I am not familiar with Chomsky's layout, I just know of terms such as generative grammar, etc. Is he still doing Linguistics? I only hear of him re politics.I prefer Chomsky over intrinsic content.
I am not familiar with Chomsky's layout, I just know of terms such as generative grammar, etc. Is he still doing Linguistics? I only hear of him re politics.I prefer Chomsky over intrinsic content.
I don't think so. Does German, e.g., have any subject matter that distinguishes it from other languages? I think no; it is used to communicate ideas, like any other human language. Some of the content of German may be unique to living conditions in Germany, but not otherwise, I believe.Don't make me search my book ... It was sooo long ago ... I have problems with 'intrinsic'. Isn't that true for any language which has a meaning?
You cannot translate idioms and sayings and you cannot really translate Shakespeare. These things are a distinction and every language has them, often even dialects. Their existence is intrinsic as well.I don't think so. Does German, e.g., have any subject matter that distinguishes it from other languages? I think no; it is used to communicate ideas, like any other human language. Some of the content of German may be unique to living conditions in Germany, but not otherwise, I believe.
Because there are aspects of life in Germany that do not have a parallel in other countries (e.g., English) , so that the abstract grammatical layout is different to account for this. German language is used to communicate ideas experienced by Germans. Maybe t o make my point more clear ( although I may be wrong) you need an interpretation and assign semantics to a (formal)language. Would you say German language is a Syntactic construct or Semantic, Mixed, etc? I say German language is used as a means to describe the German experience.You cannot translate idioms and sayings and you cannot really translate Shakespeare. These things are a distinction and every language has them, often even dialects. Their existence is intrinsic as well.
Agreed, choosing the right representation in a context is key, and IMO underrated as a means to finding solutions to a problem.A 2-D vector field also has the properties required to be an algebraically defined number. I don't recall them all; operators with identities and inverse operators, existence of zero, etc. That is why so many concepts in analysis can be treated as a complex number or as a 2-D vector. In electronics, for example, the amplitude and phase of a wave, voltage, impedance, etc. are all expressed as a complex number in analysis when they could also be understood as a vector. It is really about choosing a notation that facilitates stealing analysis techniques from another perspective in the world of math.
Howdy,Let me ask the other way around...[snip]...
Is it possible that are there other extensions to ℝ that provide solutions to other types of non-polynomial problems? For example, if I were to define a number j such that sin(j) = 2, everybody would most likely find that pretty useless (otherwise someone else would have thought of that already), but is it possible that there are other extensions beyond ℂ that would be useful for something, or do the mathematicians think that there's nothing beyond ℂ, and that's really the final frontier?
If Drak's reply does not answer your 'extensions' question, there are many forms and fields beyond complex numbers. Set theory and group theory that you mention in previous posts, have marvelous extensions that I am currently learning, along with attempts at learning more abstract algebras and geometries.You might be interested in quaternions, octonions, etc. They are extensions of complex numbers just like complex numbers are extensions of the real numbers.
What about the words that do have a correspondence in their meaning, but don't exist anyway? E.g. there is no German word for "sophisticated" and no English word for "schweigen", although the situations in which they are used do exist equally. Then by your definition, those words are intrinsic.Because there are aspects of life in Germany that do not have a parallel in other countries (e.g., English) , so that the abstract grammatical layout is different to account for this.
I am kindof reaching, but aren't C,C#, Java, etc. all means/tools for dealing with the problem of how to program? Each language is used to address the same content/issue but it is just that, a vehicle for it. But everyday language starts being an obstacle to discuss these issues.What about the words that do have a correspondence in their meaning, but don't exist anyway? E.g. there is no German word for "sophisticated" and no English word for "schweigen", although the situations in which they are used do exist equally. Then by your definition, those words are intrinsic.
But can't the language aspect be studies separately and independently-from the content? We can make a purely syntactic analysis of Mathematical language as a first-order language. These discussions get weird after a while and seemingly drift of, so sorry if I am not making too much sense.On the Language sub-theme:
If one accepts that language derives from culture expressed in an environment -- Inuit language has n words for 'snow' comes to mind from Anthropology -- then the expression "Mathematics is the language of Science" not only describes math as a language but science as its environment (culture).
The importance of ##\mathbb{C}## is that it is an algebraically closed field. In terms of numbers, that's part of the reason that further extensions are not normally considered numbers. But, of course, you have complex vectors and complex matrices etc. And, as others have said, there are quaternions etc.Is it possible that are there other extensions to ℝ that provide solutions to other types of non-polynomial problems? For example, if I were to define a number j such that sin(j) = 2, everybody would most likely find that pretty useless (otherwise someone else would have thought of that already), but is it possible that there are other extensions beyond ℂ that would be useful for something, or do the mathematicians think that there's nothing beyond ℂ, and that's really the final frontier?
They really became important for expressing the solutions to cubic polynomials (see here). Basically, cubic polynomials always have (at least) one real root and it is possible to say how to compute it, but you sometimes need complex numbers in the intermediate steps. For example, cubic polynomials of the form $$x^{3} + p x + q$$ have a root given by the Cardano formula, $$x = \Biggl(- \frac{q}{2} + \sqrt{\frac{q^{2}}{4} + \frac{p^{3}}{27}}\Biggr)^{1/3} + \Biggl(- \frac{q}{2} - \sqrt{\frac{q^{2}}{4} + \frac{p^{3}}{27}}\Biggr)^{1/3} \,.$$ That root is real, but depending on the values of ##p## and ##q## it's possible for the term ##\frac{q^{2}}{4} + \frac{p^{3}}{27}## under the square roots to be negative. In that case the Cardano formula can still be meaningful and gives the correct (real) result, but only if you accept it containing intermediate expressions ##\Bigl(a \pm b \sqrt{-1}\Bigr)^{1/3}## with imaginary parts that end up cancelling out.One way/reason for defining i as it has been done is that it allows for finding roots of polynomial of the type x^2+b . Formally, this is a field extension R[X]/(x^2+1), from which it follows that x^2+1 =0 , so that x^2 = -1. This gels together when having x=i so that x^2 =-1. So the construction/definition of i is not, in this sense, just formal, aesthetic.
Or if you're familiar with the properties of Pauli matrices, $$\begin{eqnarray*}Interesting! I'd never run into this before.
Just to test this, I tried it on examples of addition and multiplication, using 2 + 3i and 5 - 2i
Addition
##\begin{bmatrix} 2 & -3 \\ 3 & 2\end{bmatrix} + \begin{bmatrix}5 & 2 \\ -2 & 5 \end{bmatrix} = \begin{bmatrix}7 & -1 \\ 1 & 7 \end{bmatrix}##
The last matrix represents 7 + 1i, which is the sum of 2 + 3i and 5 - 2i
Multiplication
##\begin{bmatrix} 2 & -3 \\ 3 & 2\end{bmatrix} \begin{bmatrix}5 & 2 \\ -2 & 5 \end{bmatrix} = \begin{bmatrix}16 & -11 \\ 11 & 16 \end{bmatrix}##
Same result that you get by multiplying (2 + 3i) and (5 - 2i).
This is not true. There are extensions of ##\mathbb{C}##. The division algebras ##\mathbb{H}## and ##\mathbb{O}## have already be named. But you can also add a new transcendental number, say ##T##, so ##\mathbb{C}[T]## which is the same as the polynomial ring over ##\mathbb{C}## first becomes an integral domain, and thus has a quotient field ##\mathbb{C}(T)##, which is a strict field extension of the complex numbers. And you can repeat this process as often as you want. The correct wording is: "... has no proper algebraic field extension!" Both restrictions are necessary as the examples I mentioned show."F is algebraically closed field if every non-constant polynomial has a root"... which it says it is equivalent to saying that "F has no proper extension".
This is way over the top. To cite Pauli matrices just to have a name for ##i \sigma_y## is very far fetched. It's like starting with a factor group of ##SO(4)## just to explain a complex number. Sorry, but this is ridiculous.Or if you're familiar with the properties of Pauli matrices, $$\begin{eqnarray*}
\bigl( a \, \mathbb{I} + i b \, \sigma_{y} \bigr) \bigl( c \, \mathbb{I} + i d \, \sigma_{y} \bigr) &=& ac \, \mathbb{I} + i (ad + bc) \, \sigma_{y} - b d \, {\sigma_{y}}^{2} \\
&=& (ac - bd) \, \mathbb{I} + i (ad + bc) \, \sigma_{y} \,,
\end{eqnarray*}$$ since ##{\sigma_{y}}^{2}## is the identity. (It doesn't matter if you use ##\sigma_{y}## or ##-\sigma_{y}##.)
Oh... got it (kinda)... I always keep omitting words that end up being relevant! Thanks for the correction!This is not true. There are extensions of ##\mathbb{C}##. The division algebras ##\mathbb{H}## and ##\mathbb{O}## have already be named. But you can also add a new transcendental number, say ##T##, so ##\mathbb{C}[T]## which is the same as the polynomial ring over ##\mathbb{C}## first becomes an integral domain, and thus has a quotient field ##\mathbb{C}(T)##, which is a strict field extension of the complex numbers. And you can repeat this process as often as you want. The correct wording is: "... has no proper algebraic field extension!" Both restrictions are necessary as the examples I mentioned show.
WOW... sudden realization!! This is exciting!! For the last 20 years I meandered around them mysterious quantum formulas (without understanding anything, but fascinated on how cute they look) people would talk about SO(2), SU(3), etc... and refer to Lie algebra, multiplicative groups, ... ah!!!... no way someone with under 30 trillion neurons can understand any of that!!This is way over the top. To cite Pauli matrices just to have a name for ##i \sigma_y## is very far fetched. It's like starting with a factor group of ##SO(4)## just to explain a complex number. Sorry, but this is ridiculous.
Here's a list of "what is what" resp. "what an be viewed as what": https://www.physicsforums.com/insights/journey-manifold-su2mathbbc-part/WOW... sudden realization!! This is exciting!! For the last 20 years I meandered around them mysterious quantum formulas (without understanding anything, but fascinated on how cute they look) people would talk about SO(2), SU(3), etc... and refer to Lie algebra, multiplicative groups, ... ah!!!... no way someone with under 30 trillion neurons can understand any of that!!
Now I see - SO(2) is the same thing as a 2D rotation in ℝ^{2}, and SO(3) is a 3D rotation in ℝ^{3}. Hooray! So SO(4) may be related to 4D rotations? Or something else.. the wording is really difficult to translate a human English. Also I guess SU(3) too must be related to some other transformation-thingie with some particular property-thingie-thingie.
So SO(2) is related (related-sy, however the proper Mathematish phrase for that) to z ∈ ℂ such that ||z|| = 1! I learned something!!!