Imaginary Erdős Number - Numberphile
- Hans-Georg Lundahl
- I am of another theory than Erdös (sorry, don't known the ASCII Code for double grave accented o, which in Hungarian is longer version of ö).
As long as I stick to coauthors (by quotation or inspiration) already buried under Christian grave stones, like C. S. Lewis, J. R. R. Tolkien, a little further back G. K. Chesterton, Hilaire Belloc, still further back Cardinal Newman, still further back Riccioli (yes, I did refer to him in an essay), St Robert Bellarmine (yes, he was misquoted and I did dig up the real quote) St Thomas Aquinas, Bishop Tempier, I am OK. If I ever should have more coauthors alive, including really genial ones like Sungenis and Tas Walker, I need to worry about my theological allegiance. We do live in an age of confusion ... as even mathematics shows (giving an Erdös number the value i counts as a good pun, but not as excellent mathematical understanding).
- Andrea Langley Sharp
- I suppose this is amusing to mathematicians, but I'm totally lost on this one.
- Hans-Georg Lundahl
- Pun on "i" = called so because "imaginary number" and "i" = "conceptualised as 1 at right angles from the normal numberline". Which I still don't get how it fits definition of "i" = "defined as square root of minus 1". Except that both concepts are as such preposterous.
From me it gets a half amused "arf, arf".
- Andrea Langley Sharp
- The gymnastics of i seem purposeless to me (why assign a stupid non-value like square root -1 at all?). I assume that the form a + bi has some kind of computational usefulness, but it just seems like shorthand to me. As far as this video goes, I get that this is a pun on "imaginary", but ... I think I'm not nerdy enough to appreciate it.
- Hans-Georg Lundahl
- Well, I very totally agree with you.
Square roots are of numbers that are square numbers as numbers, not as adds or deductions. So even using -1 as an alternatvie square root for 1 ... though there it has at least a computational use.
- Juan Manuel Muñoz
- +Hans-Georg Lundahl
Take this example only within real numbers:
Let's say I have to solve x2-2=0, and I know nothing about irrational numbers. Now, I start scratching my brain about how it is possible that no numerical ratio will give me this number. Furthermore, it means that it has infinitely many decimal places, so it can never be known exactly.
This approach is wrong. I am finding uncomfortable results within the field of numerical quantities seen as proportions between countable things. But then I realise that this number is the solution to the question "how much does the diagonal of a side-1 square?" And I understand that I was too narrow-minded: numbers do not represent mere proportions: they can--and often do--represent lengths, which are not affected by anyone moaning that infinitely many decimal places are too many.
Exactly the same thing happens when discovering complex numbers:
Let's say I have to solve x2+2=0, and I know nothing about complex numbers.
Now, I start scratching my brain about how it is possible that no geometrical length will give me this number (which is i times the square root of two).
This approach is wrong. I am finding uncomfortable results within the field of numerical quantities seen as geometrical lengths in a numberline. But then I realise that this number is the solution to the question "which polar quantity, once the angle is doubled and the length is squared, will give me a 180º inclination and a length of two?" And I understand that I've been too narrow-minded again: numbers do not represent mere lengths: they can--and often do--represent polar lengths, so that the product of numbers is not only a product in the length, but there is also a directional factor which one has to take into account. This, again, is not affected by anyone moaning that negative quantities cannot be square-rooted.
- Hans-Georg Lundahl
- "Let's say I have to solve x^2-2=0, and I know nothing about irrational numbers."
It is very simple. First of all, you do not HAVE to solve an equation.
Second, the equation in question can be about relative sizes, but not about numbers, since including sqrt(2).
I think you might do well to consider my wording before you jump off at a tangent imagining I had considered sqrt(2) a rational number.
When we are concerned with NUMBERS only square numbers have square roots.
- Andrea Langley Sharp
- You both waaay lost me. Lemme simplify my i gripe.
If you want to indicate something two-dimensional, why is "x + yi" better than "(x, y)"? Seems unecessary.
In simple 2D graphics programming, the screen as a finite width (X axis), so i would be that length (for a 320x200 screen, i = 320). Indeed that's how drawing a simple pixel is coded: "x + yi" where (x, y) is the point and i is the screen width.
So for really basic 2D programming it makes sense that the a + bi notation makes sense, as it is literally how the point is located in screen memory.
But in an infinite 2D plane? Seems the assignment of i = sqrt(-1) seems arbitrary and misleading.
(I get the feeling I'm waay over my head here)
- Hans-Georg Lundahl
- When you deal with pixels, you can both get labels including i and numbers "below zero" because whoever designed the most general program that other programmers use was subscribing to that mathematical theory.
Now remember one thing: i was not developed to be a label on the y axis (and as said a superfluous one at that), it was developed I think by Gauss, to answer the question "what is the sqrt of minus one"?
And his reason for asking that question was he was observing "both minus one and plus one have plus one as their square".
I believe he was wrong on both counts.
Doesn't make the computer program wrong, just its labels are arranged according to a mathematical theory which is wrong.
Btw, having a zero in the middle rather than at one edge can be practical, means that one can count from middle outwards.
That does not mean "numberline" or "zero" or "numbers below zero" have any real existence as mathematical objects, any more than i.
Both of you, number basically means countable or countables but not just one.
Sqrt of two is a non-numeric ratio.
This means that the ratio is not between the NUMBER of sides but between their LENGTHS, plus that this ratio between their lengths does not parallel any one but fall between any two ratios of numbers, i e of countables.
A polarity is something different from a countable and from a length.
Once you state polarity, you are in artwork, like thermometers which has one value arbitrarily named zero - because when the scale was developed, one had not measured zero Kelvin degrees.
Temperature as such is comparable to lengths. No length is zero long, no length is minus anything long. But it is practical to have an arbitrary zero in computer screens and also on thermometers and that question is not just simply mathematical, it is applied maths.
- +Hans-Georg Lundahl
Where'd you get this definition of number? If you take number to mean "an abstract label with which one counts," then you're not understanding the term correctly. Those "non-numeric ratios" of which you speak are CALLED "irrational numbers." If you claim that they are not numbers, then you have an incorrect understanding of the word "number." It's like saying "Oh, Australia's not a continent because I think it's too tiny." The rest of the world considers Australia large enough to be a continent. Similarly, the rest of the world considers irrational and complex numbers manipulable enough to be considered numbers. A number is an abstract mathematical object with which one quantifies, measures, and models. Suppose we wish to measured the displacement of a point in a plane from the origin. To do this, we need complex numbers. They are numbers because they are used to measure displacement, just as the natural numbers are numbers because they are used to measure counts.
- Hans-Georg Lundahl
- "Those "non-numeric ratios" of which you speak are CALLED 'irrational numbers.'"
These days, yes. I know. I consider it a real misnomer.
"If you claim that they are not numbers, then you have an incorrect understanding of the word "number."
Or those framing the modern terminology have so.
"It's like saying "Oh, Australia's not a continent because I think it's too tiny." The rest of the world considers Australia large enough to be a continent."
19th C. Australia was a Part-of-the World - the same which now is called Oceania - and Ulimaroa was the largest island in it.
Pluto was a planet and is no longer so. Remember?
As to the word continent, I consider it quite feasible to use it for only Eurasia with Africa - and consider the rest as a kind of "Earthsea" (island world).
However, of this continent (which has four corners - the SE could extand as far out as Ulimaroa as the NW extends to British Islands), there are Three Parts of the World - Asia, Africa and Europe. And of Earthsea, so to speak, one could consider Americas as distinct from Oceania.
When terminology changes, sometimes it is just a question of LIMIT.
But in the question of number, it is also a question of basic definition.
Numbers go 1, 2, 3, 4 ...
Geometry goes point, line, surface, body : each except point being indefinitely divisible into infinitesimals. And each except body being a division.
There are TWO objections I have against the new terminology:
- it blurs distinction between geometry and arithmetic
- it gives the very false impression that before one used the misnomer "irrational numbers" one tried to prove pi or sqrt of two were rational numbers, when "number" was in fact not at all the most basic classification.
For instance, for π or sqrt of two, the older terminology was that a square and a circle had diagonals (and a diameter counted as a diagonal) that were incommensurable with the circumferences.
If you have two sizes like 1 ft 3' and 9', the one is not a multiple of the other, but they do share a common measure the largest of which is 3 inches. This example has a ratio of 5:3. It is commensurable and therefore has a numeric ratio - usually called simply ratio. 5 in this case is not a number per se, but a proportion to the common unit of three inches. 3 also. But these two are at least expressible as numbers. They are the same ratio (except for the one being in sizes and the other in real numbers) as 5 apples to 3 apples.
This will never happen with π or sqrt of two.
The older terminology for this was either to say the diagonal was incommensurable or to call it an "irrational ratio". Ratio because like the 5:3 of 1 ft 3' and 9' it is a proportional relation. It could for instance be repeated as 2 ft 1' to 1 ft 3' (25:15 = 15:9 = in both cases basic ratio 5:3). But irrational because it cannot be adequately and exactly expressed like a ratio with a numeral on each side of a line or of a colon. Which is why I think non-numeral ratio might do as well.
So, if you want to know when "irrational numbers" as a phenomenon was discovered, you must not ask when that phrase came into vogue, but rather look for the older phrase "irrational ratio" or the older word "incommensurable".
Some think irrational ratios were only discovered when one renamed them "irrational numbers", that is like saying the Swedish geography book from 19th C. in which "Australia" = Oceania and "Ulimaroa" = Australia minus van Diemens land and Tasmania = "largest island of Australia" prove that only what you call islands had as yet been discovered while the continent Ulimaroa/Australia was either undiscovered or incompletely discovered and thought to be smaller than it was.
"Similarly, the rest of the world considers irrational and complex numbers manipulable enough to be considered numbers."
Irrational ratios did not start to be manipulable when they were renamed irrational numbers. They did not become one whit more manipulable by being reclassified as numbers.
"A number is an abstract mathematical object with which one quantifies, measures, and models."
A number is an abstract mathematical object which is counted.
A size is an abstract mathematical object which is measured (by being compared to a standard size).
A relation or especially a ratio (for additions and subtractions are also relations) is an abstract mathematical concept applicable to both sizes and numbers.
"Suppose we wish to measured the displacement of a point in a plane from the origin. To do this, we need complex numbers."
If you are too lazy to find out the hypothenuse of a triangle or if what you want is a coordinate system, you need coordinates in two axes.
What you do NOT need is pretending one axis is the square root of left-hand or nether polarity of other axis.
As Andrea Langley Sharp so sharply observed.
- +Hans-Georg Lundahl There are two main ideas here. The first is the relation between terminology and actual properties. The second is whether it is pertinent to consider incommensurables to be numbers. I agree with you that a name obviously has no effect on the actual properties of an object. Regardless of whether incommensurables are numbers, they cannot be expressed as a ratio between any two integer lengths, almost all pairs of lengths in the natural world are incommensurable, etc. And, obviously, the Europeans discovered Australia the moment the first European explorer set foot there, even though he didn't yet call it Australia. Likewise, Columbus discovered America (for the Europeans), even though he never did recognize it as a continent. So I agree that "a rose by any other name would smell as sweet" applies here to some degree. So, if you want to say that there's a fundamental difference between natural numbers and everything else that is sometimes called a number, then I'll agree with you. Natural numbers are used to count; other, more general kinds of numbers are not. Just know that what you term "number," other people term "natural number."
However, I disagree with you when you say that it is a mistake to treat incommensurables and points on the plane as numbers. I agree that it is important to remember irrationals' origins as incommensurables, but why can't they be numbers as well as incommensurables? If we let them be numbers, then we can apply to them the ordinary arithmetic operations. Such application obviously has meaningful and useful results, but if they were not considered numbers, then they could not be added or multiplied with ordinary numbers. The same goes for complex numbers. If we let i be the square root of -1, then complex multiplication becomes an incredibly useful operation: as Juan Manuel Munoz said above, this allows us to create a point in the plane whose modulus is the product of those of two input points and whose angle to the positive x-axis is the sum of those of the two input points, algebraically. And think about things like the Riemann Hypothesis. This states that all non-trivial zeroes of the Riemann Zeta Function have real part 1/2. If complex numbers were not considered to be numbers at all, then the Riemann Hypothesis would not be statable (one can hardly apply a polynomial to a non-number). Yet, its truth would have enormous consequences for the study of primes, one consequence being a big O upper bound on the error between pi(n) (the function counting the number of primes less than or equal to a given natural number n) and Li(n) (the log-integral function). And, consider, as another example, Andrew Wiles' famous proof of Fermat's Last Theorem. In it, he had to use p-adic numbers, which are not even, in general, complex numbers, let alone natural numbers. Yet, using these numbers, he proved a statement solely about natural numbers: that there exists no quadruple of positive natural numbers (x, y, z, n) such that x^n + y^n = z^n for n>2. If p-adic numbers were not considered numbers, then such a proof, which undoubtedly relied on adding and multiplying them like normal numbers, would not have been possible. So, why do you think that math would be better off if only natural numbers were treated like numbers? Explain how such things as the Riemann Hypothesis, Euler's Formula, and Fermat's Last Theorem could be treated in this system.
- Hans-Georg Lundahl
- Glad you agree Roggeveen and James Cook had discovered [for our culture] the piece of land we now mostly call Australia [or was it Tasmania for Roggeveen?] before the day when a Swedish geography book listed the largest islands as:
- 1) Ulimaroa
- 2) Greenland ...
- (was 3 Madagascar or Sumatra or sth?)
"Regardless of whether incommensurables are numbers, they cannot be expressed as a ratio between any two integer lengths, almost all pairs of lengths in the natural world are incommensurable, etc."
I never disputed that.
It is ten years now that I have had people lecture me on that while I never even disputed it.
"Just know that what you term "number," other people term "natural number.""
You mean what I and most people in the world call "number" most or even all mathematicians call "natural number".
"I agree that it is important to remember irrationals' origins as incommensurables, but why can't they be numbers as well as incommensurables? If we let them be numbers, then we can apply to them the ordinary arithmetic operations."
I have nothing against practically once in a while treating them as numbers while they are not.
Logarithms is at its most basic, as on a slide rule, doing arithmetic with the means of geometry. Graph plotting and seeking equations for lines, circles, parabolic and hyperbolic figures etc. is at its most basic level doing geometry with the means of arithmetic.
I have nothing against either of them.
I am only saying there is a parallelism between the number 3 and the geometric ratio 3:1, the number 4 and the geometric ratio 4:1, but 3 and 4 are used in different ways.
This parallelism is of course behind the ideas of graph plotting and of logarithms. But parallelism is not identity.
"If we let them be numbers, then we can apply to them the ordinary arithmetic operations."
Even if we don't let them be numbers, we can still do so, because of the parallelism mentioned.
" If we let i be the square root of -1, then complex multiplication becomes an incredibly useful operation: as Juan Manuel Munoz said above, this allows us to create a point in the plane whose modulus is the product of those of two input points and whose angle to the positive x-axis is the sum of those of the two input points, algebraically."
I still do not get how a plane with two axes can have one axis as the square root of same length on the other axis' as long as it is the negative polarity of it.
Let us say a computer engineer wants a four dimension or four parameter coordinate system. How is calling a point "2 + 2i + 2j +2k" (with j = sqrt of minus i, with k = sqrt of minus j, obviously) superior to calling it "2x + 2y + 2z + 2w"?
Suppose the superioroty is this, that if one parameter is zero, you don't mark it. A two plane coordinate could have a point that is JUST 1 or JUST i, because the other coordinate is on the conventional zero value of the grid. Why is complex numbers better than calliing the one JUST x or the other JUST y, if shortness is the requirement (btw, I am not sure it is always a good one)?
"If complex numbers were not considered to be numbers at all, then the Riemann Hypothesis would not be statable (one can hardly apply a polynomial to a non-number)."
One can obviously apply a polynomial to sth fictionalised as a number - while still conisdering it fiction, not number.
Saying God created the world of Narnia allows CSL to make lots of theological statements that bypass the division of Old and New Covenant (or somewhat so) and therefore of Judaism and Christianity (and therefore of the real status of Judaism and the real presence of the Christian Church) in this world.
Must a theologian state as a positive belief that God really DID create Narnia in order to have access to those points?
"And, consider, as another example, Andrew Wiles' famous proof of Fermat's Last Theorem. In it, he had to use p-adic numbers, which are not even, in general, complex numbers, let alone natural numbers. Yet, using these numbers, he proved a statement solely about natural numbers: that there exists no quadruple of positive natural numbers (x, y, z, n) such that x^n + y^n = z^n for n>2. If p-adic numbers were not considered numbers, then such a proof, which undoubtedly relied on adding and multiplying them like normal numbers, would not have been possible."
Since in logic a proof may use the reduction ad absurdum, I cannot see why a proof about real numbers could not involve fictional ones. [LIKE:]
"Even if such and such were a number, such and such real numbers could not have such and such relations."
[SHOULD BE INSERTED:]
"If p-adic numbers were not considered numbers, then such a proof, which undoubtedly relied on adding and multiplying them like normal numbers, would not have been possible."
No real contradiction.
My one concern is respect for the older terminology, so that the new one is not used as an excuse, as it is often, to pass off oldies as fogies who had not yet discovered such and such a thing PLUS admit it fits reality closer and therefore is a better clue to philosophical implications of mathematical concepts than the new terminology does. Apart from that I have nothing against the new mathematics.
"So, why do you think that math would be better off if only natural numbers were treated like numbers? Explain how such things as the Riemann Hypothesis, Euler's Formula, and Fermat's Last Theorem could be treated in this system."
I don't know any of these, but I have already taken on such a challenge about logarithms.
As an exponent if really such has to be a natural number, the usual statement of what a logarithm is makes no sense.
But first of all, any decimal fraction is still a fraction as written out. It is a ratio. Of course any written out ratio will only approximate the logarithm (except when it's just a real exponent of the base), but there you go.
I restated logarithms so that 2=10^x/y MEANS (as a short cut means a full statement) 2^y=10^x. And it worked as a means for me to get started in working out a few simple logarithms.
- III (at the end of a longer thread I didn't read)
- Anonymity is an important tool in the quest for truth.
- +CowLunch Well, not really. They weren't doing this to quest for truth, they were doing this for fun.
- +CowLunch You are correct in some contexts: if you are looking for information that is illegal to possess, for example. But what if the truth you are looking for is something as simple as: "Who the heck wrote this?"
- Hans-Georg Lundahl
- +IamGrimalkin hope they enjoyed it.
Hope they enjoyed laughing at i so much they get trouble going back to taking it seriously.
+omp199 Or sth as simple (in itself at least) as "who reviewed the paper before it got published" or the somewhat more complex one "who reviewed this unpublished paper I know about so it didn't get published"?
- Belle La Victorie
You want honesty? Then you best remove evil from this world.
People like you are the reason why totalitarian dictatorships happen.
Evil people will lie no matter what laws are placed and they will ABUSE those laws to fit their own agenda.
You want to force people to use their real names? What happens when someone evil stalks them, attacks them, tries to destroy them?
It happens all the time TODAY and some people NEED a fake identity in order to EXIST.
Ever heard of witness protection?
Yeah... I wonder how bad things will get when THOSE people are not allowed to use the identities to PROTECT themselves.
The problem with YOU is that you think that people don't have a right to protect themselves.
YOU are a controlling psychopath.
Change that username of yours into a real name NOW.
YOU clearly want OTHER people to follow rules that YOU yourself are not willing to follow.
- +Belle La Victorie You are imagining that I am saying things that I have never said. It would probably take up a lot more space just to correct all your faulty assumptions than it would to add anything new and relevant to the discussion, and I'm sure we've already taken this discussion too far off the topic as it is, so I'll just bow out of this conversation now.
- Austin Cook
the purpose of the peer reviewing process is to make sure that people cant claim that things are true without proving it regardless of whether they are under a pseudonym or not.
- Hans-Georg Lundahl
- The purpose of making sure someone can't claim things without proving it is not as well served by prepublication review as by free review in freedom of reply.
So, not only does pre-publication review block one claim which is proven and would be accepted as such by a wider readereship than the pre-publication review committee, while another claim passes through their narrowminded uncritical acceptance of such a bad claim, while further blocking pertinent replies to it.
But also the use of pseudonyms covers up if such a thing has happened due to personal favour and disfavour between writer and reviewer.
+Belle La Victorie
Have you noticed that here where he is on youtube, he is not being published after a peer review only.
So, his pseudonym isn't thwarting a peer review honesty.