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Source: (consider it)
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Thread: i
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Alan Cresswell
Mad Scientist 先生
# 31
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Posted
Inspired by another thread, I got thinking ... speculating may be a better word.
i (or for engineers, j) is the imaginary unit, the square root of -1. Since any real number multiplied by itself is positive, on that basis i shouldn't exist. It's a logically fallacious, impossible entity. Well, my speculation went, what else gets called a logically fallacious, impossible entity? Well, God gets called that by many atheists.
But, as many of you will know, i turns out to be incredibly useful. First, they were first seriously considered as they were needed to solve cubic and quartic polynomials, as such maybe just a mathematical curiosity. But, it was also discovered that trigonometric functions can be expressed as exponentials involving i, allowing complex trigonometrical functions to be expressed as much simpler exponential functions. Complex numbers (combining real and imaginary numbers) form the basis of Fourier Transforms, essential in signal processing. Complex numbers are well suited to describing oscillators and waves (with all sorts of applications in seismology, vibration analysis, fluid dynamics, electromagnetism and other fields). The mathematics of quantum mechanics, the Schrodinger equation, Hilbert spaces, the matrix mechanics of Heisenberg all utilise complex numbers. Multiplication of time by i creates a space like dimension, and thus we have space-time essential to Relativity. If you remember the beauty of Mandelbrot sets, then you can thank i since they are plotted on a complex plane. It makes you wonder, if something is that useful in such a wide range of situations ... could it be a real, rather than imaginary, entity.
So, as I continue to speculate, if the atheists are right to describe God as a logically fallacious, impossible entity just as i appears what does our experience of i tell us about whether a logically fallacious and impossible entity may, nonetheless, be real?
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Don't cling to a mistake just because you spent a lot of time making it.
Posts: 32413 | From: East Kilbride (Scotland) or 福島 | Registered: May 2001
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mr cheesy
Shipmate
# 3330
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Posted
Problem is, Alan, you're probably the only person here who could follow beyond the first sentence of your post.
[I really wish I had brought up a different example other than imaginary numbers. I hate imaginary numbers..]
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arse
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LeRoc
Famous Dutch pirate
# 3216
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Posted
One of the most fascinating thing about complex numbers I find to be the Residue Theorem. This is the one mathematical theorem that make me think "How can these things be related to each other? This is so useful, surely this can't be true??"
To me, the Residue Theorem is more than sufficient proof that God created i.
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I know why God made the rhinoceros, it's because He couldn't see the rhinoceros, so He made the rhinoceros to be able to see it. (Clarice Lispector)
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hatless
Shipmate
# 3365
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Posted
Yes, I think it's a good analogy. God does not exist. If you could list all the things in the universe, God would not be on the list. But there are things you cannot (easily) say without God. God is a word that increases our ability to communicate and express ourselves. It gives us power, in particular, to speak of things like a person's worth, or of forgiveness, great beauty, moral imperatives, etc.
God, like i, is a term that only makes sense when doing some work.
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My crazy theology in novel form
Posts: 4531 | From: Stinkers | Registered: Sep 2002
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no prophet's flag is set so...
Proceed to see sea
# 15560
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Posted
When i saw the thread's title, my first thought was something Apple had invented, maybe iGod, who maybe comes with an app iJesus.
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Out of this nettle, danger, we pluck this flower, safety.
\_(ツ)_/
Posts: 11498 | From: Treaty 6 territory in the nonexistant Province of Buffalo, Canada ↄ⃝' | Registered: Mar 2010
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Beeswax Altar
Shipmate
# 11644
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Posted
quote:
Originally posted by mr cheesy:
Problem is, Alan, you're probably the only person here who could follow beyond the first sentence of your post.
[I really wish I had brought up a different example other than imaginary numbers. I hate imaginary numbers..]
Give us some credit. Even I got nearly halfway through the second complete paragraph before getting lost and I'm horrible at math. Alan was making perfect sense until Fourier Transforms.
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Losing sleep is something you want to avoid, if possible.
-Og: King of Bashan
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HCH
Shipmate
# 14313
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Posted
Some people have commented that God invented the counting numbers (1, 2, 3, ...) and all other numbers are human inventions. The only problem with i is that someone dubbed it "imaginary one" and the name stuck.
The beauty of i is that once we have it, we have the complex numbers and any polynomial of degree n has n roots. (Yes, i itself has a square root in the complex numbers.)
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Baptist Trainfan
Shipmate
# 15128
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Posted
It was the complexity of Fourier Series (all done by hand and with slide rules or log tables) which led me to abandon maths for my final year of Mechanical Engineering.
I like your analogy, though ... and Hatless's comment. (Although I don't want God to be either "imaginary" or "negative"!)
Posts: 9750 | From: The other side of the Severn | Registered: Sep 2009
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Nick Tamen
Ship's Wayfaring Fool
# 15164
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Posted
quote:
Originally posted by no prophet's flag is set so...:
When i saw the thread's title, my first thought was something Apple had invented, maybe iGod, who maybe comes with an app iJesus.
I thought Alan was just having a moment of Hispanophonic excitement.
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The first thing God says to Moses is, "Take off your shoes." We are on holy ground. Hard to believe, but the truest thing I know. — Anne Lamott
Posts: 2833 | From: On heaven-crammed earth | Registered: Sep 2009
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Sipech
Shipmate
# 16870
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Posted
quote:
Originally posted by mr cheesy:
Problem is, Alan, you're probably the only person here who could follow beyond the first sentence of your post.
No he's not. I happen to have a Master's degree in maths. ![[Devil]](graemlins/devil.gif)
The problem with the OP is the idea that i is fallacious. It was thought to be so, but then again, so were negative numbers. The widespread use of negative numbers came from having to account for debts. Neither negative numbers nor complex numbers are logically impossible. They are the very opposite, in fact; they are logically necessary.
It is better to say that they are merely outside of our normal experience. If you are used to using numbers to count how many apples you have, then of course a negative number or a complex number of apples is absurd. What is needed is to broaden your horizon to ask what a number really is.
As an aside, it's a good test to throw to maths students: ask them what a number is.
If they give you a quick answer off the cuff, they're no good. If they suddenly look all confused and pensive, then you've got someone who can actually think through the question and realise the answer is not as simple to give as the it is to phrase the question.
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I try to be self-deprecating; I'm just not very good at it.
Twitter: http://twitter.com/TheAlethiophile
Posts: 3791 | From: On the corporate ladder | Registered: Jan 2012
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Tortuf
Ship's fisherman
# 3784
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Posted
My understanding of the beauty and function of numbers is that they can be used to describe and analyze reality. As in equations help us understand gravity and its effect on objects with mass.
Where we cannot use numbers to fully accurately describe or measure something might just be a function of not knowing what numbers to use, what formula to use, or not knowing what combination of numbers and equations to use. What comes to mind is measuring the content of barrels. Merchants could accurately measure the contents by sliding a measuring stick into a barrel so the bottom of the stick was in the barrel on one side and the top was at the opposite side of the barrel.
Mathematicians of the time had to break the barrel up into ever smaller cubes to measure content and even then could never fully know what was in the barrel because there was always some bit that could not be cubed in.
So, boring the pants off of everyone further, with God, we can use some functions (texts, experiences) to describe God, but we cannot "prove" God, or the lack of "God" using the tools we have now. Will there be tools sufficient to the task in the future? Maybe. Doubt it.
Knowing that not everything can be described with fully rational, non imaginary numbers yet, may be a good start.
Posts: 6963 | From: The Venice of the South | Registered: Dec 2002
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Crœsos
Shipmate
# 238
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Posted
quote:
Originally posted by Alan Cresswell:
It makes you wonder, if something is that useful in such a wide range of situations ... could it be a real, rather than imaginary, entity.
I think you're getting caught up in the semantics. Imaginary numbers aren't "imaginary" in the sense of having no real existence any more than irrational numbers are unamenable to human reason. They're called these things largely for historic, contingent reasons and because they have to be called something.
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Humani nil a me alienum puto
Posts: 10706 | From: Sardis, Lydia | Registered: May 2001
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mr cheesy
Shipmate
# 3330
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Posted
I am humbled by your intelligence and knowledge of mathematics, oh shipmates. ![[Overused]](graemlins/notworthy.gif)
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arse
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Enoch
Shipmate
# 14322
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Posted
As a matter of tangential curiosity, and from a position of almost total ignorance of serious maths (this question probably demonstrates that):-
Are 1 and -1 two totally different numbers, largely different but with some things in common. Or are they the same number with different polarity?
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Brexit wrexit - Sir Graham Watson
Posts: 7610 | From: Bristol UK(was European Green Capital 2015, now Ljubljana) | Registered: Nov 2008
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lilBuddha
Shipmate
# 14333
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Posted
Originally posted by Alan Cresswell:
quote:
So, as I continue to speculate, if the atheists are right to describe God as a logically fallacious, impossible entity just as i appears what does our experience of i tell us about whether a logically fallacious and impossible entity may, nonetheless, be real?
God(s)/mystical forces exist because it/they do or because they fill the gap between known and unknown. They serve a purpose, so "logically fallacious" is less than accurate. IMO, but what does a non-theist know?
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I put on my rockin' shoes in the morning
Hallellou, hallellou
Posts: 17627 | From: the round earth's imagined corners | Registered: Dec 2008
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Dafyd
Shipmate
# 5549
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Posted
quote:
Originally posted by Enoch:
Are 1 and -1 two totally different numbers, largely different but with some things in common. Or are they the same number with different polarity?
I think the former is correct in the way you're asking. Although you could probably set up your terms in such a way that it's the latter.
But 9+1 does not = 9 + (-1), which generally means they're different. Or if you're bothered by the sum 9 + (-1): 9+ (5-4) is not the same as 9 + (3-4).
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we remain, thanks to original sin, much in love with talking about, rather than with, one another. Rowan Williams
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Jack o' the Green
Shipmate
# 11091
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Posted
quote:
Originally posted by Enoch:
Are 1 and -1 two totally different numbers, largely different but with some things in common. Or are they the same number with different polarity?
They have the same absolute value. A negative number is any number whose value is less than zero, and a positive number is any number whose value is greater than zero. In other words, they are sort of 'opposites' to each other, or each other's additive inverse i.e. two numbers which added together equal 0.
Posts: 3121 | From: Lancashire, England | Registered: Feb 2006
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Carex
Shipmate
# 9643
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Posted
quote:
Originally posted by HCH:
Some people have commented that God invented the counting numbers (1, 2, 3, ...) and all other numbers are human inventions.
And there is no reason that God had to create the ordinal numbers, either. A count of things such as animals would be an important concept to communicate as language developed in pre-historic people. The extension of that to convenient representations of larger numbers would be a natural progression as civilization evolved and such numbers were needed. From there it is just a continuation of extending numbers to express more things.
I'm one who uses j regularly (that's the the electrical engineer's version of i to avoid confusion with the use of i to represent electrical current) and don't consider it to be at all imaginary, except in the specialized use of that term to describe certain types of numbers. But then, we also have specialized uses for terms such as admittance and reluctance that aren't shared with most other people either.
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LeRoc
Famous Dutch pirate
# 3216
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Posted
quote:
Enoch: Are 1 and -1 two totally different numbers, largely different but with some things in common. Or are they the same number with different polarity?
It depends a bit on what you want to do with them.
If you look at it purely from a mathematical point of view, then these are two different numbers. There are no degrees in "differentness", you cannot say that one number pair is "largely different" and another number pair is "totally different". -1 ≠ 1, and that's it.
However, in some applications it makes a certain sense to see some number pairs as more different than others.
For example, sometimes you do an algebraic calculation and you're not really interested of the value of the number that comes out. You just want to know whether it's even or odd. In this application, it makes sense to see the numbers 1; 3 as pretty much the same and 1; 2 as much more different. In the former pair, the numbers are both odd, so they are basically the same for the purpose you're using them for.
In other calculations, you're not really interested in the exact result, you just want to know the rough "order of greatness" that comes out of it. In this application, the numbers 1; 2; 3 are all pretty much the same (they're all small) and 1,037,472,274,853 is very different from that (because it's much bigger).
(There are mathematically sound ways to do both things in a formal way. If you want to know, they have to do with algebraic groups and big-O notation respectively.)
So, in some applications it makes sense to consider some number pairs as "more different" and others as "less different".
I can imagine that in calculations where you're not very much interested in the sign of your answer, it would make sense to consider -1 and 1 as pretty much the same. As others have said, you'd probably use the absolute value notation |x| for that.
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I know why God made the rhinoceros, it's because He couldn't see the rhinoceros, so He made the rhinoceros to be able to see it. (Clarice Lispector)
Posts: 9474 | From: Brazil / Africa | Registered: Aug 2002
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Jengie jon
Semper Reformanda
# 273
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Posted
I have not explored the debate but there is a discussion within the philosophers of mathematics about whether mathematical constructs are invented or discovered.
The problem and it is a real one, is that logic is nothing more than an application of these constructs. (Equivalence is still debated as the challenge to derive the constructs from logic has not been shown). Therefore if mathematical constructs are invented, logic is invented!
Jengie
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"To violate a persons ability to distinguish fact from fantasy is the epistemological equivalent of rape." Noretta Koertge
Back to my blog
Posts: 20894 | From: city of steel, butterflies and rainbows | Registered: May 2001
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balaam
Making an ass of myself
# 4543
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Posted
quote:
Originally posted by HCH:
Some people have commented that God invented the counting numbers (1, 2, 3, ...) and all other numbers are human inventions.
Modern maths was impossible until the Arabs invented the concept of 0 (zero).
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Last ever sig ...
blog
Posts: 9049 | From: Hen Ogledd | Registered: May 2003
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Chapelhead
I am
# 21
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Posted
A few odd thoughts...
I recall a university maths lecturer telling me that, when he was an undergraduate, the starting point of the maths he was taught was, "There are positive integers". This, he realised later, was an assumption - there's no proof that "1" and "2" and "3" etc exist.
So I suppose that when you get to i you have something whose existence cannot be proved even more. But it still 'works'.
I quite like i, and last Christmas 'Secret Santa' bought me this. But I prefer the transfinite numbers ("numbers bigger than infinity" would be an inaccurate but 'easy' way to understand them, for those not mathematically inclined). I recall hearing Buzz Lightyear saying "To infinity and beyond", and being slightly puzzled why people thought this funny - it sounds quite reasonable to me.
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At times like this I find myself thinking, what would the Amish do?
Posts: 9123 | From: Near where I was before. | Registered: Aug 2001
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LeRoc
Famous Dutch pirate
# 3216
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Posted
quote:
Carex: I'm one who uses j regularly
Get behind me! Why don't we have a smiley making a protective cross symbol with its fingers? ![[Smile]](smile.gif)
My view of whether the numbers are discovered or invented is: clearly we invented the numbers 1, 2, 3. But this is because we discovered that it makes sense to do this. Our Universe is such that 1+1=2 makes sense in understanding it.
This was already discussed a couple of weeks ago, but as a Christian, I believe that God created the Universe in this way. We invented mathematics, but God created the Universe such that mathematics makes sense to understand it.
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I know why God made the rhinoceros, it's because He couldn't see the rhinoceros, so He made the rhinoceros to be able to see it. (Clarice Lispector)
Posts: 9474 | From: Brazil / Africa | Registered: Aug 2002
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Schroedinger's cat
Ship's cool cat
# 64
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Posted
I think the point about mathematical concepts like i is that they do not directly map onto something in the real world. The integers do - we count things using them, we can count 1 cushion, for example - it doesn't make them any more extant, but they relate to something we can see and identify. 1 cushion is a concept that we can relate to, and so we tend to accept that "1" is something extant.
Concepts like i do not have a sensory-world-analogue, so we treat these with more suspicion. We instinctively struggle with comprehending this, even if we can accept it as something that is useful, and an idea that exists within a conceptual paradigm (like, the world of Maths). The thing is, it is no more or less extant that those things we can visualise. They are just less intuitive.
In the same way, I think, God is not something/one we can relate to the physical, sensory world, by his nature. That doesn't mean he is any less real, just that his "reality" is not one that is limited by our sensory perception.
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Blog
Music for your enjoyment
Lord may all my hard times be healing times
take out this broken heart and renew my mind.
Posts: 18859 | From: At the bottom of a deep dark well. | Registered: May 2001
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mark_in_manchester
not waving, but...
# 15978
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Posted
If 'j' (I was a kind of electrical engineer, and 'i' was reserved for current) does your head in, what about quaternions?
They are 3D complex numbers (think of a 3D Argand diagram with no real axis, where i, j and k refer to 90deg rotations in each of the three orthogonal planes).
Then
i^2=j^2=k^2=ijk=-1 !!
I really miss teaching this stuff, Fourier transforms and all that - it's not hard, really. But I don't at all miss having to try to find ways to bring in money to a university, on the back of it!
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"We are punished by our sins, not for them" - Elbert Hubbard
(so good, I wanted to see it after my posts and not only after those of shipmate JBohn from whom I stole it)
Posts: 1596 | Registered: Oct 2010
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Crœsos
Shipmate
# 238
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Posted
quote:
Originally posted by balaam:
quote:
Originally posted by HCH:
Some people have commented that God invented the counting numbers (1, 2, 3, ...) and all other numbers are human inventions.
Modern maths was impossible until the Arabs invented the concept of 0 (zero).
Indians, actually. (Or rather one of the small states in what is now northwestern India.) The system of what we now call 'Arabic numerals' is called thus because the Arabs were the first people to have the brilliant idea of stealing them from India. Credit where credit is due.
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Humani nil a me alienum puto
Posts: 10706 | From: Sardis, Lydia | Registered: May 2001
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mark_in_manchester
not waving, but...
# 15978
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Posted
Ooooh - to connect with the 'God' ideas in the OP - if anyone attracted by Alan's excellent thread who likes Fourier Transforms would like to discuss Dooyeweerd (Dutch Christian philosopher, died in 1970s) and his Fourier-like decomposition of all reality (man!) into something like a set of orthogonal basis functions he calls the 'modal aspects' - which specifically include the dimension of faith - then do say and maybe I'll start another thread.
It's a bit specialist, but maybe Alan has prised up a rotting plank and revealed scurrying mathematicians within the ship...
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"We are punished by our sins, not for them" - Elbert Hubbard
(so good, I wanted to see it after my posts and not only after those of shipmate JBohn from whom I stole it)
Posts: 1596 | Registered: Oct 2010
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mark_in_manchester
not waving, but...
# 15978
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Posted
Enoch asked:
quote:
Are 1 and -1 two totally different numbers, largely different but with some things in common. Or are they the same number with different polarity?
An engineer would say they have the same magnitude, but opposite _phase_ - which is similar but a slightly more flexible concept to your polarity.
So if my 2 kids were on a pair of swings next to each other, I might swing them both to a given height and call this '1'. If they moved exactly together (same frequency, and peaking at just the same point in time) they both have magnitude 1 swinging, and where they are at *just* a particular point in time we could find write down with a 'sin' function. But very often we don't care - we just need to know how big the swing is (the magnitude). This is like saying mains voltage is 240v...but if you touch it quick, it might be less or even zero...or it might not! A bit like jumping under the swings...time it right!
If my kids move in opposite directions (one peaks back, when one peaks forwards) then we say they are 180deg out of phase. Then an engineer would say one has magnitude 1, and the other has magnitude -1.
It gets more fun if one peaks *just* as the other moves through (say) the rest position...where she'll end up when I get bored and go and read the paper. Then the first is moving back through this position *just* as the second peaks.
We say there is 90deg of phase between the two of them - but they still have the same frequency and magnitude 1.
Imagine we could multiply the motion of kid 2 by a magic something to shift her motion relative to kid 1 by 90deg. If we did it again, she'd be shifted by 180deg. This is a shift of polarity, and magnitude 1 becomes magnitude -1.
Therefore 'magic something'x'magic something' = -1
So 'magic something' ^2 = -1
So 'magic something' = sq root (-1)
Sorry folks - I just don't get the opportunity any more...
[ 15. September 2015, 18:21: Message edited by: mark_in_manchester ]
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"We are punished by our sins, not for them" - Elbert Hubbard
(so good, I wanted to see it after my posts and not only after those of shipmate JBohn from whom I stole it)
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Jay-Emm
Shipmate
# 11411
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Posted
quote:
Originally posted by Crœsos:
The system of what we now call 'Arabic numerals' is called thus because the Arabs were the first people to have the brilliant idea of stealing them from India. Credit where credit is due.
you mean Edison copied even that
![[Eek!]](eek.gif)
Another way 1 and -1 fundementally differ is that 1 is the identity for multiplication, and -1 isn't. (That is 5*1 is still 5, 5*-1 is no longer 5).
Although because 5*-1*-1=5, -1 is still in some ways like 1 [in particular when dealing with primes and irreducables, 5*2 and -5*-2 don't count as seperate factorisations, but once you deal with fractions everything's a unit, anyway].
[ 15. September 2015, 18:22: Message edited by: Jay-Emm ]
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LeRoc
Famous Dutch pirate
# 3216
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Posted
quote:
mark_in_manchester: Dooyeweerd (Dutch Christian philosopher, died in 1970s) and his Fourier-like decomposition of all reality (man!)
I've only vaguely heard of him. This is so brilliantly funny that it must be true
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I know why God made the rhinoceros, it's because He couldn't see the rhinoceros, so He made the rhinoceros to be able to see it. (Clarice Lispector)
Posts: 9474 | From: Brazil / Africa | Registered: Aug 2002
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mr cheesy
Shipmate
# 3330
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Posted
If there are an infinity of integers between 0 and infinity, and an infinity of fractions between 0 and 1.. how come there are not an infinity of infinity of possible fractions?
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arse
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LeRoc
Famous Dutch pirate
# 3216
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Posted
quote:
mr cheesy: If there are an infinity of integers between 0 and infinity, and an infinity of fractions between 0 and 1.. how come there are not an infinity of infinity of possible fractions?
The "kind of infinity" (mathematicians call it "cardinality") of the integers is actually the same as the "kind of infinity" of fractions. There is a rather beautiful proof of that.
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I know why God made the rhinoceros, it's because He couldn't see the rhinoceros, so He made the rhinoceros to be able to see it. (Clarice Lispector)
Posts: 9474 | From: Brazil / Africa | Registered: Aug 2002
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Jack o' the Green
Shipmate
# 11091
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Posted
quote:
Originally posted by LeRoc:
quote:
Carex: I'm one who uses j regularly
Get behind me! Why don't we have a smiley making a protective cross symbol with its fingers?
My view of whether the numbers are discovered or invented is: clearly we invented the numbers 1, 2, 3. But this is because we discovered that it makes sense to do this. Our Universe is such that 1+1=2 makes sense in understanding it.
This was already discussed a couple of weeks ago, but as a Christian, I believe that God created the Universe in this way. We invented mathematics, but God created the Universe such that mathematics makes sense to understand it.
I would say something very similar. I think the label which best describes my position is 'Divine Conceptionalist'.
Posts: 3121 | From: Lancashire, England | Registered: Feb 2006
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Humble Servant
Shipmate
# 18391
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Posted
quote:
Originally posted by mr cheesy:
If there are an infinity of integers between 0 and infinity, and an infinity of fractions between 0 and 1.. how come there are not an infinity of infinity of possible fractions?
It's because infinity is not a number. It is a concept that behaves very differently from the numbers. If infinity were a number then infinity +1 would be a larger number, which would mean infinity was not infinite.
Posts: 241 | Registered: Apr 2015
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LeRoc
Famous Dutch pirate
# 3216
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Posted
My internet connection is a bit better now, so I can expand a bit more on this.
In a sense, we can say there are just as many integers as there are fractions. This is a bit tricky, since the answer to both is 'infinity', but there exists a mathematical way of comparing different kinds of infinity with each other.
A thing about integers is that we can put them on a line: 1, 2, 3 ... This is how we count. What we can show is that we can put all fractions on a line in the same way.
The way to do this is shown in this image. The arrows form a path in which all fractions can be put after each other on a line. (You may have to skip the ones in red.)
Since we can put all fractions on a line, we can number them 1, 2, 3 ... Every integer corresponds to one fraction. This shows that in a sense, there are as many fractions as there are integers.
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I know why God made the rhinoceros, it's because He couldn't see the rhinoceros, so He made the rhinoceros to be able to see it. (Clarice Lispector)
Posts: 9474 | From: Brazil / Africa | Registered: Aug 2002
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Hedgehog
Ship's Shortstop
# 14125
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Posted
quote:
Originally posted by LeRoc:
quote:
mr cheesy: If there are an infinity of integers between 0 and infinity, and an infinity of fractions between 0 and 1.. how come there are not an infinity of infinity of possible fractions?
The "kind of infinity" (mathematicians call it "cardinality") of the integers is actually the same as the "kind of infinity" of fractions. There is a rather beautiful proof of that.
But there are "larger" infinities out there, as Chapelhead mentioned: the transfinite numbers. A poor university prof tried to make me understand them some 30+ years ago, and, but for Chapelhead's post, I would have forgotten their name. IIRC, the infinite number of decimals between 0 and 1 is a larger infinity than the infinite number of fractions.
And there is a mathematical proof of that, too. Because it isn't the sort of thing you'd say without proof.
But I don't remember it because it was 30+ years ago and I had no use for the knowledge until right now.
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Jay-Emm
Shipmate
# 11411
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Posted
The version I've seen (and it's a beautiful combination with the other set) follows on from the proofs where you put the natural numbers the odd numbers, the integers, the squares, the fractions, etc in some order (so you can say what the 79th fraction in the infinite list is).
It supposes you could make the list, and then creates a real number not on the list.
the diagonal proof
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LeRoc
Famous Dutch pirate
# 3216
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Posted
quote:
Hedgehog: But there are "larger" infinities out there, as Chapelhead mentioned: the transfinite numbers.
Yes there are.
quote:
Hedgehog: IIRC, the infinite number of decimals between 0 and 1 is a larger infinity than the infinite number of fractions.
And there is a mathematical proof of that, too.
Yes there is. The proof of that is a bit more complex.
quote:
Hedgehog: But I don't remember it because it was 30+ years ago and I had no use for the knowledge until right now.
See? Your maths teacher told you that it would come in handy some day
[ 15. September 2015, 21:15: Message edited by: LeRoc ]
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Jack o' the Green
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# 11091
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Posted
Is that the same or similar to 'countable' and 'uncountable' infinities (e.g. the infinite of integers, and the infinite set of real numbers?)
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Sober Preacher's Kid
Presbymethegationalist
# 12699
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Posted
quote:
Originally posted by mr cheesy:
Problem is, Alan, you're probably the only person here who could follow beyond the first sentence of your post.
[I really wish I had brought up a different example other than imaginary numbers. I hate imaginary numbers..]
Speak for yourself.
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Dafyd
Shipmate
# 5549
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Posted
quote:
Originally posted by Jack o' the Green:
Is that the same or similar to 'countable' and 'uncountable' infinities (e.g. the infinite of integers, and the infinite set of real numbers?)
That's the same. The fractions (and indeed all the algebraic numbers - that is the fractions, plus square roots, cube roots, etc.) called a countable infinity because you can put them into a list that you can count your way through.
The real numbers are an uncountable infinity because you can't put them into a list to count your way through them.
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Adeodatus
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# 4992
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To begin with, I'd like to suggest that while i isn't a "real" number (in the mathematical definition of "real"), it's every bit as real or unreal as any other number. One interpretation of i is that on a piece of 2-dimensional graph paper, if 1 is 1 unit along the x-axis, i is 1 unit along the y-axis. That's pretty real.
quote:
Originally posted by Alan Cresswell:
But, as many of you will know, i turns out to be incredibly useful.
Is this like saying that if God didn't exist, it would be necessary to invent him? That whether he exists or not, "God" is necessary because of his usefulness?
It's an interesting idea, and my private personal creed has for a long time been like that of C.S.Lewis' Puddleglum, whom I'm going to quote here at greater length than he's usually given, and hope it isn't too long a quote for the Hosts -
quote:
"Suppose we have only dreamed, or made up, all those things - trees and grass and sun and moon and stars and Aslan himself. Suppose we have. Then all I can say is that, in that case, the made-up things seem a good deal more important than the real ones. Suppose this black pit of a kingdom of yours is the only world. Well, it strikes me as a pretty poor one. And that's a funny thing, when you come to think of it. We're just babies making up a game, if you're right. But four babies playing a game can make a play-world which licks your real world hollow. That's why I'm going to stand by the play world. I'm on Aslan's side even if there isn't any Aslan to lead it. I'm going to live as like a Narnian as I can even if there isn't any Narnia."
My faith is a devotion to an idea that perhaps someone one had, of something better and brighter. I suppose you could accuse my faith of being mere utopianism.
But utopianism isn't the real problem. The real problem with faith is that it either has to be real, or it has to be better than real. And we find ourselves in a world where people of faith so often give their lives to making real life worse for others, not better. This is my problem with faith - that it can be as ugly as it is beautiful, and God as harmful as he is useful.
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Alan Cresswell
Mad Scientist 先生
# 31
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quote:
Originally posted by Crœsos:
quote:
Originally posted by Alan Cresswell:
It makes you wonder, if something is that useful in such a wide range of situations ... could it be a real, rather than imaginary, entity.
I think you're getting caught up in the semantics. Imaginary numbers aren't "imaginary" in the sense of having no real existence any more than irrational numbers are unamenable to human reason. They're called these things largely for historic, contingent reasons and because they have to be called something.
Of course, and the word "imaginary" was first used in a derogatory sense - because, as Sipech said in correcting a point in my OP, they were thought to be fallacious, and a useless mathematical curiosity. Sometimes derogatory names stick (in the other thread I also mentioned "Big Bang" as a derogatory term that endured).
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Alan Cresswell
Mad Scientist 先生
# 31
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Posted
quote:
Originally posted by Schroedinger's cat:
Concepts like i do not have a sensory-world-analogue, so we treat these with more suspicion. We instinctively struggle with comprehending this, even if we can accept it as something that is useful, and an idea that exists within a conceptual paradigm (like, the world of Maths). The thing is, it is no more or less extant that those things we can visualise. They are just less intuitive.
In the same way, I think, God is not something/one we can relate to the physical, sensory world, by his nature. That doesn't mean he is any less real, just that his "reality" is not one that is limited by our sensory perception.
Which is a great way of expressing what I was grasping for in my speculative OP.
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Don't cling to a mistake just because you spent a lot of time making it.
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orfeo
Ship's Musical Counterpoint
# 13878
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Posted
quote:
Originally posted by Schroedinger's cat:
I think the point about mathematical concepts like i is that they do not directly map onto something in the real world. The integers do - we count things using them, we can count 1 cushion, for example - it doesn't make them any more extant, but they relate to something we can see and identify. 1 cushion is a concept that we can relate to, and so we tend to accept that "1" is something extant.
Concepts like i do not have a sensory-world-analogue, so we treat these with more suspicion. We instinctively struggle with comprehending this, even if we can accept it as something that is useful, and an idea that exists within a conceptual paradigm (like, the world of Maths). The thing is, it is no more or less extant that those things we can visualise. They are just less intuitive.
In the same way, I think, God is not something/one we can relate to the physical, sensory world, by his nature. That doesn't mean he is any less real, just that his "reality" is not one that is limited by our sensory perception.
This is true of lots of things. We have abstract nouns for emotions, for example. You can't use your senses to directly detect love, anger, hope, glory or justice.
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Schroedinger's cat
Ship's cool cat
# 64
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Posted
orfeo - maybe (and if those help you grasp God then fine). I do think that our sensory world can "see" these things in action - our physical bodies know love, hate; we see acts of justice and mercy. We use words that convey these ideas that are within our sensory remit.
Even with completely abstract concepts like "an idea forming", we "see" this in the expressions and actions of a person. i is more like that actual biological and neurological processes of "an idea forming".
And Alan likes my explanation of his idea? wow. I win.
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Full of Chips
Shipmate
# 13669
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Posted
Of course there are two square roots of -1.
If i^2 = 1, then (-i)^2 = -1 x i x -1 x i = 1 x i^2 = -1
So (-i) is also a square root of -1.
The question is, which of the two square roots of -1 did we label i (or j) in the first place?
The mathematics of complex numbers is based on an unknowable choice which creates two identical systems pi out of phase with each other.
I expect we make unknowable choices in our understanding of God, resulting in belief systems out of phase with each other too.
[ 16. September 2015, 18:57: Message edited by: Full of Chips ]
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LeRoc
Famous Dutch pirate
# 3216
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Posted
quote:
Full of Chips: The question is, which of the two square roots of -1 did we label i (or j) in the first place?
(It doesn't matter.)
quote:
Full of Chips: I expect we make unknowable choices in our understanding of God, resulting in belief systems out of phase with each other too.
Nice one! ![[Big Grin]](biggrin.gif)
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I know why God made the rhinoceros, it's because He couldn't see the rhinoceros, so He made the rhinoceros to be able to see it. (Clarice Lispector)
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Full of Chips
Shipmate
# 13669
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Posted
Quote:
Le Roc:
quote:
Full of Chips: The question is, which of the two square roots of -1 did we label i (or j) in the first place?
(It doesn't matter.)
Well I agree it doesn't matter if you just want to use complex numbers. It does matter however if you care about whether mathematical systems you are constructing are natural in the sense of being categorically unique as opposed to merely one of several isomorphic objects that could result dependent on some choice.
I freely admit though that even in advanced mathematical texts of that nature, it generally results only in a little footnote that says "*subject to a choice of square root of -1".
It also matters (though perhaps only to pedants) if you want to say, as Alan did, that "i is the square root of -1".
[ 16. September 2015, 19:31: Message edited by: Full of Chips ]
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