Are you saying that if I have 1 apple, and I give you the apple, I don’t have zero apples left? I have some number of apples that is really small but is not zero?
We invented numbers so we could count things. One apple minus one apple leaves no apples. QED. Here endeth the lesson.
This is almost (but not quite) a complete non sequitur. It doesn’t matter as much as you imply if the term “mathematics” has a generally accepted definition.
Within mathematics, many terms have accepted definitions. That’s how we do mathematics. And those definitions ARE generally accepted and used. If you choose not to use the commonly accepted definitions, provide your own (hopefully well defined and not too hand-wavy) and we can go from there. That’s how math works. Otherwise, go with the standard definitions and proceed from there.
In any event, tossing out word salads with nebulously defined terms is no way to do math. One cannot draw any real conclusions that way, which is the rather the point. Once we accept some definitions and postulates, we can do some real math.
Here is a longer excerpt from Wiki’s Mathematics article.
Bleak?
BTW, I cannot decide whether it is nifty or bleak that
2[sup]ℵ[sub]0[/sub][/sup] = 𝓒[sup]ℵ[sub]0[/sub][/sup] = (2[sup]ℵ[sub]0[/sub][/sup])[sup]ℵ[sub]0[/sub][/sup] = 2[sup]ℵ[sub]0[/sub]⋅ℵ[sub]0[/sub][/sup]
even though 𝓒 is so much larger than ℵ[sub]0[/sub], let alone 2.
Is there any advantage to rigidly defining mathematics ? Keep it open, so as not to exclude computer science, applied maths, mathematical physics, biology, astronomy, logic, philosophy, and any other “excursions” that might attract a skilled mathematician. Any of these may and do lead to new and beautiful mathematics.
[Moderating]
EastUmpqua, if you’re going to quote Wikipedia, please cite your source.
[/Moderating]
As for definitions, how about this one: Mathematics is the field of study which includes theorems.
And forget about 1-1. The whole point of Planck’s constant is that Planck’s constant is different from 2 times Planck’s constant.
In 1960 Eugene Wigner wrote an article entitled “The Unreasonable Effectiveness of Mathematics in the Natural Sciences”. It’s not just that basic formulas for Physical laws are remarkably simple (e.g., no 46 line equation for any of Newton’s laws), but that the type of equations involved suggest additional Physical properties to search for.
Since 1960, things have only gotten more like this. Over and over we see things where “Wow, that’s a nice formula that explains a lot and predicts things we should look for.” The Eightfold Way is one of my favorite examples of this.
The thing to note is that many of these formulas involve real numbers and function over real numbers. In certain places integers appear. If the real numbers in these formulas weren’t truly real numbers but approximations to discrete values we would have noticed a long time ago.
Note that trying to explain the photoelectric effect with a continuous function on real numbers lead to contradictions which in turn lead to quantum theory. And that was over a hundred years ago.
If the nice tidy formulas says that real numbers go here, then real numbers go here.
Although this is probably the most common set of rules for physics in recent history.
Lie(G) = {X ∈ M(n;ℂ)|exp(tX) ∈ G for all t in ℝ}
Which is quite related to the OP, but not in the rationals. Comparing continuous regions of ℝeal numbers involves infinitesimals and rotation groups in the ℂomplex space.
It seems to me that if you consider quantum field theory, which is still considered bog-standard physics to be taught to students, the underlying spacetime still forms a continuum over the real numbers. Yet, this is a highly successful theory. Maybe in the regime where quantum theory is valid it does not matter if the underlying geometry is not truly continuous on a really, really small scale.
ETA similarly in that there appear to be physical symmetries described by Lie groups over the real numbers.
Symmetry groups and observables in QFT are based on Lie group-Lie algebra correspondence. You still get to use real based variables but the SU(2) group is in complex space.
Sure, but any complex manifold is also a “continuum”, a real manifold if you forget about the extra structure. SU(2) is a real Lie group (not complex, to be clear!!). This is in response to the contention that something must be unphysical about the use of such ideal mathematical objects as real numbers and complex numbers; I am saying maybe so, but it seems like theories such as the Standard Model are remarkably accurate to the extent that they are valid.
Yes you are correct I was too terse, and communicated the wrong thing. To be clear I was providing evidence that the representation theory with ℂ is because nice basis that doesn’t exist in SU(2)
SU(2) + iSU(2) = SL(2;ℂ)
I don’t know how to explain my point without involving taylor series and why cos(iѲ) = cosh(Ѳ) and having Euler characteristic as 0 is important.
But if we couldn’t compare the size of arbitrary continuous sets, neither Relativity nor Quantum Mechanics would be here. Everyone is focused on the Plank length when really properties like spin are more important.
The Plank length is the result of a lot of math related to infinitesimals(infinities) and continuous sets. The Plank length is a restriction on observation, not the math. The operators are continuous.
The math of the probabilities is by it’s very nature tied to infinities.
I wish I could figure out a way to share how cool the math behind invarance is.
To clarify U(n) preserves the inner product on ℂ[sup]n[/sup]
SU(n) ⊂ U(n) ⊂ GL(n, ℂ)
U(n) are made up of complex n×n unitary matrices, but use use the reals as inputs and outputs. Unitary matrices must satisfy:
U[sup]*[/sup] = U[sup]-1[/sup]
Where the *-operation is complex conjugation
So while the algebra you interact with is still ℝeal, the backing matrices are intrinsically ℂomplex.
The typical convention of considering ℝ[sup]4[/sup] as ℂ[sup]2[/sup] or ℍ to make things easier isn’t in play here. It is the property of the group that is important for needs like invariance and Lorentz transformations. Really it is the ability to make linear transformations of infinitesimals that is needed. SU(2) being a special case, being isomorphic S[sup]3[/sup] or the unit 3-sphere lets this happen.
The subtleties are often lost with the true statement that SU(2) is a real Lie group. While SU(2) **is not **a complex Lie group, it is made of complex matrices and defined by complex operations.
Sometimes you truly “can’t get there from here”.
I didn’t address the second part.
Take a typical state:
v = a|up> + b|down>
Where v is normalized to:
|a|[sup]2[/sup] + |b|[sup]2[/sup] = 1
The projection matrix:
vv* = [a][a*,b*] = [aa* ab*]
** [a*b b*b]
Mapping v -> vv* correspondence 1:1 between the states (a:b). The states (a:b) are a complex projective line, and mapping it’s points on a sphere in 3-space is just Riemann sphere mapping. Thus the elements SU(2) correspond to the change in state induced by rotating the electron, when pictured as a rotation of the sphere match up perfectly including the 90 degree rotation properties.
Theories such as the Standard Model are remarkably accurate to the extent that they are valid, expressly because we use the mathematics that we are lucky enough to have that match up with the observations. Perhaps if our mind could visualize 4 dimensions we wouldn’t describe particles as having spin. But the correspondence with the classical spinning ball, which allows us to work on these problems simply wouldn’t work if we try and do the math with ℝ.
As momentum and position operators do not commute the complex nature of SU(2) helps us get around limits of human cognition and actually maps to the observed nature of the system. The entire concept of spin as an actual spinning ball wouldn’t work if we did everything over ℝ. I don’t know if angular momentum world work under ℝ[sup]4[/sup] but you would just have to find a 4D rotation symmetry group that has the same properties and it would be harder to conceptualize. You would have to create other rules to deal with the non-commuting observables that do work under SU(2) too.
A lot of the math is to help us with visualization and representation. But notice how most of the discomfort is also directly related to our limitations on visualization. Feel free to work out the taylor series for the cosine of a circle (cos) and the cosine of a hyperbole (cosh) and consider that in 4D spacetime the hyperbole is the shape that is rotationally invariant.
It is fairly easy to prove that “cos(iѲ) = cosh(Ѳ)” analytically using Taylor series, so perhaps considering that the fact we can’t imagine a way of measuring either “cos(iѲ)” with a ruler is an indication the limitations are with our cognitive abilities and not with the concept of complex numbers. Heck if you existed in a one dimensional line you would probably have the same problems visualizing Cartesian points.
The reality is that we can use a real manifolds and real numbers to model the system, but we cannot discard the properties of complex groups or *e *or i. Perhaps future work in the field of representation theory will change, right now Physicists have to use the tools that are already found.
To be honest the only people who seem to resist using the existing tools, which as you pointed out, work pretty darn well, are those who are insistent on limiting concepts to those that they can concretely map to concepts in their mind.
For a long time these same objections prevented the adoption of negative numbers and even the concept of zero. As we can’t visualize spacetime or probability clouds I don’t think the effort to change the math has much value. There is probably greater value to humanity with humans getting comfortable that the limitations of human visualization isn’t a barrier to deeper understanding of any topic.
I was not thinking so much of real versus complex manifolds (is that an issue?), rather of the discretized quantum geometry of space-time on small scales that occurs in quantum gravity, where spacetime is not simply a real, smooth (pseudo-Riemannian) manifold. This was in response to a question posed by Measure for Measure: “Are there serious physics models that are not continuous, i.e. discrete?”, just as an example.
Now, even there real numbers are lurking everywhere, for instance via the gauge group as you point out. Or anywhere you have a Hilbert space. It seems that real (and complex- do not think I am giving complex numbers short shrift!) numbers are not that easy to get away from in physics, even though these are, in some sense, Platonic objects. You can play with p-adic mathematical physics for a change (another example for Measure for Measure). If there was a philosophical point here, I have lost track of it… (Unreasonable Effectiveness of Mathematics and all that).
Ah sorry about that.
I consider “quantum gravity” as currently lacking “serious physics models” as there are no complete and consistent quantum theory of gravity, and only candidate models like superstrings and quantum loop gravity exist today.
Had “string theory” been called what it is “string proposal” most of this confusion wouldn’t happen. But there is still zero evidence in actually tested and accepted theories that spacetime is quantized. I will be happy if anyone finally moves past that stage.
Randomly after that post I found this Wikipedia page, which demonstrates what the platform should be
Galilean transformations not working for EM is what really drove the move in the 1800s to Lorentz transformations, more specifically specifically boosts…which being hyperbolic almost force you to use (i) if you don’t enjoy pain.
Ok. Sorry
I’m still reading the abstracts from the Lattice Field Theory Symposium link posted earlier on this thread. I have the utmost respect for the patience and humor of the folks I’ve interacted with on this message board.
Really? So why do you treat us like you do?
All I know if you have 4 apples and take one away you’d have bleem left.