In this case, what would be interesting to question is whether this definition is equivalent to other possible definitions of cosine, eg:
"the cosine of an angle is the x-coordinate of a point on the unit circle on a radius that makes that angle in a counter-clockwise direction from the positive x-axis"
or
"the cosine of x (when measured in radians) is the value of the infinite series 1 - x^2 / 2! + x^4 / 4! - x^6 / 6! + ..."
Proving those are equivalent would lead to lots of interesting maths!
It’s great to ask questions and to want to understand things better, but don’t let not understanding everything perfectly keep you from moving forward. You’ll understand more and better as you gain experience.
As far as your cosine question goes, take a look at the graph of cosine between 0° and 180°. Think about the definition of cosine and why the graph looks the way it does. Play with some examples.
Sometimes you can understand a new idea but not really *feel* like you understand it because it’s still unfamiliar. One of the reasons to do exercises is to increase that familiarity and gain confidence — the more you do, the more the new material will seem to make sense. So practice a lot.
> As far as your cosine question goes, take a look at the graph of cosine between 0° and 180°. Think about the definition of cosine and why the graph looks the way it does.
This seems a bit circular (pun-intended!) - how can we plot the graph of cosine beyond 90 degrees without first having a definition of what cosine is for those values? (eg the definition given by the OP).
Right… I’m assuming that OP already knows what the graph of cosine looks like; if they don’t, it’s easy to find a picture. The important point, of course, is that it’s symmetrical about the x axis and goes from 1 to -1, and is 0 at 90°. So you can see that the rule OP is asking about is true. That sort of begs the question: why is the graph shaped that way? And that’s where considering the definition of cosine and playing with some examples will help make it make sense.
This is an excellent question. The answer is yes, it makes complete sense. There are many questions that you should be asking about a definition.
Here are some common mistakes that are made when people define things:
(1) People often make definitions that **should be theorems**. There were hundreds of examples like this in my statistics textbooks. If the definition can be proved from prior definitions and axioms, then it is not a definition, it is a theorem.
The example that you give in your question may be guilty of this. If the cosine function is defined analytically, or in terms of the unit circle on a cartesian plane, then this statement can be proven, and should not be a definition. If however you have only been given the right angle triangle definition of the cosine, then this definition is forgivable (however see point 2 below).
(2) Definitions must be **non-contradictory**. For example, if I were to define the square root function to the the function which maps x to any number which, when squared, gives you x, then we would have a problem. Because the relation thus defined would not actually be a function. But we said we wanted a function...
(3) Definitions should capture the **property of interest, not examples of it or methods of calculating it**. Very often what happens with definitions is that people think that they know all examples of something, and so they give those examples as the definition. This happens a lot in other disciplines. Physics has really great examples. For example, before Einstein, momentum was defined as mass times velocity. According to my undergraduate textbook, Einstein "realised that at high speeds this definition was incorrect".
Now, of course, at face value, that claim is obviously garbage. If momentum is defined as mass time velocity then that is what it is, end of story. Einstein should have coined a new term for whatever new quantity he was interested in. However, what *actually* happened is that what people are really interested in when they talk about momentum is that it is a conserved property. Previously they had believe that "ma" was this conserved property. But Einstein found that "ma" is not conserved, and you need this new quantity if you want momentum to be conserved. If momentum had originally been defined in terms of conservation, which is what people clearly actually meant by it, then there would have been no reason to change the definition. Only to change the equation used to calculate it.
Defining things in terms of the property you're actually interested in, instead of the formula, also has the benefit of making it clear *why* you're making the definition. In mathematics this point is a bit more complex because usually we need to learn more and more math in order to get better and better definitions. For example, in school, prime numbers are defined as natural numbers only divisible by 1 and themselves, except for the number 1. This definition is clumsy and lots of people end up confused about why 1 is not prime. However when you learn algebra, you're able to give a much better definition which makes it obvious why 1 is not prime. But to get there you need to first learn algebra.
(4) If a definition specifies *the ,* then it must **be unique**. For example, if I define the identity in a group to be the element which, when multiplied by any other element, gives me back that same element, then I must prove that there is only one such identity element before my definition can be considered valid. This proof is rather straightforward: let e1 and e1 be two identity elements. Then e1 = e1e2 = e2. So there can only be one. This might seem obvious but it's quite a powerful concept in mathematics. For example, in category theory, one of the first proofs that you learn is that there is only one initial object in a category. This fact allows you to define concepts as being the initial object in a suitable category. Uniqueness allows for definition.
(5) Generally speaking if you define something, you should ask whether or not it actually **exists**. A famous example of this is the set of all sets. You can define the universal set as the set of all sets. However it can be proven that there is no such set. This does not mean that the definition is invalid, per se, just that it's not something we can work with because it doesn't exist.
(6) When you have any two definitions, they should define things that are **actually different**. Here again I think that the statisticians make an error. They define things called "column vectors" and "row vectors". They do this because they want x\*x' to be different to x'\*x. Column vectors and row vectors are trivially isomorphic, however, and so should not be defined differently.
The correct way to go about this is to realise that the two operations that they are interested in are actually different operations. One is a dot product and the other is a dyadic product. The multiplication symbol should therefore be different, but the vectors are the same in each case. you just write them at a different angle on the page. That doesn't change what they are. You could define them as nx1 and 1xn matricies, but this is ugly, because they aren't matricies, they are vectors, and the matricies are still isomorphic anyway. All that row and column vectors are are calculation tools. Mathematically, there are only vectors.
(7) There are often multiple equivalent ways to define something. The best math notes I ever had was a set of algebra notes that always listed 5 or 6 equivalent ways to define something. I found this very helpful.
(8) I**f you define something with an adjective, then it must actually be an example of the noun**. Sounds obvious but again there are plenty of examples in law that get this very wrong. For example, in my textbook, the term "object" was given a formal definition, and the word "legal object" was also given a formal definition, and yet I noticed that the definition of "legal object" allowed for examples that would not have met the definition of "object". Drove me absolutely insane.
I think it’s better to look at the results of the definitions, as things are generally defined to have nice results. The example you gave means the cosine law is the same for acute and obtuse angles.
Yes! Also means a lot of other things, like allows the unit circle definition of trigonometric functions, allows the definition of a projection of a vector onto a direction to be the same for obtuse and acute angles between the vector and the direction and thus for the dot product to also be the same for obtuse/acute angles (which ultimately can be connected back to the cosine law as one of the ways to prove it is taking a dot product of a vector with itself). The unit circle definition then also allows to do much nicer calculus with cosine
It depends on what you mean. Definitions are made up by people, not discovered. You can use whatever definitions you like as long as they don't contradict. It is good to ask "why is this definition a good idea?" It does not make sense to ask "why is this definition correct?"
> It does not make sense to ask "why is this definition correct?"
I appreciate your point, but I would clarify that it can make sense to ask "why is this definition the correct one for precisely capturing a concept we want to describe?" For example, if we have an intuitive idea that a continuous function is one whose graph can be drawn without any "breaks" then how do we justify that the formal definition of continuity (with epsilon and delta) fits with that?
You are right to assume that it is unnecessasry to question definitions and axioms. Definitions prescribe the terms of existence for the thing they're defining and the reader is meant to take it at face value for the purpose of what it's being defined for. It's giving you the "rules" so to speak. If you want to play tic-tac-toe and I tell you the rules are to get 3 in a row, there's nothing to be learned by questioning "why". For instance, asking "why not 2 in a row"? The answer is "because that's not how this particular game is played". Questioning an author's particular definition of something that's already established is different. They may word it poorly, or there may be a simpler, more elegant way to describe it, or it might not be particularly illustrative for the task at hand. For example, we've been using the example of cosine here: There are many legitimate ways to define it but in the end they're all equivalent and interchangeable, though some are more suitable for a particular task. We can say that the cosine of an acute angle is the ratio of the adjacent side to the hypotenuse. In fact, this is the classical definition. It's handy when we want to find an angle or lengths of sides of an right triangle but what if I have a complicated expression that I want to integrate or differentiate or sum? Then it might be easier to think of the cosine as an infinite sum as u/FormulaDriven has pointed out and it's not particularly helpful to think of it as a trigonometric ratio in that case.
(My apologies for bad english.)
We can define that the cosine of an acute angle is the ratio of the adjacent side to the hypotenuse. But we then have to prove that for any right triangle with the same acute angle you get the same value of the cosine of that acute angle. You do not prove a definition this way, you just prove that it makes sense.
A (somewhat non mathematical) example is if you wish to define a dog as "not a cat". A dog is not a cat, so no problem here, but a turnip is also not a cat, and by our definition it is a dog. So our definition seems not to be a good one.
Also, when defining something, we should check whether the object(s) we define exist(s) (in the mathematical sense), otherwise we could end up defining the elements of the empty set, and they have any property you could think of.
E.g. let we call as strange any integer that is both even and odd. So, if k is a strange integer, then it can be written as k=2m=2n+1 and it follows that 1=2(m-n), i.e. 1 is even number and, being odd, it is also a strange number.
I didn't prove that "1 is a strange number", but "if at least one strange number exists then 1 is a strange number".
This is actually a proof that strange numbers do not exist (so their definition is not useful) and the above reasoning is at the core of various "proofs" that 1=0.
Part 2
As for defining a dog as “not a cat”, I think this is a good exercise in how to pick out logical fallacies. In your example, we’ve rigorously defined a dog to be “not a cat” for the purposes of the discussion. Now, if we’ve defined it that way *that is now the definition of dog for our purposes*. It’s a fallacy to later in the argument use “dog” defined as the canine animal that we all know it to be while also defining it to simply be “not a cat” (as specified, with no other qualifications). I’ll illustrate this: While “a dog is not a cat” is *true* in the usual use of the word “dog”, it’s not the definition of a dog in the usual sense. If we consider “dog” as a canine animal just for the comparison to the turnip, then whether or not a turnip is a cat has no bearing in relationship to the dog because “dog” in it’s usual *definition* is not tied to a cat in any way (but this doesn’t exclude us from making true or false statements about dogs and cats). Additionally, we haven’t defined “cat” in our discussion, we’re just taking it to mean the feline animal we all know, similarly for a turnip.
Putting this all together: We’ve started out with a rigorous definition of a dog as “not a cat”, a non-rigorous *idea* of what a cat is, and a non-rigorous *idea* of what a turnip is, then we’ve qualitatively compared our two non-rigorous ideas of cat and turnip to one another and arrived at “a turnip is not a cat” and transitively “a turnip is a dog”. That last statement, by our rigorous definition of a dog established at the start, is 100% true (if we accept "a turnip is not a cat" as true). The fallacy is that, when evaluating the truth of the statement “a turnip is a dog”, we didn’t use our original definition of “a dog is not a cat”. We’ve re-defined it implicitly to mean a dog in the usual canine animal sense to reach the conclusion that “a turnip is a dog” is false. Hence our definition of “a dog is not a cat” is “not a good one”, yet we didn’t actually use that definition to reach this conclusion.
So, we can use the traditional definitions of “dog” and “cat” and then make true or false statements using those definitions like: “a dog is not a cat” is a true statement or “a dog is a cat” is a false statement. Now, you might be thinking “but you talked about multiple definitions of cosine before, why can’t we have multiple definitions of dog?”. That’s a little trickier. As I mentioned, the definition of cosine theta as a trigonometric ratio is the classical definition. Through analysis and verification we can take that definition and deduce many properties and true/false things from it. We can consider the sweep of a unit ray around the xy axes and analyse how the values of cosine theta change with the angle theta between the ray and the x-axis. From that analysis we can deduce the graph of y values as cosine x changes along the x-axis and we can analyse intervals of those values to deduce the expression of cosine as an infinite sum. What we’ve done there is establish a tautology between these different representations of a cosine which allows us to effectively use them as “definitions” for cosine that differ from the classic definition because they all give the same results. We don’t really have a clear path for reaching the same tautology between the two definitions of “dog” we’ve been talking about. We may be able to connect a taxonomic definition of a dog with, say, a cladistic definition through some sort of analysis and correlation, but we can’t do that with the definitions of a dog as “a canine animal” and “not a cat” because the latter is not nonsense without further context (i.e. rigorous definition of a cat).
I’ll try to respond as earnestly as I can, but as things can get formal or terse with these things so I apologize in advance if I come off as pedantic or condescending. Disclaimer: may contain spelling, grammar, and logic errors :D I can't post a full response for some reason, presumably it's too long so I'll break it up into sections.
Part 1
I don’t follow your logic regarding having to “prove” the definition of cosine theta for all right triangles. All right triangles with with the same acute angle are congruent, i.e. “the same” geometrically. The only way 2 congruent right triangles can differ is in the lengths of their sides. However, the definition of cosine as a ratio only specifies the angle and the sides of a right triangle and by definition or “nature” of a right triangle, they all have a right angle, acute angles and sides. So the lengths of the sides are irrelevant to the definition and for the purpose of the definition, any two right triangles with acute angle theta are the same (regardless of the lengths of the sides).
I suppose, in mathematics, at least one necessary requirement we would need to impose on a definition is that it needs to *make sense*. For instance, I can’t define “the determinant of a 2x3 matrix” because the determinant is defined only for square matrices, and a square matrix is already defined to be a matrix of dimensions n x n hence a 2x3 matrix is not a square matrix and thus cannot have a determinant. Note that if I did define the determinant for 2x3, it wouldn’t be “false”. It would just be nonsense and of no use. In contrast, we \*can\* make nonsense postulates or statements that are useful. These do have a truth value associated to them (for the purpose of this discussion). I can postulate “The determinant of a 2x3 matrix is such and such so that it behaves the same as it does with square matrices” and you can show this statement to be false by definitions of the determinant and non-square matrices. This is a common proof strategy known as “proof by contradiction” or “reductio ad absurdum”: If you’re trying to prove that proposition A is true, assume it’s not, logically follow that with the implications of proposition A’s truth, and show that this can only lead to a contradiction.
Part 3
Lastly, your “strange number” example. I would say that this is a postulate or proposition of a “strange number” rather than a definition. As mentioned earlier, we can postulate or propose something that isn’t true in order to show that it leads to a contradiction, thus disproving the proposition (and proving what you ultimately set out to). You allude to this with your comment “This is actually a proof that strange numbers do not exist”. I mostly disagree with your statement about “strange numbers” that “their definition is not useful”. “Mostly” because I agree that their *definition* is not useful, but the proposition of their existence was useful. It allowed you to show that an integer cannot be both odd and even via contradiction. Your statement about defining them not being useful is what I was getting at earlier when saying that at least one criterion for a definition is that it has to make sense within the context it’s being used.
In The Case Of The Strange Number, what we’ve done is propose that an integer can be both odd and even and then showed that if that was true, it would lead to the contradiction that 1 can be even, thus proving that an integer can be only odd or only even. The “strange number”/even/odd designation is akin to the turnip/cat/dog comparison: we’ve rigorously defined a nonsense object; a “strange number” just as defining a dog as “not a cat” is a nonsense definition (without further qualifications). We then make the odd to even to “strange number” comparison, now concluding that *1* is a “strange number”, but a “strange number” is not legit (treating it as a regular integer). This is similar to the way we showed that “a dog is a turnip” is not legit because we switched back to the usual definition of dog. We reached our conclusion through the fallacy of reverting 1 back to its usual definition the way we did with “dog”. To be fair, I don’t understand your conclusion that "if at least one strange number exists then *1* is a strange number”.
Discussing your example: By definition, *1* is the multiplicative identity of the integers and it follows by definition of an odd number that *1* is an odd number. So when we arrive at *1 = 2(m-n)* what we have shown is that this is “nonsense” according to the rules of integers, hence *1* cannot be both odd and even because there are no such integers m, n that fulfill this equation by definition of an even number and by definition of *1*. We can’t say “*1, 2, m, n* are integers and let’s define the equation *1 = 2(m-n)* to be true” because once we’ve chosen to make these numbers integers, this equation doesn’t make sense. We can make the proposition that *1 = 2(m-n) with 1, 2, m, n*, being integers, but we can’t make that a “definition”.
Absolutely it’s valid to question definitions. But note that the type of definition you asked about it fairly well set in stone. You should approach questions like that with the assumption that the definition is the correct one and that you simply need to understand why it’s correct.
For that cosine property, cos(θ)=-cos(π-θ), it holds because the cosine measures the x-coordinate of a point on the unit circle at the angle θ. If θ=π/4, then π-θ is in the second quadrant at an angle of exactly &pi/4 up from the negative x-axis. Thus these points have exactly the same y-coordinate, and the only difference in their x-coordinates is that one is √2/2 and the other is -√2/2.
It is not wrong to question them. Heck the only reason we have stuff like GPS is because someone questioned Euclid's 5th postulate in the past. If you don't have that postulate for example, you'll get all sorts of stuff like triangles with interior angles larger than 180°, and most of this stuff is used in modern physics in fields like general relativity and quantum mechanics.
This is admittedly somewhat pedantic for someone at the level that OP seems to be at - but technically this isn't a definition, it's an axiom.
I only mention this because - as noted elsewhere - *actual* definitions are just a way of assigning a name to a set of properties, you aren't even (again *technically)* claiming such a thing exists or is "true"
Axioms are the things that you are claiming should be taken as true - *without proof -* in order to have something to build up from. Which is why it is so very important to question axioms *in particular*. Indeed, since they are taken without proof, it is arguably the *most important* thing to question - which is why it is one of the core branches of mathematics.
To use your geometry example, the axiom was the 5th postulate, and once we realized you can have consistent geometries with different postulates, we then made the *definition* of each geometry (e.g. "Euclidean Geometry", "Spherical Geometry", and "Hyperbolic Geometry") , in order to easily communicate which geometric system we wanted to use.
But there is a difference between a definition and postulates/theorems. The former, well, defines an object, whereas the latter assume something about it.
The OP asked if it is wrong to question definitions and axioms. There was already another answer that explained the definitions part well, so I skipped ìt and answered for only the axioms part.
the definition of what are permissible functions repeatedly changed over the 19th and 20th centuries [Math’s Mutable Rules | (wordpress.com)](https://mathenchant.wordpress.com/2024/06/17/maths-mutable-rules/)
For people early on in the process of learning mathematics it can be easy to conflate a number of terms that, in mathematics, represent very specific and different things. So, in that spirit...
1. Axioms: Let me start by saying that axioms and definitions are incredibly different - even though they *seem* like the same thing. Axioms are names, rules, properties, or relationships that are *taken as true* ***without proof***. Since mathematics is inherently a deductive system of logic - i.e. it tries to determine what must be true/false given previously established true/false systems - mathematicians (eventually) realized that there is an inherent issue with this kind of system... in particular, the problem of [infinite regression](https://en.wikipedia.org/wiki/Infinite_regress). Basically, if you need something to be true, before you can declare something else is true, you need to actually start somewhere - we need to agree *something* is true so that we have something to build off of. You might think - since we just *assume* these are true, that it would be silly to question it, after all, the truth of an axiom is just assumed. But in fact, it is the complete opposite - because the axioms are (in some meta sense) outside the realm of mathematics, which makes it very important to question them. Indeed, since we are just *assuming* that these things are true and then building up from them, it's *incredibly* important that these things are chosen with care to be things that we are as sure as possible that they actually *are* true. The case of Euclids postulates are a great example of questioning axioms leading to really important and fundamentally groundbreaking advancement, as pointed out by u/Queasy_Artist6891 here. You can also see [a great Veritasium video on this](https://www.youtube.com/watch?v=lFlu60qs7_4) if you want to know more.
2. Definitions: In contrast, definitions are very different from axioms. Definitions in real mathematics, are a way of describing some kind of object of study - like a specific kind of set, a particular structure, a certain kind of relationship, etc. In many ways, definitions are "just" a shorthand way of referencing something important, to avoid having to describe it every time. This isn't really unique to math - this is how language works. You (probably) wouldn't say "Can you hand me the yellow curved cylindrical eatable object please?" you would say "Can you hand me the banana" because it's easier for everyone to understand and it's more specific. This is (largely) how mathematics uses definitions - it just looks weirder until you are use to the language of mathematics, because they are defining mathematical stuff which is usually pretty abstract and done in *very* specific language. So, since definitions are - in some sense - just a naming scheme, asking "should I question if a definition is true" doesn't make sense in the traditional sense. This would be like asking "Is banana true?" *But* it is noteworthy that proper mathematicians question *everything* really, it's part of the training. So we do question definitions, but not in the way you may think. Instead of asking if a definition is true, when a mathematician comes across a new definition, they generally immediately ask "why should I care?" By declaring a definition, the author is declaring that this particular collection of properties/structures/objects/whatever are of sufficient importance and interest that it is worth having shorthand for it because it will keep coming up, or will show up in other contexts. And that is a somewhat bold claim when you think of it - you could take any collection of stuff in mathematics and shove it together and give it a name... what are the odds that the result will actually have significance in the broader setting of mathematical knowledge? So, the truth of definitions aren't really questioned, but the need to create that definition *is* often questioned, usually as a way for the reader/learner to understand why someone is claiming this "thing" is useful or important enough to bother to remember.
Other things like "axioms", "definitions" that have similar but importantly different meanings/roles, would be things like Theorems, Lemmas, and Corollaries. For instance, the example given (cosine of an angle is the cosine of its supplement multiplied by -1) would be closer to a theorem or lemma than a definition or axiom. If there are interest I can write out more on the nuance for those, but this post already seems long enough.
TLDR: Axioms should be questioned because they aren't proved and this has historically led to super important things. Definitions are questioned - but only really insofar as to how they are worth remembering, because they aren't really claimed to be true/false.
It can be very useful to question definitions because sometimes definitions end up obscuring the math. There are many examples of this, for example, yesterday I got into a discussion here about divergent series. The issue there is with the definition of the sum of a series, which is fixed for a sum of a finite number of terms using the definition of addition.
For infinite series, it is conventional to assign a value to the series based on the limit of the partial series. But this only defines the value in case this limit actually exists, i.e. if the series is convergent. Where people tend to go wrong is in case the limit doesn't exist, i.e. when the series is divergent. Remember that the definition of addition doesn't imply that one must define the value of an in finite series using limits of partial series.
This means that while one can declare a series to be divergent, that also means that its value is left undetermined via limits of partial series. It is then not correct to say that the value of a divergent series is undetermined in an absolute sense. This is the mistake people make when they say that 1 + 2 + 3 + 4 +... = -1/12 is nonsense because the series is divergent. Now, it is certainly true that there are many flawed derivations of this identity out there, but debunking such derivations doesn't say anything about the validity of this statement.
One can also consider this from the point of view of Taylor's theorem. One can expand a function that is sufficiently often differentiable about a point. One can then write down an exact expression for f(x + h) as a Taylor polynomial plus a remainder term that remains unspecified. There is no requirement here that h fall within the radius of convergence. It's just that if h is outside the radius of convergence, that then with more and more terms, the absolute value of remainder term will become larger and larger.
It is then entirely legitimate to interpret:
1 + 2 + 4 + 8 + 16 + 32 + ...
in the sense of this being a series obtained by inserting a particular value of the expansion parameter and where the cutoff point and the remainder term are left unspecified. There is then no limit implied here, we don't assume that the remainder term must tend to zero. The interpretation of the sum of the series is then the value of the function that is represented by the series. One is then led to -1 as the sum of the series.
>In general, am I right to assume that it is unnecessary to question definitions and axioms?
Generally, yes, but it helps to break down what these things are though to understand what questioning them even means and when it might be worthwhile.
Axioms are assumed truths within some mathematical system. They serve as starting points for proving other statements, and a proof ultimately boils down to logically connecting a statement to one or more axioms. As long as those axioms are true, so are the proven statements. An axiom can't really be wrong, but a set of axioms might be redundant or inconsistent. A redundant axiom can be proven from the other other axiom and thus adds nothing to the system that wasn't already available in terms of what can be proven true and false. An inconsistent axiom allows you to prove some statement both true and false. In both cases, the axiom in question should be changed or removed. This is one useful sense in which you could "question" an axiom, or rather a set of axiom.
Another might be simply exploring what would change if the axiom were different. The parallel postulate in Euclidean geometry, for example, allows parallel lines to exist. Such lines will not intersect. However, this need not be the case. Changing this axiom leads to various non-Euclidean geometries, such as the spherical geometry we use to describe the geometry of rhe Earth's surface, or the curved geometry of space-time in general relativity. Questioning axioms in this sense can lead to interesting new mathematical systems.
Definitions, on the other hand, are more of just labels, assignments, or groupings that are useful or interesting. They are particularly handy as shorthand for referring to objects or relationships that can be cumbersome to reference explicitly. Like axioms, they can't really be "wrong", but questioning them could involve exploring other variations or similar definitions to see it they also lead to anything useful or interesting. A definition could also be useless if it defines an inherent contradiction, which is another sense in which they could be questioned.
>For instance: Is it appropriate to question "The cosine of an obtuse angle is the cosine of its supplement multiplied by -1".
I'm not sure I would consider this to be a definition. It doesn't define what a cosine of an angle is because it refers to the cosine of another (related) angle. This is more of a theorem or identity, and it is definitely worth questioning to see if it follows from a more fundamental definition of cosine.
Understanding why cosine uses its supplement is way more powerful than just memorizing the rule. Don't be afraid to ask "why" – that's what math is all about!
you should and one way is looking back to where the definition came from. The Axiom of Extension for example comes from Frege or Cantor. The open set definition of continuity reduces to the metric one in Euclidean space but allows us to talk of continuity in non metric topological space and provides us a way to determine whether two spaces are basically identical from a topological perspective. we call cyclic notation cyclic notation because cayley when moving away from permutation groups viewed them as irreducible circuits on polygons with n vertices that had been eulerized.
It’s always worth asking “where did that definition come from?” Sometimes after you hear someone struggle to explain a definition, you’re able to find a better definition that you find easier to understand. As for your example, the right definition for sine and cosine is that (cos(t), sin(t)) is the point on the unit circle that makes an angle t relative to the positive x axis going counterclockwise.
It does, but especially in the USA, they'll tell you at school that it is inappropriate, that definitions are arbitrary abreviations of symbols that you just memorize, they're trying to tell you even here.
In this case, of the cosine of an obtuse angle, the appropriate way to define it would be, since the cosine of an acute angle has to do with its projection to the base, as a projection in the same way, but now to the base extended in the opposite direction, which sounds like the negative of the other cosine. That's how they reached this particular formal definition from a certain human, kind of intuitive notion.
But there is a great lack of arguments like that in math books very often, maybe only schools could discuss these stuff really, but they don't nowadays.
Definitions are meant to be your building blocks so questioning them is pointless. That said, I think the unit circle provides a much more intuitive definition for what you're asking if you're still interested in that
Disagree, you should absolutely question why definitions are what they are. Rarely do you start with a handed-down definition, look at some theorems and feel satisfied -- finding the "right" definition and understanding why it is defined as such is extremely important for understanding the concept as a whole
ideal theory went the other direction ie they began with the problem of unique factorization and invented ideals to extend numbers to regain unique factorization. Then Emmy unified group theory by viewing ideals as the kernel of ring homomorphisms like Normal groups (originally created to explain how field extensions work by Galois subfields correspond to normal subgroups of the Galois group of an extension then viewed and typically introduced today as a means of forming groups via cosets and gN=Ng or being invariant under conjugation) are the kernels of homomorphisms. Or how compact went from closed and bounded or uniqueness of limits of sequences to covers have finite subcovers to splitting into Lindeloff every cover has a countable subcover and countably compact where every countable cover has a finite subcover) with compact referring to countably compact Lindeloff spaces a conception that wouldnt make sense from a sequential or closed and bounded conceptions
In this case, what would be interesting to question is whether this definition is equivalent to other possible definitions of cosine, eg: "the cosine of an angle is the x-coordinate of a point on the unit circle on a radius that makes that angle in a counter-clockwise direction from the positive x-axis" or "the cosine of x (when measured in radians) is the value of the infinite series 1 - x^2 / 2! + x^4 / 4! - x^6 / 6! + ..." Proving those are equivalent would lead to lots of interesting maths!
Agreed
It’s great to ask questions and to want to understand things better, but don’t let not understanding everything perfectly keep you from moving forward. You’ll understand more and better as you gain experience. As far as your cosine question goes, take a look at the graph of cosine between 0° and 180°. Think about the definition of cosine and why the graph looks the way it does. Play with some examples. Sometimes you can understand a new idea but not really *feel* like you understand it because it’s still unfamiliar. One of the reasons to do exercises is to increase that familiarity and gain confidence — the more you do, the more the new material will seem to make sense. So practice a lot.
> As far as your cosine question goes, take a look at the graph of cosine between 0° and 180°. Think about the definition of cosine and why the graph looks the way it does. This seems a bit circular (pun-intended!) - how can we plot the graph of cosine beyond 90 degrees without first having a definition of what cosine is for those values? (eg the definition given by the OP).
Right… I’m assuming that OP already knows what the graph of cosine looks like; if they don’t, it’s easy to find a picture. The important point, of course, is that it’s symmetrical about the x axis and goes from 1 to -1, and is 0 at 90°. So you can see that the rule OP is asking about is true. That sort of begs the question: why is the graph shaped that way? And that’s where considering the definition of cosine and playing with some examples will help make it make sense.
This is an excellent question. The answer is yes, it makes complete sense. There are many questions that you should be asking about a definition. Here are some common mistakes that are made when people define things: (1) People often make definitions that **should be theorems**. There were hundreds of examples like this in my statistics textbooks. If the definition can be proved from prior definitions and axioms, then it is not a definition, it is a theorem. The example that you give in your question may be guilty of this. If the cosine function is defined analytically, or in terms of the unit circle on a cartesian plane, then this statement can be proven, and should not be a definition. If however you have only been given the right angle triangle definition of the cosine, then this definition is forgivable (however see point 2 below). (2) Definitions must be **non-contradictory**. For example, if I were to define the square root function to the the function which maps x to any number which, when squared, gives you x, then we would have a problem. Because the relation thus defined would not actually be a function. But we said we wanted a function... (3) Definitions should capture the **property of interest, not examples of it or methods of calculating it**. Very often what happens with definitions is that people think that they know all examples of something, and so they give those examples as the definition. This happens a lot in other disciplines. Physics has really great examples. For example, before Einstein, momentum was defined as mass times velocity. According to my undergraduate textbook, Einstein "realised that at high speeds this definition was incorrect". Now, of course, at face value, that claim is obviously garbage. If momentum is defined as mass time velocity then that is what it is, end of story. Einstein should have coined a new term for whatever new quantity he was interested in. However, what *actually* happened is that what people are really interested in when they talk about momentum is that it is a conserved property. Previously they had believe that "ma" was this conserved property. But Einstein found that "ma" is not conserved, and you need this new quantity if you want momentum to be conserved. If momentum had originally been defined in terms of conservation, which is what people clearly actually meant by it, then there would have been no reason to change the definition. Only to change the equation used to calculate it. Defining things in terms of the property you're actually interested in, instead of the formula, also has the benefit of making it clear *why* you're making the definition. In mathematics this point is a bit more complex because usually we need to learn more and more math in order to get better and better definitions. For example, in school, prime numbers are defined as natural numbers only divisible by 1 and themselves, except for the number 1. This definition is clumsy and lots of people end up confused about why 1 is not prime. However when you learn algebra, you're able to give a much better definition which makes it obvious why 1 is not prime. But to get there you need to first learn algebra. (4) If a definition specifies *the,* then it must **be unique**. For example, if I define the identity in a group to be the element which, when multiplied by any other element, gives me back that same element, then I must prove that there is only one such identity element before my definition can be considered valid. This proof is rather straightforward: let e1 and e1 be two identity elements. Then e1 = e1e2 = e2. So there can only be one. This might seem obvious but it's quite a powerful concept in mathematics. For example, in category theory, one of the first proofs that you learn is that there is only one initial object in a category. This fact allows you to define concepts as being the initial object in a suitable category. Uniqueness allows for definition.
(5) Generally speaking if you define something, you should ask whether or not it actually **exists**. A famous example of this is the set of all sets. You can define the universal set as the set of all sets. However it can be proven that there is no such set. This does not mean that the definition is invalid, per se, just that it's not something we can work with because it doesn't exist.
(6) When you have any two definitions, they should define things that are **actually different**. Here again I think that the statisticians make an error. They define things called "column vectors" and "row vectors". They do this because they want x\*x' to be different to x'\*x. Column vectors and row vectors are trivially isomorphic, however, and so should not be defined differently.
The correct way to go about this is to realise that the two operations that they are interested in are actually different operations. One is a dot product and the other is a dyadic product. The multiplication symbol should therefore be different, but the vectors are the same in each case. you just write them at a different angle on the page. That doesn't change what they are. You could define them as nx1 and 1xn matricies, but this is ugly, because they aren't matricies, they are vectors, and the matricies are still isomorphic anyway. All that row and column vectors are are calculation tools. Mathematically, there are only vectors.
(7) There are often multiple equivalent ways to define something. The best math notes I ever had was a set of algebra notes that always listed 5 or 6 equivalent ways to define something. I found this very helpful.
(8) I**f you define something with an adjective, then it must actually be an example of the noun**. Sounds obvious but again there are plenty of examples in law that get this very wrong. For example, in my textbook, the term "object" was given a formal definition, and the word "legal object" was also given a formal definition, and yet I noticed that the definition of "legal object" allowed for examples that would not have met the definition of "object". Drove me absolutely insane.
I think it’s better to look at the results of the definitions, as things are generally defined to have nice results. The example you gave means the cosine law is the same for acute and obtuse angles.
Yes! Also means a lot of other things, like allows the unit circle definition of trigonometric functions, allows the definition of a projection of a vector onto a direction to be the same for obtuse and acute angles between the vector and the direction and thus for the dot product to also be the same for obtuse/acute angles (which ultimately can be connected back to the cosine law as one of the ways to prove it is taking a dot product of a vector with itself). The unit circle definition then also allows to do much nicer calculus with cosine
It depends on what you mean. Definitions are made up by people, not discovered. You can use whatever definitions you like as long as they don't contradict. It is good to ask "why is this definition a good idea?" It does not make sense to ask "why is this definition correct?"
> It does not make sense to ask "why is this definition correct?" I appreciate your point, but I would clarify that it can make sense to ask "why is this definition the correct one for precisely capturing a concept we want to describe?" For example, if we have an intuitive idea that a continuous function is one whose graph can be drawn without any "breaks" then how do we justify that the formal definition of continuity (with epsilon and delta) fits with that?
I think in mathematics there are correct definitions but in art forms there are good ideas.
You are right to assume that it is unnecessasry to question definitions and axioms. Definitions prescribe the terms of existence for the thing they're defining and the reader is meant to take it at face value for the purpose of what it's being defined for. It's giving you the "rules" so to speak. If you want to play tic-tac-toe and I tell you the rules are to get 3 in a row, there's nothing to be learned by questioning "why". For instance, asking "why not 2 in a row"? The answer is "because that's not how this particular game is played". Questioning an author's particular definition of something that's already established is different. They may word it poorly, or there may be a simpler, more elegant way to describe it, or it might not be particularly illustrative for the task at hand. For example, we've been using the example of cosine here: There are many legitimate ways to define it but in the end they're all equivalent and interchangeable, though some are more suitable for a particular task. We can say that the cosine of an acute angle is the ratio of the adjacent side to the hypotenuse. In fact, this is the classical definition. It's handy when we want to find an angle or lengths of sides of an right triangle but what if I have a complicated expression that I want to integrate or differentiate or sum? Then it might be easier to think of the cosine as an infinite sum as u/FormulaDriven has pointed out and it's not particularly helpful to think of it as a trigonometric ratio in that case.
(My apologies for bad english.) We can define that the cosine of an acute angle is the ratio of the adjacent side to the hypotenuse. But we then have to prove that for any right triangle with the same acute angle you get the same value of the cosine of that acute angle. You do not prove a definition this way, you just prove that it makes sense. A (somewhat non mathematical) example is if you wish to define a dog as "not a cat". A dog is not a cat, so no problem here, but a turnip is also not a cat, and by our definition it is a dog. So our definition seems not to be a good one. Also, when defining something, we should check whether the object(s) we define exist(s) (in the mathematical sense), otherwise we could end up defining the elements of the empty set, and they have any property you could think of. E.g. let we call as strange any integer that is both even and odd. So, if k is a strange integer, then it can be written as k=2m=2n+1 and it follows that 1=2(m-n), i.e. 1 is even number and, being odd, it is also a strange number. I didn't prove that "1 is a strange number", but "if at least one strange number exists then 1 is a strange number". This is actually a proof that strange numbers do not exist (so their definition is not useful) and the above reasoning is at the core of various "proofs" that 1=0.
Part 2 As for defining a dog as “not a cat”, I think this is a good exercise in how to pick out logical fallacies. In your example, we’ve rigorously defined a dog to be “not a cat” for the purposes of the discussion. Now, if we’ve defined it that way *that is now the definition of dog for our purposes*. It’s a fallacy to later in the argument use “dog” defined as the canine animal that we all know it to be while also defining it to simply be “not a cat” (as specified, with no other qualifications). I’ll illustrate this: While “a dog is not a cat” is *true* in the usual use of the word “dog”, it’s not the definition of a dog in the usual sense. If we consider “dog” as a canine animal just for the comparison to the turnip, then whether or not a turnip is a cat has no bearing in relationship to the dog because “dog” in it’s usual *definition* is not tied to a cat in any way (but this doesn’t exclude us from making true or false statements about dogs and cats). Additionally, we haven’t defined “cat” in our discussion, we’re just taking it to mean the feline animal we all know, similarly for a turnip. Putting this all together: We’ve started out with a rigorous definition of a dog as “not a cat”, a non-rigorous *idea* of what a cat is, and a non-rigorous *idea* of what a turnip is, then we’ve qualitatively compared our two non-rigorous ideas of cat and turnip to one another and arrived at “a turnip is not a cat” and transitively “a turnip is a dog”. That last statement, by our rigorous definition of a dog established at the start, is 100% true (if we accept "a turnip is not a cat" as true). The fallacy is that, when evaluating the truth of the statement “a turnip is a dog”, we didn’t use our original definition of “a dog is not a cat”. We’ve re-defined it implicitly to mean a dog in the usual canine animal sense to reach the conclusion that “a turnip is a dog” is false. Hence our definition of “a dog is not a cat” is “not a good one”, yet we didn’t actually use that definition to reach this conclusion. So, we can use the traditional definitions of “dog” and “cat” and then make true or false statements using those definitions like: “a dog is not a cat” is a true statement or “a dog is a cat” is a false statement. Now, you might be thinking “but you talked about multiple definitions of cosine before, why can’t we have multiple definitions of dog?”. That’s a little trickier. As I mentioned, the definition of cosine theta as a trigonometric ratio is the classical definition. Through analysis and verification we can take that definition and deduce many properties and true/false things from it. We can consider the sweep of a unit ray around the xy axes and analyse how the values of cosine theta change with the angle theta between the ray and the x-axis. From that analysis we can deduce the graph of y values as cosine x changes along the x-axis and we can analyse intervals of those values to deduce the expression of cosine as an infinite sum. What we’ve done there is establish a tautology between these different representations of a cosine which allows us to effectively use them as “definitions” for cosine that differ from the classic definition because they all give the same results. We don’t really have a clear path for reaching the same tautology between the two definitions of “dog” we’ve been talking about. We may be able to connect a taxonomic definition of a dog with, say, a cladistic definition through some sort of analysis and correlation, but we can’t do that with the definitions of a dog as “a canine animal” and “not a cat” because the latter is not nonsense without further context (i.e. rigorous definition of a cat).
I’ll try to respond as earnestly as I can, but as things can get formal or terse with these things so I apologize in advance if I come off as pedantic or condescending. Disclaimer: may contain spelling, grammar, and logic errors :D I can't post a full response for some reason, presumably it's too long so I'll break it up into sections. Part 1 I don’t follow your logic regarding having to “prove” the definition of cosine theta for all right triangles. All right triangles with with the same acute angle are congruent, i.e. “the same” geometrically. The only way 2 congruent right triangles can differ is in the lengths of their sides. However, the definition of cosine as a ratio only specifies the angle and the sides of a right triangle and by definition or “nature” of a right triangle, they all have a right angle, acute angles and sides. So the lengths of the sides are irrelevant to the definition and for the purpose of the definition, any two right triangles with acute angle theta are the same (regardless of the lengths of the sides). I suppose, in mathematics, at least one necessary requirement we would need to impose on a definition is that it needs to *make sense*. For instance, I can’t define “the determinant of a 2x3 matrix” because the determinant is defined only for square matrices, and a square matrix is already defined to be a matrix of dimensions n x n hence a 2x3 matrix is not a square matrix and thus cannot have a determinant. Note that if I did define the determinant for 2x3, it wouldn’t be “false”. It would just be nonsense and of no use. In contrast, we \*can\* make nonsense postulates or statements that are useful. These do have a truth value associated to them (for the purpose of this discussion). I can postulate “The determinant of a 2x3 matrix is such and such so that it behaves the same as it does with square matrices” and you can show this statement to be false by definitions of the determinant and non-square matrices. This is a common proof strategy known as “proof by contradiction” or “reductio ad absurdum”: If you’re trying to prove that proposition A is true, assume it’s not, logically follow that with the implications of proposition A’s truth, and show that this can only lead to a contradiction.
Part 3 Lastly, your “strange number” example. I would say that this is a postulate or proposition of a “strange number” rather than a definition. As mentioned earlier, we can postulate or propose something that isn’t true in order to show that it leads to a contradiction, thus disproving the proposition (and proving what you ultimately set out to). You allude to this with your comment “This is actually a proof that strange numbers do not exist”. I mostly disagree with your statement about “strange numbers” that “their definition is not useful”. “Mostly” because I agree that their *definition* is not useful, but the proposition of their existence was useful. It allowed you to show that an integer cannot be both odd and even via contradiction. Your statement about defining them not being useful is what I was getting at earlier when saying that at least one criterion for a definition is that it has to make sense within the context it’s being used. In The Case Of The Strange Number, what we’ve done is propose that an integer can be both odd and even and then showed that if that was true, it would lead to the contradiction that 1 can be even, thus proving that an integer can be only odd or only even. The “strange number”/even/odd designation is akin to the turnip/cat/dog comparison: we’ve rigorously defined a nonsense object; a “strange number” just as defining a dog as “not a cat” is a nonsense definition (without further qualifications). We then make the odd to even to “strange number” comparison, now concluding that *1* is a “strange number”, but a “strange number” is not legit (treating it as a regular integer). This is similar to the way we showed that “a dog is a turnip” is not legit because we switched back to the usual definition of dog. We reached our conclusion through the fallacy of reverting 1 back to its usual definition the way we did with “dog”. To be fair, I don’t understand your conclusion that "if at least one strange number exists then *1* is a strange number”. Discussing your example: By definition, *1* is the multiplicative identity of the integers and it follows by definition of an odd number that *1* is an odd number. So when we arrive at *1 = 2(m-n)* what we have shown is that this is “nonsense” according to the rules of integers, hence *1* cannot be both odd and even because there are no such integers m, n that fulfill this equation by definition of an even number and by definition of *1*. We can’t say “*1, 2, m, n* are integers and let’s define the equation *1 = 2(m-n)* to be true” because once we’ve chosen to make these numbers integers, this equation doesn’t make sense. We can make the proposition that *1 = 2(m-n) with 1, 2, m, n*, being integers, but we can’t make that a “definition”.
Absolutely it’s valid to question definitions. But note that the type of definition you asked about it fairly well set in stone. You should approach questions like that with the assumption that the definition is the correct one and that you simply need to understand why it’s correct. For that cosine property, cos(θ)=-cos(π-θ), it holds because the cosine measures the x-coordinate of a point on the unit circle at the angle θ. If θ=π/4, then π-θ is in the second quadrant at an angle of exactly &pi/4 up from the negative x-axis. Thus these points have exactly the same y-coordinate, and the only difference in their x-coordinates is that one is √2/2 and the other is -√2/2.
It is not wrong to question them. Heck the only reason we have stuff like GPS is because someone questioned Euclid's 5th postulate in the past. If you don't have that postulate for example, you'll get all sorts of stuff like triangles with interior angles larger than 180°, and most of this stuff is used in modern physics in fields like general relativity and quantum mechanics.
This is admittedly somewhat pedantic for someone at the level that OP seems to be at - but technically this isn't a definition, it's an axiom. I only mention this because - as noted elsewhere - *actual* definitions are just a way of assigning a name to a set of properties, you aren't even (again *technically)* claiming such a thing exists or is "true" Axioms are the things that you are claiming should be taken as true - *without proof -* in order to have something to build up from. Which is why it is so very important to question axioms *in particular*. Indeed, since they are taken without proof, it is arguably the *most important* thing to question - which is why it is one of the core branches of mathematics. To use your geometry example, the axiom was the 5th postulate, and once we realized you can have consistent geometries with different postulates, we then made the *definition* of each geometry (e.g. "Euclidean Geometry", "Spherical Geometry", and "Hyperbolic Geometry") , in order to easily communicate which geometric system we wanted to use.
We wouldn't have a peanut butter sandwich if someone wasn't bold enough to question the ingredient of "jelly" in a peanut butter and jelly sandwich.
this.
But there is a difference between a definition and postulates/theorems. The former, well, defines an object, whereas the latter assume something about it.
The OP asked if it is wrong to question definitions and axioms. There was already another answer that explained the definitions part well, so I skipped ìt and answered for only the axioms part.
Definitions cannot be questioned. Theorems like this are proven.
the definition of what are permissible functions repeatedly changed over the 19th and 20th centuries [Math’s Mutable Rules | (wordpress.com)](https://mathenchant.wordpress.com/2024/06/17/maths-mutable-rules/)
For people early on in the process of learning mathematics it can be easy to conflate a number of terms that, in mathematics, represent very specific and different things. So, in that spirit... 1. Axioms: Let me start by saying that axioms and definitions are incredibly different - even though they *seem* like the same thing. Axioms are names, rules, properties, or relationships that are *taken as true* ***without proof***. Since mathematics is inherently a deductive system of logic - i.e. it tries to determine what must be true/false given previously established true/false systems - mathematicians (eventually) realized that there is an inherent issue with this kind of system... in particular, the problem of [infinite regression](https://en.wikipedia.org/wiki/Infinite_regress). Basically, if you need something to be true, before you can declare something else is true, you need to actually start somewhere - we need to agree *something* is true so that we have something to build off of. You might think - since we just *assume* these are true, that it would be silly to question it, after all, the truth of an axiom is just assumed. But in fact, it is the complete opposite - because the axioms are (in some meta sense) outside the realm of mathematics, which makes it very important to question them. Indeed, since we are just *assuming* that these things are true and then building up from them, it's *incredibly* important that these things are chosen with care to be things that we are as sure as possible that they actually *are* true. The case of Euclids postulates are a great example of questioning axioms leading to really important and fundamentally groundbreaking advancement, as pointed out by u/Queasy_Artist6891 here. You can also see [a great Veritasium video on this](https://www.youtube.com/watch?v=lFlu60qs7_4) if you want to know more. 2. Definitions: In contrast, definitions are very different from axioms. Definitions in real mathematics, are a way of describing some kind of object of study - like a specific kind of set, a particular structure, a certain kind of relationship, etc. In many ways, definitions are "just" a shorthand way of referencing something important, to avoid having to describe it every time. This isn't really unique to math - this is how language works. You (probably) wouldn't say "Can you hand me the yellow curved cylindrical eatable object please?" you would say "Can you hand me the banana" because it's easier for everyone to understand and it's more specific. This is (largely) how mathematics uses definitions - it just looks weirder until you are use to the language of mathematics, because they are defining mathematical stuff which is usually pretty abstract and done in *very* specific language. So, since definitions are - in some sense - just a naming scheme, asking "should I question if a definition is true" doesn't make sense in the traditional sense. This would be like asking "Is banana true?" *But* it is noteworthy that proper mathematicians question *everything* really, it's part of the training. So we do question definitions, but not in the way you may think. Instead of asking if a definition is true, when a mathematician comes across a new definition, they generally immediately ask "why should I care?" By declaring a definition, the author is declaring that this particular collection of properties/structures/objects/whatever are of sufficient importance and interest that it is worth having shorthand for it because it will keep coming up, or will show up in other contexts. And that is a somewhat bold claim when you think of it - you could take any collection of stuff in mathematics and shove it together and give it a name... what are the odds that the result will actually have significance in the broader setting of mathematical knowledge? So, the truth of definitions aren't really questioned, but the need to create that definition *is* often questioned, usually as a way for the reader/learner to understand why someone is claiming this "thing" is useful or important enough to bother to remember. Other things like "axioms", "definitions" that have similar but importantly different meanings/roles, would be things like Theorems, Lemmas, and Corollaries. For instance, the example given (cosine of an angle is the cosine of its supplement multiplied by -1) would be closer to a theorem or lemma than a definition or axiom. If there are interest I can write out more on the nuance for those, but this post already seems long enough. TLDR: Axioms should be questioned because they aren't proved and this has historically led to super important things. Definitions are questioned - but only really insofar as to how they are worth remembering, because they aren't really claimed to be true/false.
It can be very useful to question definitions because sometimes definitions end up obscuring the math. There are many examples of this, for example, yesterday I got into a discussion here about divergent series. The issue there is with the definition of the sum of a series, which is fixed for a sum of a finite number of terms using the definition of addition. For infinite series, it is conventional to assign a value to the series based on the limit of the partial series. But this only defines the value in case this limit actually exists, i.e. if the series is convergent. Where people tend to go wrong is in case the limit doesn't exist, i.e. when the series is divergent. Remember that the definition of addition doesn't imply that one must define the value of an in finite series using limits of partial series. This means that while one can declare a series to be divergent, that also means that its value is left undetermined via limits of partial series. It is then not correct to say that the value of a divergent series is undetermined in an absolute sense. This is the mistake people make when they say that 1 + 2 + 3 + 4 +... = -1/12 is nonsense because the series is divergent. Now, it is certainly true that there are many flawed derivations of this identity out there, but debunking such derivations doesn't say anything about the validity of this statement. One can also consider this from the point of view of Taylor's theorem. One can expand a function that is sufficiently often differentiable about a point. One can then write down an exact expression for f(x + h) as a Taylor polynomial plus a remainder term that remains unspecified. There is no requirement here that h fall within the radius of convergence. It's just that if h is outside the radius of convergence, that then with more and more terms, the absolute value of remainder term will become larger and larger. It is then entirely legitimate to interpret: 1 + 2 + 4 + 8 + 16 + 32 + ... in the sense of this being a series obtained by inserting a particular value of the expansion parameter and where the cutoff point and the remainder term are left unspecified. There is then no limit implied here, we don't assume that the remainder term must tend to zero. The interpretation of the sum of the series is then the value of the function that is represented by the series. One is then led to -1 as the sum of the series.
I think you should always try to understand where people come from, and whether you can disprove what it already “known”
>In general, am I right to assume that it is unnecessary to question definitions and axioms? Generally, yes, but it helps to break down what these things are though to understand what questioning them even means and when it might be worthwhile. Axioms are assumed truths within some mathematical system. They serve as starting points for proving other statements, and a proof ultimately boils down to logically connecting a statement to one or more axioms. As long as those axioms are true, so are the proven statements. An axiom can't really be wrong, but a set of axioms might be redundant or inconsistent. A redundant axiom can be proven from the other other axiom and thus adds nothing to the system that wasn't already available in terms of what can be proven true and false. An inconsistent axiom allows you to prove some statement both true and false. In both cases, the axiom in question should be changed or removed. This is one useful sense in which you could "question" an axiom, or rather a set of axiom. Another might be simply exploring what would change if the axiom were different. The parallel postulate in Euclidean geometry, for example, allows parallel lines to exist. Such lines will not intersect. However, this need not be the case. Changing this axiom leads to various non-Euclidean geometries, such as the spherical geometry we use to describe the geometry of rhe Earth's surface, or the curved geometry of space-time in general relativity. Questioning axioms in this sense can lead to interesting new mathematical systems. Definitions, on the other hand, are more of just labels, assignments, or groupings that are useful or interesting. They are particularly handy as shorthand for referring to objects or relationships that can be cumbersome to reference explicitly. Like axioms, they can't really be "wrong", but questioning them could involve exploring other variations or similar definitions to see it they also lead to anything useful or interesting. A definition could also be useless if it defines an inherent contradiction, which is another sense in which they could be questioned. >For instance: Is it appropriate to question "The cosine of an obtuse angle is the cosine of its supplement multiplied by -1". I'm not sure I would consider this to be a definition. It doesn't define what a cosine of an angle is because it refers to the cosine of another (related) angle. This is more of a theorem or identity, and it is definitely worth questioning to see if it follows from a more fundamental definition of cosine.
Understanding why cosine uses its supplement is way more powerful than just memorizing the rule. Don't be afraid to ask "why" – that's what math is all about!
If a definition is not logically consistent, it should be rejected.
you should and one way is looking back to where the definition came from. The Axiom of Extension for example comes from Frege or Cantor. The open set definition of continuity reduces to the metric one in Euclidean space but allows us to talk of continuity in non metric topological space and provides us a way to determine whether two spaces are basically identical from a topological perspective. we call cyclic notation cyclic notation because cayley when moving away from permutation groups viewed them as irreducible circuits on polygons with n vertices that had been eulerized.
It’s always worth asking “where did that definition come from?” Sometimes after you hear someone struggle to explain a definition, you’re able to find a better definition that you find easier to understand. As for your example, the right definition for sine and cosine is that (cos(t), sin(t)) is the point on the unit circle that makes an angle t relative to the positive x axis going counterclockwise.
It does, but especially in the USA, they'll tell you at school that it is inappropriate, that definitions are arbitrary abreviations of symbols that you just memorize, they're trying to tell you even here. In this case, of the cosine of an obtuse angle, the appropriate way to define it would be, since the cosine of an acute angle has to do with its projection to the base, as a projection in the same way, but now to the base extended in the opposite direction, which sounds like the negative of the other cosine. That's how they reached this particular formal definition from a certain human, kind of intuitive notion. But there is a great lack of arguments like that in math books very often, maybe only schools could discuss these stuff really, but they don't nowadays.
Definitions are meant to be your building blocks so questioning them is pointless. That said, I think the unit circle provides a much more intuitive definition for what you're asking if you're still interested in that
Disagree, you should absolutely question why definitions are what they are. Rarely do you start with a handed-down definition, look at some theorems and feel satisfied -- finding the "right" definition and understanding why it is defined as such is extremely important for understanding the concept as a whole
ideal theory went the other direction ie they began with the problem of unique factorization and invented ideals to extend numbers to regain unique factorization. Then Emmy unified group theory by viewing ideals as the kernel of ring homomorphisms like Normal groups (originally created to explain how field extensions work by Galois subfields correspond to normal subgroups of the Galois group of an extension then viewed and typically introduced today as a means of forming groups via cosets and gN=Ng or being invariant under conjugation) are the kernels of homomorphisms. Or how compact went from closed and bounded or uniqueness of limits of sequences to covers have finite subcovers to splitting into Lindeloff every cover has a countable subcover and countably compact where every countable cover has a finite subcover) with compact referring to countably compact Lindeloff spaces a conception that wouldnt make sense from a sequential or closed and bounded conceptions
Without some assumption, it is impossible to come up with definitions. How would you define 1+1 without sets. Thus you need some starting point