Questions about visual/physical representations of imaginary numbers.

[SamBridge]

 

What about Thinking of i as rotation? Not a physical object, but a physical transform.

On the imaginary plane 1*i=1i or a 90 degree transform,1i*i=-1 another 90,-1*i=-1i and one more multiply and you're back to 1 for a full 360.

 

I tried that, but the only problem with that is that it requires an imaginary axis for the rotation to make sense, it's just like using a word in it's definition. For Example: What does Ostentatious mean? "It's when you act ostentatiously". See the problem? Or saying x=x+1

Edited January 15, 2013 by SamBridge

[Amaton]

... if mathematics was itself a reduced form of logic, then we wouldn't need all these various mathematics systems, we should just be able to use 1 logic to deduce what is logically happening...

 

Okay, forget imaginary apples. ^This seems to be the root of your inquiry. I don't think the standard and current mathematics can satisfy the fundamental challenge underlying your questions... Namely, the existence of a mathematics which does physically coincide with reality as opposed to one which is just an abstract logical reduction applied when subjectively fit. The search for a "naturalistic" mathematics, so to speak. Would you consider your thoughts deeply rooted in philosophy? Or is it more practical and I'm just overthinking it (?). Because the context at hand isn't clear to me.

[SamBridge]

 

Okay, forget imaginary apples. ^This seems to be the root of your inquiry. I don't think the standard and current mathematics can satisfy the fundamental challenge underlying your questions... Namely, the existence of a mathematics which does physically coincide with reality as opposed to one which is just an abstract logical reduction applied when subjectively fit. The search for a "naturalistic" mathematics, so to speak. Would you consider your thoughts deeply rooted in philosophy? Or is it more practical and I'm just overthinking it (?). Because the context at hand isn't clear to me.

I would say I'm trying to use logic and evidence to extrapolate a physical meaning which by nature is philosophical because it cannot be scientifically tested at this point.

[x(x-y)]

It can't be descried using only English words, you would have to see it for yourself. The best way I can describe it, an honorable apple looks like it has a slightly scarred and randomly rugged surface and patches of discoloration, it looks as though it has been through a lot, and accomplished what it was meant to, that's about all I can put into terms of words.

 

That was one of the most hilarious things I've read so far this year - thank you.

 

Predict what "i" apples looks like.

 

You still haven't got the idea have you? i apples does not make any sense, neither mathematically nor physically - perhaps it does philosophically, but philosophy is a load of pretentious twoddle most of the time anyway.

[SamBridge]

That was one of the most hilarious things I've read so far this year - thank you.

I got plenty more if you want, I think I figured out how to model certain 4-dimensional properties as integrals, of course by using my imagination.

 

You still haven't got the idea have you? i apples does not make any sense, neither mathematically nor physically - perhaps it does philosophically, but philosophy is a load of pretentious twoddle most of the time anyway.

(This is in the philosophical section now by the way) But saying "i" apples doesn't make sense without any logical proof that it can't make sense just because we don't know what it looks like doesn't make sense. You could say the same thing about "infinite" apples, yet we still have a concept of what infinite apples would physically mean or look like, we even use infinity in math regardless of the fact that it's not a number.

[x(x-y)]

Yes, infinity is used in mathematics but the same cannot be said of the arbitrary object set of "apples". Why do you even think i apples should make some sense? It just doesn't, as Bignose said you can't just put words together (or words and the imaginary unit in this case) and expect them to make sense.

[SamBridge]

Yes, infinity is used in mathematics but the same cannot be said of the arbitrary object set of "apples". Why do you even think i apples should make some sense? It just doesn't, as Bignose said you can't just put words together (or words and the imaginary unit in this case) and expect them to make sense.

If I said "apples i row like look", that doesn't make sense, but "i apples means something" isn't a random string of words. The value "i" clearly exists since it's found in so many mathematical patterns that describe nature, just as the value "1" exists, it's only natural that we should be able to find the thing that it's suppose to represent. I can still have a beautiful apple or an elegant apple, I can still picture an honorable apple, how about you just take the traits of what you expect an honorable person to look like and apply them to an apple?

Edited January 16, 2013 by SamBridge

[x(x-y)]

I can still have a beautiful apple or an elegant apple, I can still picture an honorable apple, how about you just take the traits of what you expect an honorable person to look like and apply them to an apple?

 

Because even if you can do it tenuously through the use of pretentious language and silly poetical nonsense, you cannot do it in mathematics - it just does not make any sense to slap the word apples in front of the imaginary unit i and expect to know what this so-called "quantity" (in your opinion it's a quantity) is. The closest you can get is probably just by saying you have some purely imaginary complex number z of the form z = ai - call a "apples" if you like (it still wouldn't make any sense to, and I don't understand what your motivation behind doing so would be) , but it is just a real coefficient of the imaginary unit (i.e. the imaginary part of z).

[SamBridge]

Because even if you can do it tenuously through the use of pretentious language and silly poetical nonsense, you cannot do it in mathematics - it just does not make any sense to slap the word apples in front of the imaginary unit i and expect to know what this so-called "quantity" (in your opinion it's a quantity) is.

Well it shouldn't make sense to do that with infinity, and if I take it a step further it shouldn't make sense to do that with ANY symbol because we made those symbols up and assigned our own meanings to them, you shouldn't be able to just slap meanings onto made up symbols, From that standpoint, the whole of mathematics sounds like a bunch of bs, even though it's allowed us to do a lot. Obviously "i" is a value, it's a value of something and that something can be found as represented by patterns extrapolated from nature, it's no different than any other symbol that we call a "number", it represents some kind of value, and what it represents obviously applies to reality in some way.

 

 

 

Edited January 16, 2013 by SamBridge

[x(x-y)]

Obviously "i" is a value, it's a value of something and that something can be found as represented by patterns extrapolated from nature, it's no different than any other symbol that we call a "number", it represents some kind of value, and what it represents obviously applies to reality in some way.

 

Of course, I understand the applications of complex numbers to reality - I have studied, and continue to study, quantum mechanics, electromagnetism, optics and waves, and so I can appreciate the importance (and absolute necessity in the case of QM) of complex numbers in these fields of Physics. However, I maintain the fact that sticking the word apples after i doesn't give any sort of viable, sensible physical quantity and simply does not make sense. What do you actually think "i apples" does mean?

[SamBridge]

What do you actually think "i apples" does mean?

That's what I hope to find out. "i" like a value, and like any value it represents something, we know what most other values look like at least in some manner, I don't see how it is scientifically impossible to do the same with "i".

Edited January 17, 2013 by SamBridge

[x(x-y)]

That's what I hope to find out. "i" like a value, and like any value it represents something, we know what most other values look like at least in some manner, I don't see how it is scientifically impossible to do the same with "i".

 

That's because, for all intents and purposes, i doesn't actually exist (it is the imaginary unit after all). Sure the complex number based arithmetic works perfectly, and beautifully as it gives us some of the most elegant equations in Physics and Mathematics, however no matter how hard you try - you cannot perform a square root function on a negative number and obtain a real value, you will always obtain an "imaginary number"; but, as I alluded to above, this is perfectly fine and it clearly works (it's logical).

[SamBridge]

That's because, for all intents and purposes, i doesn't actually exist

Neither does "2". Do you see the symbol "2" just randomly floating through space? No, the number 2, like the number "i", is something we made up, and this is true for anything in math. Did you catch that math is in of itself proven to not be a form of logic? That it was created with it's own axioms? All numbers, regardless of whatever you want them to represent are symbols that we made up with meanings that we made up. The universe is its own thing and it will function however it wants independent from out understanding of it.

Edited January 17, 2013 by SamBridge

[Amaton]

Okay. It's obvious that this is very deeply rooted in philosophy. I'm just throwing up that, as far as I know, " i apples " has no standard or conventional meaning.

 

All numbers, regardless of whatever you want them to represent are symbols that we made up with meanings that we made up.

 

Interesting thought. Consider this: triangles exist. How many sides does a triangle have? Surely, this quantitative entity exists in some way, transcendent of reality or not. Whether or not "3" physically exists, can be empirically observed, etc. dives into the philosophy part. Also, consider the laws of physics, much of which are described in differential equations. These are harder to ignore than mere numbers, as differentiation is more of a process dealing with the relationship of expressions.

 

Thinking more from this point, I think this topic touches a lot on the existence (or lack) of an objective, naturalistic mathematics... independent of our axiological constructions. And also on more elementary concepts like empiricism.

[SamBridge]

Okay. It's obvious that this is very deeply rooted in philosophy. I'm just throwing up that, as far as I know, " i apples " has no standard or conventional meaning.

 

Which is why I'm investigating it.

 

Interesting thought. Consider this: triangles exist. How many sides does a triangle have? Surely, this quantitative entity exists in some way, transcendent of reality or not. Whether or not "3" physically exists, can be empirically observed, etc. dives into the philosophy part. Also, consider the laws of physics, much of which are described in differential equations. These are harder to ignore than mere numbers, as differentiation is more of a process dealing with the relationship of expressions.

The word "triangle" is still a word created by humans, the universe is its own thing, it works however it wants to work, we simply try to label patterns we see in it using math. Can you prove a triangle has 3 sides using only algebra? You can't, the word "triangle" has some outside meaning from the operations of math, it us humans who recognize some collection of sides as being distinguishable, and this meaning has a meaning outside of the universe. In reality there isn't actually a circle when you think you draw one on paper, there's clusters of atoms which you can draw straight lines between.

So, I guess we have to define, what is a value really? What really is a number? And why does it work the way it works? We can't completely answer that because numbers are just extensions of axioms. We say "1" means some thing, but that thing isn't actually found in reality, what we find in reality is physical objects. You say you find "1" apple, but really what it is, is a composite of many many atoms, which are something like oscillating fields, so really, "what is 'i' apples?" is a lot more complex than what it's being credited for, you can't just say it can't represent something when we don't really even understand what exactly a value is in the first place. Let's say some unit has a value of 1, but what does that mean beyond just that statement? Whatever the answer is, that's something we make up, using language and symbols.

Edited January 17, 2013 by SamBridge

[x(x-y)]

This is just getting silly now, this thread no longer belongs in mathematics.

[SamBridge]

I said it should be moved into philosophy a while back, but what I am saying is true, it is a fact that mathematics has it's own set of axioms and is not directly equal to physical objects or logic itself whether you like it or not. What does the value "1" actually mean? If you say there's "1" of something, define what it means without using the value "1". It's not going to be easy, and that's because it's closely related to an axiom, and axioms by definition are not provable entities, they are assumptions of which to base other logical conclusions from.

Edited January 17, 2013 by SamBridge

[imatfaal]

SamBridge. Yes - if you wanna get to a more basic level you can take a look at Peano axiomata. I think logicians have worked from even simple axiomata but frankly there does come a point where it is all becoming fairly disconnected with everyday expereince and those truths we actually do hold to be self evident.

[SamBridge]

SamBridge. Yes - if you wanna get to a more basic level you can take a look at Peano axiomata. I think logicians have worked from even simple axiomata but frankly there does come a point where it is all becoming fairly disconnected with everyday expereince and those truths we actually do hold to be self evident.

But the "truths" as you call it are based on those fundamental axioms, exploring something doesn't make it more connected or disconnected to reality, it will always have whatever correlation it always has. Numbers are all things we assigned meanings too, we don't know exactly what they mean on their own, and perhaps they have no meaning, but we didn't define "i" to have no meaning, it means something because we assign it to mean something, and it appears to pop up in equations describing patterns of reality.

Edited January 18, 2013 by SamBridge

[imatfaal]

But the "truths" as you call it are based on those fundamental axioms, exploring something doesn't make it more connected or disconnected to reality, it will always have whatever correlation it always has. Numbers are all things we assigned meanings too, we don't know exactly what they mean on their own, and perhaps they have no meaning, but we didn't define "i" to have no meaning, it means something because we assign it to mean something, and it appears to pop up in equations describing patterns of reality.

 

No they are not based on those fundamental axiomata - those axiomata are a post hoc rationalisation of the things that most people accept without question. If you are interested in logic and reduction then feel free to knock yourself out with Peano axiomata, Russel, and Godel etc (and I do) - but it is disingenuous to claim that they are the cause and root of the general understanding of arithmetics. If you wish to you can reduce it all to a feeling of self existence, a rejection of solipsism and work from there to "deducing the existence of rice pudding and income tax". But generally the fundamental level in the world at large is that of the natural numbers and we work from there - a more basic level of reasoning does not defeat the utility of a more abstracted one; ie your axiomatic base is at an arbitrary level.

[SamBridge]

 

No they are not based on those fundamental axiomata - those axiomata are a post hoc rationalisation of the things that most people accept without question. If you are interested in logic and reduction then feel free to knock yourself out with Peano axiomata, Russel, and Godel etc (and I do) - but it is disingenuous to claim that they are the cause and root of the general understanding of arithmetics. If you wish to you can reduce it all to a feeling of self existence, a rejection of solipsism and work from there to "deducing the existence of rice pudding and income tax". But generally the fundamental level in the world at large is that of the natural numbers and we work from there - a more basic level of reasoning does not defeat the utility of a more abstracted one; ie your axiomatic base is at an arbitrary level.

I agree that mathematical systems are logical, but I know they are not logic themselves. In order for numbers to work, we have to assign our own arbitrary meaning to them, the universe obviously isn't going to do it for us, we didn't have mathematics before we invented it, we have to assume what the term "1" or "2" actually is, because without us to distinguish it, it has no meaning whatsoever. However, we do not have to assign meaning to gravity in order for gravity to continue to pull things, we don't have to assign meaning to rocks in order for them to be placed where they are, but we have to assign meaning to numbers in order for them to work at all, we have to invent them for them to work, and this tells me that math is not the universe, because the universe will work however it work regardless of if we assign meaning to it, but this is not true of math, so math cannot be a real part of the universe.

That's just what my point is, but I think numbers in some way have something to do with reality, but I can't quite explain it because of the dilemma I mentioned above. The math is obviously useful, it can make predictions, but it still wouldn't exist unless we decided to invent it, so how exactly do we nail down how involved it is in reality?

Edited January 22, 2013 by SamBridge

[imatfaal]

I agree that mathematical systems are logical, but I know they are not logic themselves. In order for numbers to work, we have to assign our own arbitrary meaning to them, the universe obviously isn't going to do it for us, we didn't have mathematics before we invented it, we have to assume what the term "1" or "2" actually is, because without us to distinguish it, it has no meaning whatsoever.

yes, yes and possibly

 

However, we do not have to assign meaning to gravity in order for gravity to continue to pull things, we don't have to assign meaning to rocks in order for them to be placed where they are,

ah but we do. Rocks do what rocks do - but gravity is another thing entirely. A thousand years ago rocks fell because the predominance of the elements of earth and water within them meant they fell (as opposed to fire and air which rose), with Newton we learnt they fell as there is a force between masses, now we know that mass tells space-time how to curve, and space-time tells mass how to move. Gravity used to pull things - now it is just a manifestation of objects following the shortest path.

 

but we have to assign meaning to numbers in order for them to work at all, we have to invent them for them to work, and this tells me that math is not the universe, because the universe will work however it work regardless of if we assign meaning to it, but this is not true of math, so math cannot be a real part of the universe.

That's just what my point is, but I think numbers in some way have something to do with reality, but I can't quite explain it because of the dilemma I mentioned above. The math is obviously useful, it can make predictions, but it still wouldn't exist unless we decided to invent it, so how exactly do we nail down how involved it is in reality?

 

I do not agree that maths requires the meaning we place upon it - and in fact I think it is better in someways without the meaning. You take your arbitrary starting level and construct a valid logical edifice - that we can assign meaning to this and further use this abstract model to accurately parallel real world phenomena is brilliant, but not essential. You seem to be grasping for the underlying meaning - "how involved it is in reality?" - a sine qua non of the universe; we will not reach that point merely get closer and closer, and you can bet your last dollar it will be described in the language of mathematics

[SamBridge]

 

ah but we do.

I'm sorry to put it bluntly, but "nope", completely wrong. With our current scientific model, gravity could not have worked in the way you went later on to describe. Gravity as humans had observed still made bodies attract, we just weren't aware of it, and the universe did not care that we lived in that ignorance, yet Earth did not being to fall apart. That point does not work the way you want it to, probably because what I'm saying is true, You're trying to use what I'm saying about mathematics about real objects, but my point works on math and not real objects because math isn't a real thing. The universe exists independent of our understanding of it, but numbers can't exist without a living thing to invent them.

We can give gravity any name we want, but unlike math, it will exist outside of that name.

 

I do not agree that maths requires the meaning we place upon it - and in fact I think it is better in someways without the meaning. You take your arbitrary starting level and construct a valid logical edifice - that we can assign meaning to this and further use this abstract model to accurately parallel real world phenomena is brilliant, but not essential. You seem to be grasping for the underlying meaning - "how involved it is in reality?" - a sine qua non of the universe; we will not reach that point merely get closer and closer, and you can bet your last dollar it will be described in the language of mathematics

But math is just axioms of how symbols work. If I keep asking about why a number works, I eventually after all that number theory and we say "if we mark '1' as a and 2 as a' then via the transitive property..." I can still ask "but why does it work just because we mark 1 as "a"? and eventually I arrive to the point where I can only say "because that's just how we decided to make it work". I can assure you that any mathematical theorem has to be based off of some kind of axiom.

Edited January 24, 2013 by SamBridge