at what point in the decimal representation of the surreal does it depart from and differ from sqrt(2), keeping in mind that it cannot be the next largest real number, because that real number is infinitely far away? — tim wood

I don't know. Is there a way we can construct that?

As I understand it, the surreals are not only between all the reals, but also beyond them. They are basically the reals, which are all finite numbers, plus also transfinite numbers, and the reciplocals of transfinite numbers, infinitesimals. Basically, in the surreal numbers, you can do "one over infinity" and get something... not quite exactly zero, but closer to zero than any real number. (But then, it's not exactly just "infinity" that you're dividing by either, but a specific infinite, or transfinite, number).

Also, as I understand it, the surreals are the biggest possible ordered set: every position of every element in every ordered set no matter its size can be represented with a surreal number. There is a particular way of constructing the surreal numbers which is basically equivalent to an infinite process that enumerates every position in an order and every position in between each of those positions and so on and so forth, and in the process of listing out all of those positions, you end up constructing the equivalents of all of the real numbers, but also things that are not equivalent to real numbers, that are in between them and further down the list than any of them. I don't know all the details of that off the top of my head, but I recall it having something to do with building a branching tree of some sort: you start with one element before another, and the other after it, then construct an element before and after each of those, and then before and after each of those, and so on forever. Every node in that tree is a surreal number.

Also, as I understand it, the surreals are the biggest possible ordered set — Pfhorrest

But aren't the surreals a proper class rather than a set?

Probably. I'm in a rush to get to sleep. Someone will set me straight by morning.

There are Youtube videos, including with John Conway, credited with discovering them. From these one learns a little. The idea of construction (as best I understand it), is a notation {l} or {0 l 1} or {-2 l -1} where by the notation is meant a number strictly between the left and the right.

In mathematics you can say, "Let's suppose..", this being the equivalent of "once upon a time" or "let's pretend." Let's suppose, for example, parallel lines intersect - and you get non-Euclidean geometry. The idea is that within the tale there's a consistent logic, and maybe it's worthwhile. And no doubt there is much nonsense that never sees the light of day. But there is also the question of whether the things supposed actually exist. I know, the "existence" of mathematical entities - even the word in that context - is contentious. But if integers and rationals and reals can be reasonably said to exist, whatever that means, then do surreals enjoy the same kind of substantial existence?

For example, I can say, suppose that between every pair of integers there is an integer. If I'm clever maybe I can work out a system on this "rule." But the idea itself is absurd. The "supposed" integers don't exist. So the question is if surreals have this deficient form of conjectural existence, or do they share the more substantial existence of the reals?

And then there is the question, if the reals just are the continuum, how do the surreals fit in? Are they everywhere on the number line, or only in some places.

@Nagase references this book, Varieties of Continua, Stewart Shapiro and Geoffrey Hellman (2018). Amazon provides an interesting "peek inside" option.



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For example, I can say, suppose that between every pair of integers there is an integer. If I'm clever maybe I can work out a system on this "rule." But the idea itself is absurd. The "supposed" integers don't exist. So the question is if surreals have this deficient form of conjectural existence, or do they share the more substantial existence of the reals? — tim wood

That rule is self-contradictory, so that kind of “number” can’t exist. But you could define a new kind of number — not the integers — which satisfied a property like that. (The rationals, maybe?) Some of that new kind of number would be equivalent to integers, but others wouldn’t.

The surreals are like that relative to the reals. And they exist exactly as much as any other mathematical entity exists, which as you say is a can of worms, but we don’t have to go into it here. Whatever kind of existence mathematical objects may have, there’s no reason to exclude the surreals from it.

The trouble conceptualizing where on the number line they fit is little different from trying to mash the reals in between the rationals. With the rationals you can already find infinitely many numbers between any two numbers, so where do the reals fit? They do though. Same with the surreals fitting in between (and beyond) the reals.

Sure. But how? How do you even begin to write down a surreal number? Or maybe they cannot even be approached? As to the "existence" of this or that number, if I own three sheep and at the end of the day my sheepdog returns some other number of sheep, then I know I have a problem. Three sheep and all's well. In this sense the number three exists. And mutatis mutandis the rationals and the reals. The ground for their existence being either or both of concrete practical application or by construction. But the rule of construction for the surreals seems to be just "let's pretend" - well maybe not quite that, but close.

If there's something more substantial than that, and there probably is, then what is it and how does it work? Reals can be constructed easily enough geometrically - is there any geometric construction for surreals?

The concept of infinitesimals goes back at least to Leibniz and Newton. Modern day non-standard analysis incorporates these ideas in a legitimate mathematical model. One can prove basic theorems in calculus using infinitesimals, and there have been textbooks that have done that. I once considered teaching an experimental calculus course this way but decided against it. I have heard that for some students calculus is more understandable taught in this non-standard way. I view infinitesimals as metaphysical entities that have achieved a kind of actuality.

The set theory aspects are something else, and seem to appeal to philosophers. Have at it! :cool:

The situation with surreals vs. reals is a little different than that with reals vs. rationals. Though it may seem as if rationals completely fill their number line, being as you can fit arbitrarily many points between any two points (i.e. they are densely ordered), rationals are actually full of holes, in a sense. You can construct a sequence of rational numbers that definitely converges to... something. But that something is not a rational number. There are lots and lots of such holes between rationals - in a way, the rational number line almost entirely consists of holes. And that is where real numbers come into the picture: they fill those holes.

With real numbers the situation is different: they are complete. Any converging sequence of real numbers most definitely converges to a real number. There are no holes to fill - at least not in that sense. You have to work a little harder to find what those smug bastards are lacking: you have to violate the Archimedean property.

Surreal numbers are on the number line, unlike complex numbers, which are not. That is, surreal numbers are not complex numbers — tim wood

Every real number is a complex number, but not vice-versa. There are certain types of surreal numbers that are complex: s = a+bi , where a and b are infinitesimals:

Wiki: A surcomplex number is a number of the form a+bi, where a and b are surreal numbers and i is the square root of −1.[9][10] The surcomplex numbers form an algebraically closed field (except for being a proper class), isomorphic to the algebraic closure of the field generated by extending the rational numbers by a proper class of algebraically independent transcendental elements. Up to field isomorphism, this fact characterizes the field of surcomplex numbers within any fixed set theory.

You guys are going down a rabbit hole here. :nerd:

There are certain types of surreal numbers that are complex: s = a+bi , where a and b are infinitesimals: — jgill

Is it not more accurate to say that some surcomplex numbers are surreals, or that there is a complex extension of the surreals?

Speaking of which, I need to get my suroctonions tetrated soon...

Is it not more accurate to say that some surcomplex numbers are surreals, or that there is a complex extension of the surreals? — Pfhorrest

Picky, picky, picky! :smile:

In over fifty years of complex analysis mathematics I don't think I ever really thought of these critters.

The Dirac Delta function (0 everywhere except at x=0, there infinite) can be thought of in terms of infinitesimals, here in terms of alpha:

$\int F(x)\delta_\alpha(x) = F(0)$

The Dirac Delta function (0 everywhere except at x=0, there infinite) can be thought of in terms of infinitesimals — jgill

It's funny how ubiquitous the delta function still is in physical and engineering mathematics, and yet it is completely non-kosher from the point of view of standard analysis. It was so useful that it survived Weierstrass's reforms, which did away with non-rigorous infinitesimals of the early calculus.

May I try this: every point on the number line corresponds to a real number, and to every real number there corresponds a point on the number line. So just where are, what are, the surreals? Are there points on the number line that cannot be characterized by a real number but can be associated uniquely with surreal that is not a real?

It's funny how ubiquitous the delta function still is in physical and engineering mathematics, and yet it is completely non-kosher from the point of view of standard analysis — SophistiCat

One way to make it kosher is to consider it a generalized function. I never worked with those either.

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

So just where are, what are, the surreals? — tim wood

On the finite number line, infinitesimals are everywhere and nowhere. If r is an infinitesimal then 2+r lies to the right of 2 but to the left of any real number greater than 2. Your job, Tim, should you accept the assignment, is to find it and neutralize it! If you fail we will disavow any knowledge of it. :worry:

On the finite number line, — jgill

The finite number line is a fiction. It may be useful for some things, but to insist that it is somehow 'real' and try to make meaningful inferences from that is meaningless.

The finite number line is a fiction. It may be useful for some things, but to insist that it is somehow 'real' and try to make meaningful inferences from that is meaningless. — A Seagull

I'll pass that on to my colleagues. What a bitter disappointment. :sad:

The finite number line is a fiction. It may be useful for some things, but to insist that it is somehow 'real' and try to make meaningful inferences from that is meaningless. — A Seagull
I'll pass that on to my colleagues. What a bitter disappointment. :sad: — jgill

The truth doesn't have to be disappointing. For many it is exhilarating.

On the finite number line, infinitesimals are everywhere and nowhere. If r is an infinitesimal then 2+r lies to the right of 2 but to the left of any real number greater than 2. — jgill

Surreals by any other name are just an infinitesimals? I am under that impression that infinitesimals were just very useful fictions. And that they've been around for a while, Conway's surreals on the other hand being relatively a much newer and more recent discovery.

As to finding it, no problem, my great-grand niece already has the formula for that: "It's over dere!" Never fails. Set coordinates, flashlights on titrate. Illuminate when ready, Gridley.

Is it sense to suppose that if the decimal representation of a real number like sqrt(2) has - how many digits? - N digits? Then a surreal has a larger transfinite number of digits?

Surreal Numbers

That's pretty cool. .

Nice video! I watched some of his others - all good!

What I get from it is that the surreals that are extensions of the reals are "simply" such numbers whose decimal expansions require more than N digits, i.e., ω, ω+1, ω+2,...ω+ω.... digits, and off into the endless transfinite. Seems simple enough - which usually means I don't got it....

Surreals by any other name are just an infinitesimals? — tim wood

No. I'm old enough that I knew Leibniz, and am familiar only with his basic ideas about infinitesimals. Surreals by Conway are much more elaborate. One can teach a calculus course using only the basic elements of infinitesimals. IMO the surreals are for set theorists and philosophers. Purer mathematics than anything I've done. The real and complex numbers are challenging enough for me!

Well, "the number line" in its usual sense is just a visual metaphor for the real numbers (it will do for the rationals as well, though see above about "holes"). So in that sense, no, the number line is not missing anything. You have to work harder to motivate things like infinitesimals and hyperreals. And then you have to work even harder just to reproduce all the things that we can already do with real numbers, like addition and multiplication.

One way to make it kosher is to consider it a generalized function. I never worked with those either.

https://en.wikipedia.org/wiki/Generalized_function — jgill

Sure, I shouldn't be surprised that these nasties have long since been tamed, just like infinitesimals and infinities were earlier.

Seems to me, in my novice terms, that the left part describes some sequence with the surreal number as its lower limit, and the right number describes some sequence with the surreal number as its upper limit, and that further this mode of representation is powerful enough to produce all the other sorts of numbers... except imaginary numbers, but presumably they could be incorporated with a bit of fiddling. A PhD for someone...

Because they use limits the surreals will presumably be unavailable to those who do not agree that 0.999...=1.

all the other sorts of numbers... except imaginary numbers, but presumably they could be incorporated with a bit of fiddling. A PhD for someone... — Banno

Those would be the surcomplex numbers discussed above, numbers of the form (a + bi) where a and b are surreal numbers.

And as there are extensions of the complex numbers into more than just two dimensions, hypercomplex numbers including most notably four-dimensional quaternions and eight-dimensional octonions (beyond which they lose most of the properties that make numbers useful as numbers), you can have surhypercomplex numbers like the suroctonions that I mentioned earlier, numbers of the form (a + bi + cj + dk + el + fm + gn + ho), where a through h are surreal numbers and i through o are the imaginary number and the next seven hypercomplex units beyond it.

(And tetration is the operation after exponentiation, the fourth item in the series {addition, multiplication, exponentiation ...}, hence my joke about "I need to get my suroctonions tetrated").

:up:

And as there are extensions of the complex numbers into more than just two dimensions, hypercomplex numbers including most notably four-dimensional quaternions and eight-dimensional octonions (beyond which they lose most of the properties that make numbers useful as numbers), — Pfhorrest

This process can be continued indefinitely as in the Cayley-Dickson construction.

There's another generalization of the process called Clifford algebras.

I also wanted to mention in passing that the question has been raised as to "where you fit" the extra numbers when you go from the reals to the hyperreals to the surreals and so forth. I believe that the answer is that the claim that "the line in my mind" is the same as any particular mathematical version of a line, is a belief and not a fact that could ever be proven. Is Euclid's line the same thing as the set of real numbers? We take as an unspoken axiom that it is; but if we remember that this is just an assumption, we can resolve our confusion over where the extra points go.

In fact there are a lot of real lines. The constructive mathematicians, who don't believe in noncomputable real numbers, have their constructive line. There are various flavors of the intuitionistic line. The hyperreals form the hyperreal line. It's clearly not the same as the real line, since there are no infinitesimals on the real line.

The surreal numbers are a totally ordered proper class; and if they're totally ordered, we can imagine lining them up in order and calling that the surreal line. But it's not the same line as the standard real line or any of the other many alternative models of the real line. It's a bit of a category error to ask where the extra points go. It's a completely different model of the continuum. That's my understanding, anyway.

I believe that the answer is that the claim that "the line in my mind" is the same as any particular mathematical version of a line, is a belief and not a fact that could ever be proven. Is Euclid's line the same thing as the set of real numbers? We take as an unspoken axiom that it is; but if we remember that this is just an assumption, we can resolve our confusion over where the extra points go. — fishfry

This is the key point. It is a mistake to try and make the numbers, which represent no specific spatial properties, into a line, which represents a spatial dimension. This is the incompatibility between the non-dimensionality of the numbers, and the dimensionality of the line, which is very similar to the incompatibility between the one-dimensional straight line, and the two-dimensional curved line. They are incommensurate.

What this indicates is that it is a mistake to try and represent spatial existence with distinct dimensions.

The surreal numbers are a totally ordered proper class; and if they're totally ordered, we can imagine lining them up in order and calling that the surreal line. But it's not the same line as the standard real line or any of the other many alternative models of the real line. It's a bit of a category error to ask where the extra points go. It's a completely different model of the continuum. That's my understanding, anyway. — fishfry

I believe that representing numeration as any sort of line is a fundamental problem. The problem is that a line is meant to represent something continuous, and numbers represent discrete units. The smallest possible spatial unit must occupy all three dimensions of space. So reducing space in this way, attempting to make the smallest possible discrete unit represented by a number, something with one dimension, is a hopeless enterprise because "one dimension" doesn't represent any sort of reality. Therefore an "infinitesimal" must be an infinitesimal point, having position in all three dimensions of space, rather than just a spot on a one dimensional line.

Is Euclid's line the same thing as the set of real numbers? We take as an unspoken axiom that it is; but if we remember that this is just an assumption, we can resolve our confusion over where the extra points go. — fishfry

@aletheist will be along shortly, I am sure, invoking the ghost of Charles Sanders Peirce and insisting that Euclid's line is not a collection of points at all. He would have a point, at least to the extent that it isn't a given that a line is identical with a particular collection of points.

Thanks for the shout-out! I am actually fine with the quoted statement by @fishfry; I have consistently acknowledged that the real numbers are an adequate (though approximate) model of a continuous line for many (perhaps most) mathematical and practical purposes.