra makes sense, after all. One can get away with a truly rudimentary understanding in that context.
Don't get me wrong - I think we do high school students a real disservice by never exposing them to the argumentation and proof of mathematics as an intellectual discipline rather than just a toolbox handed down from the Gods. But rather than expecting a subtle understanding of functions, we really expect the opposite: an extremely naive and intuitive understanding of functions and R, which to me at least seems very reasonable.
A hand-wavy idea of a function as being an assignment of points in one set (pretty much always thought of as a subset of R) to points in another set (pretty much always thought of as a subset of R) is more or less sufficient
The idea that a function is just an assignment of points in one set to points in another is a deep concept. Most students come in to calculus on the step before that, where a function is primarily a formula.
It took a good two hundred years for mathematicians to get completely away from thinking of functions as formulas to thinking of functions as assignments of points, but now we breeze over it like it's nothing and call that the "naive and intuitive understanding".
Excuse me kind sir, I don't understand why you're making it seem like functions are this deep, mysterious thing. I'm currently a student finishing my calculus 2 course with an A and I think functions are fairly intuitive and easy to understand. Am I missing something here? I feel like everyone learns what functions are since middle school.
Most students entering calculus 1 (and plenty of students leaving calc 1 and even calc 2) basically think of functions as being formulas. That is, they think of every function as coming with an explicit mathematical calculation; they may have some idea that functions are more general, but they haven't really internalized it yet. A classic calculus example is the characteristic function of the rationals---the function which is 1 on the rationals and 0 on the irrationals. Many calculus students have trouble viewing it as a function, and a great deal of trouble applying any of the concepts of calculus (like continuity and differentiability) to it.
If you were taught that since middle school, that's downright unusual. There are some really good schools out there, but it's not standardly taught (at least, not well enough to sink in) even at many good schools.
That may come easily to you. It comes easily to a lot of people with the right mathematical inclinations (and if you're getting an A in calculus 2, that may well include you). But it's important to remember that we're not representative of the bulk of students who take calculus.
Most students entering calculus 1 (and plenty of students leaving calc 1 and even calc 2) basically think of functions as being formulas. That is, they think of every function as coming with an explicit mathematical calculation; they may have some idea that functions are more general, but they haven't really internalized it yet. A classic calculus example is the characteristic function of the rationals---the function which is 1 on the rationals and 0 on the irrationals. Many calculus students have trouble viewing it as a function, and a great deal of trouble applying any of the concepts of calculus (like continuity and differentiability) to it.
Nah, not like that for me. I went to a public middle school, and, although they didn't explain the "real" definition of functions, they made sure that we internalized that the following function (as a series of coordinates):
(8, 10)
(9, 28)
(10, 30)
(11, pi)
was just as much of a function as f(x)=26x. No set theory, but good enough IMO for differentiability and whatnot. Piecewise functions in trig/pre-calc sort of required this level of understanding. I do wish they were more rigorous, but I definitely didn't have to redefine/define functions for myself in calc (we didn't do set theory there either :( ), and of what I understand, neither did anybody in my class. But then again, our district is weird, so this might be different from the rest of the US.
Edit: added an informative statement in parentheses.
Didn't a lot of stuff that is now regarded as "basic" like complex numbers, infinitesimals, transcendental numbers, set theory (with that "set of all sets" problem), go pretty much like in the comic for many decades?
I mean, I don't think this was nearly so controversial as, e.g. Cantor or Gรถdel. After all, Gauss was one of the first people to have his hands in it, and when Janos Bolyai sent him a letter, Gauss was basically like "meh I did that stuff like 10 years ago bruh".
claird ยท 14 points ยท Posted at 23:08:47 on December 5, 2015 ยท (Permalink)
Gauss was notoriously "... like 'meh I did that stuff like 10 years ago bruh'" with everything. The anecdotes which fit that description are several.
phafy ยท -4 points ยท Posted at 23:32:46 on December 5, 2015 ยท (Permalink)
The anecdotes which fit that description are several.
There are several anecdotes which fit that description.
TheFlying ยท 15 points ยท Posted at 20:13:54 on December 5, 2015 ยท (Permalink)*
You're right in a sense. Bolyai published the Tentamen in 1831 and according to this source the idea was popularized in 1868. It should be noted that is after Lobachevsky died, but the delay of acceptance is understandable considering difficulty of communication and subject matter.
The reason I was thinking of non-euclidean geometry was this heartbreaking letter from Farkas Bolyai to his son Janos which I could not find online but can be seen in one of my favorite texts Euclidean and non-Euclidean geometries Development and History by Marvin Jay Greenberg:
" You must not attempt this approach to paralells. I know this way to its very end. I have traversed this bottomless night, which extinguished all light and joy of my life. I entreat you, leave the science of parallels alone... I thought I would sacrifice myself for the sake of truth. I was ready to become a martyr who would remove the flaw from geometry and return it purified to mankind. I accomplished monstrous, enormous labors; my creations are far better than those of others and yet I have not yet achieved satisfaction... I turned back when I saw that no man can reach hte bottom of the night. I turned back unconsoled, pitying myself and all mankind.
I admit I expect little from the deviation of your lines. It seems to me that I have been in these regions; that I have traversed past all reefs of this infernal Dead Sea and have always come back with broken mast and torn sail. The ruin of my disposition and fall date back to this time. I thoughtlessly risked my life and happiness- aut Caesar aut nihil (either a Caesar or nothing)."
Well I would say so, but I have never spent my entire life devoting considerable resources and talent to a tree that bore no fruit. I can't imagine how difficult it must be to come to the point where you have nothing left to give and realized that you have never produced anything of value.
And he was writing this to his son, to scare him from the beast so it makes sense he might hyperbolize to show him he was serious.
[deleted] ยท 4 points ยท Posted at 22:51:25 on December 5, 2015 ยท (Permalink)
I think that is an inherent potential bottleneck to the progression of academic fields. Like could you imagine if something far more powerful and appropriate and just better in every way came along to replace the biological taxonomy? So many people have their lives invested in the old way, all the literature is in the old way, etc... resulting in a lot of resistance unrelated to the merits of the discovery itself.
Anyhow that's a marvelous bit of prose, thanks for posting it.
Basically, before the 'modern' idea of mathematics developed in the early 20th century, which took a very, very broad, inclusive, and general view of mathematics, it seems that most things were subject to this sort of debate. Thankfully, that has largely subsided now. E.g., everyone agrees that non-standard analysis is correct and valid, and functionally equivalent to standard analysis, and the only real debate surrounding it is whether it's useful in that regard. If it had shown up in, say, the 1860s instead of the 1960s, there may have been decades of debate as to whether it was 'correct' or just inane and worthless.
My signals and systems text covers the development of the Fourier Series and Transform over about a few generations of mathematicians. It played out pretty similar to this comic.
A lot of the concepts were pretty generally understood, but people argued quite a bit that you couldn't create a signal with discontinuities (like a square wave) with a series of sinusoids or complex exponential.
Yes, you are correct. You may be interested in this lecture by eminent logician Kurt Gรถdel. In it, he describes what he calls the "leftward (empirical) upheaval", in which the very things you cited, particular Russell's paradox, caused great divisiveness in the mathematical community. To quote him again, he says that, due to these "supposed contradictions", many mathematicians regarded "only a certain portion of mathematics, larger or smaller, according to their temperament" as representative of any truth. The rest was discarded as purely hypothetical.
Here is a reading of this same lecture, may be easier than reading it. It's nice for driving or while exercising, etc. I hope that you enjoy it =)
docmedic ยท 3 points ยท Posted at 22:31:43 on December 5, 2015 ยท (Permalink)
a lot of stuff
One, math wasn't agreed upon rigorously or standardized until somewhat recently, made worse because communication back then was slow. Two, math research and results encompass so much that even that lot of stuff is tiny.
So I think the comic doesn't describe how math works (as the title claims), just a select number of results, some of which happen to be huge (which of course would generate bigger backlash). How math works right now is pretty damn cordial.
More important to math history were philosophical positions such as intuitionism, logicism and formalism. Such issues used to be very hotly debated among mathematicians. Underlying these debates were results similar to the ones shown in the comic. For example, the existence of continuous functions that were not differentiable gave some 19th century mathematicians the fits. Charles Hermite famously said he rejected such findings with "fright and horror". Cantor's findings were even more severely rejected by some, though none could find flaws with his reasoning. The attacks really did become personal. All of this 19th century upheaval led to enormous introspection over math and its foundations. As the comic noted, some did really want to restructure math to make such results wrong, e.g. the debate over the axiom of choice. It may have been a time of "great insights", but it was not always a happy one, nor was their validity always readily accepted.
[deleted] ยท 3 points ยท Posted at 22:59:48 on December 5, 2015 ยท (Permalink)
As an aside, is it just me or is intuitionism/constructivism enjoying an uptick in popularity? Maybe I'm seeing it by way of computer science and buzz about type theory/HoTT
Interesting. The way I was told it was: The discovery of irrational numbers by the Pythagorean's was shocking to them. As such they valued this as a "secret" and were upset when one of them divulged this fact outside of their sect. There is a myth that as devine retribution for this divulgence, the one who divulged it was drowned at sea. https://en.wikipedia.org/wiki/Hippasus
You're correct, but I still see as similar to the situation in the comic. The reason it was valued as secret is because they outwardly preached that everything could be expressed as a ratio of whole numbers. The discovery of an irrational number conflicted with that notion, so they tried to bury it.
Wait, what? As an atheist who's largely ignorant of the philosophy of math, I would definitely at a glance think that math was discovered - that what was going on was pinning down underlying truths of the way the universe works. It seems bizarre to me that that would be conflated with theism any more than physics or chemistry or biology. Like, Darwin (and others later) discovered the mechanisms and rules of natural selection, but that definitely doesn't imply a theistic perspective...?
[deleted] ยท 5 points ยท Posted at 19:52:36 on December 5, 2015 ยท (Permalink)
[deleted]
claird ยท 2 points ยท Posted at 23:11:26 on December 5, 2015 ยท (Permalink)
... because it's too serious.
Said differently: practitioners, for the most part, live their daily lives with rather dramatic disregard for foundations.
Without having to mention creationism at all, we can still discuss whether math is discovered or invented. I mean, not in any seriously meaningful way (because it shouldn't matter). But there are decent reasons to believe that math, in parts, are indeed invented rather than discovered.
I would say that the axioms (necessary rules to describe any mathematical objects/structures) are chosen/invented. The consequences of axioms (theorems) are discovered.
jroot ยท 34 points ยท Posted at 17:09:18 on December 5, 2015 ยท (Permalink)
I resist your argument and deny it's truth. I shall take this stance with me to the grave. Also, Your mother's a whore.
How antithetical is it when I can think of 3 examples off the top of my head without even thinking hard?
Cantor Cardinals as mentioned above.
Definition of a Function.
Incompleteness.
Non-Euclidean geometries
McPhage ยท 19 points ยท Posted at 20:29:30 on December 5, 2015 ยท (Permalink)
Four... I can think of four examples: Cantor Cardinals, Definition of a Function, Incompleteness, Non-Euclidean Geometries, and the Existence of Zero. Five...
...Look, I'll come in again.
[deleted] ยท -5 points ยท Posted at 20:08:21 on December 5, 2015 ยท (Permalink)
Uhhhh, you should probably go re-read your mathematical history.
Godel and Cantor were both hounded their whole lives by mathematicians who wouldn't accept their proofs, solid though they were. If I remember correctly Cantor went crazy because of it and Godel became a recluse after Hilbert and others attacked him for pretty much decimating his program and what it stood for to its core (by showing that it would never be complete).
Same thing with non-Euclidean geometries although I don't remember the exact stories behind that.
Its nice to think that mathematicians and scientists are above such petty emotional squabbles and that we trust empirical proof above all else, but historically this has most definitely, 100%, not been the case and to argue otherwise is a weird position imo.
[deleted] ยท 7 points ยท Posted at 22:53:11 on December 5, 2015 ยท (Permalink)
No it was a great tool utilized most effectively first by Gauss and Riemann but had been around for quite a while. Thats the whole point of this comic, today these things are accepted without even thinking about it, but when new ideas like removing the parallel postulate are introduced, people get emotional about it and do push back (even if those conversations get lost in the history of time as people accept the proofs, as it should be)
However, as the history of Mathematics is not my specialty I can't remember the specific sources but I'm def sure its not as clear cut as you suggest where people just immediately accept a proof without emotion.
For Godel: most of what I know is from biographies of Godel and survey books on Incompleteness and various things I've read relating to the connection with the halting problem and Turing Machines that I'm sure have mentioned it. I could list about 20 books here, but I can't remember the specific sources.
As to the rest of my knowledge of mathematical history, most of it comes from the classic four volume set http://www.amazon.com/The-World-Mathematics-Four-Volume-Set/dp/0486432688 that I found for a great deal in a used book store a long long time ago. There are copious examples of new ideas and the push back generated due to emotional fixations beyond those offered in proofs.
Ideally I'd like to live in your world too in all kinds of scientific and rational endeavors, but that doesn't seem to be the way most of the world works and to pretend that mathematics is immune is ignoring a lot of history and human psychology imo.
Theorems with easily understood proofs are not subjective or controversial. The really interesting part, though, is deciding which axioms to include (like the axiom of choice) or exclude (the parallel postulate) or how things should be defined.
Edit: Instead of "easily understood", I really mean easily verified.
Yeah, I really thought the turn was going to be something along the lines of "oh wait, I am thinking of politics" or something like that. But, I guess it does historically follow.
Although I would argue that's not a very good joke if we need that historical context to appreciate it. Then again, they can't all be Wiener jokes.
That didn't stop Einstein from throwing quantum tantrums.
elenasto ยท 15 points ยท Posted at 01:48:52 on December 6, 2015 ยท (Permalink)
That was about an interpretation though, not about the theory itself. Einstein didn't dispute the quantum theory itself, but believed that there was an underlying mechanism which we are missing and which would get rid of the pesky uncertainties. He was perfectly fine using the mathematics involved. Also to be noted that bell's inequalities were after his death
ummwut ยท 1 points ยท Posted at 03:54:41 on December 6, 2015 ยท (Permalink)
Or other physicists over parity violation in weak interactions.
But who's to say the experiment is even correct, or useful? It's by no means as clear-cut as you make it out to be.
elenasto ยท 2 points ยท Posted at 01:51:50 on December 6, 2015 ยท (Permalink)
This is philosophical question and not something I wish to argue about here. I only wanted to say in my OP, that according to philosophy science follows, this would not work in physics.
You are welcome! I find the transition from formal mathematics (in the 19th c. sense of involving manipulation of formulas, or approximately the layperson idea of math) to the modern conceptual variety (roughly, Riemann onwards) a really exciting part of history. If there is anything that can usefully thought of as a paradigm-shift in math, I think it is this.
magus145 ยท 2 points ยท Posted at 16:30:49 on December 6, 2015 ยท (Permalink)
I'll second the thanks on this great read.
Stories and narratives purportedly from history, told by non-historians, are so often dangerously about now rather than then.
It's good to keep in mind that most "math history" you hear along the way is anecdotal at best, and purposefully propagandist at worst. Always do your research and don't trust all sources cough Bell cough.
In other news, Emmy Noether remains a BAMF.
MxM111 ยท 66 points ยท Posted at 16:41:22 on December 5, 2015 ยท (Permalink)
That's how the mathematicians work. The math is just fine.
Great. Now, all we need is to be able to do math without involving mathematicians in any way, and we'll be all set!
MxM111 ยท 3 points ยท Posted at 16:05:07 on December 6, 2015 ยท (Permalink)
Observing the development of Mathematica software, we are moving in that direction.
Vtempero ยท 10 points ยท Posted at 16:46:09 on December 5, 2015 ยท (Permalink)
How exactly any model of the world exists without people caring about it.
jlink005 ยท 16 points ยท Posted at 21:24:00 on December 5, 2015 ยท (Permalink)
"A scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die and a new generation grows up that is familiar with it." -Max Planck
And has recently started adding hover-over jokes (a la XKCD) as well. Three jokes for the price of one
B-Con ยท 13 points ยท Posted at 21:49:39 on December 5, 2015 ยท (Permalink)
That's recent? Oh good. I've been reading it for years (since it started?) and had noted very distinctly that it didn't have hover-over jokes on the comic itself. It did eventually do the red button hover-over, so I thought that was the equivalent.
Then very recently I happened to hover the mouse over the comic and saw some alt text. I didn't know when it started and was mortified that maybe I'd missed years worth.
Meapalien ยท 12 points ยท Posted at 21:55:57 on December 5, 2015 ยท (Permalink)*
More funny comics by both absolute number and percent than xkcd...
Apothsis ยท 41 points ยท Posted at 21:49:09 on December 5, 2015 ยท (Permalink)
Having witnessed an actual fistfight between my Advisor when I was a Candidate, and his rival, over a conjecture that they have been arguing about for 30 years, I can confirm, this is exactly how Math works.
jzapate ยท 17 points ยท Posted at 22:53:16 on December 5, 2015 ยท (Permalink)
Do you remember the conjecture?
Apothsis ยท 23 points ยท Posted at 23:23:00 on December 5, 2015 ยท (Permalink)
Not really. I think it really reduced to "FUCK YOU! FUCK YOU AND DIE!" more than anything usable to the Body of Knowledge.
[deleted] ยท 11 points ยท Posted at 23:06:42 on December 5, 2015 ยท (Permalink)
I remember reading about math challenges between the 1500's Italians that had some serious consequences (like losing a professorship etc).
mearco ยท 6 points ยท Posted at 20:27:57 on December 6, 2015 ยท (Permalink)
Yeah professors basically had to verify their mathematical abilities every few years through challenges, like solving cubic equations. This led to them not disclosing their methods so they had less competition.
[deleted] ยท 6 points ยท Posted at 01:52:34 on December 6, 2015 ยท (Permalink)*
[deleted]
Apothsis ยท 8 points ยท Posted at 01:56:23 on December 6, 2015 ยท (Permalink)
You have not lived until you have seen a gathering of "Great minds" shoving each other and throwing extremely ineffective punches.
I love this comic, but can I be the pedant on this thread for the sake of the college algebra and calculus 1 people who will see this and use it to excuse their own misunderstanding?
The mathematical formula you are being taught in class is nothing like what the proof of the rigorous mathematics behind it. Mathematicians didn't just come up with the formula for a derivative out of thin air, and plug in different values of x.
The development of mathematics is a beautiful subject that is heavily trivialized in early math classes. Don't worry about not understanding a formula right up front, but the worst thing you can do (concerning your grades and your understanding of math) is to learn mathematics as blind symbol manipulation.
Me neither. The problem is that this is just really far from my perception of how mathematical research works nowadays. It should rather be called "How math used to work."
thismynewaccountguys ยท 202 points ยท Posted at 16:49:28 on December 5, 2015 ยท (Permalink)
Cantor and the cardinal numbers being a prime example.
[deleted] ยท 72 points ยท Posted at 17:40:52 on December 5, 2015 ยท (Permalink)
Prime cardinals. Now there's a thought
UniformCompletion ยท 62 points ยท Posted at 17:45:20 on December 5, 2015 ยท (Permalink)
c2 = c for any infinite cardinal, so the prime cardinals are just the usual primes.
[deleted] ยท 30 points ยท Posted at 00:32:37 on December 6, 2015 ยท (Permalink)
I say your mother's a whore
With Loathing,
/u/pt2091
[deleted] ยท 38 points ยท Posted at 18:58:01 on December 5, 2015 ยท (Permalink)
I swear, for a sec I read that as (Speed of Light)2 = (Speed Of Light), before remembering that you were talking about infinite cardinals.
SKRules ยท 71 points ยท Posted at 21:19:33 on December 5, 2015 ยท (Permalink)
Just work in natural units and you're fine.
caks ยท 59 points ยท Posted at 22:20:48 on December 5, 2015 ยท (Permalink)
Classic mathematician ignoring dimensional analysis.
vingnote ยท 49 points ยท Posted at 00:13:57 on December 6, 2015 ยท (Permalink)
It's funny because your flairs should be exchanged.
zanotam ยท 7 points ยท Posted at 23:39:17 on December 5, 2015 ยท (Permalink)
Just do it at the end to add further reasons why the thing you're not sure actually works totally should work because you even checked the units.
jimprovost ยท -1 points ยท Posted at 15:50:43 on December 6, 2015 ยท (Permalink)
... Found the engineer.
[deleted] ยท 3 points ยท Posted at 16:20:49 on December 6, 2015 ยท (Permalink)
Physics major, actually.
[deleted] ยท 3 points ยท Posted at 22:34:34 on December 5, 2015 ยท (Permalink)
Touche2
PthariensFlame ยท 2 points ยท Posted at 18:20:21 on December 5, 2015 ยท (Permalink)*
Now, hold on. You could define a prime cardinal as a cardinal that has no factors other than one and itself. Then, in particular,
[; \aleph_0 ;]is prime, and I'm sure there are others.EDIT: Well, I'm dumb. See the replies. :(
tfaal ยท 18 points ยท Posted at 18:33:37 on December 5, 2015 ยท (Permalink)
Aleph_0 is not prime since Aleph_0 = Aleph_0*2, and thus 2 is a factor of Aleph_0.
magus145 ยท 8 points ยท Posted at 20:25:54 on December 5, 2015 ยท (Permalink)
If instead you define prime cardinals to be the ones that can't be expressed as the product of a finite number of strictly smaller cardinals (a different, reasonable generalization), then (in ZFC) every infinite cardinal is prime!
[deleted] ยท 6 points ยท Posted at 22:33:26 on December 5, 2015 ยท (Permalink)
So then let's try to think of a formulation of prime for infinite cardinals which doesn't give a trivial result.
UniformCompletion ยท 14 points ยท Posted at 23:34:24 on December 5, 2015 ยท (Permalink)
Under the presence of the axiom of choice, the cardinals have a very trivial multiplicative structure, since the product of two cardinals is just their maximum unless they are both finite. So a study of multiplication of cardinals boils down to studying chains, and we have ideas like limit cardinals but nothing that is really anything like arithmetic.
As I just mentioned in another comment, there may be some nontrivial things we can formulate about cardinal products without the axiom of choice, but we will never be able to exhibit any examples.
[deleted] ยท 2 points ยท Posted at 22:03:14 on December 5, 2015 ยท (Permalink)
That requires the axiom of choice, IIRC.
UniformCompletion ยท 2 points ยท Posted at 23:14:17 on December 5, 2015 ยท (Permalink)
So there might be a theory of prime cardinals with the absence of the axiom of choice, but it still won't be possible to exhibit any non-trivial ones.
jtaentrepreneur ยท -3 points ยท Posted at 19:35:17 on December 5, 2015 ยท (Permalink)
Isn't c only 1 then? I don't understand how that's infinite.
arnet95 ยท 17 points ยท Posted at 19:41:21 on December 5, 2015 ยท (Permalink)
If c is an infinite cardinal, then c2 = c. It happens to also hold for c = 1. If you don't know what an infinite cardinal is, take a look here.
F-0X ยท 1 points ยท Posted at 14:40:10 on December 6, 2015 ยท (Permalink)
It's becasue everything is one, man.
UniformCompletion ยท 6 points ยท Posted at 19:47:57 on December 5, 2015 ยท (Permalink)
c2 = c for an infinite cardinal c means that, for any infinite set S, there is a bijection between S and SรSโ.
Since a cardinal number is the "size" of a set, this is the usual convention for multiplying cardinal numbers.
Yakone ยท 2 points ยท Posted at 01:32:18 on December 6, 2015 ยท (Permalink)
We define cardinal arithmetic for cardinals. It's the same for the small numbers but extends to cardinals of all sizes.
[deleted] ยท 0 points ยท Posted at 19:41:32 on December 5, 2015 ยท (Permalink)
c2 = c does not necessarily imply that c = 1, c could also be 0. But both of these inferences only make sense in the context of the real numbers, and the set of real numbers doesn't include any infinite elements.
UlyssesSKrunk ยท -13 points ยท Posted at 19:36:18 on December 5, 2015 ยท (Permalink)
c2 = c for c ∈ {1}
โด 1 is an infinite cardinal
QED
bowtochris ยท 3 points ยท Posted at 20:17:23 on December 5, 2015 ยท (Permalink)
I think that prime ordinals are a more reasonable idea. Then, for any limit ordinal a, a + 1 would be prime and a + p would be prime for any prime p. Not sure if that's all of them.
magus145 ยท 6 points ยท Posted at 22:11:28 on December 5, 2015 ยท (Permalink)
https://en.wikipedia.org/wiki/Ordinal_arithmetic#Factorization_into_primes
There are others, and a + p isn't prime since (a + 1)(p) is a nontrivial factorization.
faore ยท 37 points ยท Posted at 01:02:30 on December 6, 2015 ยท (Permalink)
In fact this is essentially the only example. The comic is kind of irritating because more than any scientist, a mathematician can actually revolutionise his beliefs very quickly by reading a single proof.
[deleted] ยท 7 points ยท Posted at 13:01:03 on December 6, 2015 ยท (Permalink)
Nowadays, yes. That's mostly thanks to the modern mindset that as long as your definitions/axioms are all logically consistent and your proofs are all sound, there is no such thing as being "incorrect" in mathematics. It's an idea we take for granted, but it only really started taking shape in the 19th century, and it didn't fully solidify until the 20th century.
thismynewaccountguys ยท 7 points ยท Posted at 01:47:44 on December 6, 2015 ยท (Permalink)
I can't think of any other examples but are there definitely no more?
faore ยท 6 points ยท Posted at 02:49:26 on December 6, 2015 ยท (Permalink)
I'm sure there are others but none many people have heard of
Vonbo ยท 3 points ยท Posted at 16:59:09 on December 6, 2015 ยท (Permalink)
Greeks (specifically, the Pythagorean school) around 500 BC believed every number could be expressed as a ratio of two integers. According to legend, Hippasus, a member of the pythagorean school, proved the irrationality of โ2 while on a sea trip. His fellow pythagoreans then threw him overboard for betraying their beliefs, where he drowned.
This story is not factual though, and most likely just a tale with a bit of truth to it.
dfqteb ยท 3 points ยท Posted at 20:46:38 on December 6, 2015 ยท (Permalink)
But this is hardly relevant to the current culture within the academic maths community.
goodcleanchristianfu ยท 2 points ยท Posted at 16:12:03 on December 6, 2015 ยท (Permalink)
According to Mandelbrot, his work was treated as mathematically irrelevant cool art until the Fractal Geometry of Nature came out.
00zero00 ยท 1 points ยท Posted at 20:22:33 on December 6, 2015 ยท (Permalink)*
https://en.wikipedia.org/wiki/Dan_Shechtman#Work_on_quasicrystals
InSearchOfGoodPun ยท 0 points ยท Posted at 20:44:19 on December 6, 2015 ยท (Permalink)
I completely agree. I have yet to see anyone give an interesting and important example of this phenomenon other than the creation of set theory. It's a stupid comic.)
(And even the example of Cantor seems rather overblown.)
Tiervexx ยท 29 points ยท Posted at 00:55:39 on December 6, 2015 ยท (Permalink)
Yeah, but set theory was just about the only thing that really shocked the math world that badly. In general, mathematicians are much better than other intellectuals at accepting new proofs.
almightySapling ยท 15 points ยท Posted at 09:25:52 on December 6, 2015 ยท (Permalink)
Didn't someone pitch a fit when Weierstrass demonstrated his nowhere differentiable continuous function? Basically was like "nah man, that's dumb so it doesn't count".
Tiervexx ยท 3 points ยท Posted at 17:09:17 on December 6, 2015 ยท (Permalink)*
I could be mistaken, but I was under the impression that was mostly a delightful shock, rather than a "OMG THE WORLD IS BURNING!" Maybe an individual mathematician lost their mind but as a whole I don't think it shocked them that bad.
almightySapling ยท 1 points ยท Posted at 20:15:52 on December 6, 2015 ยท (Permalink)
Yeah, it was an individual, I just don't remember who. Math history isn't my forte.
akjoltoy ยท 19 points ยท Posted at 04:35:36 on December 6, 2015 ยท (Permalink)
Yeah that's because of math. Not mathematicians.
Mathematicians are just as petty as any scientists.
It's just that math is purely abstract. That makes it more malleable.
Tiervexx ยท 2 points ยท Posted at 17:09:38 on December 6, 2015 ยท (Permalink)
agreed.
HarlequinNight ยท 9 points ยท Posted at 17:34:19 on December 5, 2015 ยท (Permalink)
Don't bring primes into this!
BitchinTechnology ยท -1 points ยท Posted at 22:09:32 on December 5, 2015 ยท (Permalink)
Really? Aren't those just positive whole numbers?
thismynewaccountguys ยท 20 points ยท Posted at 23:37:58 on December 5, 2015 ยท (Permalink)
Cardinal numbers also include 'transfinite numbers' such as the size of the set containing all the natural numbers and the size of the set containing all the real numbers. When Cantor first came up with the idea and proved for instance, that there are more real numbers than natural numbers (colloquially, that there are different sizes of infinity), he was met with resistance and scorn from much of the mathematical establishment in particular Kronecker. Cantor had a really tough time of it and suffered severe bouts of depression as a result and ended up dying impoverished. Of course, now these ideas are some of the first things you will learn in an elementary set theory class in university.
[deleted] ยท 2 points ยท Posted at 22:22:27 on December 5, 2015 ยท (Permalink)
He's referring to infinite cardinals
[deleted] ยท 28 points ยท Posted at 23:44:44 on December 5, 2015 ยท (Permalink)
Seeing "Your mother's a whore" written in flowy cursive was the funniest thing to happen to me today.
Eurynom0s ยท 3 points ยท Posted at 13:06:50 on December 6, 2015 ยท (Permalink)
Sean Connery, Oxford mathematician.
[deleted] ยท 381 points ยท Posted at 16:39:10 on December 5, 2015 ยท (Permalink)
I feel like this would have been funnier, and more accurate, with modern physics replacing math.
redrumsir ยท 124 points ยท Posted at 16:50:22 on December 5, 2015 ยท (Permalink)
Agreed. It's not really true with math. It's nearly antithetical to math and I would argue that stronger if it hadn't taken so long for the notion of limits to be developed (which did settle quite a few disagreements).
le_epic ยท 212 points ยท Posted at 17:38:03 on December 5, 2015 ยท (Permalink)
Didn't a lot of stuff that is now regarded as "basic" like complex numbers, infinitesimals, transcendental numbers, set theory (with that "set of all sets" problem), go pretty much like in the comic for many decades? I guess infinitesimals fall under the "solved with the notion of limits" category. (I'm just a curious layman with only the vaguest notions of both History and Math, I might be completely wrong, it's a genuine question.)
elseifian ยท 137 points ยท Posted at 18:03:29 on December 5, 2015 ยท (Permalink)
Not just the fancy stuff, the definition of a function took centuries to get right, but we expect high school students to just understand the subtleties without even teaching the concept directly.
Roller_ball ยท 78 points ยท Posted at 19:36:06 on December 5, 2015 ยท (Permalink)
Even the concept of 0 took a pretty long time.
PostFunktionalist ยท 40 points ยท Posted at 19:05:46 on December 5, 2015 ยท (Permalink)
Yeah what's the fuckin deal with that. Took me 'til college to learn what functions and the number systems were.
UlyssesSKrunk ยท 100 points ยท Posted at 19:52:17 on December 5, 2015 ยท (Permalink)
You guys are making me worried. What subtleties of the definition of a function do I not know? WHAT AREN'T THEY TELLING ME?
magus145 ยท 86 points ยท Posted at 20:21:28 on December 5, 2015 ยท (Permalink)
A function is a set F consisting of a triple of sets: a domain X, a codomain Y, and a subset of X x Y that has exactly one element of the form (x,y) for each x in X.
This definition treats functions as static objects (sets) rather than somehow dynamic processes, and so allows rigorous analysis of sets of functions, or letting the domain elements be other functions.
It also makes rigorous the intuitive concept of what counts as a "rule", which is important if you want things like nowhere continuous functions or weirder things like non-computable functions.
saarl ยท 15 points ยท Posted at 22:48:50 on December 5, 2015 ยท (Permalink)
Why is it a set containing a triple of sets and not just a triple of sets?
AllNewSadness ยท 23 points ยท Posted at 23:41:17 on December 5, 2015 ยท (Permalink)
It is just a triple of sets. Some people (like me) prefer to just think of as a set of ordered pairs, without the triple part with the domain and codomain.
magus145 ยท 26 points ยท Posted at 00:15:32 on December 6, 2015 ยท (Permalink)
This is a bad way to formalize functions as it becomes impossible in the object language to define surjectivity.
AllNewSadness ยท 2 points ยท Posted at 05:08:19 on December 6, 2015 ยท (Permalink)
This is the standard way in set theory, where it's much more convenient. In other areas the definition with triples is probably better. However, putting the domain in the definition seems completely redundant to me.
DR6 ยท 3 points ยท Posted at 08:55:19 on December 6, 2015 ยท (Permalink)*
It's not redundant at all. f(x) = x2 with domain R and domain N are drastically different functions: with domain N it's injective and monotonically increasing, with domain R it isn't.
almightySapling ยท 4 points ยท Posted at 09:34:06 on December 6, 2015 ยท (Permalink)
But it is. The definition of a function survives just fine as a collection of ordered pairs (x,y) such that no two have the same x value.
It is true that f:N->N by f(x)=x2 is very different than f:R->R by the same formula... but they are also completely different as sets. In practice, it is useful to know and state a donain and range explicitly, but "behind the scenes" the ordered pairs tell you everything you need to know unambiguously.
DR6 ยท 4 points ยท Posted at 09:44:36 on December 6, 2015 ยท (Permalink)
... well, you can make the domain redundant like that, but you still need the codomain. Taking again f(x) = x2 , f : R -> R+ and f : R -> R are again different functions with different properties(one is surjective, the other isn't) and you can't tell from the ordered pairs alone.
almightySapling ยท 3 points ยท Posted at 09:50:27 on December 6, 2015 ยท (Permalink)
Yeah, I just mentioned elsewhere that I personally don't see surjectivity as a property of the function itself but rather as a property of the pair (f,Y), such that a function is surjective onto its image but not surjective onto any superset. I like the malleability there, no reason to restrict the function to one range.
AllNewSadness ยท 1 points ยท Posted at 17:02:58 on December 6, 2015 ยท (Permalink)
Yap. I actually did write "However, putting the domain in the definition seems completely redundant to me".
AllNewSadness ยท 1 points ยท Posted at 17:03:40 on December 6, 2015 ยท (Permalink)*
You're considering a function as a rule here, not a set of pairs. But I see /u/almightySapling already explained this
guilleme ยท 4 points ยท Posted at 01:57:44 on December 6, 2015 ยท (Permalink)
Agreed with this!! Also, the property of the set being ordered is rather important and makes intuitive why something like x = y2 + 3 is not a function (although it is a well-founded relation (actually a pretty standard conic), so something like wolfram alpha or desmos.com should be able to plot it).
Think of it like this: a function is a set of pairs of numbers (x,y) where there is only one (one and only one) pair that has an specific number on the place of the 'x'. This allows us to speak about the order of the pairs of the function.
magus145 ยท 6 points ยท Posted at 00:09:05 on December 6, 2015 ยท (Permalink)
Well, we want a function to be a single object, and in set theory, the only objects are sets. So it IS a triple of sets, but a triple of sets is itself, a set. (And not just a set of the three elements since we really want it to be an ordered triple of sets.)
docmedic ยท 5 points ยท Posted at 22:36:09 on December 5, 2015 ยท (Permalink)
Out of curiosity, is there something inherently illogical/ill-defined about defining functions in the dynamic process manner, if one's goal did not include discussions about nowhere continuous or non-computable functions?
ZirconCode ยท 7 points ยท Posted at 23:45:00 on December 5, 2015 ยท (Permalink)
I think mostly the point is that we can define functions in terms of sets, which are define by even lower levels all the way down do ZF(C) and L.PA, logic and such. It makes for a better model. It would be like defining 2 as a separate number rather than as 1+1 (or the number following 1).
docmedic ยท 3 points ยท Posted at 00:14:53 on December 6, 2015 ยท (Permalink)
I see. Rereading the definition of function from wikipedia:
is it really necessary to specify that F is a triplet of sets, rather than just a subset of a Cartesian product satisfying that one condition?
magus145 ยท 3 points ยท Posted at 00:32:02 on December 6, 2015 ยท (Permalink)
Formally, yes, as I explained earlier. If a function is just a set of ordered pairs, how do you write a well formed formula in set theory to identify the codomain? After all, the element (1,1) is in the set R x R as well as R x R+.
So how else could you formally define surjective functions?
ZirconCode ยท 2 points ยท Posted at 00:46:18 on December 6, 2015 ยท (Permalink)
But the set R x R+ is in R x R so wouldn't you have the same problem? Couldn't you say the element would be surjective on (1,1) and not (1,1...2) for example?
magus145 ยท 2 points ยท Posted at 03:07:10 on December 6, 2015 ยท (Permalink)
I'm not sure I understand your notation. R x R+ is a subset of R x R, not an element. A function can't be surjective on a single point, and I have no idea what (1, 1...2) is supposed to mean.
ZirconCode ยท 1 points ยท Posted at 10:52:08 on December 6, 2015 ยท (Permalink)
Right, sorry about that. You could say The function with element (1,1) would be surjective on {1} x {1} but not {1} x {1,2}. So just as {1} subset of {1,2}, so is R+ with R. Why can't we define surjective to make sense this way, rather than have to assign a couple of extra sets to a functions definition? What am I missing here?
Thanks =)
magus145 ยท 2 points ยท Posted at 14:35:41 on December 6, 2015 ยท (Permalink)
You could, but it depends on what properties of functions you want to be intrinsic.
If you define a function to just be its set theoretic graph, then being a bijection (i.e., being invertible) is no longer an intrinsic property of functions but now depends on what set you embed their ranges into.
This makes cardinality arguments way more laborious as well as formalizing notions in category theory, where the codomain is prior to the Hom Sets.
ZirconCode ยท 1 points ยท Posted at 14:56:14 on December 6, 2015 ยท (Permalink)
Ah, now I see what you mean, thanks for taking your time to explain!
magus145 ยท 2 points ยท Posted at 15:17:03 on December 6, 2015 ยท (Permalink)
No problem! Thanks for sticking it out and being polite!
Krestiv ยท 2 points ยท Posted at 13:20:47 on December 6, 2015 ยท (Permalink)
I don't see why one would need to define surjectivity as a function property though. How would one formally define continuous functions? Even if you include the domain and the codomain in the definition of a function, you have no way of telling whether it is continuous for you haven't included the topology yet.
You may have a function which maps certain elements to certain elements. Then you may state "If the codomain has this topology, then this function is continuous." in the same way you may state "If this is the entire codomain, then the function is surjective."
Or, you could try to include the topology, if available, in the function itself. Then you still wouldn't know whether the function is monotonically increasing since you don't have an order on the domain and codomain. Like this, there are many properties of functions that depend on more than just what the codomain set itself is.
So why should surjectivity get special treatment over the other bazillion properties? We could just define f(x) = x2 as a subset of R x R, the simplest way possible. Then f would be surjective from R to R+ and not surjective from R to R.
magus145 ยท 2 points ยท Posted at 15:14:27 on December 6, 2015 ยท (Permalink)*
Of course you could define "function" that way, since what you described is what in standard set theory is called "the graph of the function", and I've been denoting G, the third element of my ordered triple that makes up a function. It's a perfectly well defined set.
Why shouldn't we use the word "function" for that set?
Let me answer from a categorical perspective. For now, I'll just consider concrete categories, so our objects are all "sets with extra structure" and our morphisms are all "set functions with extra properties".
Your question is essentially "Why should the morphisms in the category of Set include the data of domain and codomain?" There is a "top down" answer and a "bottom up" answer.
In category theory, the domain and codomain objects are philosophically prior to the morphisms between them. That is, every morphism lives in a Hom Set, Hom(A,B), so you can't even talk about a function without first taking about which Hom set it lives in. Since this perspective has been so useful in unifying and clarifying the concepts of maps with stricture, it's both convenient and philosophically satisfying to build this data straight into the set theory itself.
So what is a topological space? From this perspective, it's a set with the additional structure of an associated set of subsets called a topology. What are the appropriate morphisms that preserve this structure? Continuous maps. What then become invariants of this category, i.e., what properties should be the same when two topological spaces are homeomorphic but not literally represented by the same set? Things like compactness, connectedness, separability properties, etc.
Similarly, in group theory, the objects are groups, the maps are group homomorphisms, and the invariants are things like order, simplicity, being nilponent, being hyperbolic, etc.
Now, consider set theory, i.e., the category Set. It's the most basic concrete category as all other ones have a forgetful functor into Set. So its objects are just called sets, its morphisms "set functions" or just "functions". Pick a different definition of "function" and you get a different category with different invariants.
So what ARE the invariants of set theory? When should two sets be considered isomorphic (basically the same up to relabeling) when they aren't literally the same set? What invariant should be an invariant for ALL mathematical objects (in concrete categories)?
The answer is cardinality. The notion of a size of a set should be the only invariant of Set. It should be an invariant of every other concrete category (Top, Grp, Met, ...), and set functions should preserve it. In order for this to happen, invertibility (i.e. bijectivity) needs to be a concept intrinsic to the set function.
This is why bijectivity and hence surjectivity is more foundational to functions than continuity or other properties that depend on other structure. The only structure on a set is the containment binary relation, and the only invariant defined just from this structure is cardinality.
So functions with different codomains but identical graphs need to be different functions, or you would lose even this, and have no category at all (composition wouldn't be well-defined).
docmedic ยท 1 points ยท Posted at 03:19:56 on December 6, 2015 ยท (Permalink)
I see, so what you want is for f != g in the set sense if they're not the same function. So rather than associating with f two sets and an element of a cartesian product, you set every element of f to contain that information. That is, let X and Y be sets. Then f = {(X, Y, (x, y)) | x in X and y in Y} with the usual restriction on (x,y) is a function. So if the domain or codomain differ, then no element of either set will be in the other.
magus145 ยท 1 points ยท Posted at 04:57:23 on December 6, 2015 ยท (Permalink)
You've got the conceptual idea right exactly, but that set formalism is wrong. You need an ordered triple of sets (X, Y, G), where G = {(x,y) | x in X} is a subset (not element) of X x Y.
Using the standard way of encoding ordered triples, it would literally look like:
f = {{{{X}, {X,Y}}},{{{X}, {X,Y}},G}}
Of course, that looks so ugly, and so we define the notation (X, Y, G) as soon as possible.
docmedic ยท 1 points ยท Posted at 05:14:04 on December 6, 2015 ยท (Permalink)*
Fair enough, but it should still be possible to write as
f = {(X, Y, (x,y))| (x,y) in G a subset of X x Y with the usual property}
which seems to be more explicit on what elements of f consists of, if I understand correctly.
Or is it in set theory that functions are singletons represented by a triplet of sets (X,Y,G)?
magus145 ยท 1 points ยท Posted at 05:43:57 on December 6, 2015 ยท (Permalink)
You wouldn't want to define it that way because then the points on the graph of the function would be in the same set as the entire domain. Then if you said something like "For each element of f....", you'd need to specifically exclude two points that shouldn't be in there. It's a category error.
As for your second point, I think the answer is "yes". Set theoretically, a function IS the ordered triple (X,Y,G) (which is itself a set).
It's not a singleton though, because an ordered triple is not encoded that way.
docmedic ยท 1 points ยท Posted at 06:28:46 on December 7, 2015 ยท (Permalink)
I guess I don't see it. For instance, f(x) = 3x from R to R has the element (R, R, (0,0)), specifying the domain, codomain, and a point. Which two points aren't on that function?
If we consider f = (R, R, G), then wouldn't an element specify an element of R, R, and G, like (3, 4, (0,0))?
magus145 ยท 2 points ยท Posted at 13:00:09 on December 7, 2015 ยท (Permalink)
Ah, I see the confusion. The function does not consist of ordered triples, rather it is a SINGLE ordered triple (X, Y, G), where G is itself a set consisting of all the ordered pairs.
So f: R -> R, f(x) = 3x gets encoded as f = (R, R, {(0, 0), (1,3), (2,6), (pi, 3pi), ...}).
So a single point on the function still refers to an ordered pair on the graph, i.e., in the set G, like (2,6).
Only the function itself, globally, is encoded as the ordered triple of data. Each point in the function is still a regular old ordered pair of elements.
almightySapling ยท 1 points ยท Posted at 09:45:31 on December 6, 2015 ยท (Permalink)
I guess personally surjectivity is something I don't care about to much. No reason I can't talk about the image of f living inside of any superset, why need to restrict to one of them?
But, I can see in most fields surjectivity is a way more natural property that we would want associated with the function itself, I still don't see why it needs to be a triple... Just give the range and graph, the domain can be ddetermined from the graph.
magus145 ยท 1 points ยท Posted at 14:40:34 on December 6, 2015 ยท (Permalink)*
I mean, definitions are choices, and we could make different choices to formalize different aspects of intuitive ideas.
It's useful in set theory to have functions be distinct when they have distinct codomains, so that we can talk about a function being a bijection rather than it depending on external information (like what set you embed the range into). Universal mappings properties become WAY more difficult to formulate in the object language.
Similarly, yes, you could rederive the domain from the graph, but:
It's way more complicated to write a wff that does that rather than just project onto the first coordinate of an ordered triple, and
From the categorical perspective, the domain and codomain are philosophically prior to the function, so it makes sense to declare them both first in your collection of data that makes up a morphism.
Of course, the informal math won't change if you change the formalism in a consistent way. But for historical and formal reasons, these were the definitions that eventually became standard for convenience.
bananasluggers ยท 1 points ยท Posted at 06:18:21 on December 6, 2015 ยท (Permalink)
Cartesian product of what? Saying its a subset of a Cartesian product of X and Y is the same data as just giving X and Y. You still need to give X and Y and the ordered pairs -- three sets.
You might ask 'why not just give the ordered pairs?' but then you couldn't talk about surjective functions anymore.
docmedic ยท 1 points ยท Posted at 08:41:28 on December 6, 2015 ยท (Permalink)
Right, got that like 8 hours ago.
magus145 ยท 4 points ยท Posted at 00:40:05 on December 6, 2015 ยท (Permalink)
Let me give another example of a reasonable question that distinguished the two approaches.
How many functions from the reals to the reals are there?
Obviously infinitely many, but if you want a more precise cardinality answer, you'll need a more rigorous set theoretical definition of function. After all, there are only countably many "rules" you could write in English, but there are uncountable many such functions (in fact, beth2 which is a bigger cardinality than R).
magus145 ยท 3 points ยท Posted at 00:25:29 on December 6, 2015 ยท (Permalink)
Informally, no. And most people, even professional mathematicians writing rigorous proofs, are working informally.
Formally, in the sense of formal languages, mathematical logic, and foundations, we'd like to ground all of mathematics in a single framework so that we can compare their metamathematical relationships and properties. In the case of set theory, everything is a set: numbers, functions, operators, groups, spaces, etc. Everything has be be reducible (or at least interpretable) as a set. So functions need to be sets.
That doesn't mean that if you're, say, a number theorist, you need to worry about how exactly the set theorists are building the natural numbers in ZFC, but it's important that it can be done.
As for functions specifically, a rigorous definition is critical to actually understanding their properties. If you informally think of them as dynamic processes and rules, the second you start to do any functional analysis, you'll run into foundational existence and convergence issues.
[deleted] ยท 6 points ยท Posted at 03:42:08 on December 6, 2015 ยท (Permalink)
Sorry but this is not a subtlety of functions......
This how they are defined in set theory. This matters close to none for practicing mathematicians
magus145 ยท 4 points ยท Posted at 05:14:46 on December 6, 2015 ยท (Permalink)
Do most mathematicians care about the formal definition of an ordered pair and how a function is represented by a set? No.
Do most mathematicians care about the fact that a function has a domain, codomain, and graph, and that if you change any of them, you change the function? And furthermore, that any graph that passes the vertical line test qualifies? Absolutely! And that requires a formal definition of a function more rigorous than "a rule" or "a formula".
[deleted] ยท -2 points ยท Posted at 05:21:40 on December 6, 2015 ยท (Permalink)
why not explain that rather than the unenlightening set theoretic version?
teganandsararock ยท 5 points ยท Posted at 00:02:23 on December 6, 2015 ยท (Permalink)
it's important to mention that the function is itself the subset of X x Y, not just that it "consists of" those three things.
magus145 ยท 13 points ยท Posted at 00:14:09 on December 6, 2015 ยท (Permalink)
The standard definition (e.g. in Enderton) is that relations (and thus functions) are a single ordered triple of sets rather than just the subset of X x Y (which is often called the graph of the function). When we informally speak of elements of the relation, we mean elements of the graph, but the relation formally contains the extra data of the domain and codomain separately. Otherwise, the codomain would be formally indistinguishable from the range, and all functions would be surjrctive.
teganandsararock ยท 1 points ยท Posted at 06:12:45 on December 7, 2015 ยท (Permalink)
i guess it depends on your definition then. i learned the subset definition, but it makes sense to talk about it that way too.
AgAero ยท 1 points ยท Posted at 06:13:12 on December 7, 2015 ยท (Permalink)
So a function is not simply a mapping of elements of one set onto another one? Never heard it put quite that way. Interesting.
magus145 ยท 2 points ยท Posted at 12:53:48 on December 7, 2015 ยท (Permalink)
Informally, yes, it's a mapping of one set into another set.
Formally, every mathematical object needs to be encoded as a set, so we need a way to represent functions as sets so that we have a single framework to ask about existence, or to rigorously ground functions of functions.
PostFunktionalist ยท 18 points ยท Posted at 20:10:12 on December 5, 2015 ยท (Permalink)
The two big things for me were "set theory is a thing" and "functions are objects in their own right." Before that functions were just sort of these floaty magic things. Being able to make a set of Every Function over a single set... That's A Power I Never Thought Possible
[deleted] ยท 4 points ยท Posted at 22:39:03 on December 5, 2015 ยท (Permalink)
A function's just a specific case of a relation, bro!
bowtochris ยท 1 points ยท Posted at 20:24:17 on December 5, 2015 ยท (Permalink)
So, a set is a small enough collection of things (most mathematicians never consider collections that are too big while doing math, so you probably won't either). A set X is a subset of a set Y if everything in X also belongs to Y. The product of two sets A and B is the set of all order pairs (a,b) with a in A and b in B. A relation R between A and B is a subset of A*B. We say that a is related to b (or aRb) if (a, b) is in R. A function f from A to B is a relation where each in A is related to exactly one thing in B. Instead of writing afb, we write f(a) = b.
[deleted] ยท 29 points ยท Posted at 19:39:42 on December 5, 2015 ยท (Permalink)
Except we don't expect high school students to understand much of any subtlety - quite the opposite in fact. Our standard math curriculum is more or less designed to impart a very rough understanding sufficient for basic physics and engineering courses. Students are expected to have a reasonable intuition for what a real number is, how functions work, etc. but are not at all expected to understand anything deep.
elseifian ยท 10 points ยท Posted at 21:17:05 on December 5, 2015 ยท (Permalink)
How functions work is deep: the concept of a function as an abstract rule, rather than a formula, took several hundred years for mathematicians to pin down. But we barely teach it: students often see nothing but elementary functions until calculus, when suddenly we expect them to actually understand how functions work.
makemeking706 ยท 22 points ยท Posted at 22:03:31 on December 5, 2015 ยท (Permalink)
It is deep, but we aren't teaching that deepness. That's what OP is trying to convey.
Speaking of deepness, have you guys read Mistborn? It's pretty good, but also lacks a lot of subtly, much like how we teach high school math. Sanderson's writing has definitely improved since then.
elseifian ยท 5 points ยท Posted at 22:26:10 on December 5, 2015 ยท (Permalink)
Yes, I'm saying that we don't teach that depth. The question is whether we expect students to understand that depth without being taught it, and the answer is that we do. The standard pre-calculus and calculus curriculum, formulaic as it is, typically includes the difference between definedness, continuity, and differentiability; expects students to understand domain, codomain, and range; and expects students to be able to use the chain rule and fundamental theorem of calculus on novel or arbitrary functions.
But none of those concepts make sense unless you actually understand what a function is. So while we're not really teaching functions, we are teaching things that depend on functions. (We then turn around and teach those topics half-heartedly and formulaically, because what else are we going to do with students who haven't been given the necessary background to understand it?)
[deleted] ยท 9 points ยท Posted at 23:08:42 on December 5, 2015 ยท (Permalink)*
These are sufficiently robust concepts, though, that one can use them very naively and find it very difficult to get into a logical jam. Arguably, this is why the modern definitions of functions, continuity, etc. were only settled upon in the 19th century: it takes work to find the edge cases where naive intuition breaks down.
A hand-wavy idea of a function as being an assignment of points in one set (pretty much always thought of as a subset of R) to points in another set (pretty much always thought of as a subset of R) is more or less sufficient to build a similarly hand-wavy understanding of continuity, limits, and differentiability which is entirely serviceable for most practical situations. And differentiating arbitrary functions is in practice more or less formal string manipulation; this is why differential algebra makes sense, after all. One can get away with a truly rudimentary understanding in that context.
Don't get me wrong - I think we do high school students a real disservice by never exposing them to the argumentation and proof of mathematics as an intellectual discipline rather than just a toolbox handed down from the Gods. But rather than expecting a subtle understanding of functions, we really expect the opposite: an extremely naive and intuitive understanding of functions and R, which to me at least seems very reasonable.
elseifian ยท 8 points ยท Posted at 23:33:29 on December 5, 2015 ยท (Permalink)
The idea that a function is just an assignment of points in one set to points in another is a deep concept. Most students come in to calculus on the step before that, where a function is primarily a formula.
It took a good two hundred years for mathematicians to get completely away from thinking of functions as formulas to thinking of functions as assignments of points, but now we breeze over it like it's nothing and call that the "naive and intuitive understanding".
Bleakfall ยท 3 points ยท Posted at 00:13:41 on December 6, 2015 ยท (Permalink)
Excuse me kind sir, I don't understand why you're making it seem like functions are this deep, mysterious thing. I'm currently a student finishing my calculus 2 course with an A and I think functions are fairly intuitive and easy to understand. Am I missing something here? I feel like everyone learns what functions are since middle school.
elseifian ยท 3 points ยท Posted at 01:17:23 on December 6, 2015 ยท (Permalink)
Most students entering calculus 1 (and plenty of students leaving calc 1 and even calc 2) basically think of functions as being formulas. That is, they think of every function as coming with an explicit mathematical calculation; they may have some idea that functions are more general, but they haven't really internalized it yet. A classic calculus example is the characteristic function of the rationals---the function which is 1 on the rationals and 0 on the irrationals. Many calculus students have trouble viewing it as a function, and a great deal of trouble applying any of the concepts of calculus (like continuity and differentiability) to it.
If you were taught that since middle school, that's downright unusual. There are some really good schools out there, but it's not standardly taught (at least, not well enough to sink in) even at many good schools.
That may come easily to you. It comes easily to a lot of people with the right mathematical inclinations (and if you're getting an A in calculus 2, that may well include you). But it's important to remember that we're not representative of the bulk of students who take calculus.
thelaxiankey ยท 2 points ยท Posted at 02:52:19 on December 6, 2015 ยท (Permalink)
Nah, not like that for me. I went to a public middle school, and, although they didn't explain the "real" definition of functions, they made sure that we internalized that the following function (as a series of coordinates): (8, 10) (9, 28) (10, 30) (11, pi) was just as much of a function as f(x)=26x. No set theory, but good enough IMO for differentiability and whatnot. Piecewise functions in trig/pre-calc sort of required this level of understanding. I do wish they were more rigorous, but I definitely didn't have to redefine/define functions for myself in calc (we didn't do set theory there either :( ), and of what I understand, neither did anybody in my class. But then again, our district is weird, so this might be different from the rest of the US.
Edit: added an informative statement in parentheses.
makemeking706 ยท 1 points ยท Posted at 21:59:29 on December 5, 2015 ยท (Permalink)
The subtleties? No, just the dumb down version of definition itself, and some manipulation.
linusrauling ยท 28 points ยท Posted at 17:55:30 on December 5, 2015 ยท (Permalink)
You're correct.
TheFlying ยท 23 points ยท Posted at 18:22:01 on December 5, 2015 ยท (Permalink)
Or how about non-Euclidean geometry? Or the way Cantor was horribly mistreated? From my understanding of math history this comic is very true.
clutchest_nugget ยท 12 points ยท Posted at 19:50:23 on December 5, 2015 ยท (Permalink)
I mean, I don't think this was nearly so controversial as, e.g. Cantor or Gรถdel. After all, Gauss was one of the first people to have his hands in it, and when Janos Bolyai sent him a letter, Gauss was basically like "meh I did that stuff like 10 years ago bruh".
claird ยท 14 points ยท Posted at 23:08:47 on December 5, 2015 ยท (Permalink)
Gauss was notoriously "... like 'meh I did that stuff like 10 years ago bruh'" with everything. The anecdotes which fit that description are several.
phafy ยท -4 points ยท Posted at 23:32:46 on December 5, 2015 ยท (Permalink)
There are several anecdotes which fit that description.
TheFlying ยท 15 points ยท Posted at 20:13:54 on December 5, 2015 ยท (Permalink)*
You're right in a sense. Bolyai published the Tentamen in 1831 and according to this source the idea was popularized in 1868. It should be noted that is after Lobachevsky died, but the delay of acceptance is understandable considering difficulty of communication and subject matter.
The reason I was thinking of non-euclidean geometry was this heartbreaking letter from Farkas Bolyai to his son Janos which I could not find online but can be seen in one of my favorite texts Euclidean and non-Euclidean geometries Development and History by Marvin Jay Greenberg:
" You must not attempt this approach to paralells. I know this way to its very end. I have traversed this bottomless night, which extinguished all light and joy of my life. I entreat you, leave the science of parallels alone... I thought I would sacrifice myself for the sake of truth. I was ready to become a martyr who would remove the flaw from geometry and return it purified to mankind. I accomplished monstrous, enormous labors; my creations are far better than those of others and yet I have not yet achieved satisfaction... I turned back when I saw that no man can reach hte bottom of the night. I turned back unconsoled, pitying myself and all mankind.
I admit I expect little from the deviation of your lines. It seems to me that I have been in these regions; that I have traversed past all reefs of this infernal Dead Sea and have always come back with broken mast and torn sail. The ruin of my disposition and fall date back to this time. I thoughtlessly risked my life and happiness- aut Caesar aut nihil (either a Caesar or nothing)."
clutchest_nugget ยท 5 points ยท Posted at 20:37:17 on December 5, 2015 ยท (Permalink)
Wow, that's heavy stuff. Bolyai the Elder sure took his math seriously. Seems a bit dramatic though, don't you think?
TheFlying ยท 15 points ยท Posted at 20:42:51 on December 5, 2015 ยท (Permalink)
Well I would say so, but I have never spent my entire life devoting considerable resources and talent to a tree that bore no fruit. I can't imagine how difficult it must be to come to the point where you have nothing left to give and realized that you have never produced anything of value.
And he was writing this to his son, to scare him from the beast so it makes sense he might hyperbolize to show him he was serious.
[deleted] ยท 4 points ยท Posted at 22:51:25 on December 5, 2015 ยท (Permalink)
I think that is an inherent potential bottleneck to the progression of academic fields. Like could you imagine if something far more powerful and appropriate and just better in every way came along to replace the biological taxonomy? So many people have their lives invested in the old way, all the literature is in the old way, etc... resulting in a lot of resistance unrelated to the merits of the discovery itself.
Anyhow that's a marvelous bit of prose, thanks for posting it.
TwoFiveOnes ยท 1 points ยท Posted at 23:03:41 on December 5, 2015 ยท (Permalink)
Can you clarify what it was he was trying to accomplish though? Was he trying to prove or disprove the independence of the parallel axiom?
popisfizzy ยท 16 points ยท Posted at 18:49:28 on December 5, 2015 ยท (Permalink)
Basically, before the 'modern' idea of mathematics developed in the early 20th century, which took a very, very broad, inclusive, and general view of mathematics, it seems that most things were subject to this sort of debate. Thankfully, that has largely subsided now. E.g., everyone agrees that non-standard analysis is correct and valid, and functionally equivalent to standard analysis, and the only real debate surrounding it is whether it's useful in that regard. If it had shown up in, say, the 1860s instead of the 1960s, there may have been decades of debate as to whether it was 'correct' or just inane and worthless.
MostlyTolerable ยท 8 points ยท Posted at 05:10:41 on December 6, 2015 ยท (Permalink)*
My signals and systems text covers the development of the Fourier Series and Transform over about a few generations of mathematicians. It played out pretty similar to this comic.
A lot of the concepts were pretty generally understood, but people argued quite a bit that you couldn't create a signal with discontinuities (like a square wave) with a series of sinusoids or complex exponential.
rocker5743 ยท 4 points ยท Posted at 21:52:37 on December 5, 2015 ยท (Permalink)
Had a teacher tell us that negative numbers weren't accepted until around the 1700's before we started into complex numbers this semester.
clutchest_nugget ยท 6 points ยท Posted at 19:49:04 on December 5, 2015 ยท (Permalink)*
Yes, you are correct. You may be interested in this lecture by eminent logician Kurt Gรถdel. In it, he describes what he calls the "leftward (empirical) upheaval", in which the very things you cited, particular Russell's paradox, caused great divisiveness in the mathematical community. To quote him again, he says that, due to these "supposed contradictions", many mathematicians regarded "only a certain portion of mathematics, larger or smaller, according to their temperament" as representative of any truth. The rest was discarded as purely hypothetical.
Here is a reading of this same lecture, may be easier than reading it. It's nice for driving or while exercising, etc. I hope that you enjoy it =)
docmedic ยท 3 points ยท Posted at 22:31:43 on December 5, 2015 ยท (Permalink)
One, math wasn't agreed upon rigorously or standardized until somewhat recently, made worse because communication back then was slow. Two, math research and results encompass so much that even that lot of stuff is tiny.
So I think the comic doesn't describe how math works (as the title claims), just a select number of results, some of which happen to be huge (which of course would generate bigger backlash). How math works right now is pretty damn cordial.
UniversalSnip ยท 1 points ยท Posted at 01:50:21 on December 6, 2015 ยท (Permalink)
don't forget the parallel postulate
[deleted] ยท -7 points ยท Posted at 18:48:42 on December 5, 2015 ยท (Permalink)
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Marcassin ยท 3 points ยท Posted at 20:39:01 on December 5, 2015 ยท (Permalink)
More important to math history were philosophical positions such as intuitionism, logicism and formalism. Such issues used to be very hotly debated among mathematicians. Underlying these debates were results similar to the ones shown in the comic. For example, the existence of continuous functions that were not differentiable gave some 19th century mathematicians the fits. Charles Hermite famously said he rejected such findings with "fright and horror". Cantor's findings were even more severely rejected by some, though none could find flaws with his reasoning. The attacks really did become personal. All of this 19th century upheaval led to enormous introspection over math and its foundations. As the comic noted, some did really want to restructure math to make such results wrong, e.g. the debate over the axiom of choice. It may have been a time of "great insights", but it was not always a happy one, nor was their validity always readily accepted.
[deleted] ยท 3 points ยท Posted at 22:59:48 on December 5, 2015 ยท (Permalink)
As an aside, is it just me or is intuitionism/constructivism enjoying an uptick in popularity? Maybe I'm seeing it by way of computer science and buzz about type theory/HoTT
theotherborges ยท 1 points ยท Posted at 19:27:26 on December 5, 2015 ยท (Permalink)
Tell that to the Pythagoreans who rejected the existence of irrational numbers.
redrumsir ยท 1 points ยท Posted at 19:39:58 on December 5, 2015 ยท (Permalink)
Interesting. The way I was told it was: The discovery of irrational numbers by the Pythagorean's was shocking to them. As such they valued this as a "secret" and were upset when one of them divulged this fact outside of their sect. There is a myth that as devine retribution for this divulgence, the one who divulged it was drowned at sea. https://en.wikipedia.org/wiki/Hippasus
theotherborges ยท 5 points ยท Posted at 19:58:50 on December 5, 2015 ยท (Permalink)
You're correct, but I still see as similar to the situation in the comic. The reason it was valued as secret is because they outwardly preached that everything could be expressed as a ratio of whole numbers. The discovery of an irrational number conflicted with that notion, so they tried to bury it.
Jess_than_three ยท 0 points ยท Posted at 19:42:48 on December 5, 2015 ยท (Permalink)
Wait, what? As an atheist who's largely ignorant of the philosophy of math, I would definitely at a glance think that math was discovered - that what was going on was pinning down underlying truths of the way the universe works. It seems bizarre to me that that would be conflated with theism any more than physics or chemistry or biology. Like, Darwin (and others later) discovered the mechanisms and rules of natural selection, but that definitely doesn't imply a theistic perspective...?
[deleted] ยท 5 points ยท Posted at 19:52:36 on December 5, 2015 ยท (Permalink)
[deleted]
claird ยท 2 points ยท Posted at 23:11:26 on December 5, 2015 ยท (Permalink)
... because it's too serious.
Said differently: practitioners, for the most part, live their daily lives with rather dramatic disregard for foundations.
almightySapling ยท 2 points ยท Posted at 10:08:50 on December 6, 2015 ยท (Permalink)
Without having to mention creationism at all, we can still discuss whether math is discovered or invented. I mean, not in any seriously meaningful way (because it shouldn't matter). But there are decent reasons to believe that math, in parts, are indeed invented rather than discovered.
I would say that the axioms (necessary rules to describe any mathematical objects/structures) are chosen/invented. The consequences of axioms (theorems) are discovered.
jroot ยท 34 points ยท Posted at 17:09:18 on December 5, 2015 ยท (Permalink)
I resist your argument and deny it's truth. I shall take this stance with me to the grave. Also, Your mother's a whore.
makemeking706 ยท 3 points ยท Posted at 22:07:38 on December 5, 2015 ยท (Permalink)
Actually, we were looking for "what is calculus?"
Eurynom0s ยท 1 points ยท Posted at 13:04:45 on December 6, 2015 ยท (Permalink)
Sean Connery is a mathematician?
daidoji70 ยท 14 points ยท Posted at 19:40:56 on December 5, 2015 ยท (Permalink)
How antithetical is it when I can think of 3 examples off the top of my head without even thinking hard?
Cantor Cardinals as mentioned above. Definition of a Function. Incompleteness. Non-Euclidean geometries
McPhage ยท 19 points ยท Posted at 20:29:30 on December 5, 2015 ยท (Permalink)
Four... I can think of four examples: Cantor Cardinals, Definition of a Function, Incompleteness, Non-Euclidean Geometries, and the Existence of Zero. Five...
...Look, I'll come in again.
[deleted] ยท -5 points ยท Posted at 20:08:21 on December 5, 2015 ยท (Permalink)
[deleted]
daidoji70 ยท 21 points ยท Posted at 20:21:28 on December 5, 2015 ยท (Permalink)
Uhhhh, you should probably go re-read your mathematical history.
Godel and Cantor were both hounded their whole lives by mathematicians who wouldn't accept their proofs, solid though they were. If I remember correctly Cantor went crazy because of it and Godel became a recluse after Hilbert and others attacked him for pretty much decimating his program and what it stood for to its core (by showing that it would never be complete).
Same thing with non-Euclidean geometries although I don't remember the exact stories behind that.
Its nice to think that mathematicians and scientists are above such petty emotional squabbles and that we trust empirical proof above all else, but historically this has most definitely, 100%, not been the case and to argue otherwise is a weird position imo.
[deleted] ยท 7 points ยท Posted at 22:53:11 on December 5, 2015 ยท (Permalink)
[deleted]
daidoji70 ยท 3 points ยท Posted at 23:10:18 on December 5, 2015 ยท (Permalink)
No it was a great tool utilized most effectively first by Gauss and Riemann but had been around for quite a while. Thats the whole point of this comic, today these things are accepted without even thinking about it, but when new ideas like removing the parallel postulate are introduced, people get emotional about it and do push back (even if those conversations get lost in the history of time as people accept the proofs, as it should be)
However, as the history of Mathematics is not my specialty I can't remember the specific sources but I'm def sure its not as clear cut as you suggest where people just immediately accept a proof without emotion.
For Godel: most of what I know is from biographies of Godel and survey books on Incompleteness and various things I've read relating to the connection with the halting problem and Turing Machines that I'm sure have mentioned it. I could list about 20 books here, but I can't remember the specific sources.
As to the rest of my knowledge of mathematical history, most of it comes from the classic four volume set http://www.amazon.com/The-World-Mathematics-Four-Volume-Set/dp/0486432688 that I found for a great deal in a used book store a long long time ago. There are copious examples of new ideas and the push back generated due to emotional fixations beyond those offered in proofs.
Ideally I'd like to live in your world too in all kinds of scientific and rational endeavors, but that doesn't seem to be the way most of the world works and to pretend that mathematics is immune is ignoring a lot of history and human psychology imo.
adamcrume ยท 3 points ยท Posted at 19:31:42 on December 5, 2015 ยท (Permalink)
Theorems with easily understood proofs are not subjective or controversial. The really interesting part, though, is deciding which axioms to include (like the axiom of choice) or exclude (the parallel postulate) or how things should be defined.
Edit: Instead of "easily understood", I really mean easily verified.
makemeking706 ยท 1 points ยท Posted at 22:06:20 on December 5, 2015 ยท (Permalink)
Yeah, I really thought the turn was going to be something along the lines of "oh wait, I am thinking of politics" or something like that. But, I guess it does historically follow.
Although I would argue that's not a very good joke if we need that historical context to appreciate it. Then again, they can't all be Wiener jokes.
yangyangR ยท 1 points ยท Posted at 23:41:30 on December 5, 2015 ยท (Permalink)
Well sometimes that is the case. See the relationship between Laplace and Fourier.
cultofmetatron ยท 1 points ยท Posted at 02:55:16 on December 6, 2015 ยท (Permalink)
complex/imaginary numbers illicited this response when it first came out.
elenasto ยท 10 points ยท Posted at 18:40:38 on December 5, 2015 ยท (Permalink)
In physics it is the experiment which is the final arbiter though. Your math could be right, even beautiful but that wouldn't make the theory right
Ostrololo ยท 21 points ยท Posted at 21:17:44 on December 5, 2015 ยท (Permalink)
That didn't stop Einstein from throwing quantum tantrums.
elenasto ยท 15 points ยท Posted at 01:48:52 on December 6, 2015 ยท (Permalink)
That was about an interpretation though, not about the theory itself. Einstein didn't dispute the quantum theory itself, but believed that there was an underlying mechanism which we are missing and which would get rid of the pesky uncertainties. He was perfectly fine using the mathematics involved. Also to be noted that bell's inequalities were after his death
ummwut ยท 1 points ยท Posted at 03:54:41 on December 6, 2015 ยท (Permalink)
Or other physicists over parity violation in weak interactions.
abuttfarting ยท 7 points ยท Posted at 22:28:03 on December 5, 2015 ยท (Permalink)
But who's to say the experiment is even correct, or useful? It's by no means as clear-cut as you make it out to be.
elenasto ยท 2 points ยท Posted at 01:51:50 on December 6, 2015 ยท (Permalink)
This is philosophical question and not something I wish to argue about here. I only wanted to say in my OP, that according to philosophy science follows, this would not work in physics.
invisiblerhino ยท 1 points ยท Posted at 22:01:32 on December 6, 2015 ยท (Permalink)
Indeed, check out these measurements of some important quantities in particle physics as a function of time
abuttfarting ยท 3 points ยท Posted at 22:27:15 on December 5, 2015 ยท (Permalink)
Yup, this is pretty much a textbook description of Kuhn's paradigm shifts (apart from the anomalies).
00zero00 ยท 2 points ยท Posted at 20:22:50 on December 6, 2015 ยท (Permalink)*
https://en.wikipedia.org/wiki/Dan_Shechtman#Work_on_quasicrystals
[deleted] ยท 0 points ยท Posted at 18:08:06 on December 5, 2015 ยท (Permalink)
I think this fits pretty well with Galios and his work.
irock007-king ยท -1 points ยท Posted at 23:45:30 on December 5, 2015 ยท (Permalink)
yeah haha
punning_clan ยท 25 points ยท Posted at 16:52:15 on December 5, 2015 ยท (Permalink)
Funny. But, I'll just leave this fascinating article* (PDF) here.
*McClarty, Theology and it's discontents: The Origin Myth of Modern Mathematics
brianterrel ยท 12 points ยท Posted at 17:55:46 on December 5, 2015 ยท (Permalink)
Thank you, that was a delightful read.
punning_clan ยท 10 points ยท Posted at 20:17:29 on December 5, 2015 ยท (Permalink)
You are welcome! I find the transition from formal mathematics (in the 19th c. sense of involving manipulation of formulas, or approximately the layperson idea of math) to the modern conceptual variety (roughly, Riemann onwards) a really exciting part of history. If there is anything that can usefully thought of as a paradigm-shift in math, I think it is this.
magus145 ยท 2 points ยท Posted at 16:30:49 on December 6, 2015 ยท (Permalink)
I'll second the thanks on this great read.
Stories and narratives purportedly from history, told by non-historians, are so often dangerously about now rather than then.
It's good to keep in mind that most "math history" you hear along the way is anecdotal at best, and purposefully propagandist at worst. Always do your research and don't trust all sources cough Bell cough.
In other news, Emmy Noether remains a BAMF.
MxM111 ยท 66 points ยท Posted at 16:41:22 on December 5, 2015 ยท (Permalink)
That's how the mathematicians work. The math is just fine.
Cosmologicon ยท 10 points ยท Posted at 05:28:48 on December 6, 2015 ยท (Permalink)
Great. Now, all we need is to be able to do math without involving mathematicians in any way, and we'll be all set!
MxM111 ยท 3 points ยท Posted at 16:05:07 on December 6, 2015 ยท (Permalink)
Observing the development of Mathematica software, we are moving in that direction.
Vtempero ยท 10 points ยท Posted at 16:46:09 on December 5, 2015 ยท (Permalink)
How exactly any model of the world exists without people caring about it.
jlink005 ยท 16 points ยท Posted at 21:24:00 on December 5, 2015 ยท (Permalink)
CunningTF ยท 98 points ยท Posted at 16:05:29 on December 5, 2015 ยท (Permalink)
SMBC always delivers.
nopenopenopenoway ยท 40 points ยท Posted at 16:35:45 on December 5, 2015 ยท (Permalink)
always hides the best joke under the button.
AndreasVesalius ยท 28 points ยท Posted at 20:40:46 on December 5, 2015 ยท (Permalink)
And has recently started adding hover-over jokes (a la XKCD) as well. Three jokes for the price of one
B-Con ยท 13 points ยท Posted at 21:49:39 on December 5, 2015 ยท (Permalink)
That's recent? Oh good. I've been reading it for years (since it started?) and had noted very distinctly that it didn't have hover-over jokes on the comic itself. It did eventually do the red button hover-over, so I thought that was the equivalent.
Then very recently I happened to hover the mouse over the comic and saw some alt text. I didn't know when it started and was mortified that maybe I'd missed years worth.
Meapalien ยท 12 points ยท Posted at 21:55:57 on December 5, 2015 ยท (Permalink)*
I edit old comments
makemeking706 ยท 10 points ยท Posted at 22:09:15 on December 5, 2015 ยท (Permalink)
Can't tell if this is an excellent troll, or legitimate.
AndreasVesalius ยท 6 points ยท Posted at 22:45:15 on December 5, 2015 ยท (Permalink)
I think he is. If you hit random enough you'll find comments that seem to be commenting on the comic from the present
frog971007 ยท 1 points ยท Posted at 05:19:26 on December 23, 2015 ยท (Permalink)
This is a good example of a comment with a newer red button panel, though it doesn't have a hover-over.
redminx17 ยท 3 points ยท Posted at 00:40:32 on December 6, 2015 ยท (Permalink)
I .... I never knew there was another joke hiding there. Cheers, mate
AcellOfllSpades ยท 2 points ยท Posted at 09:26:21 on December 6, 2015 ยท (Permalink)
There's hovertext in the recent ones too!
[deleted] ยท 1 points ยท Posted at 05:27:58 on December 6, 2015 ยท (Permalink)
HA! Thank you... I had always noticed that button, but didn't know it hid another joke.
AcellOfllSpades ยท 1 points ยท Posted at 09:26:24 on December 6, 2015 ยท (Permalink)
There's hovertext in the recent ones too!
[deleted] ยท 4 points ยท Posted at 17:12:04 on December 5, 2015 ยท (Permalink)
[deleted]
AcellOfllSpades ยท 19 points ยท Posted at 17:18:46 on December 5, 2015 ยท (Permalink)
True, but he puts them out every day and they're generally pretty high quality. A couple of duds are fine.
Jess_than_three ยท 5 points ยท Posted at 19:03:27 on December 5, 2015 ยท (Permalink)
More funny comics by both absolute number and percent than xkcd...
Apothsis ยท 41 points ยท Posted at 21:49:09 on December 5, 2015 ยท (Permalink)
Having witnessed an actual fistfight between my Advisor when I was a Candidate, and his rival, over a conjecture that they have been arguing about for 30 years, I can confirm, this is exactly how Math works.
jzapate ยท 17 points ยท Posted at 22:53:16 on December 5, 2015 ยท (Permalink)
Do you remember the conjecture?
Apothsis ยท 23 points ยท Posted at 23:23:00 on December 5, 2015 ยท (Permalink)
Not really. I think it really reduced to "FUCK YOU! FUCK YOU AND DIE!" more than anything usable to the Body of Knowledge.
omegasavant ยท 5 points ยท Posted at 14:33:06 on December 6, 2015 ยท (Permalink)
Relevant SMBC
[deleted] ยท 11 points ยท Posted at 23:06:42 on December 5, 2015 ยท (Permalink)
I remember reading about math challenges between the 1500's Italians that had some serious consequences (like losing a professorship etc).
mearco ยท 6 points ยท Posted at 20:27:57 on December 6, 2015 ยท (Permalink)
Yeah professors basically had to verify their mathematical abilities every few years through challenges, like solving cubic equations. This led to them not disclosing their methods so they had less competition.
[deleted] ยท 6 points ยท Posted at 01:52:34 on December 6, 2015 ยท (Permalink)*
[deleted]
Apothsis ยท 8 points ยท Posted at 01:56:23 on December 6, 2015 ยท (Permalink)
You have not lived until you have seen a gathering of "Great minds" shoving each other and throwing extremely ineffective punches.
GetOffMyLawn_ ยท 18 points ยท Posted at 19:59:41 on December 5, 2015 ยท (Permalink)
Kuhn's "Structure of Scientific Revolutions" in a nutshell. Excellent book BTW.
abuttfarting ยท 7 points ยท Posted at 22:28:57 on December 5, 2015 ยท (Permalink)
I cannot recommend that book highly enough. Changed my whole look on science.
[deleted] ยท 10 points ยท Posted at 22:40:32 on December 5, 2015 ยท (Permalink)*
Is this not true for physics students?
edit: I think for CS the last frame would be "Go download the latest version"
webchimp32 ยท 4 points ยท Posted at 17:06:22 on December 5, 2015 ยท (Permalink)
Just watched a doc on James Clark Maxewell last night, pretty much how it went down with his theories.
IForgetMyself ยท 4 points ยท Posted at 17:46:57 on December 5, 2015 ยท (Permalink)
Boltzmann too.
[deleted] ยท 4 points ยท Posted at 17:54:26 on December 5, 2015 ยท (Permalink)
What was it called? I would love to check it out.
webchimp32 ยท 13 points ยท Posted at 18:00:17 on December 5, 2015 ยท (Permalink)
James Clerk Maxwell: The Man Who Made the Modern World
It's a new one on the BBC, so you might need to approach it in a sideways manner.
linusrauling ยท 8 points ยท Posted at 17:56:49 on December 5, 2015 ยท (Permalink)
By coincidence, I just did Fourier Series.
theblackhand ยท 3 points ยท Posted at 20:06:15 on December 5, 2015 ยท (Permalink)
It's back on your site! No more navigating there from imgur to see the button! Hooray!
Moonpiles ยท 4 points ยท Posted at 15:54:46 on December 5, 2015 ยท (Permalink)
This is great.
[deleted] ยท 2 points ยท Posted at 16:26:26 on December 5, 2015 ยท (Permalink)
Very fuckin true.
Source: I dabble.
Jeffreyrock ยท 1 points ยท Posted at 20:21:19 on December 5, 2015 ยท (Permalink)
Love Step 4
russian_proofster ยท 1 points ยท Posted at 21:39:42 on December 5, 2015 ยท (Permalink)
proofs?
Revenchule ยท 1 points ยท Posted at 00:31:56 on December 6, 2015 ยท (Permalink)
My old college algorithms class in a nutshell.
Glitch29 ยท 1 points ยท Posted at 04:00:57 on December 6, 2015 ยท (Permalink)
My guess is the Axiom of Choice. But it could have been so many different things.
jaskamiin ยท 1 points ยท Posted at 18:15:29 on March 31, 2016 ยท (Permalink)
I love this comic, but can I be the pedant on this thread for the sake of the college algebra and calculus 1 people who will see this and use it to excuse their own misunderstanding?
The mathematical formula you are being taught in class is nothing like what the proof of the rigorous mathematics behind it. Mathematicians didn't just come up with the formula for a derivative out of thin air, and plug in different values of x.
The development of mathematics is a beautiful subject that is heavily trivialized in early math classes. Don't worry about not understanding a formula right up front, but the worst thing you can do (concerning your grades and your understanding of math) is to learn mathematics as blind symbol manipulation.
hot4math ยท 1 points ยท Posted at 06:16:09 on April 5, 2016 ยท (Permalink)
Ha, this shit cray!
agumonkey ยท 1 points ยท Posted at 18:50:30 on December 5, 2015 ยท (Permalink)
Of the sadness of education of things.
ps: we can define school as a Field of some sort ? diffusion of bs.
InSearchOfGoodPun ยท -4 points ยท Posted at 18:20:10 on December 5, 2015 ยท (Permalink)
I find this neither accurate nor funny.
๐๏ธ Frigorifico ยท 4 points ยท Posted at 01:11:17 on December 6, 2015 ยท (Permalink)
That's a good opinion, you shouldn't get downvoted for that
InformationOverflow ยท 4 points ยท Posted at 10:43:05 on December 6, 2015 ยท (Permalink)
Me neither. The problem is that this is just really far from my perception of how mathematical research works nowadays. It should rather be called "How math used to work."
Eurynom0s ยท 0 points ยท Posted at 13:08:18 on December 6, 2015 ยท (Permalink)
Well, your mother's a whore.
With loathing, Eurynom0s
[deleted] ยท -18 points ยท Posted at 17:08:58 on December 5, 2015 ยท (Permalink)
[deleted]
Alexanderdaawesome ยท 3 points ยท Posted at 19:03:56 on December 5, 2015 ยท (Permalink)
Maybe in your case.
[deleted] ยท -47 points ยท Posted at 16:38:55 on December 5, 2015 ยท (Permalink)
[removed]
nerga ยท 24 points ยท Posted at 16:50:33 on December 5, 2015 ยท (Permalink)
Off the top of my head I know Cantor was basically shunned as a crack pot. Now his ideas are taught in every undergrad curriculum.
bystandling ยท 7 points ยท Posted at 19:41:41 on December 5, 2015 ยท (Permalink)
And Mandelbrot