taeto

2 hours ago, Conjurer said:

It just so happens that finding the derivative is the only mathematical situation that is known where you can actually divide by zero and get a correct reasonable finite answer. 

I think that if you look carefully enough, you will find that the derivative is calculated using hyperreals by dividing by an infinitesimal \(\Delta x\) which is nonzero. Only after dividing do you extract the standard part.

2 minutes ago, Conjurer said:

You can also write a program that determines the calculation of pi to however many places you want.  It is still said that pi continues randomly on forever.  That just means that the sequence of numbers do not repeat the same way forever.  

Your version of something being random is very different from everyone else's. If I could write a computer program which predicts with absolute 100% certainty the winning lottery numbers in next weeks lottery, and it would be correct every time, would you still persist in a belief that the lottery numbers are drawn every week completely at random? Most people would say that they are completely predictable.

7 minutes ago, Conjurer said:

It appears to continue on forever repeating randomly.  It becomes infinitesimally close to a rational number, since each digit after the decimal point is a smaller and smaller description that approaches a rational number.  

Actually it does not continue "randomly", since you can write a computer program which will tell you the \(n\)'th digit after the decimal point if you input \(n.\)

And the only real number that \(1/\pi\) "becomes infinitesimally close to" is the number \(1/\pi\) itself. All other real numbers have a positive real number distance. For the simple reason that the decimal expansion that you are talking about is precisely the decimal expansion of \(1/\pi\) and nothing else. Also, it is only the real numbers that have usual decimal expansions, other hyperreals do not.

4 hours ago, Lauren1234 said:

Sorry it should be 

Let A={e^xsin(x),e^xcos(x),sin(x),cos(x)}

That makes sense. The meaning is that \(A\) is a set of four functions. The role of \(x\) remains a little ambiguous. If we tread lightly, we can surmise that \(x\) is a coordinate, and its range is the reals.

Do you see what is the span(\(A\))\(=V\)? Is it true that if a function \(f\) is any one of the four functions in \(A,\) then \(f'\) belongs to \(V?\) And what if \(f\) is any function in \(V?\)

2 minutes ago, Conjurer said:

Infinitesimals have to continue on forever past the decimal point.  Then that makes them irrational numbers, because they cannot be expressed as a fraction.  That is the definition of an irrational number.  

A hyperreal would just be a number that continues on forever without a decimal point.  If you put that number on the bottom of a fraction it would make it an infinitesimal, because you would be dividing 1 by an infinite number of numbers.  The process would never stop, so you would end up with an infinite number of numbers past the decimal.  Therefore, it would become an infinitesimal or irrational number.

To "continue forever past the decimal point" is not quite the same as being irrational. The decimal expansion of the rational number \(1/3\) is \(.333\ldots \). 

You seem a little too hung up on the representation of numbers. The numbers themselves do not "have decimal points" nor do they "continue forever". These kinds of descriptions only apply to whatever representation we choose, which starting from elementary school happens to be mostly the decimal point representation. But the actual properties of numbers do not depend on the way in which we happen to choose a representation of them. In the decimal notation we represent \(1/3\) as \(.333\ldots \) whereas in ternary notation we would represent the exact same number as \(.1\). 

14 minutes ago, studiot said:

I think perhaps there have been some transciption errors in the OP?

Something is wrong, I may not have pinpointed it exactly. The notation is suspect; \(A\) is a set of what kind of elements, and then what is span(\(A\))? It makes sense if we read the OP as \(A = \{\exp,\exp \cos, \exp \sin, \cos, \sin \}\), and taking span(\(A\)) to be the space of all linear combinations with real coefficients of those five functions. Then the only thing to prove is that the image of the span under \(T\) is contained in the span. Hence my original comment. 

22 minutes ago, Ejup Dermaku said:

A given circle with area A = 1 has a radius = 1/sqrt(pi). In this case there exist a square with the sides of length = 1 which has an area equal to 1. This problem is referred  as a "squaring the circle". Due to the "irrational" and "transcendental" nature of number pi , squaring of circle is not possible to be constructed only by ruler and compass. However, I've read in an old mathematical book, that a construction is possible only in case when circle area A=1, without any further explanation given. Is there any one who can support this claim? 

That is not what "squaring the circle" means. Given a circle of area 1, yes, there also does exist a square also of area 1. That is not a problem. The problem is that from a line segment of length equal to the radius (or equivalently the diameter) of such a circle, it is not possible only using ruler and compass to construct a line segment to make a side of a square of the same area as the circle.

The claim in your old book does not make immediate sense. It is true that if you are given a line segment of unit length, then you can quite obviously construct a square of unit area. But having been additionally given a circle of unit area would not be helpful in any way to do it.

23 hours ago, Lauren1234 said:

Define
T:V→V,f↦df/dx.
How do I prove that T is a linear transformation?

The assignment is wrong. It is like asking how to prove that \(f: [0,1]\to [0,1],\, x\mapsto 2x\) defines a linear function \(f\), when \([0,1]\) means the closed interval of real numbers between \(0\) and \(1.\) 

8 hours ago, gib65 said:

What happens if you multiply the infinitely small hyperreal number e by the infinitely large hyperreal number R? Do you get a real number?

I am not an expert, so if I say something incorrect, wtf may want to correct it. But i am pretty sure that it only works in one possible way. Namely among the infinitesimals you have to pick a standard one, say call it \(\varepsilon,\) and likewise among the infinite hyperreals you pick an \(\omega,\) and then you arrange that multiplication works so that \(\varepsilon \omega = 1,\) that is to say, they are reciprocals. And in general, if \(a\) is real and nonzero, then \(a\varepsilon\) and \(\omega /a \) would be other possible, and just as good, choices. And you do always get a real number by multiplying an infinitesimal \(a\varepsilon\) by an infinite \(b\omega\) when \(a\) and \(b\) are real.

What I think is perhaps a little interesting is that once you have made your choices of \(\varepsilon\) and \(\omega,\) then because you can multiply all hyperreals, there is an even much smaller infinitesimal (it is still called that, right?) \(\varepsilon^2\) and en even larger infinite \(\omega^2\) which you could have picked in place of \(\varepsilon\) and \(\omega,\) without anything working any different. Which indicates that there are even more brutal ways of making smaller infinitesimals than just by dividing by \(2\) or any other large real. The exact converse situation would occur if we were to introduce new hyperreals \(\sqrt{\varepsilon}\) and \(\sqrt{\omega}\) which are hugely larger, respectively smaller, than \(\varepsilon\), and \(\omega\), themselves.  It would mean creating an extension of our original field, vaguely similar to how we create \(\mathbb{C}\) as an extension of \(\mathbb{R}\) by adding \(\sqrt{-1}.\) Except in the case of hyperreals, it looks to me like the new extension field is pretty much the same as the one we had already. 

37 minutes ago, Conjurer said:

things like a human consciousness is more likely to exist compared to the current Big Bang model, bringing into question the possibility of a preexisting godlike entity existing before the universe was even created.   

Presumably because if some entity with a human consciousness existed before the universe was created, then that entity might have been responsible for creating the universe? You yourself, would you say that you possess human consciousness? And could you create an entire universe? Or is there any difficulty to do so?

Quite. But I read in another thread something to the effect that if someone has gotten convinced to believe something other than by facts, then any facts presented to them may not change their beliefs. What we do experience is actually a little stronger than that, namely that they are increasingly likely to avoid confronting facts. Of course "Impossible to see - the future is" (Yoda, ~4.7B BC).

On 1/8/2020 at 11:57 PM, StringJunky said:

Unless a YT video is by or of a person I know knows their stuff I don't generally class it as an educational portal; it is for entertainment. The thing with Wikipedia is to check the references.

Sound advice. But I try to think of myself when once I was just a high school student, and pitifully naive. Back then we did not have the massive presence of pseudoscientific material which exists now. It scares me to think of such a vulnerable mind, somewhat interested in scientific subjects, and not yet able to tell the difference between actual evidence and stuff that is purely made up. How will you react to hundreds of anti-science propaganda videos with "easy refutations" of just about every scientific topic. And as I started saying, it is kind of okay that YT keeps videos available that are propaganda, misinformative, conspiratorial and presenting made up evidence. What bothers me is that some such videos get clearly marked by Youtube as being "Educational". YT videos by Ken Wheeler get placed by Youtube in the "Educational" category. 

An "e" is missing or "warp" should be "wasp". But "y"?

16 hours ago, Huckleberry of Yore said:

Keep in mind that those functions emit pseudo random numbers with a flat distribution.  To achieve a Gaussian distribution you would use the Central Value Theorem (already mentioned) by adding a series of rand() return values which in the limit should be normal.

There is a classical algorithm, called Box–Muller transform, to convert sampling from a "flat" (uniform) distribution, say uniform over the interval \( [0,1] \), to a Gaussian distribution:

en.wikipedia.org/wiki/Box–Muller_transform

I feel like opening another thread, because this phenomenon bothers me increasingly.

Quite a lot of people, if they are not sure about the exact explanation of a topic, they may go to Wikipedia, or if they have more time, sit down and watch an educational youtube video. Most of us will have to admit to doing this, on occasion.

The problem is what kind of experience this may bring to you. Mostly I am personally interested in mathematical topics. I can read a Wikipedia page and identify mistakes in maths, and make corrections, so that is fine, it seems. Except when the work required to do so begins to exceed my capacity. I cannot spend every working hour on this project. I have to live with the fact that increasing amount of Wikipedia content consists of inaccuracies and misinformation. This is bad enough, but...

Youtube is even worse, because you can obviously not enter into a video and correct its mistakes. You may make a comment to the video. Then you either end up as the 180'th commenter, and the poster may answer that, oh yeah, I said it wrong there, it should really be etc. in the best case scenario. Or you get no response, or you get a nonsensical response that makes it obvious that the poster of the video really has no idea of the subject.

Youtube does not distance itself from videos that argue that gravity does not exist, irrational numbers do not exist, or the earth is flat as a pancake and the sun is a little electric powered lamp that moves around above the surface. This still is kind of okay, because posters, however misguided, present accounts of the world as they see it individually. Which makes it a document of the diverse cultural and non-scientific attitudes to topics some of which happen to be scientific. 

I do however have a problem with the youtube categories, that allow exactly such videos to be labelled as "Educational". I can provide examples, if necessary, but I do not imagine they are hard to come by.

Am I wrong in thinking that such a practice is unethical? I could be just old-fashioned in this respect. But to me, there is a certain responsibility attached to certifying, as it were, content as having educational value. 

1 hour ago, Country Boy said:

Are you saying that you are an unruly schoolchild?

If that is what I am saying, then yes.

1 hour ago, studiot said:

My maths dictionary has

Predicative  adjective -  (Logic)

(Of a definition) given in terms that do not require quantification over entities of the same type as that which it is thereby defined.

Part of Russell's solution to the paradoxes of self-reference as contained in hy type theory was to require all definitions to be predicative.

Thanks! 

If we define in the ordinary way a group \( (G,+) \) so that it satisfies the axiom (which I seem unable to typeset properly)

\( \exists 0 \forall x : x+0 = 0+x = x,\)

then we agree that the definition is impredicative, since the universal quantification is over all group elements.

It seems that in type theory you define a group \( (G,+,0) \) so that it satisfies the axiom

\( \forall x : x+0 = 0+ x = x,\)

and this definition is predicative, since 0 is not in the scope of the quantifier? A statement like \(0+0=0\) now becomes a theorem that does not follow immediately, since you cannot replace \(x\) by \(0\) in the definition of \(0\). So can you prove it?

As I began by saying, I got intrigued by a publication by Edward Nelson on "Predicative Arithmetic", in which the first few paragraphs state that usual induction is "impredicative".

Now induction is a basic defining property of natural numbers and their arithmetic. The more intricate behaviors of natural numbers are applied in everyday situations, especially when we want to transfer or extract amounts of money via electronic means, and also in ways of communicating messages confidentially, familiar by the difference between having an "http" or an "https" at the front of the name of an internet page like this. And for something to be "impredicative" is somehow indicative of it being in some way defective, almost circularly defined.

Wikipedia says: "Roughly speaking, a definition is impredicative if it invokes (mentions or quantifies over) the set being defined, or (more commonly) another set that contains the thing being defined. There is no generally accepted precise definition of what it means to be predicative or impredicative."

Clearly this clashes two-fold with the idea that induction is impredicative. First, induction is a collection of axioms, not a set. Second, none of the axioms of induction refers to induction or to any sets that are defined by induction.

I have tried to figure out if the axiom of the neutral element of a group is impredicative. The axiom says that if G is a group with addition +, then 0 is a neutral element of G if 0+x=x+0=x is true for every element x of G. All people whom I asked seem to agree that this is an impredicative definition, since by saying "every element x" you also include the element 0 itself, and therefore the definition already refers to the element which it defines. The best answer that I got so far says that you have to define not what a group (G,+) is, but you have to define what a group (G,+,e) with neutral element e is. Then the definition becomes predicative, which is to say, now it is alright, since you no longer get to choose e among the elements of G, but it is already fixed. 

Do you guys see what I am missing? I still do not know why arithmetic is supposed to be "impredicative", nor why group theory is somehow "wrong". 

10 minutes ago, Country Boy said:

How dare you speak about my aunt like that!

If used to dozens of unruly schoolchildren daily, then she is probably not taking too much offence. 

The mathematical idea of how a line, or more generally a curve, is thought of in physics and mathematics is something we got from the work of Euclid, a greek mathematician who lived about 2300 years ago. Accordingly a line is formally just the same as the collection of points that lie on it. In particular where you seem to think of "point" and "position" as two different things, they are actually just the same thing, in the usual understanding. We have other words like "distance" and "length" to talk about where points are located with respect to each other.

Other than that, your reasoning is fairly spot on. If we take a line segment, and we consider any finite collection of different points on it, then between any two points \(x\) and \(y\) there is at least one additional point, no matter how close \(x\) and \(y\) are to each other. The `midpoint' between them is one such point that one is typically taught in basic geometry to construct with a ruler and compass. Therefore any finite collection cannot contain all points of the line segment, and this is what it means to say that the set of points is infinite.

Euclid didn't really consider concepts such as countability of points, which is a topic that came later. His arguments basically apply in the same way to both the `rational line' on which all points can be described by coordinates that are rational numbers, and to the `real line', with more points, including some that have irrational coordinates, like \(\sqrt{2}\) and \(\pi\) and others. And the `algebraic line' which is inbetween the two and contains \(\sqrt{2}\) but not \(\pi\). The rational and algebraic lines have countably many points, the real line uncountably many. Almost always when people speak of a line without qualification they mean the real line.

On 11/24/2019 at 10:02 PM, Andrew26 said:

If 1/infinity is the number that when multiplied by infinity equals 1, then the number is 1/infinity. 

Yes, you are right. 

And if \(1/\infty\) is a number, other than \(-1\) or \(+1\),  that when multiplied by infinity produces \(1\), then your aunt is yellow, has four wheels and is a schoolbus.

Anything follows from a false antecedent, whether it is true or false does not matter.

If you follow standard definitions, then \(a/\infty = 0\) is true for every real number \(a\). And \(0\) multiplied by \(\infty\) is most often defined to be \(0\). When you combine those usual definitions you get \( (a/\infty)\times \infty = 0\) for all real values \(a\). Always \(0\), never \(1\).

If you have non-standard definitions, you have to present them, that will be fine.

I watched the video, and it comes down to \( \pi \approx 6\phi^2/5.\) The RHS is about \(3.141640786\) so it is not too bad.

An author of a paper on Angle Trisection in last year's issue of the same journal, The Journal of Advances in Mathematics, is convinced that the correct value of \(\pi\) is exact \(3.14\). Don't know whether they will clash, or settle on some average as a compromise  

Surely depends on a lot of parameters, the spectrum of the local star(s), atmosphere of the planet, the chemical composition of life and the strategies for extracting energy out of light, and of course we are referring to our own peculiar perception of colors, probably not universal, although at least we can pin wavelengths on each of the colors familiar to us. We can only wait for the explanation from the OP why purple is a good guess.

11 minutes ago, StringJunky said:

Vegetation will evolve photosynthetically  to the average wavelengths of their light source.

Is it not correct to say that even if the light source is uniformly white, that is, it contains different wavelengths equally, then in the visible spectrum plants would be most likely to reflect green light? I saw it argued somewhere that red light can have a wavelength that is twice that of blue light. So that if a plant can sample red light, then a blue photon just means a double portion, so to speak. Whereas if the plant eats green light, then there is no other visible light that resonates enough to be edible to it. Loosely speaking. 

On 12/31/2019 at 11:34 PM, mathematic said:

3. Looks strange.  Why purple?

Maybe due to the special chemistry of the hypothesized photosynthesizing cells or molecules, perhaps?!