Is mathematics invented or discovered

Great Debates
61

It is closed under exponentiation, that is what E was for. The exponential as it is usually defined for complex numbers is really just the limit if a Taylor series of algebraic polynomials. And incidentally, I am curious how you consider such quantities as e^i, without this kind of redefinition of exponentiation that fundamentally changes what it means to raise a number to a power, from the usual way we think of it. Its easy to define x^3 = xxx, and from that use roots and limits get to any real raised to any other real, but to go from that to raising to an imaginary power of are going to have to extend the definition of exponentiation in a quite artificial (but very useful) way that probably invloves Taylor series.

I think a good example of what I am talking about would be the Fourier transform which I’m sure you know because it is fundemental to physics and also makes very strong use of complex numbers. However from a mathematical point of view, all it really is is a change of basis. Rather like moving from Cartesian to polar coordinates.

A vector is still the same thing no matter what coordinate system I use. If I want to talk about adding vectors I’m better off using Cartesian coordinates. But if I want to rotate vectors, then I am better off using polar coordinates. That doesn’t mean that I couldn’t do it without polar coordinates, just that its a pain in the ass relative to doing it with Taylor series.

The Fourier transformation is the same thing. It provides an easy way to represent functions that makes the convolution of two functions trivial to calculate, while without it doing such calculations is a pain in the ass. But all it is really doing is making the symbol manipulation easier.

62

Should have been

Too early in the morning for math, need coffee.

63

And after you implement workarounds to implement all of the lost properties of complex numbers to represent them in the form of 2‑tuples and 4-tuples to track state through matrix operations you add a lot of complexity while only really changing the representation of complex numbers.

So you add a lot of required state tracking in order to complicate the way complex numbers are written.

You are not removing the properties of complex numbers, you are creating a ton of work and complexity to hide the fact that the concept of complex numbers is unsettling for some reason.

Nothing about ordered pairs removes the need for a field where a + bi, a + b real where i[sup]2[/sup] = −1.

In the complex plane addition still has to be the same adding the Cartesian coordinates and multiplication still has to be a rotation.

Resorting to existentialization does not change that fact that the field of real numbers is insufficient to describe the wave function.

I get that in:

z -> z[sup]1/2[/sup]

Where z is point (a, b)∈R[sup]2[/sup] works in most cases but it also fails in others.

As an example the rules for fractional exponents for reals do not hold.

−1 = i[sup]2[/sup] =√−1√−1 != √(−1)(−1) = √1 = 1

And you have your log quirks:

log(−1) = log(e[sup]iπ[/sup]) = iπ + 2nπi = (2n + 1)πi, for integer n

And if α is any complex number and z is a non-zero positive complex number

z[sup]α[/sup] = e[sup]α log z[/sup]

Will obviously be a problem if α is not rational or when log(z) > 2πi where it becomes problematic.

As you are far more experienced in complex analysis I don’t need to explain the value of analytic continuation to you. As you mentioned the properties of e, sine, cosine, and i are pretty dang useful in Taylor polynomials; but those properties are also are useful with the unit circle in the complex plane.
For the set of complex numbers z:

z = e[sup]it[/sup] = cos(t) + i sin(t) = cis(t), for all t

Is Euler’s formula and the quantum phase factor.

When you try to check that vectors and matrix are unitary without i things are going to get really hard really fast.

These relationships are the most accurate models we have ever produced of our physical world, but without the simple addition of “i[sup]2[/sup] = −1” they become so computationally complex due to the required rules to avoid that simple axiom that some may be past our abilities.

I fully agree that in complex analysis or linear algebra using a 2-tuple when possible helps the readability, but “i[sup]2[/sup] = −1” not existing in the set of Real numbers is a problem when trying to describe waves and other very natural phenomena.

I get that mathematics is about abstract quantities, and rules. But the interesting point related to this thread is that the multiplicative group the complex set with absolute value 1, or the unit circle of the complex plane is the critical emergent property here.

If you read through and do the math on the collage course above and consider the emergence unit complex numbers maybe it will help explain why I am claiming that your claim is a fallacy of division.

I may try again and see if we can get the SDMB to add one of the math extensions as the limits of written language really make this concept hard to share.

64

Math is invented, and it is used to discover truths about the real world.

You can invent any form of math you like. When I was just learning math addition in elementary school, I learned that 9+9=18. For some reason, I also knew that 9x9=81. I extrapolated from that that multiplication is just addition than reversing the digits. That was a math that I invented. It can be used, you can do operations on it, you can do algebra. You just can’t use it to relate to anything in the real world.

Another form of useless math is lunar arithmetic. Once again, it can be self consistent, have rules, you can make predictions and check them, within the rules of the math, but doesn’t do anything to describe the real world.

What we think of as math is the math that was invented for the purpose of describing the world. When we invent new maths with complex numbers, we can then discover ways of relating those to the world, but not all maths that we invent will have that property.

65

That sounds a lot like, “Axioms are invented, proofs are discovered”.

66

I think we can bring our two-party discussion regarding complex numbers to a close by my agreeing with everything that you say here with the exception that I don’t actually find complex numbers at all unsettling. I think they are great and are just as real as real numbers, Abelian groups and integers, and any other concept we use in mathematics.

Through Physics they are an extremely useful way of handling and describing many of the things that happen in physics. There are other ways of defining the same things that generate all the same results but aren’t nearly as elegant. You say that means that complex numbers are therefore in some sense a fundamental part of our universe while I say they are just useful way to represent a description of our universe.

This is all a lengthy digression into what comes down to the question of whether the mathematics in physics is a description of our physical laws, or is it the laws themselves. My view is the former.

Or as I originally wrote

My goal with the alternative but equivalent fromulations of the complex numbers was to argue that since the same physical laws could be represented by mathematics in multiple ways, we could reason that the math was just a description.

67

Oh yeah? I’d like to see you try.

It seems to me that the problems with your “multiplication” are internal, not (just) external. It’s not just that it doesn’t relate to anything in the real world; it isn’t inherently, mathematically useful or interesting or fruitful. You can’t really “do algebra” with it. It doesn’t even have an identity element.

68

It looks like gemetric algebra and planar bivectors/volumetric trivectors would work for fixing the issue surrounding the dependance on Cauchy integral formula for the inside of a disk to avoid singularities, and this would allow me to replace most of Maxwell’s equations with a real form.

This is way out of my wheelhouse though as some other problems like keeping calculations of superposition would seem to require green’s function from geometric calculus which re-introduces i but it may be possible and I have hit the limits of my knowledge here in a very concrete fashion.

So I will eat a bit of crow on some of my statements above on where I claimed complex numbers were required in some of the cases but have not found solutions in the others.

But yes a rose by any other name… If I was smart enough for the purposes I am talking about, were I smart enough, I would just recreate a new dialect.

The crazy thing is that some rare individuals can ‘invent’ math to describe our world for properties they yet cannot observe and a century later when technology allows we can build experiments that prove them right. Especially when those solutions exceed our ability to visualize them.

69

Do you say that because you would be interested in the extroplations of such a system, or because you do not think it is possible to define an operator to perform such an operation?

As I said, I didn’t really develop this into a robust mathematical system, but rather, it was a few minutes of musing when I was 6 or 7.

Point is, is that it was a mathematical system that was invented, but did not relate to the real world. I was simply using it as an example of that. Any other flaws it has, it also has the flaw that it does not relate to the real world. Nothing that you’ve said has contradicted anything that I have said.

Algebra is simply symbols and rules for those symbols. It is the algebra that we use on a daily basis that has an identity element. That you cannot imagine a form of math that does not have an identity element does not mean that you cannot state, as an axiom, “There are no elements that leave other elements unchanged, when combined with them.”

Certainly makes the math hard, and not all that useful (not useful at all), but it can be created.

If you feel that in order to exist, a math system must be useful, interesting, or fruitful, check out the link on Lunar Arithmetic, and tell me how it is “inherently, mathematically useful or interesting or fruitful.”

I posit that there are an uncountable infinity of different systems of math, and only a very small fraction of those are useful, interesting, or fruitful.

70

I say that because I don’t think it can be developed into a robust mathematical system—but if I’m wrong about that, I’d be interested to see it.

Not only does your definition of “multiplication” lack most of the properties of ordinary multiplication, it doesn’t have any other properties to make up for it. Or at least, so it seems to me. If it seems otherwise to you, then an attempt on your part to actually work out some of those properties might serve as a revelation to one or the other of us.

71

It probably cannot be developed into a robust system. That doesn’t mean that it is not a form of math that was invented. In Lunar Arithmetic, you can’t even do subtraction, how robust is that?

What I asked was not whether it could be developed into a robust system, what I asked was whether you didn’t think that you could define an operator to perform the operations as described.

Sounds like it isn’t a very useful, interesting, or fruitful mathematical system to me. It also, as I said previously and was my only reason for bringing it up, is utterly useless in describing the real world. (With the one exception of 9 times 9 which actually does get you the “right” answer.)

72

Here is an example of the English language predicting all the phenomena that Newtonian mechanics predicts—Hartry Field’s famous nominalization of the latter, i. e. its re-expression without reference to mathematical entities as a defense of nominalism, which is one position denying objective existence to mathematical objects.

(I’m really just subscribing to this thread with this post; I hope to get around to adding something more substantial to it later…)

73

Proposal:

mathematical relationships exist independently of any observer. They can be discovered.

How to express them is something which requires someone to express them. These are invented.
That two items plus two items equals four items was true before anybody figured out counting or addition. But the symbols “2, 4, +, =, two, plus, four, equal, dos, cuatro, más, igual a (etc.)” needed to be invented in order to be able to transmit the concepts; being able to transmit concepts along greater distances required the invention of writing.
So, some parts of math “just are” (geometry; trig). Others are invented as a way to express the discovered math in a way that’s relatively brief (mathematical symbols, algebra inasmuch as it’s a way to write math). The book which explains algebra for the first time is, while not very thick, a lot thicker than the chapter explaining “algebraic concepts” in any of the schoolbooks any of us used, because the author needed to put everything in longhand and was conscious of his readers not being able to just raise their hands and asking for clarification.

In an example from a different field but which is closely linked to algebra, H[sub]2[/sub]O existed long before any living cells contained water. The formula for water, like the name “water”, was invented.

74

QM without complex numbers has been formulated by Stückelberg in the 60s (although one needs to introduce a superselection rule by postulating a special operator that commutes with everything).

Moreover, many aspects of QM become much more intuitive if formulated in terms of Clifford algebras, rather than restricting oneself to imaginary numbers. QM itself, however, doesn’t really care what number system you use.

75

Well, QM being more intuitive in terms of Clifford algebras requires one to know what a Clifford algebra is :slight_smile: I don’t, so in my case I’ll need either to stick to imaginary numbers or for someone to explain Cliffords.

(mind you, it’s possible that I know them but just not by that name)

76

I find it odd that the discussion in this thread seems to be mixing the question of whether mathematics is invented or discovered with talk of how it is useful for describing physical entities in the world, as to me those are completely separate ideas.

If you rigorously define a mathematical entity and its properties, and then you prove by deductive logic that these properties imply other additional properties, then you have learned something true, regardless of whether or not that mathematical entity is at all useful for describing some entity or phenomenon in the physical world. Specifically, you have learned that if something were to exist with the properties you originally proposed, it would also necessarily have the additional properties you proved. “Discovery” is not limited to gaining knowledge of what exists, but also includes gaining knowledge of what is possible.

It seems to me that mathematics is at least partly discovered, and also partly invented - and that this applies regardless of whether the mathematics in question is actually useful for anything.

77

Don’t include me in your “we” when I’m clearly not part of it. I know what’s a constant and what’s a function of the measurement system/ coordinates / units used. And I already told you that 90º is not a constant, it’s a function of the system used. The angular value is a constant; the name of it being 90º or 100º is a function of the measurement system.

78

Yes, thank you. This was a big part of my issue with k9bfriender’s posts. He kept talking about how something was “utterly useless in describing the real world,” which felt like a non sequitur.

79

Those additional rules are the rub though. I am not saying complex numbers are ideal, just that replacing them with something else 1:1 isn’t changing anything.

Heck i would replace English with Finnish in a second if I could as Finnish is 100% phonetic. But that wouldn’t change much except for the convenience of working with the system.

That said I have used it through Dirac, and if you look into spin and Lie groups you will see that while it may hide i from the formulas it still goes complex.

Not saying it is bad, just doesn’t get rid of all of the complex numbers. That said Clifford Spaces are nice in some contexts.

80

How do we test that it isn’t ‘invented’ if we don’t look outside the human mind?