[CogitoCogito]

The continuity of real numbers provides a clean theoretical basis for continuity of functions. I personally view it as more of a theoretical tool that seems to do pretty well and instead steer clear of the philosophical questions.

One thing that I think is important to note though is that the jump from real numbers to complex numbers is nothing compared to the jump from integers/rational numbers/etc. to real numbers. The complex numbers come about by simply adding one dimension whereas the real numbers come about from an abstract "completion" of (say) the rational numbers in a very specific mathematical sense.

My point is that deriding "imaginary numbers" (as many do) is total nonsense if you accept real numbers.

[j2kun]

> The complex numbers come about by simply adding one dimension

Complex numbers are also best thought of (in my opinion) as an abstract completion of the reals under the operation of taking roots of polynomials.

Because otherwise, what would be the difference between the real plane and the complex numbers? They are topologically identical, after all. There has to be something more substantial than simply adding a dimension.

[nilkn]

If by real plane you just mean the standard vector space R^2, the answer is algebraic rather than topological. R^2 as a vector space has an addition operation, but no multiplication operation. The complex plane is obtained by simply picking the appropriate multiplication operator.

In general, completing the rationals into the reals is more complex than constructing the complex plane from the real numbers. For the latter, you just need to adjoin a single element (sqrt(-1)), enforce existing arithmetic rules, and the rest falls into place. For the former, you can't just adjoin a single new element like sqrt(2). Doing so will get you the ring (actually field) Q[sqrt(2)], but not R.

If you take R and adjoin two special new elements (sqrt(-1) and the point at infinity), you do obtain a topologically different result: the Riemann sphere. This sphere is in many ways the more natural domain for complex analysis than the complex plane.

[jacobolus]

Complex numbers are (isomorphic to) a certain subset of linear operators on a 2-dimensional Euclidean vector space which rotate and/or scale the vectors in the plane without skewing or anisotropically stretching them. (We usually also include a zero operator here; depending on use case we sometimes omit it (the “punctured plane”), or sometimes add a point at infinity (the “Riemann sphere”).)

If you want you can write them down as matrices acting on vectors in an orthonormal basis by matrix multiplication:

  [a -b]
  [b  a]

Or if you prefer you can consider i to be a unit bivector in the plane, with a complex a + bi acting on vectors by Clifford’s geometric product.

Or if you want you can write them using a length and an angle measure, and use high school trigonometry to figure out how they apply to vectors.

The difference between the real plane and the complex numbers is that for two vectors in the plane, the product is not a vector. (Indeed, if you use Clifford’s geometric product, then the product of two vectors is a complex number (scalar + bivector).)

[bzbarsky]

They're also isomorphic as abelian groups under addition. At least if you assume the axiom of choice. See eg. http://math.stackexchange.com/questions/925706/is-it-true-th...

[ThrustVectoring]

Agreed. The jump from real to complex numbers is about as difficult to explain as the jump from natural numbers to integers. Integers are the numbers you need for "subtracting numbers gives you a number". Complex numbers are the numbers you need for "factoring polynomials with numeric coefficients gives you numeric roots". There's a similar argument for real numbers that you hinted at, but I don't understand it well enough to give a simple explanation for it, so I pretty much always hand-wave over it.

[j2kun]

For reals, it's "Real numbers are the numbers you need to ensure every convergent sequence of rational numbers has a terminating point"

Convergence is determined in the "Cauchy" sense by having a vanishing distance between subsequent sequence entries, so as not to rely on the (potentially nonexistent) limit.

[chadcmulligan]

I thought you could have calculus using 'computable numbers' that is reals that have a rule. It's been on my list for a while to look into it - there's some notes here for the curious http://math.stackexchange.com/questions/963061/can-the-set-o...

[Animats]

The continuity of real numbers provides a clean theoretical basis for continuity of functions.

Exactly. Real numbers are an invention to make mathematics simpler and cleaner. If you don't have continuous reals, it takes extensive case analysis to prove relatively simple things for, say, a binary floating point representation. It's possible to do that; Boyer and Moore did work like that to formalize floating point and check its correctness. (That work was funded by AMD, after the famous Pentium divide bug.)

Newtonian mechanics assumed that the physical universe is described by real numbers. But today, it seems that time, length, and mass are all quantized. There's thus not a physical basis for the existence of reals.

Maybe reals should be viewed merely as a useful convenience for analysis, not some fundamental part of mathematics.

"God created the integers. All the rest is the work of Man." - Kronecker

[codethief]

> But today, it seems that time, length, and mass are all quantized. There's thus not a physical basis for the existence of reals.

This is actually not the case. Space and time are not regarded as quantized by the majority of physicists and there's also absolutely no evidence suggesting that. To the contrary, such quantization would lead to inconsistencies – e.g. with relativity – pretty quickly.

Fur further reading:

https://physics.stackexchange.com/questions/9076/does-quantu...

https://physics.stackexchange.com/questions/67899/is-time-qu...

[prmph]

An interesting idea that follows from this: what other kinds of "numbers" might we come up with if we relax our logical blinders?

I have this concept of "materialization" and wonder if there is a formal mathematical term for it. Complex numbers are actual, in the sense that they can be used in calculations that finally would give us a number we can make sense of (materialization), even if we cannot actually imagine a complex quantity.

In the same way, what if we invent a new class of real numbers that are quantized (such that there exists a smallest quantity x)?

What if we propose the existence an "imaginary" algorithm (call it omega, if you like) that can decide the halting problem?

These mathematical objects do not have to "exist" for them to be useful. I think there is a whole lot of new and interesting math that would be unlocked by dispensing with our logical blinders.

[johncolanduoni]

> What if we propose the existence an "imaginary" algorithm (call it omega, if you like) that can decide the halting problem?

Although our mathematical systems rely on the existence of computable procedures (that halt) to e.g indicate whether a formula is well formed or verify a proof, they have no trouble talking about uncomputable functions. There actually ends up being a hierarchy, as once you allow "solve the halting problem for this Turing machine" as an operation the expanded set of computable functions is now unable to solve its own halting problem: https://en.wikipedia.org/wiki/Arithmetical_hierarchy

> These mathematical objects do not have to "exist" for them to be useful. I think there is a whole lot of new and interesting math that would be unlocked by dispensing with our logical blinders.

These kind of things are actually pretty well studied. Some interesting examples are:

* The hyperreal numbers which include infinitely many distinct numbers which are infinite or infinitesimal: https://en.wikipedia.org/wiki/Hyperreal_number

* The surreal numbers which include all ordered fields as a subfield (reals, complex numbers, hyperreals, etc.): https://en.wikipedia.org/wiki/Surreal_number

* The quaternions and octonions, which along with complex numbers are the only finite-dimensional algebras over the real numbers that can have "division" and "absolute value" in the usual sense: https://en.wikipedia.org/wiki/Hurwitz%27s_theorem_(compositi...

In general, mathematicians are really good at investigating things of the form "Okay we have structure X with properties A, B, C. What if we get rid of C? How about B? How about B and a weaker version of A? ...".

[scatters]

> What if we propose the existence an "imaginary" algorithm (call it omega, if you like) that can decide the halting problem?

This is called an oracle (https://en.wikipedia.org/wiki/Oracle_machine). We can posit oracles for solving computable problems in constant time (e.g. factoring the product of two arbitrarily large primes) as well as for solving uncomputable problems (e.g. halting problems).

Oracles are a great tool for studying complexity and computability, since oracle machines have their own complexity and computability limits; an oracle machine for the halting problem can determine whether a simple Turing machine will halt, but cannot determine whether it itself will halt (this is called a Turing jump). Thus oracle machines for halting problems form a class hierarchy, which Post's theorem shows is precisely the arithmetic hierarchy.

[CogitoCogito]

> An interesting idea that follows from this: what other kinds of "numbers" might we come up with if we relax our logical blinders?

It depends on exactly what you mean by this, but I would argue that we do this absolutely everywhere. For example, matrices have similar properties to numbers. You can add/subtract them, you can multiply them, you can (sometimes) divide them. Depending upon how you restrict your set of matrices, ab might equal ba or it might be the case that ab makes sense while ba does not (i.e. not only is it not commutative, but it doesn't even make sense to multiply the other way).

> In the same way, what if we invent a new class of real numbers that are quantized (such that there exists a smallest quantity x)?

Here's an example of something similar:

https://en.wikipedia.org/wiki/Dual_number

Basically you're adding in a number smaller than everything else that squares to 0.

> What if we propose the existence an "imaginary" algorithm (call it omega, if you like) that can decide the halting problem?

I'm no logician, but I believe that they would refer to this sort of thing as "model theory". I.e. you're basically taking proofs (sequences of logical statements) and studying different logical systems in which these statements make sense. For example, you could extend your theory/model to take as an axiom that there exists an algorithm to solve the halting problem (though this would be stupid since you can already prove within most theorems that the halting problem is impossible...i.e. you would have a self-contradictory axiomatic system).

> These mathematical object do not have to "exist" for them to be useful. I think there is a whole lot of new and interesting math that would be unlocked by dispensing with our logical blinders.

I 100% agree. Math is a tool that we invent to better understand the world and our thoughts (as well as do more). However, you'll probably have to track down a true logician if you were to go down that rabbit hole...

[prmph]

Started reading about dual numbers after reading your reply; interesting concept

[zamalek]

Given my experience when learning about complex numbers, I would rather say:

> The complex numbers come about by simply adding back the missing dimension

Mathematics makes so much more sense with complex numbers. Trigonometry a great example of this.

[hammock]

This makes sense. It feels analogous to economics (and economic models) - something fake we make up to be able to create functions and theories and so on

[CuriouslyC]

Unfortunately, physicists seem to believe that infinity and the infinitesimal are real things.

[CogitoCogito]

What do you mean by this statement?

[CuriouslyC]

Infinite values (e.g. the density of a black hole, the size of the universe) and infinitesimals (e.g. continuous space-time) are assumed to be real, rather than inaccurate but useful approximations.

[tejon]

Not a physicist but I thought the Planck length specifically denies infinitesimality? And one of the proposed solutions to the black hole information problem is that due to local relativistic effects, they never actually reach singularity in finite time.

[psyc]

The Planck length isn't a minimum length or size, and doesn't have a whole lot of real significance other than as it results from a particular choice of units.

[CogitoCogito]

Assumed to be real by whom? Not all physicists believe the same thing.

Also why is this unfortunate? Why does it matter if they do or do not "believe" it? Are they able to make useful predictions with their models? Are they able to better understand physics? If so, why worry about their personal beliefs?

[CuriouslyC]

If we want to put physics on the faith table, I'm cool with whatever anyone wants to believe, and more power to them. But you can't be objectively right on the faith table - that's the price of admission.

If we want to be "true" and fully rational then we need to try to accurately represent our degree of knowledge about the world. Thus we shouldn't be making strong statements about things with an absence of evidence.

[sirclueless]

I wouldn't say assuming infinite density of black holes involves an absence of evidence. It is a hypothesis made in advance of evidence, and it leads to specific predictions that should be falsifiable, and in that sense it's considered scientifically sound.

As for implications that can't be falsified, for example ones that (to butcher Douglas Adams) "rather involve being on the other side of the event horizon," scientists can and do feel free to disregard those. Contradictory assumptions by scientists do not imply that one of them is "wrong" unless their theories imply contradictory observable phenomena. In which case there is probably a fruitful experiment to be done.

[gertef]

Rationals are conintuous, but not a continuum. you don't need real numbers to have continuous functions.

http://math.stackexchange.com/a/672151

[CogitoCogito]

> Rationals are conintuous, but not a continuum.

No. In fact, your claim is contradicted by your own link. As stated by Asaf Karagila in the comments:

> Continuity is a property of functions. You seem to ask why the rational numbers are not connected (or path connected).

But you are correct that you don't need real numbers to describe continuous functions. As you link points out, one is a property of spaces (or domains or whatever you want to call it) and the other is a property of function.

However, talking about continuous functions between complete spaces (basically "complete" is what makes the real numbers "real") is extremely natural and basically goes hand in hand with continuous functions. It really ties together a lot of the theory if you're talking about metric continuity.

Regardless, you also don't _lose_ anything by talking about real numbers. You can of course use it as a tool to develop a theory and then choose to apply the results to the rational numbers (or algebraic, etc.).