Numbers

[Speakpigeon]

It seems an interesting coincidence to me that humans seem to have had the intuition of the continuity of space and time even before they could develop, crucially independently from that intuition, a basic, unsophisticated, arithmetic that only much later, broadly in the 19th century, could be shown to snug tightly into that intuition with the first formal expression of the notion of limit. And it's also only much later that more sophisticated mathematicians started to go beyond our basic intuitions, with for example the Complex numbers. And yet, I don't know that anybody has ever imagined any non-Real number that would nonetheless be conceived of as properly belonging to a continuous interval between two integers for example. As if we can't go beyond our basic intuition about the continuity of space and time. I believe it should be possible to invent such a number in the abstract (as for the Complex) even though our intuition seems to say such a number couldn't possibly represent any real point in space or time. Is it just that mathematicians are stopped by their very human intuition, or is it really impossible to invent that kind of number? The later would be a real shocker to me.
EB

You can expand the continuum using the hyperreals and the surreals (don't know so much about the latter, but am consistently told they are mind-blowingly cool). The hyperreals consist both of non-real infinite numbers, and non-real numbers that are the sum of an ordinary real with an infinitesimal part. They can be used to do calculus without using limits. Where a standard analyst would take a limit, a hyperreal analyst will just drop the infinitesimal part and see what real is left. I think historians broadly agree that the hyperreal approach is more closely aligned with how Newton understood calculus. The standard approach is more aligned with the Greeks.

If (some?) non-real numbers "are the sum of an ordinary real with an infinitesimal part", how come they're non-real?

The hyperreals, in one "construction", arise by looking at the first-order axioms of the reals and saying: I can always consistently say there is a number bigger than any given numeral, so I must be able to consistently say that there is a number bigger than all of them. This gives you an infinite number, and since we're still looking at the axioms for reals, we note that we've still got all the normal algebraic rules and, in particular, we can take the reciprocal of our infinite number and get an infinitesimal.

The bit about being "able to consistently say that there is a number bigger than all of them" seems like artistic license to me. Creative but not necessarily too accurate.

So, as I see it, there's little doubt we could always find a greater number than any given number. There's a straightforwardly algorithmic solution to that. However, I wouldn't know where to start for finding a number bigger than all numbers. And to talk of all given numbers is definitely not to talk of all numbers.

So, in my view, no, it doesn't work, or at least, it doesn't do what it says it's doing. So, yes, it's artistic license.

The surreals, as I understood, arise by taking the basic idea that Dedekind came up with in constructing the reals and then going completely insane with it.

Right, but Picasso, Dali, Breton, Duchamp, Magritte, Prévert, Ernst, and a few others got there first!


I'm also not clear that these non-Real numbers "would nonetheless be conceived of as properly belonging to a continuous interval between two integers for example".
EB

 

[Phil Scott]

If (some?) non-real numbers "are the sum of an ordinary real with an infinitesimal part", how come they're non-real?

For the same reason that non-real complex numbers are the sum of an ordinary real number with an imaginary part.

The hyperreals, in one "construction", arise by looking at the first-order axioms of the reals and saying: I can always consistently say there is a number bigger than any given numeral, so I must be able to consistently say that there is a number bigger than all of them. This gives you an infinite number, and since we're still looking at the axioms for reals, we note that we've still got all the normal algebraic rules and, in particular, we can take the reciprocal of our infinite number and get an infinitesimal.

The bit about being "able to consistently say that there is a number bigger than all of them" seems like artistic license to me. Creative but not necessarily too accurate.

So, as I see it, there's little doubt we could always find a greater number than any given number. There's a straightforwardly algorithmic solution to that. However, I wouldn't know where to start for finding a number bigger than all numbers. And to talk of all given numbers is definitely not to talk of all numbers.

So, in my view, no, it doesn't work, or at least, it doesn't do what it says it's doing. So, yes, it's artistic license.

So little faith

If you want something a bit more formal: take the axioms of ordered fields and consider additional axioms of the form:

∞ > 1 + 1 + ... + 1

where "∞" is a constant. Any finite set of such axioms can be consistently added to the axioms of the real ordered fields. This means that all of these axioms can be consistently added: any possible contradiction would have to be a contradiction of one of the finite sets, since a proof of contradiction can only use finitely many axioms at a time (this general property is the "Compactness" of first-order logic). Since the resulting theory with the totality of axioms is consistent, it must have a model (this is the "Completeness" of first-order logic). This means there is an ordered field containing a number bigger than any numeral: an infinite number ∞. Moreover, there are infinitely many such infinite numbers, since we're in a field. You can take any integer n, and obtain distinct infinite numbers n∞ and ∞/n. The set of such numbers is bounded below, but lacks a limit. We also have infinitesimals since we're in a field. Just take 1/∞. As before, we can obtain infinitely many such infinitesimals by n/∞ and 1/(n∞). The set of such numbers is bounded above, but lacks a limit. This means that we're in an ordered field, but not an Archimedean one. We also have non-real, non-infinitesimal numbers like 1 + 1/∞.

Thus: I can always consistently say there is a number [∞] bigger than any given numeral [an expression of the form 1 + 1 + ... + 1, and thus all smaller numerals], so I must be able to consistently say that there is a number [∞] bigger than all of them [by the Compactness Theorem].

I'm also not clear that these non-Real numbers "would nonetheless be conceived of as properly belonging to a continuous interval between two integers for example".
EB

Take the integers 1 and 2. We have 1 < 1+1/∞ < 1.5 + 1/∞ < 2 - 1/∞ < 2. In other words, we have a bunch of non-real numbers between 1 and 2.

 

[Kharakov]

You can expand the continuum using the hyperreals and the surreals (don't know so much about the latter, but am consistently told they are mind-blowingly cool).

Geometric representation?

The hyperreals consist both of non-real infinite numbers, and non-real numbers that are the sum of an ordinary real with an infinitesimal part. They can be used to do calculus without using limits. Where a standard analyst would take a limit, a hyperreal analyst will just drop the infinitesimal part and see what real is left. I think historians broadly agree that the hyperreal approach is more closely aligned with how Newton understood calculus. The standard approach is more aligned with the Greeks.

It appears to me that inverse Cauchy products (with "symmetric zeros" such as 1-1) work for differentiation in some cases, but they get excessively complex, and give you the same answer as the limit approach which is easier to write out.

Example:
Inverse Cauchy product diagram:

Show

Inverse Cauchy product has terms r_0... r_n, we only use A=1 and B=-1, so drop the C....:
\(\begin{tabular}{ l | c c c c c } & a & b & c & d & e ... \\ \div & A & B & C & D & E ... \\ \hline r_0 & r_0 A & r_0 B & r_0 C & r_0 D & r_0 E \\ r_1 & r_1 A & r_1 B & r_1 C & r_1 D & \\ r_2 & r_2 A & r_2 B & r_2 C & & \\ r_3 & r_3 A & r_3 B & & & \\ r_4 & r_4 A & & & & \\ \end{tabular} r_0=\frac{a}{A} r_1=\frac{b-B r_0}{A} r_2=\frac{c- C r_0 - B r_1}{A} .... \)

Say you want the derivative of f(x) = x^2

Instead of an infinitesimal or limit, simply use 1-1, multiplications are Cauchy products, with positions (terms??) preserved:
("/" symbolizes inverse Cauchy below)
f'(x) = [(x+1-1)^2 - x^2]/ (1-1)

("/" symbolizes inverse Cauchy below)
= [ (x^2 -x^2) + 2x + (1-2x) -2 +1 ] / (1-1)

Group like positioned terms, so x^2 and -x^2 cancel out (they are 0). 2x is a second termer, 1-2x is a 3rd spot, -2 is 4th, +1 is 5th..
\(\begin{tabular}{ l | c c c c c } \times & x & 1 &-1 & \\ \hline x & x^2 & x &-x \\ 1 & x & 1 & -1\\ -1 & -x & -1 & 1 \\ \end{tabular} \,\, \)\(- \,\, \begin{tabular}{ l | c c c c c } \times & x & 0 &0 & \\ \hline x & x^2 & 0 &0 \\ 0 & 0 &0 & 0\\ 0 &0 & 0 &0 \\ \end{tabular} \) =0 + 2x (-2x+1) -2 + 1

So you end up with:

\(\begin{tabular}{ l | c c c c c } & 0 & 2x & 1-2x & -2 & 1 ... \\ \div & 1 & -1 & 0 & 0& 0 ... \\ \hline 0 & 0 & 0 & \\ 2x & 2x & -2x &\\ 1 & 1 & -1 & \\ -1 & -1 & 1 & \\ 0 & 0 & 0 & \end{tabular} \)

In the end you can either collapse the product (just consider 2x) or you can accept that f'(x) =0 + 2x + 1 -1

It's more complicated when you do f(x) = x^3.. you end up doing the inverse Cauchy ("/" symbolizes inverse Cauchy below) of:

[0 + 3x^2 + (3x-3x^2) + (1-6x) + (-3+3x) + 3 - 1] / (1-1)


You can arbitrarily tack on extra 1-1s to then original function, you just have to do the inverse Cauchy with more terms then... go ahead and try:

Show

f(x)= x^2 again:
("/" symbolizes inverse Cauchy below)
[x^2-x^2 2x (1-2x) (2x-2) (-2x+3) -4 3 -2 1] / (1 -1 + 1 -1) = 0 + 2x +1 -1 +1 -1

 

[beero1000]

Clever.

Not exactly how we ordinarily think of an infinite space, but I can see it works. And I remember programming something like the Pac-Man's space myself and this definitely feels somewhat similar to a curved space, or more exactly a cube.
EB

The two dimensional case is the study of surfaces and topologists exhibit examples often by thinking about pacman worlds. More specifically, they start with a square and then say what the rules are on hitting a boundary. In Pacman's world, the left and right sides wrap around. You can embed this three dimensionally as Pacman moving around the outer surface of a ring. There would be some distortion due to the curvature, but it's not something you would notice at small enough scales.

Now in Asteroids, the world wraps in top and bottom as well. You embed this in three dimensions as flying along the surface of a doughnut.

Doughnut world truly lacks any edges. You can fly anywhere on the surface without hitting the boundary. This is conceptually true even in the 2d embedding. We might think there is a boundary at the edge of the screen, but you can always translate and bring that boundary to the centre. It is the edge of the screen which is artificial, rather than a real part of the world (and in Asteroids, the world is continually translated to keep the player in the centre of the screen).

You can do more interesting wrappings too: take Pacman again, and put a cool twist on the horizontal wrapping: have it so when you cross either side of the screen, you flip Pacman's y coordinate. Pacman's world now embeds on a band with a twist in it: a Moebius band.

What if we do the same for Asteroids: keep the vertical behaviour but flip the y when crossing the left and right edges? Now the world has no 3d embedding. It can only be realised in four dimensions. You might have already heard of this bizarre object.

What is common in these cases is that, at small enough scales, they start looking like flat planes. All manifestations of this idea turn out to be fully classifiable. I haven't studied the 3d case of surfaces (manifolds), where locally the world looks like ordinary 3d space, but am told it is the same there and would enjoy reading beero's link.

Found it: https://talkfreethought.org/showthr...inite-universe&p=426708&viewfull=1#post426708

Basically, while relativity describes the local behavior of spacetime, we can also talk about the global geometry of the universe. As far as we can tell from experiment, besides the local distortions described by relativity, the universe is globally isotropic, homogeneous, and has zero curvature. So for the spatial universe we are looking at Euclidean 3-manifolds. These were completely classified in the 20's.

There are 18 Euclidean 3-manifolds, 8 of which are non-orientable. Physicists have checked the cosmic background for evidence of non-orientability but haven't found any, so we conclude that the universe is orientable. Of the 10 orientable candidates, 6 are compact and 4 are non-compact. I believe that most cosmologists conclude that the universe is compact, though I'm not sure of the evidence for that.

 

[lpetrich]

Getting back to the original subject, Phil Scott posted in #6 how to get the integers from the natural numbers (nonnegative integers). I will now concern myself with how to obtain them. In 1889, Giuseppe Peano stated an axiomatic formulation of them, and I will state his axioms here.

First, axioms of equality:

1. It is reflexive: x = x
2. It is symmetric: x = y implies y = x
3. It is transitive: x = y and y = z implies x = z
4. It is closed: if x = y for x in some set S, then y must also be in S.

Now for Peano's axioms. The set of natural numbers I will call N, and their successor function S.

1. There is a number in N, a number which will be called 0.
2. For every x in N, S(x) is also in N.
3. For all x and y in N, x = y is equivalent to S(x) = S
4. There is no x in N such that S(x) = 0
5. Mathematical induction: for some predicate function (one that returns true or false) f such that f(0) is true, and if for every x in N, f(x) being true implies the truth of f(S(x)), then f(x) is true for all x in N.

Mathematical induction can also be stated in set-theoretic fashion. If there is some set T that has 0 in it, and if for every x in N, x being in T implies S(x) being in T, then every x in N is in T.

The first few natural numbers are 0, 1 = S(0), 2 = S(S(0)), 3 = S(S(S(0))). Notice that this is a unary representation, a base-1 one.

Addition one gets with these axioms:

• x + 0 = x
• x + S = S(x + y)

Thus, S(x) = x + 1

Multiplication with these axioms:

• x*0 = 0
• x*S = (x*y) + x

Ordering: x <= y means that there is some a in N such that x + a = y

Addition and multiplication have familiar properties, thus making the natural numbers a semiring with unity. Both addition and multiplication are commutative monoids, both being associative and both having an identity within N: 0 and 1.

 

[lpetrich]

One can go further and get Peano's axioms from set theory. One needs a concept of cardinality or number of elements in a set. If there exists a bijection between two sets, then those sets have the same cardinality. In fact, Bertrand Russell once proposed defining a natural number as the set of all sets that have that cardinality.

So all we need are some exemplar sets. We can construct them with:
0 = {} the empty set
S(X) = union of X and {X}

Thus,
1 = {{}}
2 = {{}, {{}}}
3 = {{}, {{}}, {{}, {{}}}}
etc.

One can define addition, multiplication, exponentiation, and factorial in set-theoretic fashion.

Addition: if X and Y are disjoint (X intersect Y = {}), then card(X union Y) = card(X) + card(Y)

Multiplication: about the set of all Cartesian products (x,y) for x in X and y in Y, it has cardinality card(X)*card(Y)

Exponentiation: about the set of all functions f where for every i in Y, f(i) is in X, it has cardinality card(X)^card(Y)

Factorial: the number of all self-bijections of a set

 

[lpetrich]

One can even do ordering with sets: if X is a subset of Y, then card(X) <= card(Y).

But if X is a proper subset of Y, meaning X != Y, then card(X) < card(Y) is only guaranteed if X is finite.

Getting into the philosophy of mathematics again, infinite sets caused a lot of controversy in the late 19th and early 20th centuries. The controversy was not over the results but about the legitimacy of the operations involved. Like whether or not an infinite set is a legitimate mathematical object.

 

[Kharakov]

First, axioms of equality:

1. It is closed: if x = y for x in some set S, then y must also be in S.

Put coherently: if x is part of a set, such as the Integers (...-2,-1,0,1,2...), and y=x, y is also part of that set (if x is an integer, y is an integer).

 

[Bomb#20]

We are allowed to think and use our rational capacities.

We can rationally examine this notion of a real infinity.

We can look at something like space and ask is it possible to have the smallest amount of space?

What volume of space could we have where theoretically we could not have a smaller volume of space?

What would that smallest theoretical volume be?

This requires thinking, I warn you.

...
If you totally ignore all my questions you are just wasting my time.

I am saying there is no such thing as the smallest theoretically possible volume. It is not a rational concept.

You confuse my arguments about movement with my points here.

To have any movement it must be a movement greater than zero. Any movement greater than zero is a finite movement.

So to have movement you must make some first finite movement. You cannot make an infinitely small movement. There is no such thing.

My apologies for not reading your post with fresh eyes. I read your questions in the context of everything else I've ever seen you write on the topic.

If you're now saying there is no smallest amount of space, that's great; and you can disregard my last post. Sorry I misunderstood you. But if you're now saying there is no smallest amount of space, then you need to understand, that's the same thing as saying that space is infinitely divisible. You're in effect now saying that every cubic centimeter of space in the real world really contains an actual infinity of smaller volumes. That's because if the number of volumes a cubic centimeter can be broken down into were finite, then one of those finitely many finite volumes has to be the smallest. So you are reversing your long-held and intensely-insisted-upon position that the real world cannot possibly contain actual infinities. Is that really what you mean to be saying? If so, bravo! You are finally making progress!

 

[Kharakov]

 

[untermensche]

You're in effect now saying that every cubic centimeter of space in the real world really contains an actual infinity of smaller volumes.

No I am not.

Now we have to talk about physical limits.

Anything physical has limits.

You can't divide any volume infinitely for real. At least it can't be demonstrated that this is possible.

You can only imagine it.

 

[Speakpigeon]

Own goal?
EB

 

[lpetrich]

Looking at the various kinds of numbers, they gain properties and lose properties on the way.

Natural numbers: Peano's axioms, set theory

• Addition and multiplication both commutative monoids, identities are 0 and 1
• Multiplication distributive over addition
• Ordering

Integers: use Grothendieck construction on the natural numbers: x = (x1,x2), satisfying

• Equality: x = y if x1+y2 = x2+y1
• Ordering: x <= y if x1+y2+a = x2+y1 for some a in N
• Addition: x + y = (x1+y1,x2+y2)
• Additive inverse: - x = (x2,x1)
• Addition is a commutative group
• Multiplication: x*y = (x1*y1 + x2*y2, x1*y2 + x2*y1)
• No smallest number

Rational numbers: use field-of-fractions construction on the integers: x = (x1,x2), satisfying

• Equality: x = y if x1*y2 = x2*y1
• Addition: x + y = (x1*y2+x2*y1, x2*y2)
• Multiplication: x*y = (x1*y1, x2*y2)
• Additive inverse: - x = (-x1,x2)
• Multiplicative inverse: 1/x = (x2,x1)
• Multiplication is a commutative group if one omits zero
• Ordering: x > 0 if x1 and x2 have the same sign, x < 0 if they have opposite signs
• Density: rational numbers are dense, meaning that one can find a rational number that is arbitrarily close to any other rational number

Real numbers: all Cauchy sequences of rational numbers. Real numbers are Cauchy-closed: Cauchy sequences of them are also real numbers.

Algebraic numbers: all solutions of polynomial equations with rational-number coefficients. Algebraic numbers are algebraically closed, closed over algebraic extension: using algebraic-number coefficients in the polynomial equations.

Real algebraic numbers are not closed over algebraic extension, and real numbers are not either. So the algebraic numbers are complex ones, with form (real algebraic number) + i*(real algebraic number) where i = sqrt(-1). Likewise, the algebraic extension of real numbers has the form (real number) + i*(real number) -- the complex number. Like the (complex) algebraic numbers, complex numbers in general are algebraically closed. There is a remarkable theorem that states that i is the only new kind of number that one needs for algebraic closure.

Complex numbers can only be partially ordered. If one tries to order them, one will always end up with each order position having an infinite number of complex numbers in it.

 

[Phil Scott]

One can even do ordering with sets: if X is a subset of Y, then card(X) <= card(Y).

Though it's more natural to say that card(X) <= card(Y) exactly when there is an injection from X to Y.

Cool posts, anyway.

Getting into the philosophy of mathematics again, infinite sets caused a lot of controversy in the late 19th and early 20th centuries. The controversy was not over the results but about the legitimacy of the operations involved. Like whether or not an infinite set is a legitimate mathematical object.

Also to get into the philosophy of maths, the idea that numbers are something like Frege cardinals had enough purchase in the early 1900s to get anthropologists seriously thinking about it. You can spin the story that counting has always been about one-one correspondence. Take the fable of the Shepherd who discovers counting by keeping a bag of stones, and when the sheep come in for the evening, he drops a stone into a bucket for each sheep that passes. If he runs out of sheep before he runs out of stones, then there are sheep missing. If he runs out of stones exactly as the last sheep comes in, all the sheep come back.

The shepherd is checking for one-one functions between pebbles and sheep, comparing the size of the set of sheep with the size of the set of pebbles using the definition above. Eventually, his people invent spoken numerals, and he is able to get rid of his pebbles and instead count off the sheep that come back in the evening. Now he is forming one-one correspondences between a set of sheep and a set of spoken sounds.

In this story, numbers are whatever you use for forming one-one correspondences, which is exactly what the set theorists came up with as well. Frege (not Russell, IIRC), wanted to say that every number was literally the abstraction of this process, but it doesn't work, so we talk instead of exemplars.

Personally, I have a particular soft spot for the Church numerals: 0 is the identity function. It doesn't do anything. 1 is the function that does it. 2 is the function that does it, and does it again.

The definitions of addition and multiplication you get out of this are beautiful. In the beginning, there was the lambda, and the lambda was with Church, and the lambda was Church.

 

[Bomb#20]

You're in effect now saying that every cubic centimeter of space in the real world really contains an actual infinity of smaller volumes.

No I am not.

Now we have to talk about physical limits.

Anything physical has limits.

You can't divide any volume infinitely for real. At least it can't be demonstrated that this is possible.

You can only imagine it.

Okay, you appear prima facie to be making inconsistent claims. So help me make sense of them. Earlier you asked, "We can look at something like space and ask is it possible to have the smallest amount of space?". Well, you tell me. What is your answer? Is it possible to have the smallest amount of space, or isn't it?

 

[untermensche]

You're in effect now saying that every cubic centimeter of space in the real world really contains an actual infinity of smaller volumes.

No I am not.

Now we have to talk about physical limits.

Anything physical has limits.

You can't divide any volume infinitely for real. At least it can't be demonstrated that this is possible.

You can only imagine it.

Okay, you appear prima facie to be making inconsistent claims. So help me make sense of them. Earlier you asked, "We can look at something like space and ask is it possible to have the smallest amount of space?". Well, you tell me. What is your answer? Is it possible to have the smallest amount of space, or isn't it?

No inconsistency. You simply are calling two things the same thing.

There are two smallest things.

The theoretically smallest volume: Does not exist. A figment of the imagination.

The physically smallest volume: Something real and constrained. The smallest volume that could still be considered space. A quantum of space. To halve it is impossible.

 

[lpetrich]

Where can I find out more about the Church numerals? Is that a lambda-calculus construction?

The lambda calculus is essentially doing everything with functions, and only with functions. One can get Boolean values and operations out of it rather easily, and also Peano's axioms.


Another kind of number is computable numbers, numbers that can be approximated to arbitrary precision using a Turing machine run for a finite number of steps. This includes all the algebraic numbers and nearly all the transcendental numbers that we are likely to be familiar with. An exception is Chaitin's constant, related to whether or not a Turing machine will halt. Since there is no Turing machine that can work that out in all cases, Chaitin's constant is thus not a computable number.


Next is definable numbers, numbers that can be described in some finite-sized way. They include all the computable numbers, and also numbers like Chaitin's constant.


How many there are of the various kinds of numbers is rather interesting.

The cardinality of the natural numbers is aleph-zero or A-0. That is also the cardinality of the integers, the rational numbers, the real algebraic numbers, the real computable numbers, and the real definable numbers, but not the real numbers in general, as Georg Cantor had proved. There are infinitely more of them than there are of the other kinds of numbers that I'd listed.

 

[Kharakov]

What is it called when you approach real x from infinity? Is it just the limit as infinity-->x?


Is there a modern justification for rule systems that allow infinity-(infinity, shifted left 2.41428... places) = some specific value (1+sqrt(2))?

 

[untermensche]

Nothing extends FROM infinity.

That is impossible to define.

To define any progression you have to define a starting point.

You can only extend out infinitely from a defined point. And you can extend out in infinite directions. In mathematics, not the real world.

 

[Phil Scott]

Where can I find out more about the Church numerals? Is that a lambda-calculus construction?

Yeah, it's just the standard lambda calculus encoding.

The lambda calculus is essentially doing everything with functions, and only with functions. One can get Boolean values and operations out of it rather easily, and also Peano's axioms.

Talking about Peano's axioms in the context of lambda calculus means you're talking about the formal deductive system, which means you're talking about a typed lambda calculus. There are many such calculi, and the one I'm most familiar with is the Simple Theory of Types which is mostly what Church originally intended, though this presentation is particularly beautiful and massively cut down from Church's mishmash of lambdas and quantifier logic. I've only seen how to get the Peano axioms out of this with the axiom of infinity. Maybe you can get the same axioms without it, like you can for ZF without the axiom of infinity, but I haven't seen it.