The almost ubiquitous sine wave, and a cycle take pi units of time.
2*pi radians is also half way around the unit circle which is heavily related to trig and many other types of math.
The NUMBER pi is under our system is philosophically inconvenient, but don’t confuse the issue we personally have with irrational numbers with the mistaken belief that it is not manifested and observable in nature.
The philosophy in play here does not relate to the existence of pi, but or acceptance of a location on the number line that doesn’t match our non-universal, invented number system in a pretty way.
BTW, another example on why this claim is too broad, measuring things using numbers is math unless you narrow the invented definition to match your argument.
The philosophy invented portion is that narrow choice of what is included as being math.
Something in this thread is pure sophistry, but not in my objections.
All of these, again, are abstract concepts which emerged long after the notion of a geometrically perfect circle.
I’m not claiming that it’s not valid to interpret and model various natural phenomena using abstractions like the geometrically perfect circle. I’m just pointing out that the conceptualization of the geometrically perfect circle isn’t merely a naive “discovery” of “truths that exist in nature”, because no such object is directly physically discoverable in nature in its perfect form; and especially not at the time that humans first conceptualized a geometrically perfect circle.
Mathematics is a set of rules that guide symbol manipulation, that is invented by man and in and of itself disconnected from reality.
Smart people can take a look at the real world invent definitions and axioms that provide an approximate isomorphism that maps objects and events in the real world, to objects in the space of symbol manipulation.
Using the symbol manipulative laws smart people can discover interesting new theorems regarding the symbol manipulated objects.
These can then be transformed back to provide interesting predictions about how things should work in the real world.
Note that there is no reason that a given set of axioms or definitions need necessarily have anything do to with the real world. Mathematicians are quite willing to
I would also point out that in the difference between invented and discovered is very fuzzy. Was the wheel invented or discovered? It was a device that someone in a given culture first created, so in that sense it is an invention, but that fact that round things move more easily across terrain is a fact of nature that was just waiting for someone to figure out.
So in the same way that if we ever meet aliens I would have expected them to figure out the wheel, I also would have expected that in their own possibly very different version of mathematics they would have something that is analogous to the Pythagorean theorem.
That is the interesting part in my mind, that the irrationals like 𝜋 ⅈ ℯ or hard to imagine values like ∞ and their relationships appear to be some of the most fundamental, universal, and natural concepts in math.
Yet our teaching philosophy and or our cognitive abilities also tend to place these values in our “invented” math which tends to judge them as weird and or as imaginary.
Euler’s formula: ℯ^ⅈx = sin x + ⅈ sin x,
Is pretty fundamental to most modern fields of applied math like physics and engineering.
Set x = 𝜋 and you get Euler’s identity ℯ^ⅈ𝜋 + 1 = 0
Which is so fundamental and mathematically beautiful and so easily derived given unlimited intelligence and unlimited time, that portion almost certainly directly relates to the underlying physics of our world in an objective way.
Lets be clear Euler’s formula directly impacts several practical observations like harmonic motion, Quantum particles in a box, Snell’s law, Alternating current, sound, light, harmonic analysis, spatial rotation, signal processing, probabilities, and even unproven ideas like string theory.
A more narrow and possibly answerable question is why do we tend to think of our “invented” axioms as being safe or normal while really debasing values and relations that seem to be fundamental across number systems and coordinate systems.
Unit circles and natural units are as close as we can get to empirical truth here, but remember that physics uses math to describe and predict our physical world and even if the theories that use Euler’s formula are disproved the accuracy the formulas based on this elegant relation is so good that we really don’t have a current known application that will require more precision from a superseding theory except for our own curiosity.
I think that this debate is often like grammar, axioms and conventions are focused on because they are natural in the same way that random prescriptive grammar is often focused on by those who are bothered by those who don’t follow the rules.
But outside of our education system physics in general is far more interested in the the truth than dogma. And while it is still a product of humans and humans are attracted to beauty the nearly universal application and simplicity is pretty hard to ignore here.
But one has to really focus on specific examples to make any claim like the OP offered. Prime numbers in base 10 are interesting but are an artifact of a chosen number system, but that doesn’t mean that there are not universal and real world observable in other areas like the elements of Euler’s Formula.
I am responding and expanding in this reply, don’t assume I am claiming you are making any arguments you are not.
Perhaps the problem is that not many individuals are exposed to higher forms of math that expand past the invented rules that symbols.
Pythagorean theorem is directly related to and can be co-derived from 𝜋 ⅈ ℯ in the complex plane.
Geometry and thus Trigonometry and thus the Pythagorean theorem are provable despite differences in convention.
In my hobby area the ability to fairly arbitrarily and invariantly transform between various separate and distinct coordinate systems and even between euclidean and non-euclidean geometries is a big reason why tensors or quaternions are used.
Quaternions are non-commutative and this lack of a commutative property is often a reason people who are more “authoritarian” try to avoid them. To be fair being non-commutative does add some difficulty, but that assumption that is hammered in earlier calculus class is “invented” for our conventions.
I already am witnessing too much on this thread, but I wish I could share the deeper implications related to partial derivatives etc…
As 3brown1blue is one of the best sources for teaching intuitions today I already referenced this video. But as the SDMB doesn’t have math or image support I have to use external sources.
If you just remember that multiplying by i rotates you by 90 degrees then watch it you may get what I want to share.
A should make the point that with the above mention relations you can often stay symbolic and avoid the loss in precision by truncating irrational numbers or being frustrated that you can’t point to sqrt(-2) on our invented number line.
While he doesn’t go into detail in that video the relation between imaginary numbers, orbits, the Pythagorean theorem and pi all arise from segments in that video.
Newton and Kepler’s formulas may have been “invented” but they would have probably arisen by others given enough time if just using different syntax and or conventions.
Newton’s approximations are not producing “perfect circles” but that simply doesn’t matter in the big picture.
We simply just don’t seem to teach math to a level where that bigger picture is easily visible in our current system.
Actually General Relativity and QED would exist just fine without complex numbers, and have for some 13.8 billion years before i entered the scene. It’s just that our current representation of thesenatural laws in symbols wouldn’t exist as it is today without i. But that isn’t to say that there might not be a different mathematical formulation that leads to the same conclusions but comes at if from a different way that doesn’t explicitly involve i.
We have developed the formulation we use because, it makes it easier to perform calculations and draw conclusions that seem to be analogous to how the world works. Even though its my favorite number, there is nothing about e that is fundamental to our physical universe. It just happens to be the result of exp(1) where exp is a the function that has the property that its its own derivative and exp(0)=1. Now having a function that is its own derivative makes calculations a whole lot easier, and the exponential function has the property that exp(xy)=exp(x)^y, so that for real numbers exp(x)=exp(1)^x=e^x. So its often easier to consider the exponential function exp(x)=e^x even when x is a complex number. But all of this is just notation invented because it makes the calculations easier.
You have missed my point.
The point is that what we think of as constants is based on how “obvious” they are to us: 90 degrees is just a quarter of whatever we have defined a whole turn to be in our coordinate system, and doesn’t feel “special”.
But if we are saying that Pi is something out there as a universal truth, we can ask whether 90 degrees is out there as a universal truth, since why would the universe care about what is obvious to a human?
As science is a systematic model that builds and organizes knowledge in the form of testable explanations and predictions, why not just dismiss the entire body of human understanding then? Really as far as the universe goes we really just don’t matter at all. I don’t accept that the kierkegaardian “subjectivity is truth” argument didn’t do much to give us smart phones and the internet.
You are begging the question as it is obvious that the universe exists just fine without intelligent beings, the question is if intelligent beings can accurately describe the physical world without concepts like the natural log.
But as we could talk about the issues with not being able to measure the area under the hyperbole, or how that relates to space-time where the hyperbole is the shape that is rotation-ally invariant how about rising to the challenge and freeing all of the new College QM students from this burden of “complex numbers”
I think you will fine the same thing these authors did, that to make it work you end up just re-creating the rules anyway.
Or a simpler bar.
Explain how we solve real problems that sometimes required the manipulation of square roots of negative numbers? Or how about producing a non-tortured explanation of EM propagation, where electric and magnetic components are simply the real and imaginary parts and which is which depends on your relative motion. While a gross simplification you only interact with the “real” portion of EM, but you need to account for both together.
I doubt you have the same claimed issues with using fractions or ratios that end up not being “real” like lets say one third of a population of 127 cats; or any other random representation error caused by indivisible “quanta” despite the fact that these ratios are just as “fake” and to “make the math simpler”.
I would argue that because the “invented” portion of mathematics to describe the physical universe is primarily due to the need for an effective way to describe the world, you are inventing a more complex solution to avoid something you find philosophically less correct.
I bet you would have no problem winning a Nobel if you just got rid of the ‘bra’ in QM’s use of Bra–kets. While in math postulates are not falsifiable, they are in physics, so if complex numbers are not needed why not just write a paper and falsify a couple of Postulates of quantum mechanics?
It would be nice to know a Nobel laureate, especially one who won that honor for keeping physics Real.
So I say math was discovered and endorse this view…
…but I confess I struggle with the idea of non-material things existing outside of somebody’s mind. A good starting point for myself might involve a taxonomy of stuff that exists, eg., things made of matter, ideas of one person, shared ideas, viral ideas. Then work in math (various types?) and logic.
I think you are taking my disagreement too far. I have a BA in math, PHD in statistics and started out as a physics major, and I’m not in any way downplaying the significance complex analysis in the advancement of science.
The simplest way to take what I am saying is that there is a difference between the words “the apple on my desk” and the actual apple on my desk, and that there are different ways to describe it that might not use the words “apple” and “desk”, but whichever description we use will in the end equivalently represent the same object.
Going to your question of 127 goats divided into 3 equal groups. Well back before I had any form of mathematics I would get my two sons and myself to each take turns taking one of the goats until in the end we each ended up with a set of goats and found one left over that we cooked into a stew that we put into three bowls of the same size.
But now with mathematics I can count the goats, take out a slate, write down the symbols 3 and 127, mix them around a bit until I get 42 with one left over. Now nothing I did there had anything to do with goats. But I can map this back into the physical world by telling each of my sons to take 42 goats and take 42 for myself, compute the volume of the stew made by the final goat and give a third of it to each of by sons. It still ends up in the same place but its down much more quickly.
Similar you challenged me to give a non-tortured way of describing EM propagation. And I agree that I can’t because any version I gave would appear tortured relative to the much simpler version you used thanks to the convenient invention of i.
The other example you gave of the ordered pairs being equivalent to imaginary numbers is exactly the sort of thing I was talking about. The ordered pair version of this symmetry is a perfectly valid description that doesn’t involve imaginary numbers. But doing so is awfully cumbersome and we have all of this theory around complex analysis that we might actually want to apply to these ordered pairs, so for convenience we write them as complex numbers. In fact all of complex analysis can be written without officially invoking i, if you just treat it as ring of ordered pairs of reals with a relation that results in (0,1)*(0,1)=(-1,0)
To get a real handle on what I’m talking about regarding the understanding of mathematics as pure symbol manipulation I would recommend the excellent classic Geodel Esher Bach.
To get a real handle on what I’m talking about regarding the understanding of mathematics as a tool and what is required to describe our universe I recommend the excellent classic Feynman Lectures
No offense, but you obviously didn’t make it to Quantum Mechanics or stopped after an intro course. And yes people who teach those classes actually expect the opinion you shared above as it is sold almost universally as dogma at some point during the math track.
Random example of course materials meant to address this, which every QM course track has to do because of these “assumptions” that complex numbers are imaginary and unneeded.
Your argument is mistaking the map for the territory, and yes, I fully understand there are plenty of books extolling the complexities of manipulating formalisms in pure math.
IN Quantum theory complex numbers aren’t just a convenience, they are necessary. I get that you are a smart person and a good at school, but so was Paul Dirac but he is also one of the most significant physicists of the last century.
Really it is just an issue of familiarity vs. understanding. The Feynman Lectures are pretty simple to read, and if you even work on the basic parts about the wave function you will see why intensity and phase or two numbers. Try and work through the problems where you don’t hit i and because you do have a math background you will run into the same issues that block everyone who assume that the complex plane is just purely about convenience.
I am not sure why the math track instills the idea that complex numbers aren’t “real” (not in the R context) but hopefully it helps out over there because it is an impediment for learning on the physics side.
It seems to me that noting that a thing exists that is different from oneself is discovering, while deciding to call a single instance of that thing “one” (instead of “two” or “Charlie”, etc.) is inventing.
Apologies for snark: my point is that for some things the truth is not and cannot be in the middle. Just saying (but see below).
Also, for a discussion of the reality of imaginary numbers, I’d recommend this website. This ebook, adapted from articles on the website, it works its way to an explanation of Euler’s equation. For those with a kindle it’s well worth the $10. Until then, here is his treatment on imaginary numbers, designed for a high school audience. It doesn’t quite reach the point where we can say that imaginary numbers are more than a gimmick or calculation trick, but it provides the groundwork. (For me at least: I never took college physics, although I own a couple of physics texts.)
I think this post wins the thread. Can anybody pick it apart?
The parts of math that would be the same no matter who worked on them are discovered. In some sense the base 10 system was invented, though the idea that there can be base 2, base 4, base 10, or base 12 systems was discovered.
Math doesn’t necessarily exist in some platonic sense, anymore than an accurate and concise answer key to a math book exists prior to its calculation. But as a definition the answer key is discovered, insofar as there can be but a single answer to a math problem.
What if there is more than 1 proof for a mathematical conjecture? I suspect this would be a matter of tightening up the language of my claim, but we won’t know for sure until we try.
A number like pi is something I would expect when attempting to derive the ratio between a circle’s diameter and it’s circumference. What always fascinated and confused me is how certain ratios in nature use exact integers. Why is the area of a circle pir^2, and not pir^2.1? Or pi*r^2.0184628? It always seems incredibly magical that you raise the radius by the power of precisely 2.
This system has an advantage over the real numbers in that it is algebraically closed so that every polynomial of degree K has (counting multiplicty) K solutions.
For further fun let’s define ||{a,b}|| = a^2+b^2
which allows us to define derivatives on Z->Z functions in terms of limits of infinitesimals.
Then we define E(z) as function Z -> Z that is the solution to the differential equation:
E’(z)=E(z) with E({0,0})=1
and functions S, C and R-> R are the solution to the differential equation S’‘(x)=-S(x), C’‘(x)=-C(x), but S(0)=C’(0)=0 and C(0)=S’(0)=1.
By doing some symbol manipulation I can show the interesting result that E({a,b})=E({a,0})*{S(b),C(b)}
Everything done in terms of reals and differential equations. No trigonometry, or unreal numbers. I know that all of the rest of whatever you had in your link can be written in this form, because this form is entirely equivalent to complex numbers that you are used to. I may not have had more than two years of college physics (including basic QM) but I did have a full year of graduate level complex analysis.
Now it may be interesting to note what EXP({1,0}) is or to separate out {0,1} and note that S(x)=C(x+p) where p has the interesting property that for a circle are radius R, it circumference will be 4p, but those results can be derived from this system.
I think you will find that is isn’t so much i as sqrt(-1) that is driving the physics, it is the relationship between the exponentials, phase shifts, and solutions to differential equations, that I define above that are important. Writing these in terms of a value whose square is -1, is just a very convenient notation to encapsulate all of this. But instead of real and imaginary part you could just as easily talk about the first and second component of the bivariate.
Again I don’t think we are disagreeing as much as you think we are. I’m sensing a small amount of hostility from you, I hope you aren’t sensing any from me.
No hostility, but you mentioned phase shift yet didn’t account for it in your above solution, also consider the solution you provided won’t square root property when you try to work with amplitude.
The real problem here is that algebraic closure may have worked for you in complex analysis but field theories require closure under exponentiation. I tried to format an example of this on here but gave up after losing half an hours worth work.
I do agree we aren’t that far apart, but unfortunately the lack of mathjax or some tex extension makes matrix displays painful. I would bet we could come to agreement in a small number of posts if that was an option.
