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/sci/ - Science & Math

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2030108 No.2030108 [Reply] [Original]

Math question:

Please explain imaginary numbers. If these numbers don't exist in a sense that means indescribable by reality, then what are they? I know that i technically is equal to square root of -1. But I don't understand. Why do two things that don't exist, when multiplied, result in a negative number?

If it's imaginary, how can you make use of it,since it's beyond reality?

Don't show. Please explain on a conceptual level.

>> No.2030118

Numbers don't exist

>> No.2030120

They exist every bit as much as normal numbers. Have you ever seen "2" in reality? No. Numbers are mathematical abstractions, all of them. Just because when we observe quantities we get a certain subset of all possible numbers (i.e reals instead of complexes) does not make one set less real than the other.

>> No.2030147

>>2030120
Same deal as negative numbers, fractions, and even zeros. None of these actually exist: show me -1 apples, or .5 apples, or zero apples. Just abstractions that don't really exist, and which previously were considered impossible, or as having no bearing on reality. These days i is used in many areas of important calculations, especially (if memory serves, don't quote me here) engineering.

>> No.2030157

what about 2 apples?

>> No.2030161

x = a +bi, y = c + di
let the x = (a,b) and y = (c,d)
the definition of imaginary numbers means that
xy = (ac -- bd, ac + bd )
If you use that as the definition then it implies that i is the root of negative one but the definition stands.

>> No.2030164

imaginary numbers, have imaginary rules.

true story

>> No.2030198

Is math imaginary? if so, then isn't any solution possible?

>> No.2030205

It's mostly for technical reasons. Say we find an equation that is X^2 = -36 (assume we are dealing with a decelerating object that is now moving backwards). . i acts to keep track of the fact that you are using a mathmatically impossible value.

>> No.2030217

>>2030205
But if your resorting to imaginary numbers, no matter the perceivable accuracy,something is fundamentally wrong.

>> No.2030228

Imaginary numbers are useful when doing math on systems which oscillate, like springs and capacitors. One powerful concept based on imaginary numbers is the Phasor:
http://en.wikipedia.org/wiki/Phasor

>> No.2030229

>>2030108

The important thing to remember is this is not 1000 years ago, when the primary function of math was to describe reality and solve real world problems.

These days, math is a very, very abstract thing. Essentially, these days math is a great big game of "let's pretend."

As mathematicians we talk about real numbers. As mathematicians we talk about the square root function. And we say "Oh, we can compute the square root of a positive number, but we can't do this for a negative number..."

WHAT IF, we just made up a way to define the square root of -1? LET'S PRETEND the square root of -1 exists, and let's see what things we can conclude.

As mathematicians, we could easily take as an axiom "The square root of a negative number doesn't exist." Math would still work, math would still make sense. But we don't, because as mathematicians we like to play pretend.

Also, don't get hung up on the term "imaginary" its just a term.

>> No.2030236

Lets just all sit back, and pretend math exists.

>> No.2030239

http://en.wikipedia.org/wiki/Philosophy_of_math

>> No.2030245

Imaginary numbers do exist. They are the numbers that when multiplied together result in negative numbers. Maybe the name is confusing you.

>> No.2030246

>>2030229
>Essentially, these days math is a great big game of "let's pretend."
WRONG

>> No.2030256

I can't believe there are people who think numbers don't exist. If something has a property, and that property is innate and immutable, then that thing is far more real that you are, who are just a temporary blip in existence. Numbers are some of the only truly existing things that we can experience.

>> No.2030265

>>2030217
if 3 x = 4, what is x?
4/3.
>But if your resorting to fractions, no matter the perceivable accuracy,something is fundamentally wrong.

>> No.2030270

Math is something that shouldn't be philosophical.
Math is real, and is self proving.

>> No.2030271

The most interesting property of imaginary numbers is not that you can square them to get a negative number, people just get hung up on that because that's the definition.

Imaginary numbers are primarily useful in trigonometric systems, because of Euler's formula:<div class="math">e^{ix} = \cos x + i\sin x</div>

>> No.2030277

>>2030147
>Same deal as negative numbers, fractions, and even zeros. None of these actually exist: show me -1 apples, or .5 apples, or zero apples.
All you've done there is prove that apples don't really exist.

>> No.2030278

>>2030108
imaginary numbers are real.

don't know why their called imaginary. That's why the more correct term is 'complex numbers'

>> No.2030288

Apple is something humans apply to a certain perceived object. Kind of like numbers.
Therefore:
apples = numbers = doesn't exist.

>> No.2030291

Imaginary numbers are useful in mathematics because some concepts in physical systems are against human intuition. Notice how imaginary numbers are used to describe systems on the quantum level?

Consider humans, right now, re-derived all of mathematics with a more absolute knowledge of, say, quantum mechanics. I would dare say that a large amount of math would be intuitively different.

Think about it another way. Say you are on a team who is trying to program a complex application. There is no way that your design team can create a perfect document to describe how the system will be programmed. You and your team start working on programming the more intuitive parts of the program and everything seems to be working well. Then as the system gets more complex your team finds that it is taking more "hacking" of code together to get everything to work together. By the end of the development your program, that looked elegant in the beginning, is now a mess....but it still works.

Imaginary numbers are a result of humans not having a perfect understanding of physical systems of the Universe. So you have to "hack" mathematical ideas together to get some math to work with other math. It doesn't seem elegant but, it ends up working.

>> No.2030314

>>2030291
Furthermore, it is completely possible for humans to re-derive and entire mathematical paradigm in which imaginary numbers would never appear. However, fundamental changes in the way current mathematics is presented would be needed and it simply isn't practical. Who says we even need to use negative numbers to represent a concept in mathematics? You don't really need them. However, that isn't how our interpretation of mathematics evolved and we build other ideas on top of this interpretation now.

If you took a being who had a perfect understanding of how everything in the Universe interacted but, they were unable to speak or communication with you and they were tasked with presenting their perfect interpretation of the Universe to you, they would undoubtedly present you with mathematics that would look very different. Many ideas would be the same but, it would look very different than what we humans call mathematics indeed.

>> No.2030333

I was told by my math teacher years ago that imaginary numbers are used in engineering and they explain movement of things, (like 3 pendulums swinging off of each other) or something like that, is this correct, am I just wrong, or am I getting things mixed up here

>> No.2030346

>>2030291
>>2030314
I'm not OP but I really like this explanation.

>> No.2030352

How are imaginary numbers related to fractals?

>> No.2030360

>>2030277

4chan once again widens my mind.

>> No.2030428

>>2030291
>>2030314

Let me expand(repeat) on this idea.

So math is like a language of humans. For example English is a tool we use to communicate in almost every way. It was a tool that we created to explain and describe the universe that surrounds us. English is not perfect. While enough to get us by just fine, it has its weak points: (autoantonyms:THINK (believe a truth vs. be uncertain about a fact)) and a another language may be better in perhaps describing emotions etc.. Is English
the language in which the universe operates? No.

So Math is a tool created by humans to help us navigate the physical realm. It's just our language. It has it's weak points(imaginary numbers not really imaginary). It's not the actual language in which the universe operates.

>> No.2030501
File: 101 KB, 400x347, 0.5apple.jpg [View same] [iqdb] [saucenao] [google]
2030501

>>2030147
>show me 0.5 apples
pic related

>> No.2030518

>>2030501

0.50134427

>> No.2030540

>>2030518
I have defined 0.5 to be exactly equal in proportion to 1 as this half apple is in proportion to its pre-cut self

>> No.2030548

>>2030501
The problem is that your pic isn't .5 apple, it is 1 half apple. Fractions play tricks with the mind by luring us into a false sense of ease.

>> No.2030549

>>2030501
Yeah, I was waiting for that one.

>>2030352
You know the mandelbrot set? It comes from just simple operations on the complex numbers. Each point represents a complex number, and what happens when you take it and do (((c^2 + c)^2 + c)^2 + c....

>> No.2030559

>>2030501
The problem is that your pic isn't 1 half apple, it is 0.50134427 apple.

>> No.2030612

The only numbers that really exist are natural numbers ,0 , 1, 2, 3, 4, 5, 6, ect.

The other types of numbers come from the mathematical operations. For example, negatives come from subtraction, the inverse of addition.
now we have ...-3, -2, -1, 0, 1, 2, 3......
These are your integers

Now we want irrational numbers. These come from division, the inverse operation of multiplication.
so any integer divided by any other integer yields all rational numbers.

irrational numbers are numbers that cannot be put into the form a/b where a and b are integers. Like square root 2, or pi. These arise from inverse operations as well.

You have no trouble understanding these numbers because they are familiar. The pythagoreans had trouble with irrationals like square root 2. So they pretty much banned these numbers.

Now, complex numbers come from the root operator, or fractional exponents: root 2 equals 2 to the 1/2 power. Complex numbers arise when we try to take the root of a negative. We don't know how to deal with them, so we separate them out into the real and the complex part.

In the end complex numbers, arise just like other types of numbers from mathematical logic. And that is just what it is. Complex numbers are useful because they have triginometirc tendancies and greatly simplify
sine, cos, tan equations. Esp. useful in quatum mechanics with wave equations.

>> No.2031232

bump

>> No.2031255

>>2030118
Thread ended here, why are you people still talking?

>> No.2031272

I'd believe in imaginary numbers before non-computable reals.

>> No.2031286

>>2031255
Because it's fucking wrong.

>>2030612
>The only numbers that really exist are natural numbers ,0 , 1, 2, 3, 4, 5, 6, ect.
This is full retard. If the natural numbers really exist, the the mathematical operations really exist. If the operations really exists, then so do the numbers they yield.

I can understand the version of retardation that says no numbers really exist. That's just normal retardation that most people possess. But saying that some numbers exist and others don't... that's extra-special retardation.

>> No.2031297

>>2031286
> everyone should be a realist with respect to mathematics
cool religion bro got any pamphlets?

>> No.2031305

>>2030271

Gotta love it when people are amazed at euler's identity, when it is RIGHT FUCKING THERE if you look at the definition of e^ix

>> No.2031310

>>2031305
quite possibly the dumbest thing I've ever read on /sci/

9/10 I raged a little in my mouth

>> No.2031321

>>2031310

You're just angry because I said your "Beautiful equation" is no more beautiful than 1+1=2

>> No.2031350

>>2031286
I don't understand. How exactly do numbers exist?

>> No.2031358

>>2031321
Yeah... no.

> Derivation
> The identity is a special case of Euler's formula from complex analysis
Many people know this you massive faggot.

>> No.2031374

>>2031297
Yes, but they are immaterial.

>> No.2031380

>>2031374
God is real. He's just immaterial, so you can't see him.

>> No.2031392

>>2031350
They exist because they have properties. They have properties that we can't change, no matter how much we may want to. Moreover, those properties are eternal... they can never have changed, and they never can change. The number 12 will always be composite, while the number 11 will always be prime.

At the moment of the big bang, and "before" or outside of spacetime altogether, 12 was composite and 11 was prime. Provably so. You and I were nothing during the big bang, and a little latter we'll be nothing again, but 11 will still be prime and 12 will still be composite. Guaranteed. They therefore have more reality than either of us.

>> No.2031441

>>2031392
to go even a step further, not only is this true in our universe, but in any universe that could possibily exist. Mathematics is completely independent of the laws of physics, and everything else. In any reality that could be a reality, our mathematical rules would be 100% valid.

>> No.2031499

>>2031392
>>2031441
Oh, I see. When I think of "real", I think of physical, material things. Thanks for the explanation.

>> No.2031508

>>2031392
My eyes glazeth over and I read something like
> unicorns exist because we can think about them
and I have to say that this conversation has suddenly got very, very dull.

>> No.2031520

>>2031508
mfw my whole fucking point was that numbers exist independent of whether or not we think about them.

>> No.2031534

>>2031520
like I said, cool religion

>> No.2031548

Imaginary numbers are by no means "imaginary." It's just a name. The term complex is a better descriptor IMO.
>Why do two things that don't exist, when multiplied, result in a negative number?
That's the point, they exist to fill in that gap in mathematics, and widen the scope of arithmetic and algebra in amazing ways. In no way do they "not exist."
Take the greeks for example: they found cases in trigonometry where they were forced to take the square root of non-square numbers, and they absolutely hated it. They couldn't imagine how an irrational value could possibly "exist." They only thought in fractions and ratios. We now see clearly how, even though you can't show me sqrt(5) or pi apples, these values are very real and very important to understanding nature.
The same thing applies to "imaginary" numbers. They are definitely not imaginary, they just expanded so suddenly past what we considered "real" numbers. Like sqrt(5), they fill a gap not addressed by other numbers.
The complex number system is very beautiful; it completely revolutionized how we perceived that numbers fit together. It's interesting to note that complex numbers are closed under addition, multiplication, exponentiation, etc. There's no standard operation you can perform on a complex number that will have an answer outside the complex number plane (nope, not even sqrt(i), i^i, log(i); they all have solutions that are part real and part imaginary). They complete the reals in an elegant way.

>> No.2031685

http://www.vortexmath.com/

Numbers are real morons

>> No.2031728

its important for electrical engineering/transmission lines/waves and stuff like that.

basically describes how capacitors/inductors behave in an AC circuit compared with a resistor

>> No.2031776

>>2031392
11 ain't prime if you are counting with an octal number system
11/3=3
if you assume base 8

>> No.2031786

>>2031441
Also not true.
In our universe, under perverse conditions, circles have more than 360 degrees (which monkeys with sine, cosine and such and triangles could have more/less than 180 degrees.
These effects were hinted at with relativity, but have been described more clearly since then.

Of course, we can't get near the speed of light or close to a black hole event horizon, so it's academic.

>> No.2031794

>>2031786
The surface of the earth is a "perverse condition"?

>> No.2031796

>>2031776
But the value <span class="math">11_{10}[/spoiler] exists and is prime.
<span class="math">11_8[/spoiler] is just another way of expressing <span class="math">9_{10}[/spoiler]. The way we physically represent a value has no bearing on the actuality of that value.
In the eyes of math, <span class="math">11_{10}[/spoiler] and <span class="math">11_8[/spoiler] have no relation whatsoever. We only see the correlation because we use the same symbols to represent different concepts.

>> No.2031802 [DELETED] 

>>2031776
If it was confusing to you, I meant <span class="math">11_10[/spoiler] which refers to the number eleven.

>> No.2031805

>>2031776
If it was confusing to you, I meant <span class="math">11_{10}[/spoiler] which refers to the number eleven.

>> No.2031808

>>2031796
True, but something that did need pointing out.

>> No.2031809

>>2031796
The representation of values by means of a formal axiomatic system do indeed have bearing on the "actuality" of the value, though apart from some mystery school religions and strawman characterizations of idealism I've not really heard too much about the hypostasis of numbers.

>> No.2031811

>>2031786
You're assuming that that somehow makes math invalid.
The concept behind different geometries is part of an even larger expansion governed by math.
There could exist a universe where pi is 4, and e is sqrt(5). This universe would still have math, but it's geometry would be so incredibly twisted it's not even funny.
Sine and cosine are defined on specific assumptions that are not necessarily true, but part of a broader concept that simplifies greatly in our reality.

>> No.2031820

>>2031534
Whatever you feel moved to call it, it does have the advantage of being provably true in any sound logical system, which I suspect is more than could be said of your religion.

>> No.2031822

>>2031820
> it does have the advantage of being provably true in any sound logical system
cool circular logic bro

>> No.2031830

Would probably be best for you to pick up classes that study complex numbers instead of asking here.

>> No.2031838

>>2031786
No. You don't understand what you are talking about. Every circle has 360 degrees, and pi=3.141592... in every universe.

You are apparently confused because if you, for example, follow locally straight paths over curved surfaces, which is something that you can apply to physical topologies, you can get what at each local point appears to be a triangle, but whose angles do not add to 180 degrees. This is of course because it is not a triangle at all, which is a 2D shape in a plane; rather it is a curved 3D shape following a contour. And yes, curved 3D shapes on contours follow the same mathematics in every universe according to the curvature of the contour.

>> No.2031845

>>2031811
Actually, now that I'm thinking about this... Pi and e are fundamentally pieces of the same... "thing." What does messing with their values do to math? They're different than physical constants, which are measured within this universe. They can be derived without making any measurements of the physical universe.
Also, a universe could feasibly exist with no rules at all. Would this universe be devoid of math? Things happening spontaneously without happening, no such thing as a "number" -- oh God my brain.
Who are we to say what can exist and what can't?

>> No.2031869

>>2031845
Right, nothing can change pi or e which are derived from plane and solid geometry, which doesn't exist in nature per se, yet as pythagoras said, nature imitates it. Since nature imitates it, math is useful in science. In a universe which did not imitate number, math would be have no use in science and would likely remain uninvestigated by its inhabitants.

>> No.2031881

>>2031869
A universe without humans wouldn't have plane geometry.

>> No.2031892

>>2031881
Plane geometry doesn't exist within universes. It is a set of fundamental realities of logic.

>> No.2031903

>>2031892
oh god not transcendentalism
> runs screaming from thread

>> No.2031908

>>2031869
>>2031881
I can see how math as a concept could exist outside of the universe, but those universes where physicality did not mimic math would never derive it. The concept of number could be literally inconceivable in that realm, but math would still in a sense exist.
It's also possible, then, that other "abstractions" could exist separate from math that we will never be able to conceive of, but still exist in the same sense as math.
Other very different realities that we couldn't even understand would find this other self-contained system, but we simply cannot.
Dude. And I'm not even high.

>> No.2031912

>>2031908
Math is a set of conventions.

>> No.2031919

It makes me want to explore an abstraction of pi=4 and re-derive the rest of math on that assumption.
I'm too tired, I can't even

>> No.2031929

>>2031919
pi is not the basis of our mathematical system so it doesn't seem you'd get particularly far

>> No.2031934

They have been labeled "imaginary" numbers because at the time they were introduced, they were no way to represent them, thus you had to "imagine" they were here. Now you can represent imaginary numbers as those on the vertical axis defining a plane. But it's a choice. One could chose another representation.

Negative numbers, as it has been stated, do not "exist" neither, but you can represent them as those before a chosen 0 on a straight line.

Same can be said for several mathematical concepts. Can you pick up polynoms in your house and make sauce out of it ? Do you see matrixes in your life ? Can you really see a point ? Did you ever see Australia ? You can see representation of those, but not the actual objects.

>> No.2031935

>>2030157
Thats "two apples", not "two".

"two" is the property that all pairs have in common.

>> No.2031950

>>2031935
The existence of this property is an artifact of language not some underlying metaphysic. Please don't mistake characteristic phrases with ontological status. For example,

"An apple is a physical object." This is not a statement, it is a piece of instruction---an instruction for someone who either doesn't know what apples are, or don't know what physical objects are. If it were a statement, it would be contingent, and it would require in its place some other characteristic example which would be equally as "obvious". Regression problems are not limited to deities.

>> No.2031971

>>2031950
>The existence of this property is an artifact of language not some underlying metaphysic.

"Two" exists whether or not anything else does. Its a consequence of accepting a certain set of axioms and the rules of logic. We choose these axioms and the rules of logic because they are useful and correspond to things in the real world - if you put pebbles in a bucket when sheep leave through a gate, and take pebbles out of a bucket when sheep enter through a gate, you have two more pebbles when two more sheep have left than have entered. This is why we have chosen to create a system of mathematics that, while they exist independently of reality, usefully corresponds to observable phenomena.

That said, perhaps a better definition is as follows:

"Two" is what you get when you count the number of objects in any pair.

>> No.2031974

>>2030108

Am I the only one who sees an outline of a pair of tits in OP's pic?

>> No.2031979

>>2031971
> Its a consequence of accepting a certain set of axioms and the rules of logic.
I agree. It is a consequence of agreeing to a convention. Only if no one is there to accept it, then it isn't there.

>> No.2031983

>>2031929
Hmmm, I was thinking along:
C = 2*pi*r... you either rewrite the rules of multiplication, leading somewhere I can't think about right now, or leave the rules the same and obtain new geometry?
A = pi*r^2
2*r must still be d, maybe...
New geometry = new calculus = new algebra...
I'm sure inconsistencies arise by keeping multiplication the same.
I'm sleeping now.

>> No.2031985

the imaginary unit is just an excuse for expanding the number line to a number plane. I've been wondering though, is there anything stopping us from defining a number space? Would it have any practical uses at all?

>> No.2031987

>>2031971
> if you put pebbles in a bucket when sheep leave through a gate, and take pebbles out of a bucket when sheep enter through a gate, you have two more pebbles when two more sheep have left than have entered
What you describe is how we teach people to use the word two.

>> No.2032002

>>2031985
http://en.wikipedia.org/wiki/Quaternion
I know of them, but not much about them.
I think they're 4-d, not 3-d though.

>> No.2032007

My understanding of math is that it is like language in that we come up with words to describe reality, we don't uncover words.

The universe does not conform to math, math conforms to the universe.

>> No.2032008

>>2031979
>Only if no one is there to accept it, then it isn't there.

We're having a definitional dispute here, along the lines of "if a tree falls in a forest, does it make a sound?"

When exactly is it the case that nobody is around to accept the premises of a mathematical system?

What exactly is the consequence of mathematical systems not existing in the hypothetical situation where nobody is around to accept the premises?

>> No.2032021

>>2031987
You can also construct the sheep-pebbles relation without mentioning numbers, though. Consider the following formal system:

the alphabet is - and p

a formula is well-formed if it is a series of '-', followed by one and only one 'p', followed by another series of '-'

'p' is an axiom

x is true iff the formula '-x-' is true

I have no idea what exactly I'm trying to say here, but anyhow - you don't need to invoke two to get the pebbles, buckets, and sheep.

>> No.2032429

>>2032008
> When exactly is it the case that nobody is around to accept the premises of a mathematical system?
When we're in a thread talking about how math is the same in all possible universes.

> What exactly is the consequence of mathematical systems not existing in the hypothetical situation where nobody is around to accept the premises?
Presumably people would stop making existence claims about numbers.

>> No.2032442

>>2032007
I agree with this.

>> No.2032536

>>2031979
The axioms only fix our mathematical language. The reality exists apart from the language. If you want to cal the language "math", that would be a different definition, which requires people and axioms. I'm calling "math" is the fundamental unchangeable realities described and explored through those axioms, symbols, and concepts.

Saying that math would exist without people is just like saying the moon wouldn't exist without people. Yes, you are right, if by "moon" you mean the word "moon" and the definition of "moon", or even the visual impression that the moon makes when you look at it carefully. But most people would agree that the moon existed before people because "moon" means the physical reality behind those human definitions and perceptions.

Likewise math existed before humans and before the universe, not the symbols or the axioms, but the fundamental reality which we scrutinize using those symbols and axioms.

>> No.2032543

>>2032007
It's the distinction between "numeral" and "number". "Numeral" is the language. "Number" is the reality.

>> No.2032545

>>2032536
> I'm calling "math" is the fundamental unchangeable realities described and explored through those axioms, symbols, and concepts.
They're unchangeable because we've defined them to be that way. It is our insistence on a language of certainty that gives math its air of authority, nothing more.

>> No.2032553

>>2032002
Quaternions were a profound insight and still underutilized today, although today they have their first mainstream application in computer graphics.

But our entire physics could be restated more simply in terms of quaternions:
http://world.std.com/~sweetser/quaternions/qindex/qindex.html

>> No.2032559

multiplying by -1 represents a rotation of 180 degrees (takes a number and gives you it's negative, which is a rotation of 180 about the origin, 0, of an infinite number line). Since -1 represents a rotation of 180, then the root of -1, if it existed, would represent a rotation of 90 degrees. Since this point is an image of the number one that does not exist on the line, it is called an imaginary number and given a name, i. But, even though it is only a concept, it was defined by rules within a consistent system, and so can be useful. So the root of -1 represents a phase shift of 90 degrees into the imaginary. Whether this is real or not is inconsequential since it can be used to convey information nonetheless.

>> No.2032569

>>2032545
Not at all, we can define things however we like, but it doesn't change the underlying mathematical reality. There is nothing about our definitions that force numbers to be composite or prime, other than the definitions of the words "composite" and "prime" tell us wtf we're talking about when we use those words. Just like our definitions of blue and red don't change what color something is. The color is a property of a physical object that doesn't care about our definitions. Primality is a numerical property that doesn't care about our definitions.

If I have eleven marbles, there are only two ways they can be arranged into equal groups, namely one group of eleven or eleven groups of one.

If I have twelve marbles, there are six ways they can be arranged into equal groups. Nothing about our definitions, or whether or not we even have definitions changes the properties of that quantity of marbles.

This does not arise from any physical property of marbles, but from the fundamental properties of numbers, apart from our definitions or understandings of numbers.

>> No.2032575

>>2032569
> There is nothing about our definitions that force numbers to be composite or prime, other than the definitions of the words "composite" and "prime" tell us wtf we're talking about when we use those words.
In fact, it is our definitions and axioms which create those very properties.

>> No.2032628

>>2032575
That's like saying our definition of the moon gives it its craters.

>> No.2032641

>>2032628
I am not sure I see an analytical relationship here.

>> No.2032644

Imaginary numbers are junctions with other functional number systems.

>> No.2032650

>>2032641
Our definitions of properties are not what give things their properties. As I pointed out in an example, none of our definitions contribute to the actual primality of a number. You can say that our definition of "primality" determines the primality, but that is like saying our definition of "craters" determine whether or not their are craters on the moon -- which isn't true. The definition is just what allows us to talk about it.

>> No.2032657

>>2032429
>how math is the same in all possible universes.

What do you mean by "possible universes"?

Do you mean "universes that we can imagine?" Do you mean "universes that we can describe?" Are there possible universes that we can neither imagine nor describe?

>Presumably people would stop making existence claims about numbers.

If there are no people to make existence claims about numbers, then nobody would be making existence claims about numbers. This is completely covered by the fact that there are no people in the hypothetical universe, so claiming this as a consequence of math not working in a universe with no minds adds no extra detail (that is to say that your statement is true of every universe with no minds, not just those with no minds and nonworking mathematics)

>> No.2032672

>>2032650
The difference is that primality critically depends on our axioms and definitions. In arithmetic systems which do not have multiplication, for example, there is no way to express primality. It's not that primality is still "really there" in the background (or wherever you suppose mathematical objects "exist"), waiting to be uncovered.

>> No.2032678

Math is just one other metanarrative construction, not even math is an untouched, metaphysical subject.

>> No.2032679

>>2032569
>The color is a property of a physical object that doesn't care about our definitions

Color is a subjective distinction between groups that activate different sets of neural responses in the eye. Color is in the mind, and as such, it does care about our definitions.

There's only one level of reality, and that is on the level of fundamental particle fields and microscopic forces. Everything above that is a map of varying levels of simplification, not the reality itself.

>There is nothing about our definitions that force numbers to be composite or prime, other than the definitions of the words "composite" and "prime" tell us wtf we're talking about when we use those words.

Except that our definitions of the words "number", "composite", and "prime" include all the information that someone needs in order to determine whether a given number is composite or prime. If you define "number", "composite", and "prime", you gain no new information if you figure out whether 3 is composite or prime.

>> No.2032687

>>2032678
Do you mean "not reducible to quarks, particle fields, and elementary particles" when you say "metaphysical"?

>> No.2032690

>>2032687
errr, elementary physics, not elementary particles

>> No.2032694

>>2032672
>The difference is that primality critically depends on our axioms and definitions.

Primality is completely independent of our axioms or definitions or whether we even have axioms or definitions, as I showed here: >>2032569

>In arithmetic systems which do not have multiplication, for example, there is no way to express primality.

Exactly, we'd have no way to EXPRESS it. But there would still be exactly two ways in which eleven marbles could be divided into equal groups, because of the nature of the number eleven. We don't need to even have a word for the number eleven for the number to have that property.

>It's not that primality is still "really there" in the background (or wherever you suppose mathematical objects "exist"), waiting to be uncovered.

It is EXACTLY that primality is really there, in the numbers, waiting to be discovered, and would be discovered by anyone who developed any consistent set of axioms and definitions. The possibilities of how you can group eleven marbles or twelve marbles are fundamental to those numbers.

>> No.2032705

>>2032687
I'm saying that it doesn't exist outside the operator of math. Just like ethics, math is not absolute, untouched, eternal or etheric even if we like to think so at times, nor does it exist outside of its operators (math as we know it does not exist apart from those who use it). Imaginary numbers exist because we have constructed math as such, and between every generation interpretation of the construction we are given changes. I can write 1+1=3 and it's just as valid but ofcourse nobody would take me seriously, still it is just as valid as a statement because there is no necessary connection between the symbol and the amount of something outside of us who use math, the operators.

>> No.2032708

>>2032694
> Exactly, we'd have no way to EXPRESS it. But there would still be exactly two ways in which eleven marbles could be divided into equal groups, because of the nature of the number eleven.
Again you seem to put the cart before the horse. We explain the operations on items like marbles. You are giving a characteristic example used to explain numbers. It isn't the other way, that these are the way they are because of numbers, which we discover in some deep way. There's nothing deep there (at least, not philosophically).

> waiting to be discovered, and would be discovered by anyone who developed any consistent set of axioms and definitions.
It is amazing you cannot see the circularity I am trying very hard to point out to you. You wish to restrict consideration to consistent axioms and definitions only, which then (to your apparent surprise, but not mine), *reveal* the underlying mathematical object or property. But how, indeed, are you to find such a set of axioms and definitions without enormously begging the question?

It's as if you wish to use the dictionary to prove something about reality. The dictionary has no attachment to reality. It's a completely circular construct, a self-contained world. It is only through *pointing* can we associate the word 'crater' and 'moon' to the object in the heavens.

But what you do when you repeatedly manipulate marbles or apples, or make claims about cows and portioning, and *ask* the individual to figure out the underlying connection for themselves, is not pointing to a mathematical object, but is instruction in the use of a word that has no real referent. (There are many of these. Avowals, the word 'pain', color, and such. Most "properties" are this way.)

>> No.2032710

>>2032679
>Color is a subjective distinction between groups that activate different sets of neural responses in the eye. Color is in the mind, and as such, it does care about our definitions.
Not at all. We perceive the blue light reflecting off an object in a certain way that has nothing to do with our definitions. The sky will look the same color, no matter what definitions you change. Oxygen atoms will refract light in the same frequency range, no matter what definitions you change.

>There's only one level of reality, and that is on the level of fundamental particle fields and microscopic forces.
Fields and forces are merely models that are constructed to predict observation. The properties of what we model with fields and forces perhaps have an underlying reality. But the properties of numbers definitely have an underlying reality.

>Except that our definitions of the words "number", "composite", and "prime" include all the information that someone needs in order to determine whether a given number is composite or prime. If you define "number", "composite", and "prime", you gain no new information if you figure out whether 3 is composite or prime.

So since there are an infinite number of primes and an infinite number of composites, it is your belief that the very definition of prime, composite and number contains an infinite quantity of information?

>> No.2032713
File: 155 KB, 540x1400, smbc_irrational.gif [View same] [iqdb] [saucenao] [google]
2032713

itt: this thread

>> No.2032720

>>2032694
>Primality is completely independent of our axioms or definitions or whether we even have axioms or definitions

No its not. You can define equality to be the relation "all numbers are equal to all other numbers", and its still an equivalence relation (that is, reflexive, symmetric, and transitive). Granted, the mathematics you get from this definition isn't nearly as useful as, say, ZFC with the definitions from standard mathematics, but its still valid, and you can divide 11 marbles into two equal groups of 3 marbles and 8 marbles.

Something that you seem very wedded to is the idea that two groups of objects have the same number iff you can make a one-to-one relationship between the objects of one group and the objects of another group. This idea is the result of axioms and mathematical definitions. You have to discard this if you wish to discard all axioms and mathematical definitions; if you don't have math, you cannot claim that a one-to-one relationship implies equality.

>> No.2032727

>>2032708
>We explain the operations on items like marbles. You are giving a characteristic example used to explain numbers. It isn't the other way, that these are the way they are because of numbers, which we discover in some deep way.
I'm using a characteristic example of marbles so the underlying truth can be understood. It is not a property of marbles, how eleven or twelve numbers can be grouped. Is that your belief? It is rather a property of eleven and twelve. Eleven and twelve have those properties apart from being used to count anything, which is why no matter what you count with those numbers, when you try to group those things, the possible ways you can group them is completely dependent on the property of the number. Not the other way around.

>There's nothing deep there (at least, not philosophically).
Deep is subjective. I'm making no claims about depth. I'm making claims about the fundamental nature of number, which are objective.

>You wish to restrict consideration to consistent axioms and definitions only, which then (to your apparent surprise, but not mine), *reveal* the underlying mathematical object or property. But how, indeed, are you to find such a set of axioms and definitions without enormously begging the question?
It's not to my surprise at all. It is obvious. And I am not talking at all about what is dependent on axioms or definitions. That's is what you are trying to claim, but it is false. I'm talking about what is independent of axioms and definitions. The properties of the number eleven are independent of axioms or definitions, despite the fact that you can't refer to the number without definitions.

cont..

>> No.2032730

...cont
>It's as if you wish to use the dictionary to prove something about reality.
That is the opposite of what I'm saying. That is in fact what you are claiming, since you claim that the properties of numbers arise out of our definitions.

>The dictionary has no attachment to reality.
Precisely.

>It is only through *pointing* can we associate the word 'crater' and 'moon' to the object in the heavens.
But the object is there whether we point or not, or whether we have words for it or not.

>But what you do when you repeatedly manipulate marbles or apples, or make claims about cows and portioning, and *ask* the individual to figure out the underlying connection for themselves, is not pointing to a mathematical object, but is instruction in the use of a word that has no real referent. (There are many of these. Avowals, the word 'pain', color, and such. Most "properties" are this way.)
So it is likewise your view that the moon has no color and no craters? Or it had none before life existed? That there were no photons that bounced off the moon before someone defined what a photon was?

>> No.2032743
File: 17 KB, 203x203, 1285177390158.jpg [View same] [iqdb] [saucenao] [google]
2032743

>>2032713

What comic is this? I had it bookmarked a long time ago.

>> No.2032749

>>2032720
>if you don't have math, you cannot claim that a one-to-one relationship implies equality.

All you're doing is showing that I can't explain what primality is without words definitions, which is obviously true. If all words mean "blue", and also mean "prime", and "number", then all "numbers" are "prime". But that's just playing with words, rather than using words to get to meanings. If someone had no words, but they had eleven marbles, and they wanted to arrange them in even groups on a balanced disk, so that the disk did not tip over, the properties of the number eleven would determine how they could and could not do it.

Since I have definitions, I can talk about it while they could not, but the underlying reality is the same for both of us.

>> No.2032756

>>2032730
> That is in fact what you are claiming, since you claim that the properties of numbers arise out of our definitions.
It would be, if I were suggesting that mathematical objects have an existence outside these constructs, but my very claim is that they do not.
> But the object is there whether we point or not, or whether we have words for it or not.
Some are. Moons and craters, cows and apples. Others aren't, like characters in novels. There's nothing behind the words of "Pet Semetary". There's nothing behind mathematical formalism.
> So it is likewise your view that the moon has no color and no craters? Or it had none before life existed?
We start with mind, sensation, phenomena. We investigate the world around us. This is the order of things. It seems you wish it to be the other way, for reasons I cannot fathom (but in which you are not alone). No reasonable man could agree that the moon only came into existence when we looked at it, or named it. But names do not have any relationship to existence, except in the mind. Showing that something exists outside of the mind is largely impossible, for reasons that should be clear to such a logical thinker. But our investigations have suggested this is so, and I am not prepared to disagree with those investigations. In the course of those investigations, we've created constructs for categorization. Math. Measurement. Science. These constructs do not exist, but in our minds.

>> No.2032759

>>2032710

>So since there are an infinite number of primes and an infinite number of composites, it is your belief that the very definition of prime, composite and number contains an infinite quantity of information?

Nope, since information does not add that way. If I flip a coin, do I gain more information about which face landed up when I look at the face that landed up and the face that landed down, or just the face that landed up?

Information lets us make better predictions about the state of the universe, and is measured in bits. A bit is that which is enough to change our belief about the likelihood of an event by a factor of two. An illustrative example: suppose we have a 1/2^20 chance of a lottery ticket being the winning ticket, and a box that beeps for a quarter of the losing lottery tickets and all the winning lottery tickets. If we feed all the lottery tickets through the box, we have gained two bits of information - since now we have a total of about 2^18 lottery tickets that could possibly be winners. If we run all those possible winners through the same box again, we gain no information, since there's no change in the total number of lottery tickets.

Some information is redundant, which means that if we know that information, it doesn't tell us anything that we didn't know before. So if we know that 3 is prime, does it tell us anything about whether or not 287,637 is prime? Does the primeness of 3 tell us anything about the primeness of 5? The answer is no, so "3 is prime" tells us no information that isn't redundant. Thus, the total information in the following two statements is equivalent:

The definitions of "prime", "composite", and "number"

The definitions of "prime", "composite", and "number", and the fact that "3 is prime" is true.

>> No.2032774

>>2032743
saturday morning breakfast cereal
smbc-comics.com

>> No.2032786

>>2032759
You seem like the sort of fellow that could appreciate this. Chaitin, of Chaitin's Constant, has a rhetorical "proof" that math must be incomplete. I found it so totally intuitive to understand that it had to be true, even if I didn't already know it would be true.

Roughly, he said, if mathematics as we commonly know it could be complete, then the axioms and definitions would represent an unbelievable form of compression. So there must be "incompressible" mathematical truths.

Kind of a cool rhetorical point, IMO.

>> No.2032802

>>2032710
>Not at all. We perceive the blue light reflecting off an object in a certain way that has nothing to do with our definitions. The sky will look the same color, no matter what definitions you change. Oxygen atoms will refract light in the same frequency range, no matter what definitions you change.

Deconstruction, hooooo:

>Blue light reflecting off an object in a certain way generates a pattern of neural activity that has nothing to do with our definitions. The sky will generate the same pattern of neural activities, no matter what definitions we change. Oxygen atoms will refract the same set of photons, no matter what definitions we choose

You're talking about the territory-level facts that correspond to the map-level distinctions between colors. I'm sorry if my previous use of language is unclear (blame English), but I was talking about map-level concept of color, not the territory-level facts of what photons of different color do. Basically, its the difference between how we experience color, and how we categorize things by color. If you show a blue piece of paper and a green piece of paper to a non-colorblind person, they will experience seeing different pieces of paper. If you ask a person whether a blue piece of paper is the same color as a green piece of paper, you'll get different results based on the person's language (some languages do not have separate names for "green" and "blue", so people who speak that language are less able to distinguish green things and blue things as having a different "color").

tl;dr my definition is the map, yours is the territory, and its about damn time we started using a language that differentiates cleanly between the two.

>> No.2032805

>>2032756
>Showing that something exists outside of the mind is largely impossible, for reasons that should be clear to such a logical thinker. But our investigations have suggested this is so, and I am not prepared to disagree with those investigations. In the course of those investigations, we've created constructs for categorization. Math. Measurement. Science. These constructs do not exist, but in our minds.

Yes, our constructs are products of our minds. If "math" is taken to mean our construct, then like "science" it exists only in our mind. But both these constructs are used to explore purported reality. The construct of science we use to explore the purported of reality of the physical, which we access through the senses. The construct of number we use to explore the purported reality of number which we access through logic and formalism.

Between the two, I suspect it is easier to prove the reality of number than physical reality, but any attempt to do so would obviously be attacked as relying on definitions, etc. I am content to claim that numerical reality is more real than physical reality, because physical reality is temporal and mutable, while numerical reality is not.

>> No.2032836

>>2032759
>Does the primeness of 3 tell us anything about the primeness of 5? The answer is no, so "3 is prime" tells us no information that isn't redundant.
I don't get it. Seems like it should be the other way around. Since the information is independent it is not redundant.

> Thus, the total information in the following two statements is equivalent:
>The definitions of "prime", "composite", and "number"
>The definitions of "prime", "composite", and "number", and the fact that "3 is prime" is true.
Is the number 2^9000-1 prime? I know the definitions of number, prime, and composite, but I don't know whether that particular number is prime or composite. If I were to figure it out, I would have more information than I have now.

>> No.2032863

>>2032802
I was talking about both at the same time, as the point remains the same for both: 1) We experience the color of the sky in the same way no matter what words we have for it, or whether or not we have words. 2) The quantum photon scattering due to oxygen molecules happens the same way, also independent of our definitions.

>> No.2032881

>>2032836

>I don't get it. Seems like it should be the other way around. Since the information is independent it is not redundant.

If I flip a coin and it comes up heads, its independent of whether or not 2^9000 - 1 is prime. Flipping a coin does not give any more information about whether or not 2^9000 - 1 is prime. The result of a coinflip is redundant to the truth of the statement "2^9000 - 1 is prime". I don't know any more about the truth of "2^9000 - 1 is prime" after flipping a coin.

I'm more than a little confused too, to be honest. I'm trying to be a better rationalist, though.

>> No.2032911

>>2032863
>I was talking about both at the same time

No, you weren't, because you didn't address what things we classify as which color.

>We experience the color of the sky in the same way no matter what words we have for it, or whether or not we have words.

This is not true of all visual experiences. Someone with one word for the category of all blue things and all green things will experience blue things and green things differently than someone with one word for the category of all blue things and another for the category of all green things. We know this is true because we can ask people about how they experience blue things and green things without using the words for green and blue, and get different answers based on the language that the specific person speaks.

>> No.2032922

>>2032805
It seems like number would be harder than physical reality. For example, to demonstrate physical reality I would kick you in the balls, but to demonstrate number, apparently I'd have to kick you in the balls several times, in several ways, for you to discover the underlying mathematical object that governs the testicle assault. Some might say I would be doing this to teach you a lesson, but you know that's not true.

>> No.2032940

>>2032805
Mathematical objects are made up of patterns of neurological activity or patterns of symbols in some sort of physical storage. Both of these exist within physical reality.

>I suspect it is easier to prove the reality of number than physical reality

You're mixing levels here, and suspecting that it is easier to prove the existence of the concept of number than it is to prove the existence of actual reality.

If you compare the concept of number and the concept of reality, reality is the easier one to prove (look, ma, there's what I perceive to be reality).

>> No.2032953

>>2032940
Why do you call physical reality "actual" reality? If you're going to claim that numerical reality only exists in the mind (which is false), why don't you claim that physical reality only exists in the mind?

>> No.2032964

>>2030108
nice pic.
it confirms I am thinking with mah dick

>> No.2032966

>>2032953
Personally, I do. Straight up confirmed idealist here, a la Hume. But that discussion is usually uninteresting to realists so I let it go.

>> No.2032980

>>2032953
>why don't you claim that physical reality only exists in the mind?
Even for my most strongly held hypotheses, there will still occasionally be an experiment that I do that will surprise me. There's still a point to labeling one of my expectations of how the world works as a "mental description of reality" and that which surprises me as "reality", and working to try to minimize the discrepancies between the two.

>If you're going to claim that numerical reality only exists in the mind (which is false)

I'm not claiming that numerical reality exists only in the mind. I'm claiming that it exists only in the brain, which is the sum total of the states of all the neurons in the brain, which is the sum total of all the fundamental particles in the neurons.

>> No.2032983

>>2032922
Either way, all I would experience were sensations in my mind. That doesn't prove the ultimate reality of anything.

The best evidence is that we seem to have physical reality in common. That is the only thing going for physical reality, and it also applies to numerical reality.

>> No.2032989

>>2032980
>I'm not claiming that numerical reality exists only in the mind. I'm claiming that it exists only in the brain
Then you're claiming that mathematical truths only exist within physical reality, when the reverse is true.

>> No.2033020

>>2032989

I had a related, but subtle thought.

Lets start with a definition for truth:

"snow is white" is true iff snow is white.
"the sky is blue" is true iff the sky is blue.
"Hitler killed millions of Jews" is true iff Hitler killed millions of Jews.
etc, etc, for all statements

Now, when you are talking about whether mathematical truth exists in the physical world or some other special non-deconstructable realm, are you talking about the left hand side or the right hand? That is, are you talking about the statement "If you accept ZFC and the standard definition of prime, then you must also accept that there are infinitely many primes"? Or are you talking about that which the previous statement compares itself to when determining whether or not the statement is true? Both of these, however, are products of the physical world - the statement is a collection of English-language symbols posted on /sci/, while it compares itself to whether rational minds accept that statement.

>> No.2033053
File: 133 KB, 550x500, 1221540360568.jpg [View same] [iqdb] [saucenao] [google]
2033053

Lol. trollin trollin

"math" is a system of logic.
When we say something is true / not true
it depends on the conditions.
We have axioms we take to be true, and all of math is derived as consequences of those axioms.
the name "imaginary" is incredibly badly chosen, but the point is none the less, that it doesnt exist in the countable world (you cant cut an apple into i slices),
but that does not mean it has more or less existance than any other number.

how can one say that e is not equally imaginary? because it lies on the Real number line?
bah

>> No.2033072

>>2033020
The truth of ZFC has nothing to do with whether or not rational minds accept the statement. Rational minds come along after the fact and explore such truths. We all acknowledge that the statements and the formalities are constructs. We use those constructs to investigate the immutable properties of number -- the same properties which dictated how physical processes worked before we existed -- and the same properties which existed from necessity before physical processes existed.

You keep implying that for number to exist apart from the physical, it must do so in some "special realm" which you therefore dismiss. Well, where is the "special realm" in which space exists?

Thinking of things as existing in realms is an artifact of our spatial sensory experience, rather than reason. That is an illusion of the senses.

>> No.2033080

>>2033053
>but the point is none the less, that it doesnt exist in the countable world (you cant cut an apple into i slices),

I like the point made earlier, that rather than proving that i does not exist, that proves that apples do not exist. O_O

>> No.2033082

>>2033072
Axiomatic systems are not true or false. Truth is not defineable. Axiomatic systems represent a means of transforming statements syntactically to other statements. If the transformations are permitted by the system, the transformation is called valid. A sequence of valid transformations from one form to another is called a proof.

Truth is a useless concept in mathematics. It's like asking whether the starting position in chess is true. It doesn't mean anything. It's just the starting position. Those are the rules.

>> No.2033465

>>2033082
Truth-ness is definable as the property that beliefs that agree with reality have, and beliefs that don't agree with reality don't have.

>>2033072

I'm not talking about the truth of systematic logic and ZFC. I'm talking about the truth of the *implication* "systematic logic and ZFC implies that there are infinitely many primes".

>Rational minds come along after the fact and explore such truths.

You still need a rational mind for the truths to be true. You cannot force a rock that believes "((A -> Z) /\ A) -> Z" to believe Z, since the rock can counter by claiming that you need to first convince the rock of "(((((A -> Z) /\ A) -> Z) -> Z) /\ ((A -> Z) /\ A)) -> Z ". The rock can counter argue recursively with that pattern forever: in order to have systematic logic, you have to have a belief holding system that actually concludes Z when it believes "(A -> Z) /\ A".

Systematic logic only makes sense in the context of rational minds that behave according to systematic logic. Such minds are necessary for mathematical truth, so mathematical truth cannot exist in some outside context that doesn't care about whether or not there are rational minds.

>> No.2033489

I like how people "mystified" be the complex numbers usually have no fucking idea what the reals are.

They are way more confusing and counterintuitive in many ways.

The complex numbers are easily obtained if you have the reals.

>> No.2033494

>>2033465
amazing how you can switch between correspondence theory of truth and deflationary theory of truth in the same thread without batting an eye

>> No.2033497

>>2033489
reals are fucking trash
integers are god tier
rationals and complex rationals are good tier
computable numbers are ok tier
reals are shit tier

>> No.2033510

>>2033494
actually I'm going to retract that upon rereading an earlier post, seems you've always stuck with correspondence

>> No.2033629

You too

>> No.2033664

>>2033465
anyway, further consideration here

> Truth-ness is definable as the property that beliefs that agree with reality have
That's fine, but math doesn't agree or disagree with reality, so truth doesn't apply, which is my only point.

>> No.2033759

There is so much retard in this thread that it astounds me.

>> No.2033764

>>2033497

Integers are real numbers. So integers are god tier and trash at the same time?

>> No.2033782

>>2033082
You're confusing truth with consistency.

>> No.2033783

>>2033764
don't be ridiculous, real numbers don't exist
> God created the integers, all else is the work of man
Kronecker said it, I believe it, that settles it

>> No.2033785

>>2033759
And you're the king retard.

>> No.2033789

>>2033782
No, I am not.

>> No.2033800

>>2033783
What a faggot. God is all numbers, including the ones we have no formalism for, which someone could probably prove is almost all of them. God is the way that number interacts with itself, which you can only see under one particular defined formalism at a time -- but God is all of it. Kronecker needs more pythagoras.

>> No.2033804

>>2033789
Yes you are.

>> No.2033835

>>2033800
Actually, he doesn't need more Pythagoras. You just need more Cauchy.

>> No.2033844

>>2033804
Lessee...
> my side: no mathematical truth
Support: tarski's undefineability theorem
> your side: you meant that there's no completeness
Support: none, there are complete systems

go away kid, the adults are talking

>> No.2033861

>>2033465
>You still need a rational mind for the truths to be true.
Prove it.

>You cannot force a rock that believes "((A -> Z) /\ A) -> Z" to believe Z,
The rocks believe this already. I at least have never heard one deny it. Regardless, do you think that that proposition is a random consequence of our brain chemistry? Rather, we believe that because we can tell it is true. To assert otherwise is absurdity.

>since the rock can counter by claiming that you need to first convince the rock of "(((((A -> Z) /\ A) -> Z) -> Z) /\ ((A -> Z) /\ A)) -> Z ".
I don't follow where you get that. ((((A -> Z) /\ A) -> Z) -> Z) is false.

>> No.2033862

>>2033664
>math doesn't agree or disagree with reality

Mathematical proofs that are considered to be true are merely those that convince a rational mind that holds the premises to also hold the conclusion. In this sense, math can agree with reality, as well as disagree with it (eg, an invalid proof).

>> No.2033870

>>2033844
now you're just being a moron.

>> No.2033873

>>2033835
What does Cauchy have to do with anything?

>> No.2034828

math != reality. Anyone who says otherwise is demonstrably wrong. for instance, R19 does not exist in reality.
However, math is very good at describing reality. So good, we have an enormous field devoted to this: Physics. Imaginary numbers occasionally come up in physics, and other applied math fields.
An example is plane flight. You can generate a matrix to describe the rotation of a plane, and it will potentially have complex eigenvalues. This is just one of many examples.
i=sqrt(-1)
for all x in R(eal numbers), there is a number xi which is imaginary. C(omplex numbers) are the pairwise sum of all R and I. Thats the theoretical basis.
Alternatively, you can define C as the closure of the algebraic numbers. See the fundamental theorem of algebra for more on this.
Imaginary is just a name, it doesn't mean imaginary as you would use it in normal conversation. Just like Real numbers aren't "real."

>> No.2035236

>>2033862
What you describe are syntactic manipulations according to the rules, nothing more. Truth is without a place in mathematics. There is validity, but not truth.