Rune Allnor wrote: ...> Mathematics is supposed to be based on just a few axioms. I have a vague > recollection there are some eleven (or thereabouts) such axioms. > > Does anyone know where to find those axioms?I think you mean the Zermelo-Fraenkel (ZF) axioms for set theory, which are usually extended by the axiom of choice (C or AC). The combined axioms are known by the name ZFC. There are seven of them: http://www.jboden.demon.co.uk/SetTheory/axiomsZFC.html Note that the notion of a "set" is not strictly defined. It seems to be impossible without ending in paradox hell. There are some other axioms for set theory, but none have been studied as extensively as ZFC. Also note that these axioms have to be interpreted very carefully (read the passage on Russell's paradox in the above webpage). However, this is not the basis for all maths, just set theory (which is admittedly an important chunk). Regards, Andor
Negative Frequencies
Started by ●July 15, 2003
Reply by ●July 21, 20032003-07-21
Reply by ●July 21, 20032003-07-21
Jerry Avins <jya@ieee.org> wrote in message news:<3F1418BA.822C806@ieee.org>...> No amount of math will reach a conclusion here, because the issue is not > math, but philosophy. What is clear is that a Fourier transform with > sines and cosines doesn't use negative frequencies in the analysis. > > Calculating with complex exponentials entails using negative > frequencies. That doesn't confirm the existence negative frequencies or > of complex exponentials. It simplifies manipulations while extending the > repertoire of necessary concepts.I am in Jerry's camp on this issue. Sometimes I get the feeling that there are people who think that exp(jwt) is more fundamental than sine and cosine. I don't think this is correct because one cannot define exp(jwt) without using sine and cosine. If exp(jwt) were a more basic building block its definition should not depend on sine and cosine.
Reply by ●July 21, 20032003-07-21
Jerry Avins wrote:> Randy Yates wrote: > > > > Jerry Avins wrote: > > > [...] > > > Trig functions are > > > one-to-one mappings of one scalar to another. > > > > y=cos(a) is not a one-to-one mapping from a to y. a1 != a2 does > > not imply that cos(a1) != cos(a2)....> Maybe I said it wrong. What I mean is that for every a, there is one and > only one cos(a), and that both a and cos(a) are scalars.When you talk about a map, then the mapping function is only half of the definition. You have to include argument and range (are these the proper words in English?) domain in the definition: Thus cos : [0, Pi] -> [-1,1] is bijective whereas cos : RR -> [-1,1] is surjective but not injective and cos : [0, Pi] -> RR is injective but not surjective and finally cos : RR -> RR is neither injective nor surjective (RR is the set of real numbers)
Reply by ●July 21, 20032003-07-21
"Bhanu Prakash Reddy" <itsbhanu@yahoo.com> wrote in message news:28192a4d.0307142216.4c6ee88@posting.google.com...> Hi, > Can anyone explain the concept of Negative frequencies clearly. Do > they really exist?About 30 years ago I wondered the same question (do negative frequencies exist?) Here you have one example from acoustics: Let's assume a sound wave propagating from a source at speed c to certain direction. We sample this sound wave by a microphone moving at speed 2c to the same direction. What we get? The exact time domain wave form but the signal represented in time reversed. We can define the instantaneous frequency as the phase change in time unit (f = d phi/dt). In the microphone signal this change is negative when compared with the original signal. The frequency is negative by definition. In real life we can easily listen and analyze sounds "by negative frequencies" playing recorded signals backwards. Although the signal played forwards and backwards have same spectrum they sounds differently (at least when listened by humans) Kari Pesonen
Reply by ●July 21, 20032003-07-21
Hello Rune, A while back Kurt Goedel (1931) proved that any Mathematics based on a finite set of axioms is incomplete. This has sometimes been called the uncertainty theorem of math. So there will always be things in math that are unknowable. For more see: http://www.miskatonic.org/godel.html Euclid had 5 postulates. Clay "Rune Allnor" <allnor@tele.ntnu.no> wrote in message news:f56893ae.0307202001.626f6f27@posting.google.com...> Jerry Avins <jya@ieee.org> wrote in messagenews:<3F1AC7E6.9A18EFC7@ieee.org>...> > many of the concepts that some take as > > necessary and fundamental -- as Euclid's fifth postulate once was -- are > > in fact merely postulates of convenience. Labeling them "obvious" or > > "true" is a disservice to those most inclined to ask "why". > > > > Jerry > > Mathematics is supposed to be based on just a few axioms. I have a vague > recollection there are some eleven (or thereabouts) such axioms. > > Does anyone know where to find those axioms? > > Rune
Reply by ●July 21, 20032003-07-21
Vanamali wrote:> > Jerry Avins <jya@ieee.org> wrote in message news:<3F1418BA.822C806@ieee.org>... > > > No amount of math will reach a conclusion here, because the issue is not > > math, but philosophy. What is clear is that a Fourier transform with > > sines and cosines doesn't use negative frequencies in the analysis. > > > > Calculating with complex exponentials entails using negative > > frequencies. That doesn't confirm the existence negative frequencies or > > of complex exponentials. It simplifies manipulations while extending the > > repertoire of necessary concepts. > > I am in Jerry's camp on this issue. Sometimes I get the feeling that > there are people who think that exp(jwt) is more fundamental than sine > and cosine. I don't think this is correct because one cannot define > exp(jwt) without using sine and cosine. If exp(jwt) were a more basic > building block its definition should not depend on sine and cosine.Believe it or not, exp(jwt) can be defined without any trig. It's just a bit hard to explain it that way to people with intuition as limited as mine. x� x� 3� x^4 x^5 Consider the series 1 + �� + �� + �� + �� + �� + ... !1 !2 !3 !4 !5 f(x) + f(-x) For any function f(x), the even part is ������������ and 2 f(x) - f(-x) the odd part is ������������. 2 Of course, the series is exp(x), and the even and odd parts are cosh(x) and sinh(x). When x is imaginary, ordinary trig appears. Simply from summing the terms, it is clear that exp(i�x) = cos(x) + i�sin(x). So which came first? Historically, trigonometry. Logically, the question means nothing to me. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●July 21, 20032003-07-21
Rune Allnor wrote:> > Jerry Avins <jya@ieee.org> wrote in message news:<3F1AC7E6.9A18EFC7@ieee.org>... > > many of the concepts that some take as > > necessary and fundamental -- as Euclid's fifth postulate once was -- are > > in fact merely postulates of convenience. Labeling them "obvious" or > > "true" is a disservice to those most inclined to ask "why". > > > > Jerry > > Mathematics is supposed to be based on just a few axioms. I have a vague > recollection there are some eleven (or thereabouts) such axioms. > > Does anyone know where to find those axioms? > > RuneGoogle for "Peano's Axioms"; that may be what you want. "There is a number. For every number, there is a successor. ..." Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●July 21, 20032003-07-21
"Vanamali" <vanamali@netzero.net> wrote in message news:8d4ba7e3.0307210251.6ed3268f@posting.google.com... (snip)> I am in Jerry's camp on this issue. Sometimes I get the feeling that > there are people who think that exp(jwt) is more fundamental than sine > and cosine. I don't think this is correct because one cannot define > exp(jwt) without using sine and cosine. If exp(jwt) were a more basic > building block its definition should not depend on sine and cosine.I would say that they are equally fundamental. Either can be derived from the other. For a related question, is linear or circular polarization more fundamental? It happens that there is a simple way to make linear polarizers, but quantum mechanics seems to like circular better. (Photons are spin one with a missing spin 0 state.) Circular polarization is a linear combination of linear polarization (with the right relative phase), and linear polarization is the appropriate linear combination of circular polarization. Are rectangular or polar coordinates more fundamental? -- glen
Reply by ●July 21, 20032003-07-21
Kari Pesonen wrote:> > "Bhanu Prakash Reddy" <itsbhanu@yahoo.com> wrote in message news:28192a4d.0307142216.4c6ee88@posting.google.com... > > Hi, > > Can anyone explain the concept of Negative frequencies clearly. Do > > they really exist? > > About 30 years ago I wondered the same question (do negative frequencies > exist?) > Here you have one example from acoustics: > Let's assume a sound wave propagating from a source at speed c to certain > direction. We sample this sound wave by a microphone moving at speed > 2c to the same direction. What we get? The exact time domain wave form > but the signal represented in time reversed. > We can define the instantaneous frequency as the phase change in time > unit (f = d phi/dt). In the microphone signal this change is negative > when compared with the original signal. The frequency is negative by > definition. > > In real life we can easily listen and analyze sounds "by negative frequencies" > playing recorded signals backwards. Although the signal played forwards > and backwards have same spectrum they sounds differently (at least when > listened by humans) > > Kari PesonenThere is linear or rotary motion involved in playing a tape or record backward, so direction reversal makes sense. What about the loudspeaker cone? Does that vibrate backwards? The examples above are a sort of time reversal. Does that really negate the frequencies? If so, how? Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●July 21, 20032003-07-21
Vanamali wrote:> I am in Jerry's camp on this issue. Sometimes I get the feeling that > there are people who think that exp(jwt) is more fundamental than sine > and cosine. I don't think this is correct because one cannot define > exp(jwt) without using sine and cosine. If exp(jwt) were a more basic > building block its definition should not depend on sine and cosine.There is also an other argument: it cannot be more basic, since "j" is more... complex... In fact "j" was introduced in order to solve sqrt(-1), so it extends the basic set of R and thus cannot be a building block. bye, -- piergiorgio
