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God created the real numbers

ethanheilman.com

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Re: God created the real numbers

#131

Post by Dylan16807 »

gpm wrote: post #79
Kranar wrote: post #71

Earlier quoted context omitted.

There is no black hole that is a perfect sphere. That would, at a minimum, require a body with absolutely no angular momentum which isn't in anyway feasible. Any rotating/spinning black hole will no longer be a perfect sphere.

Yeah but if you look down the axis of rotation you will have a perfect (to many decimal places anyways) circle... which was the demand.

That might be right.

But even then, the biggest black hole we think is possible measured down to the planck length gives you a number with 50 digits. And the entire observable universe measured in planck lengths is about 60 digits.

So how are you going to get a physical pi of even a hundred digits on the path toward arbitrary precision?

Re: God created the real numbers

#132

Post by blueplanet200 »

CyLith wrote: post #65
andrewla wrote: post #39

Earlier quoted context omitted.

I loosely identify with the schools of intuitinalism/construtivism/finitism. Primary idea is that the Law of the Excluded Middle is not meaningful. So yes, generally not starting with ZFC. I can't speak to "truth" in that sense. The skepticism here is skepticism of the utility of the ideas stemming from Cantor's Paradise. It ends up in a very naval-gazing place where you prove obviously false things (like Banach-Tars…

I don't understand why you believe Banach-Tarski to be obviously false. All that BT tells me is that matter is not modeled by a continuum since matter is composed of discrete atoms. This says nothing of the falsity of BT or the continuum.

All that BT tells me is that when I break up a set (sphere) into multiple sets with no defined measure (how the construction works) I shouldn't expect reassemlbing those sets should have the same original measure as the starting set.

Re: God created the real numbers

#133

Post by blueplanet200 »

axolotliom wrote: post #40
blueplanet200 wrote: post #23

Earlier quoted context omitted.

Please say more, I don't see how you can be _skeptical_ of those ideas. Math is math, if you start with ZFC axioms you get uncountable infinites. Maybe you don't start with those axioms. But that has nothing to do with truth, it's just a different mathematical setting.

> I don't see how you can be _skeptical_ of those ideas. Well you can be skeptical of anything and everything, and I would argue should be. Addressing your issue directly, the Axiom of Choice is actively debated: https://en.wikipedia.org/wiki/Axiom_of_choice#Criticism_and_... I understand the construction and the argument, but personally I find the argument of diagonalization should be criticized for using finities t…

The axiom of choice is debated as a matter of if its inclusion into our mathematics produces useful math.

I don't think it's debated on the ground of if it's true or not.

And I was imprecise with language, but by saying "math is math" I meant that there are things that logically follow from the ZFC axioms. That is hard to debate or be skeptical of. The point I was driving was that it's strange to be skeptical of an axiom. You either accept it or not. Same as the parallel postulate in geometry, where you get flat geometry if you take it, and you get other geometries if you don't, like spherical or hyperbolic ones...

To give what I would consider to be a good counterargument, if one could produce an actual inconsistency with ZFC set theory that would be strong evidence that it is "wrong" to accept it.

Re: God created the real numbers

#134

Post by lo_zamoyski »

moi2388 wrote: post #127

I am a finitist and constructionist at heart. Sure, mathematical abstractions and infinite structures are fun to play around with.. But go ahead and actually provide me the list of all naturals. You can not. Ever.

This might be conflating two concerns, though, which is the reality of numbers and their actual existence.

Compare Plato with Aristotle. Plato held that the all forms exist in some third realm, so numbers would be counted among them. Aristotle, however, held that forms exist in particular instantiations or in minds that abstracted them from reality. (Aquinas could be said to synthesize both views in the sense that forms exist in particulars and in minds, but also have their origin in God, thus making God a sort of third realm, in a way. Neo-Platonists would view the "mind of God" similarly.)

Now, in the Aristotelian view, numbers are quantities abstracted from concrete reality (indeed, quantity is one of the categories), but they are not substantial forms, as you will not see instances of numbers as substances in the world. They're abstractions of accidental forms. Furthermore, a form needn't be instantiated actually, but can exist potentially. This is how he resolves Zeno's paradoxes. You can divide a length an infinite number of times - or in a CS context, you can apply the successor function indefinitely - but only potentially; as a matter of actuality, you have not divided a length an infinite number of times.

So, for Aristotle, you have a finite plurality of things that are potentially infinitely divisible, or a finite series of actions that can be potentially infinitely repeated or whatever.

For a contemporary realist, Aristotelian treatment of math, James Franklin is worth checking out [0].

[0] https://web.maths.unsw.edu.au/~jim/structmath.html

Re: God created the real numbers

#135

Post by blueplanet200 »

andrewla wrote: post #5
empath75 wrote: post #4

Earlier quoted context omitted.

> But the Cantor vision of the real numbers is just wrong and completely unphysical. They're unphysical, and yet the very physical human mind can work with them just fine. They're a perfectly logical construction from perfectly reasonable axioms. There are lots of objects in math which aren't physically realizable. Plato would have said that those sorts of objects are more real than anything which actually exists in…

There are two things being talked about here, and worth teasing them out. On the one hand, this article is talking about the hierarchy of "physicality" of various mathematical concepts, and they put Cantor's real numbers at the floor. I disagree with that specifically; two quantities are interestingly "unequal" only at the precision where an underlying process can distinguish them. Turing tells us that any underlying…

> For example, the idea that there are the same number of integers as even integers is a stupid one that in the end does not lead anywhere useful.

Well, there are the same number. So, uh, sorry?

Re: God created the real numbers

#136

Post by adrian_b »

drdeca wrote: post #129
adrian_b wrote: post #126

Earlier quoted context omitted.

The Planck units are bogus units that do not have any significance. Perhaps our theories of time and space would break down at some extremely small scale, but for now there is no evidence about this and nobody has any idea which that scale may be. In the 19th century, both George Johnstone Stoney and Max Planck have made the same mistake. Each of them has computed for the first time some universal constants, Stoney h…

The Planck length is at least around the right order of magnitude for things to get weird. If you have the position uncertainty of something be less that ~ a Planck length, and it’s expected momentum equal to zero, by Heisenberg position momentum uncertainty, the expectation of the square of the momentum is big enough that the (relativistic) kinetic energy is big enough that the Schwartzchild radius is also around th…

The right magnitude for things to get weird must be very small, but nobody can say whether that scale is a million times greater than the Planck length or a million times smaller than the Planck length.

Therefore using the Planck length for any purpose is meaningless.

For now, nobody can say anything about the value of a Schwartzschild radius in this range, because until now nobody succeeded to create a theory of gravity that is valid at these scales.

We are not even certain whether Einstein's theory of gravity is correct at galaxy scales (due to the discrepancies non-explained by "dark" things), much less about whether it applies at elementary particle scales.

The Heisenberg uncertainty relations must always be applied with extreme caution, because they are valid in only in limited circumstances. As we do not know any physical system that could have dimensions comparable with the Planck length, we cannot say whether it might have any stationary states that could be characterized by the momentum-position Heisenberg uncertainty, or by any kind of momentum. (My personal opinion is that the so-called elementary particles, i.e. the leptons and the quarks, are not point-like, but they have a spatial extension that explains their spin and the generations of particles with different masses, and their size is likely to be greater than the Planck length.)

So attempting to say anything about what happens at the Planck length or at much greater or much smaller scales, but still much below of what can be tested experimentally, is not productive, because it cannot reach any conclusion.

In any case, using "Planck length" is definitely wrong, because it gives the impression that there are things that can be said about a specific length value, while everything that has ever been said about the Planck length could be said about any length smaller than we can reach by experiments.

Re: God created the real numbers

#137

Post by spauldo »

schuyler2d wrote: post #107

A friend of mine argued that "math is invented" rather than discovered. This seemed wrong to me and in arguing against it I found https://plato.stanford.edu/entries/nominalism-mathematics/ At least at this stage I think it relates to whether you believe "the universe"/reality is a sort of momentary collection of the currently-existing things. Vs seeing reality as the set of all things that might obstruct "me" or any…

I personally consider it invented. The universe is what it is. Math is a model we've collectively built that tries to work within the underlying rules of how logic works in the universe. Like all models, any disagreement with observation means our model is wrong.

But honestly, the whole question is akin to asking how many angels can dance on the head of a pin.

On a separate subject, that site you linked does something strange with scrolling on Firefox mobile. Hey web devs - stop screwing around with things like scrolling! Browser devs implement scrolling in a consistent manner. You aren't making the user experience better with your silly JavaScript tricks!

Re: God created the real numbers

#138

Post by moi2388 »

lo_zamoyski wrote: post #134
moi2388 wrote: post #127

I am a finitist and constructionist at heart. Sure, mathematical abstractions and infinite structures are fun to play around with.. But go ahead and actually provide me the list of all naturals. You can not. Ever.

This might be conflating two concerns, though, which is the reality of numbers and their actual existence. Compare Plato with Aristotle. Plato held that the all forms exist in some third realm, so numbers would be counted among them. Aristotle, however, held that forms exist in particular instantiations or in minds that abstracted them from reality. (Aquinas could be said to synthesize both views in the sense that fo…

If something can exist theoretically but not practically, your theory is wrong.

But go ahead and divide a length an infinite number of times then. And actually infinite, not as it tends to infinity.

Re: God created the real numbers

#139

Post by adrian_b »

dudinax wrote: post #118
adrian_b wrote: post #25

Earlier quoted context omitted.

The quantity of matter and the quantity of electricity are discrete, but work, time and space are continuous, like also any quantities derived from them. There have been attempts to create discrete models of time and space, but nothing useful has resulted from those attempts. Most quantities encountered in nature include some dependency on work/energy, time or space, so nature deals mostly in continuous quantities, o…

We do not know whether work time and space are continuous

What we know is that we use mathematical models based on the continuity of work, time and space (and on the discreteness of matter and electricity) and until now we have not seen any experiment where a discrepancy between predicted and measured values could be attributed to the falseness of the supposition that work, time and space are continuous.

Obviously this does not exclude the possibility that in the future some experiments where much higher energies per particle are used, allowing the testing of what happens at much smaller distances, might show evidence that there exists a discrete structure of time and space, like we know for matter.

However, that has not happened yet and there are no reasons to believe that it will happen soon. The theory about the existence of atoms is more than 2 millennia old, then it has been abandoned for lack of evidence, then it was revived at the beginning of the 19th century, due to accumulated evidence from chemistry, and it was eventually confirmed beyond doubt in 1865, when Johann Josef Loschmidt became the first who could count atoms and molecules, after determining their masses.

So the discreteness of matter had a very long history of accumulating evidence in favor of it.

Nothing similar applies to the discreteness of time and space, for which there has never been any kind of evidence. The only reason of the speculations about this is the analogy made with the fact that matter and electricity had been believed to be continuous, but eventually it has been discovered that they are discrete.

Such an analogy must make us keep an open mind about the possibility of work, time and space being discrete, but we should not waste time speculating about this when there are huge problems in physics that do not have a solution yet. In modern physics there are a huge amount of quantities that should be computable by theory, but in fact they cannot be computed and they must be measured experimentally. Therefore the existing theories are clearly not good enough.

Re: God created the real numbers

#140

Post by lo_zamoyski »

moi2388 wrote: post #138
lo_zamoyski wrote: post #134

Earlier quoted context omitted.

This might be conflating two concerns, though, which is the reality of numbers and their actual existence. Compare Plato with Aristotle. Plato held that the all forms exist in some third realm, so numbers would be counted among them. Aristotle, however, held that forms exist in particular instantiations or in minds that abstracted them from reality. (Aquinas could be said to synthesize both views in the sense that fo…

If something can exist theoretically but not practically, your theory is wrong. But go ahead and divide a length an infinite number of times then. And actually infinite, not as it tends to infinity.

Take the time to understand the subject matter, because your first sentence doesn't makes sense.
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