Numbers

[untermensche]

Very Wrong. As wrong as a human can be.

Boy this is easy!

And totally fucking worthless.

Except that my statement is supported by the most well-evidenced physical theories of all time, and yours is based on what you think makes sense at the time. I'm pretty sure you're completely ignorant of relativistic  Lorentz scalars, but that's really no excuse for the sheer confidence with which you are continually wrong...

Your opinion as far as I can see is supported by absolutely nothing.

Yes there are theories that exist.

Your claims that you can connect anything I say to them is empty and worthless.

I don't believe you and anybody that does simply doesn't care.

How exactly do you think the smallest possible volume of space can be measured? Not speculated on, measured.

 

[beero1000]

Very Wrong. As wrong as a human can be.

Boy this is easy!

And totally fucking worthless.

Except that my statement is supported by the most well-evidenced physical theories of all time, and yours is based on what you think makes sense at the time. I'm pretty sure you're completely ignorant of relativistic  Lorentz scalars, but that's really no excuse for the sheer confidence with which you are continually wrong...

Your opinion as far as I can see is supported by absolutely nothing.

Yes there are theories that exist.

Your claims that you can connect anything I say to them is empty and worthless.

I don't believe you and anybody that does simply doesn't care.

How exactly do you think the smallest possible volume of space can be measured? Not speculated on, measured.

Luckily for all of us, physicists are much smarter than you are, and they actually thought through the consequences of the different possibilities. They figured out that the potential discretized spacetimes imply a violation of Lorentz invariance, which in turn implies consequence that would be observable with today's technology. Specifically, the divergence from predicted results in vacuum Cherenkov radiation, particle decay, gamma ray bursts, etc, would all show as consequences of there being a 'smallest possible volume of space'. We've checked dozens of these as precisely as is currently possible (and that is VERY precisely, down to well below the Planck scale), and the consequences do not appear. You are wrong.

 

[lpetrich]

I have made a Venn diagram of the rational, algebraic, real, and complex numbers:


I omitted the integers and natural numbers for clarity. I've restored them below:

Natural numbers < Integers < Rational numbers < Real algebraic numbers < (Real numbers, Algebraic numbers) < Complex numbers

 

[lpetrich]

Back to philosophy of mathematics, the names of some kinds of numbers give hints about how past mathematicians regarded them. Positive vs. negative numbers and real vs. imaginary numbers. Implying that only positive numbers and real numbers are legitimate ones.

Karl Friedrich Gauss proposed calling positive, negative, and imaginary numbers direct, inverse, and lateral ones, but that did not catch on. Some centuries earlier, Brahmagupta called positive and negative numbers asset and debt ones, but he was not know about in the West until recent centuries.

 

[Speakpigeon]

Sorry, but this is the twenty first century. Facts are subordinate to beliefs now, so as unter believes it with all has heart, it becomes the truth.

Apparently.

Not apparently! Well, OK, yes, somewhat apparently. Still, you can't possibly fault his logic. Truth is relative "since all measurements are relative and are made by observers trapped in a specific frame of reference".

The same thing can have two different measurements. Because all measurements are relative and are made by observers trapped in a specific frame of reference.

QED.

Can't beat them? Join them.
EB

 

[Speakpigeon]

I have made a Venn diagram of the rational, algebraic, real, and complex numbers:
View attachment 15066

So, it all comes down to "potatoes", as we would call those in French.

And me who thought it was all very complicated.

EB

 

[Phil Scott]

Back to philosophy of mathematics, the names of some kinds of numbers give hints about how past mathematicians regarded them. Positive vs. negative numbers and real vs. imaginary numbers. Implying that only positive numbers and real numbers are legitimate ones.

But those are modern terms. Descartes, who did some of the best early work legitimising them, still called negative numbers "false numbers"! I don't have the reference to hand, but there were practising mathematicians in the early 1800s still saying that negative numbers are nonsensical.

 

[untermensche]

Your opinion as far as I can see is supported by absolutely nothing.

Yes there are theories that exist.

Your claims that you can connect anything I say to them is empty and worthless.

I don't believe you and anybody that does simply doesn't care.

How exactly do you think the smallest possible volume of space can be measured? Not speculated on, measured.

Luckily for all of us, physicists are much smarter than you are, and they actually thought through the consequences of the different possibilities. They figured out that the potential discretized spacetimes imply a violation of Lorentz invariance, which in turn implies consequence that would be observable with today's technology. Specifically, the divergence from predicted results in vacuum Cherenkov radiation, particle decay, gamma ray bursts, etc, would all show as consequences of there being a 'smallest possible volume of space'. We've checked dozens of these as precisely as is currently possible (and that is VERY precisely, down to well below the Planck scale), and the consequences do not appear. You are wrong.

Physicists are not smarter than me. Except a very few. And besides you are no physicist.

They have not figured any of this out.

Suppose I give you a small cube of pure carbon. Can you divide it infinitely?

How is space any different? I ask rhetorically because you won't answer because you do not know.

 

[untermensche]

Back to philosophy of mathematics...

You misuse the word philosophy.

All you are doing is discussing arbitrary mathematical theory.

You are not telling anybody what a number is.

 

[Phil Scott]

You misuse the word philosophy.

Have you still not bothered to read the wikipedia page you linked to defining "philosophy of mathematics"? You appear to be capable of googling, but not actually reading the linked content.

 

[untermensche]

You misuse the word philosophy.

Have you still not bothered to read the wikipedia page you linked to defining "philosophy of mathematics"? You appear to be capable of googling, but not actually reading the linked content.

My only real mentor, Chomsky, is a great lover of mathematical philosophy. It is a very interesting field and I am very ignorant of great parts of it. But it is not going to be advanced or even challenged by anyone here.

But the philosophical examination of a "number" is trying to get to the bottom of what a number is and that is removed from mathematical philosophy.

 

[Phil Scott]

You misuse the word philosophy.

Have you still not bothered to read the wikipedia page you linked to defining "philosophy of mathematics"? You appear to be capable of googling, but not actually reading the linked content.

My only real mentor, Chomsky, is a great lover of mathematical philosophy. It is a very interesting field and I am very ignorant of great parts of it. But it is not going to be advanced or even challenged by anyone here.

This thread was started by someone trying to share stuff that's well known in the field, but might be of general interest to non-experts. Ipetrich is keen to get back to that. You're derailing.

 

[Phil Scott]

On lambda calculus....

A common symbol used in maths is ↦ which is read "maps to". You can use it as a convenient way to talk about a function. So to talk about the function which squares its argument and adds 1, you write \(x \mapsto x^2 + 1\). Many programmers are now familiar with doing this, and would say that this expression is an "anonymous function" (meaning it's a function that hasn't been given a name). Alonzo Church used a different but completely equivalent notation, namely \(\lambda x. x^2 + 1\), where the λ symbol acts like a quantifier from logic, and so the system he developed is called "lambda calculus." For familiarity, I'll use the regular math notation and avoid the glorious lambda.

Lambda calculus is the set of rules that tell you how anonymous functions work. According to Church, the only logical thing anyone ultimately wants to do with a function is apply it to some input. So lambda calculus governs the computational rules behind the application of anonymous functions. The main rule is this one:

(x ↦ φ[x]) y = φ[y]

In other words, applying a function just means replacing the x in the function body with the input. This is incredibly simple and bone-headedly obvious, but this rule now basically gives you something that is Turing complete! Every computation that can be conceived can be expressed with anonymous functions and applications of anonymous functions. You don't actually need \(^2\) or "+". Everything can just be abstraction and application, beginning with the simplest function x ↦ x and the simplest applications (x ↦ x) (x ↦ x) = (x ↦ x). It's like how in set theory everything ends up being built somehow from the empty set.

Lambda calculus gets its power in part because the inputs to functions are so frequently other functions, and those functions are themselves functions which input more functions. You end up climbing a tower of "higher-order" functions, much like you climb towers of nested sets in set theory. Here's numbers:

0 := f ↦ (x ↦ x)
S := n ↦ (f ↦ (x ↦ f ((n f) x)))

So 0 is a function which takes some f and outputs a function which takes some x and outputs that x. More succinctly, 0 is the constant function that spits out the identity function. It's the function which ultimately says "do not do anything." S, the successor function,is roughly the function which inputs a number and says "do that number of things and then do it again."

1 is defined as the successor of 0. Here it is computed:

\(\begin{align*} S 0 &= (n \mapsto (f \mapsto (x \mapsto f\, ((n\, f)\, x)))) (f \mapsto (x \mapsto x))\\ &= f \mapsto (x \mapsto f\, (((f \mapsto (x \mapsto x))\, f)\, x))\\ &= f \mapsto (x \mapsto f\, ((x \mapsto x)\, x))\\ &= f \mapsto (x \mapsto f\, x)\\ \end{align*} \)

Thus, S 0 reduces to the function which takes some f, and outputs the function which applies it once.

 

[lpetrich]

I checked on words for zero, and they come from words for nothing and empty. That is rather reasonable, though it took a long time for Western mathematicians to accept zero as a legitimate number and not just the absence of something.


Then there are kinds of numbers that require additional postulates, numbers that are not simply extensions of the natural numbers. Numbers like infinitesimals. Such numbers were used in the earlier formulations of the calculus, until 19th cy. mathematicians like Weierstrass made them superfluous with the idea of a limit.

Thus, one finds a derivative with f'(x) = (f(x+e) - f(x))/e where e is an infinitesimal. e has the property that e² = 0, and likewise for higher powers. For limits, one does f'(x) = limit for h -> 0 of (f(x+h) - f(x))/h. That's essentially finding the slope of a curve at some point by finding the line between two points and then running those points together.

 

[untermensche]

My only real mentor, Chomsky, is a great lover of mathematical philosophy. It is a very interesting field and I am very ignorant of great parts of it. But it is not going to be advanced or even challenged by anyone here.

This thread was started by someone trying to share stuff that's well known in the field, but might be of general interest to non-experts. Ipetrich is keen to get back to that. You're derailing.

It is in the philosophy section.

Asking us for some reason to not philosophize is pretty stupid.

 

[Bomb#20]

Luckily for all of us, physicists are much smarter than you are, and they actually thought through the consequences of the different possibilities. They figured out that the potential discretized spacetimes imply a violation of Lorentz invariance, which in turn implies consequence that would be observable with today's technology. Specifically, the divergence from predicted results in vacuum Cherenkov radiation, particle decay, gamma ray bursts, etc, would all show as consequences of there being a 'smallest possible volume of space'. We've checked dozens of these as precisely as is currently possible (and that is VERY precisely, down to well below the Planck scale), and the consequences do not appear. You are wrong.

Correct me if I'm wrong, but don't the presently measurable statistical consequences of discretization you're talking about scale with the size of the hypothetical spacetime quanta? So the nonappearance of these consequences can't ever rule out spacetime quanta, but can only put an upper bound on their size? For the sake of definiteness I used the Planck volume in my posts, and I take it that was way too big; but couldn't untermensche simply always postulate grains of unknown size far smaller than whatever the demonstrable upper bound is? This would make the spaceless interior of each quantum the Gap of the Gaps.

 

[lpetrich]

Algebraic numbers are an extension of the rational numbers by including all solutions of polynomial equations with rational-number coefficients. But by multiplying by the Least Common Multiple of the coefficients' denominators, one gets an integer-coefficient polynomial. If the coefficient of the highest-power term is 1, then the polynomial is a "monic" one. If its other coefficients are integers, then the roots are called "algebraic integers".

But often useful is extending the rational numbers with the solution of only one polynomial equation or some set of polynomial equations. The result is an algebraic field called Q(roots) where roots are for those polynomial equations. Q by itself is the rational numbers ("quotient").

The simplest one of these uses the first known irrational, sqrt(2), and is called Q(sqrt(2)). Its members x = x0 + x1*sqrt(2), where both x0 and x1 are rational.

One can continue further, by adding solutions of polynomial equations in that extension field, like sqrt(sqrt(2)+1). It solves x² - (sqrt(2)+1), and it is not in Q(sqrt(2)). So one gets Q(sqrt(sqrt(2)+1)).

These extension fields for polynomial roots are called "splitting fields".

 

[Bomb#20]

I have answered that.

Possibly so; but if you've already answered somebody and he keeps asking anyway, then he evidently didn't understand how your answer answered his question, so telling him you already answered is useless. You need to answer again, in English this time instead of in untermenschese. He can't understand untermenschese, presumably because he's not as smart as physicists and physicists aren't smarter than you. I asked you a yes-or-no question, and your answer wasn't "yes", and your answer wasn't "no".

The same thing can have two different measurements. Because all measurements are relative and are made by observers trapped in a specific frame of reference.

There can be no meaningful partitions of the smallest possible volume of space. It cannot be broken apart. If you tried to somehow break it apart you would not have space anymore. You would destroy the structure.

Yes, I got that. What you appear to have already told me is that

Observer 1's partition: |_4.2_|_4.2_|
Observer 2's partition: |_2.1_| 2.1_|​

is a possible scenario. But that's not what I asked. When two observers split the same region of space as far as it can be split without destroying the structure, into individual quanta of space, is this a possible scenario?

Observer 1's partition: |_____|_____|
Observer 2's partition: |___|___|___|​

Yes or no?

 

[untermensche]

I have answered that.

Possibly so; but if you've already answered somebody and he keeps asking anyway, then he evidently didn't understand how your answer answered his question, so telling him you already answered is useless. You need to answer again, in English this time instead of in untermenschese. He can't understand untermenschese, presumably because he's not as smart as physicists and physicists aren't smarter than you. I asked you a yes-or-no question, and your answer wasn't "yes", and your answer wasn't "no".

The same thing can have two different measurements. Because all measurements are relative and are made by observers trapped in a specific frame of reference.

There can be no meaningful partitions of the smallest possible volume of space. It cannot be broken apart. If you tried to somehow break it apart you would not have space anymore. You would destroy the structure.

Yes, I got that. What you appear to have already told me is that

Observer 1's partition: |_4.2_|_4.2_|
Observer 2's partition: |_2.1_| 2.1_|​

is a possible scenario. But that's not what I asked. When two observers split the same region of space as far as it can be split without destroying the structure, into individual quanta of space, is this a possible scenario?

Observer 1's partition: |_____|_____|
Observer 2's partition: |___|___|___|​

Yes or no?

You are showing me some lines.

Not a real division of anything.

 

[lpetrich]

There is an interesting issue in relation to these extension fields: automorphisms. These are self-bijections that satisfy the fields' operation properties. Thus, f(x+y) = f(x) + f and f(x*y) = f(x) * f. For natural numbers, integers, rational numbers, and real numbers, the only automorphism is the identity one. But let us consider complex numbers. f(x+y*i) = x + y*f(i) where we must find the possible values of f(i). Since f(i)² + 1 = 0, f(i) must equal either i or -i. We get both the identity automorphism and complex conjugation. Not surprisingly, these two automorphisms form a group: Z2.

Working with Q(sqrt(2)), one finds both the identity automorphism and reversing the sign of sqrt(2). Again, the automorphism group is Z2.

There is an interesting theorem about doing a sequence of polynomials. Start with field K1. Find its splitting field with respect to polynomial P1, giving splitting field K2. Do that again with polynomial P2, giving splitting field K3. Thus, K3 is the splitting field for some combined polynomial P12 over K1.

Let the automorphism group of K2 with K1 fixed be (K2:K1). This is the group of interchanges of roots of P1 that are not in K1, the polynomial's "Galois group". Yes, him who was killed in that duel. No, he communicated his work well before that duel.

Then, group (K3:K1) has (K2:K1) as a "normal subgroup", with "quotient group" (K3:K2).

This has implications for what polynomial equations can be solved with what radicals or nth roots, since doing so implies that the breakdown of the polynomial's Galois group must contain Z and Z*, where the latter is the group of multiplication of numbers relatively prime to n. So if a polynomial's roots can be found with square roots, its Galois group must break down to Z2's, while if one can use cube roots, that breakdown must include Z3's. Z*(2) = identity group and Z*(3) = Z2.

This means that the centuries-old problems of duplicating the cube and trisecting the angle cannot be solved by ruler-and-compass methods. That is because ruler-and-compass methods only do arithmetic and square roots, and because those two problems require cube roots. Archimedes used a neusis or marked ruler for cube roots, but most of his colleagues tried to avoid that.

Solution by radicals also has the general consequence that a polynomial's Galois group must be "solvable", with decomposition that is ultimately a chain of Z's. That is the case for quadratic, cubic, and quadric (degree 4) equations. But for quintic (degree 5), the Galois group is, in general, not solvable. Thus, general quintics cannot be solved with radicals.